Proof. First take smooth compactly supported data. On a relatively compact smooth domain use the componentwise zero-Dirichlet realization of the Hodge heat equation. Its Weitzenböck formula is a connection Laplacian plus a zeroth-order curvature reaction. At a null vector \(X\) of a nonnegative Hermitian form \(A\), diagonalize \(A\) in a unitary frame. The reaction on \((X,\bar X)\) is a positive fixed multiple of \[\sum_j R_g(X,\bar X,e_j,\bar e_j)A_{j\bar j}\ge0;\] the Ricci-times-\(A\) terms vanish there. The tensor maximum principle therefore preserves the nonnegative cone. Applying the same argument to \(Cg-A\), with nonnegative boundary data, proves the upper bound by \(Cg\). The trace satisfies the scalar heat equation, including the zero boundary condition. This is the null-eigenvector mechanism in [44].

We specify the complete realization and the exhaustion step. On compactly supported smooth forms put \(D=d+d^*\). Completeness gives essential self-adjointness of \(D^2\) and hence of \(D\): a deficiency vector for \(D\) would lie in the kernel of \((D^2)^*+1\), which is zero [5]. The closed quadratic form of the resulting Hodge Laplacian is \[q(A)=\|dA\|_2^2+\|d^*A\|_2^2, \qquad V=\overline{C^\infty_c}^{\,(\|\cdot\|_2^2+q)^{1/2}}.\] For a nested smooth exhaustion \(\Omega_j\), let \(V_j\) be the same closure of forms supported in \(\Omega_j\), extended by zero. On a fixed domain, local ellipticity identifies this with the full zero boundary-value section domain. It is not an absolute or relative Hodge boundary condition. The spaces \(V_j\) are nested and their union is dense in \(V\). For \(\lambda>0\), the domain resolvent applied to an \(L^2\) form, and extended by zero, is the orthogonal projection of the complete resolvent onto \(V_j\) for the inner product \(q+\lambda\langle\cdot,\cdot\rangle\): both satisfy the same variational identity against \(V_j\). The projections therefore converge in that norm. Strong resolvent convergence and the common lower bound zero imply strong heat convergence [54]. Here \(\Delta_j\) denotes the nonnegative domain Hodge Laplacian. Zero extension of its heat means \(e^{-a\Delta_j}P_j\), where \(P_j\) is restriction followed by zero extension; it does not mean extending its generator by zero. The cone and trace assertions pass to this limit and then to bounded data by cutoff and local regularity. More explicitly, the complete Hodge semigroup has the off-diagonal estimate \[ \|\mathord{% \BeginAccSupp{method=hex,unicode,space=false,pdfliteral=direct,ActualText=D835DFCF}% \mathbf 1% \EndAccSupp{pdfliteral=direct}% }_E\mathcal H_a\mathord{% \BeginAccSupp{method=hex,unicode,space=false,pdfliteral=direct,ActualText=D835DFCF}% \mathbf 1% \EndAccSupp{pdfliteral=direct}% }_F\|_{2\to2} \le C\exp\bigl(-c\mathop{\mathrm{dist}}(E,F)^2/a\bigr). \tag{7}\] To see that this estimate needs no curvature upper bound, use the self-adjoint first-order operator \(d+d^*\). Its principal symbol has norm equal to the covector norm, so the wave evolution has finite propagation speed [39]. The Gaussian Fourier representation of \(\exp(-a(d+d^*)^2)\) then gives (7), with the inessential change of constants required by our normalization [14]. Since \(\mathop{\mathrm{Vol}}B(o,R)\le CR^{2n}\), dyadic annular decomposition shows that the cutoff evolutions of a bounded form are Cauchy in \(L^2\) on each compact, uniformly for bounded positive times. Interior parabolic estimates, whose coefficients are bounded on the compact under consideration, give smooth convergence away from \(a=0\).

Closedness is proved using the complete semigroup, not the Dirichlet operators on domains. Let \(\zeta_R\) be a compactly supported Lipschitz cutoff equal to one on \(B(o,R)\), supported on \(B(o,2R)\), and satisfying \(|d\zeta_R|\le C/R\). The complete Hodge semigroup commutes with \(d\) on the \(L^2\) domains, by the self-adjoint de Rham complex. The fixed time normalization does not affect the spectral identity \(D e^{-aD^2}=e^{-aD^2}D\), and projection to degree one higher gives the \(d\) identity. The data \(\zeta_RA\) used here belong to the full \(D\) graph domain by compactly supported \(H^1\) approximation. For bounded closed \(A\), \[d\mathcal H_a(\zeta_RA) =\mathcal H_a^{(3)}(d\zeta_R\wedge A).\] The data on the right have \(L^2\) norm at most \(CR^{n-1}\) and support outside \(B(o,R)\). Estimate (7) in degree three shows that they tend to zero on each fixed compact, uniformly for \(0<a\le a_0\). Distributional passage to the limit proves \(d\mathcal H_aA=0\). Local regularity also gives the initial value \(A\) on each compact. Kähler commutation on closed forms now gives \[\partial_a\mathcal H_aA=\partial\bar\partial\mathop{\mathrm{tr}}_g\mathcal H_aA =\partial\bar\partial H_a\mathop{\mathrm{tr}}_g A.\] Integration proves (5), first in distributions and then as an identity of smooth forms for \(a>0\).

The passage from closed form heat to a scalar potential by integrating its trace is also the mechanism in [43].

Finally put \(f_0=f-c\), where \(c\) is the constant value at infinity. This is a compactly supported Lipschitz function. Complete \(L^2\) Kähler commutation, smooth approximation, and duality give, for a compactly supported nonnegative Hermitian test form \(B\), \[\langle\partial\bar\partial H_af_0,B\rangle =\langle\partial\bar\partial f_0,\mathcal H_aB\rangle.\] The metric pairing here identifies the cone of nonnegative Hermitian forms with its dual; equivalently one pairs the current with the Hodge star of \(B\). Choose \(0\le\zeta\le1\) compactly supported and equal to one near \(\mathop{\mathrm{supp}}f_0\). Since \(\partial\bar\partial f_0\) is supported there, \[\langle\partial\bar\partial f_0,\mathcal H_aB\rangle =\langle\partial\bar\partial f_0,\zeta\mathcal H_aB\rangle \le C\int\zeta\mathop{\mathrm{tr}}_g\mathcal H_aB\,dV_g \le C\int\mathop{\mathrm{tr}}_gB\,dV_g.\] Positivity and scalar trace commutation justify the last two steps. Finally \(H_a1=1\) under nonnegative Ricci curvature, so \(\partial\bar\partial H_af=\partial\bar\partial H_af_0\). This proves (6). ◻

Proof. Take \[\chi=u^{-k}\exp(-AH/u^2).\] For fixed \(A\), (30) gives the two-sided size estimate and the gradient estimate. Expanding \(\chi^{-1}\partial\bar\partial\chi=\partial\bar\partial\log\chi+ \partial\log\chi\otimes\bar\partial\log\chi\) gives at infinity \[\chi^{-1}\partial\bar\partial\chi \le u^{-2}\{-A(g-h_t)+C_kg+o(1)g\},\] uniformly on the fixed time interval. The terms containing \(H/u^2\) or its gradient are \(o(u^{-2})g\) after \(A\) is fixed. Choose \(A>C_k+1\) and then take \(u\) sufficiently large. The negative multiple of \(g\) absorbs the remaining error and leaves \(\chi^{-1}\partial\bar\partial\chi\le Au^{-2}h_t\). On the remaining compact, smoothness and positivity of \(h_t\) give the asserted bound. Finally, \(\chi_t/\chi=-AH_t/u^2\) and (30) give \[\chi_t/\chi\le A|H|/(tu^2)+C_T/u \le C_T(1+t^{\beta-1}).\] The exponent \(\beta>0\) makes this time coefficient integrable at zero. ◻

Proof of Proposition 6. The Ricci identity \(\mathop{\mathrm{Ric}}(h_t)=\mathop{\mathrm{Ric}}(g)+\partial\bar\partial\rho\), traced with \(h_t\), proves (32), since \(\rho_t=R_{h_t}\). Its source is nonnegative. Solve the source equation with zero initial and lateral boundary data on a smooth relatively compact exhaustion, first as a weak energy solution [32], and then use local parabolic regularity. A source need not vanish at the initial lateral corner, so full corner compatibility is not asserted. This bounded-domain construction and its energy estimates pass between a finite set of coordinate charts; it requires no completeness of the evolving metric. Domain comparison makes these solutions increase, and comparison with \(\rho\) bounds them above. Their limit \(\rho_{\min}\) is the minimal nonnegative potential. Local estimates give a smooth solution for \(t>0\) and continuous zero initial values, since it is bounded by the smooth function \(\rho=O_K(t)\) on each compact. The difference \(z=\rho-\rho_{\min}\) satisfies \[0\le z\le\rho,\qquad z_t=\Delta_{h_t}z, \qquad z(\cdot,0)=0.\]

Use \(\chi\) from Lemma 5, with \(k>2n+3\), and a cutoff \(p_N=p(u/N)\) which is one on \(u\le N\) and zero on \(u\ge2N\). Set \[I_N(t)=\int_M z\chi p_N\,dV_{h_t}.\] Since \(\partial_tdV_{h_t}=-R_{h_t}dV_{h_t}\) and \(R_{h_t}\ge0\), integration by parts on the compact support of \(p_N\) gives \[ I_N'(t)\le C_T(1+t^{\beta-1})I_N(t)+E_N(t), \qquad |E_N(t)|\le C_TN^{2n-1-k}. \tag{35}\] Here the error contains the derivatives falling on \(p_N\). Its complex Hessian, including the gradient cross terms with \(\chi\), has \(g\)-norm at most \(C_TN^{-k-2}\) on \(N\le u\le2N\). The pointwise cofactor identity and \(0<h_t\le g\) imply, for any real \((1,1)\)-form \(B\), \[ |\mathop{\mathrm{tr}}_{h_t}B|\,dV_{h_t} \le C_n|B|_g\,dV_g. \tag{36}\] Indeed diagonalize \(h_t\) relative to \(g\); the coefficient multiplying each diagonal entry of \(B\) is the product of the other \(n-1\) eigenvalues, each at most one. This also covers \(n=1\). Equation (10) bounds the integral of \(z\) on the shell by \(C_TN^{2n+1}\), proving the error bound.

The mass \(I=\int z\chi\,dV_h\) is finite. Its tails tend to zero uniformly in \(t\) by \(z\le\rho\), \(dV_h\le dV_g\), and the same dyadic shell estimate. Its initial limit is zero by local convergence and this tail control. Integrating (35) from a positive initial time, sending \(N\to\infty\), and then sending that time to zero gives \(I(t)\le\int_0^tC_T(1+s^{\beta-1})I(s)\,ds\). Gronwall’s inequality gives \(I=0\), and positivity of \(\chi\) gives \(z=0\). Thus \(\rho=\rho_{\min}\).

For the tails, choose smooth spatial cutoffs \(0\le\theta_j\le1\) increasing to one, each compactly supported and eventually equal to one near each fixed compact. Let \(\rho_j\) be the minimal zero-data potential of \[F_j=(1-\theta_j)\mathop{\mathrm{tr}}_{h_t}\mathop{\mathrm{Ric}}(g).\] It is nonnegative and supercaloric. Linearity on each exhaustion domain and monotone limits give the decomposition of \(\rho\) into this exterior-source potential and the minimal potential of \(\theta_j\mathop{\mathrm{tr}}_h\mathop{\mathrm{Ric}}(g)\). The latter is at most \(T\sup_{M\times[0,T]}\theta_j\mathop{\mathrm{tr}}_h\mathop{\mathrm{Ric}}(g)<\infty\) by the scalar maximum principle on the domains. This proves \(\rho_j\ge\rho-C_j\). Monotone convergence of the nonnegative Dirichlet Green integrals, first in the domains and then for the source cutoffs, shows that \(\rho_j\downarrow0\) pointwise.

We record the initial regularity needed later. On every fixed compact neighborhood the coefficients of \(\Delta_{h_t}\) and the source \(F_j\) are smooth and uniformly parabolic through \(t=0\). The domination \(0\le\rho_j\le\rho=O_K(t)\) supplies continuous zero initial values. For clarity, this initial regularity is local in space. Extend the smooth coefficients and source from a coordinate neighborhood to a smooth uniformly parabolic Cauchy problem, and solve it with zero initial data [32]. Subtract this local particular solution from \(\rho_j\) on a smaller neighborhood. The difference is \(O(t)\) there and solves the homogeneous equation for \(t>0\). A spatially localized energy estimate from time \(\delta>0\), followed by \(\delta\downarrow0\), puts the difference in the local weak energy class through zero; the squared \(L^2\) norm of its initial value on the fixed compact is \(O(\delta^2)\) and tends to zero. Extend that difference by zero to negative times, and extend the coefficients smoothly. Its zero initial trace removes a distributional boundary term, so it solves the homogeneous equation across the initial surface. Interior regularity [32] makes it smooth there. Adding back the particular solution proves the claimed smooth one-sided jets. This argument extends the homogeneous difference, not the forced potential, by zero. In particular, \[\partial_t\rho_j(x,0)=F_j(x,0),\] and higher initial derivatives are determined recursively from \((\rho_j)_t=\Delta_{h_t}\rho_j+F_j\); they are smooth functions of \(x\). No regularity at a distant Dirichlet corner is used. Dini’s theorem on compact spacetime sets now gives the local uniform convergence in (33). If an auxiliary smooth extension across \(t=0\) is desired on a compact, it can be chosen to match these jets. Extension by zero to negative time is not asserted.

For (ii), let \(w\) be the smooth majorant from Lemma 7. Thus \(v\le w\le\rho\), and \(w\) solves equality in (34). Multiplying its equation by \(e^w\) gives the useful volume-form identity \[ \partial_t\bigl[(e^w-1)dV_{h_t}\bigr] =(\Delta_{h_t}w)dV_{h_t}. \tag{37}\] For example, the cancellation uses \(\partial_tdV_h=-R_hdV_h\) and \(e^w(1-e^{-w})R_h=(e^w-1)R_h\).

Integrate (37) against \(\chi p_N\) and move the Laplacian onto that compactly supported function. Since \(w\le e^w-1\) and \(\Delta_h\chi\le C_T\chi\), the interior term is bounded by a constant times \[M_N(t)=\int_M(e^w-1)\chi p_N\,dV_h.\] The time derivative of \(\chi\) is bounded by \(C_T(1+t^{\beta-1})\chi\). The exterior error is again \(O(N^{2n-1-k})\), using \(w\le\rho\) and (36). Moreover \[0\le(e^w-1)dV_h\le e^\rho dV_h=dV_g.\] Thus the full weighted mass is finite, its tails are uniformly small, and its initial limit is zero by dominated convergence on compacts and \(\chi\asymp u^{-k}\). The same limiting Gronwall argument as above forces it to vanish. Hence \(w=0\), and \(0\le v\le w\) proves (ii). ◻

Proof of Lemma 7. We construct the majorant on relatively compact domains and then pass to a limit. On a smooth relatively compact domain \(\Omega\) and for a fixed time less than \(T\), consider equality in (34). Zero and \(\rho\) are respectively a subsolution and a supersolution. To check the latter, use \[R_{h_t}-\Delta_{h_t}\rho=\mathop{\mathrm{tr}}_{h_t}\mathop{\mathrm{Ric}}(g)\ge0.\] The function \(\rho+2\alpha\) is also a supersolution for every constant \(\alpha>0\).

Take constant initial data \(\alpha\). On \(\partial\Omega\) choose a smooth \(\zeta:[0,\infty)\to[0,1]\), equal to one near zero and supported in \([0,1]\), and put \[\varphi(x,t)=\rho(x,t)+\alpha -e^{-\alpha}R_g(x)t\zeta(t/\delta).\] For \(\delta\sup_{\partial\Omega}R_g\le\alpha/2\) this lies between \(\rho+\alpha/2\) and \(\rho+\alpha\). It has \(\varphi(\cdot,0)=\alpha\) and \(\varphi_t(\cdot,0)=(1-e^{-\alpha})R_g\), which are exactly the value and first equation compatibility conditions.

We give the local existence and continuation argument on this fixed compact domain. We use the compatible linear Dirichlet Schauder theorem [32] and its uniform small-time inverse estimate [32]. These Euclidean-domain statements have the same compact-manifold form by the following finite-chart construction for a smooth uniformly parabolic linear operator \(L\). Choose a partition of unity \(\chi_i\) and cutoffs \(\psi_i=1\) near \(\mathop{\mathrm{supp}}\chi_i\) in finitely many interior and boundary charts. Local domains are smoothly capped outside these supports, and \(\psi_i\) vanishes near every artificial boundary. If \(E_i\) is the local zero-data inverse, then \(Sf=\sum_i\psi_i E_i(\chi_i f)\) satisfies \(LSf=f+\mathcal Rf\) for the global linear operator \(L\). The remainder consists of first- and zeroth-order commutators \([L,\psi_i]E_i(\chi_i f)\). On the source space \(f(\cdot,0)=0\), the local solutions have zero initial trace. Parabolic interpolation and the uniform inverse bounds give \(\|\mathcal R\|_{C^{\beta,\beta/2}\to C^{\beta,\beta/2}} \le C\epsilon^\theta\) on a time interval of length \(\epsilon\), with \(\theta>0\); for \(0<\beta<1\) one can take \(\theta=(1-\beta)/2\). The remainder preserves this source space. Thus \(S(I+\mathcal R)^{-1}\) is the required global inverse for small \(\epsilon\). All constants here may depend on \(\Omega\).

Extend the boundary data to a smooth function \(p\) with \[p(x,0)=\alpha,\qquad p_t(x,0)=c(x):=(1-e^{-\alpha})R_g(x) \quad\hbox{throughout }\Omega.\] For example, in a smooth boundary collar with projection \(\pi\) and cutoff \(\vartheta=1\) near the boundary, take \[p(x,t)=\alpha+t c(x) +\vartheta(x)\{\varphi(\pi(x),t)-\alpha-tc(\pi(x))\},\] and use \(\alpha+tc(x)\) outside the collar. The bracket and its first time derivative vanish at zero. Set \(L=\partial_t-e^{-p}\Delta_h\) and seek \(v_{\Omega,\alpha}=p+z\), with zero initial and lateral data for \(z\). Its equation is \[Lz=\mathcal N(z):= (e^{-p-z}-e^{-p})\Delta_h(p+z) +(1-e^{-p-z})R_h-Lp.\] For every such \(z\in C^{2+\beta,1+\beta/2}\), \(\mathcal N(z)(x,0)=0\) throughout \(\Omega\). The linear inverse is therefore applied to compatible sources. If \(z(\cdot,0)=0\), then \(\|z\|_\infty\le\epsilon\|z_t\|_\infty\); spatial interpolation and the time Hölder quotient give \[\|z\|_{C^{\beta,\beta/2}} \le C\epsilon^{1-\beta/2} \|z\|_{C^{2+\beta,1+\beta/2}}.\] On a fixed ball in the latter space, the product and composition estimates consequently give \[\|\mathcal N(z)-\mathcal N(w)\|_{C^{\beta,\beta/2}} \le C\epsilon^{1-\beta/2} \|z-w\|_{C^{2+\beta,1+\beta/2}}.\] The coefficient of every highest-derivative difference has this small factor. Also \(\mathcal N(0)\) is smooth and vanishes at \(t=0\), so its \(C^{\beta,\beta/2}\) norm tends to zero with \(\epsilon\). The linear inverse and the contraction theorem give a local Dirichlet solution. Classical comparison with the barriers gives \(0\le v_{\Omega,\alpha}\le\rho+2\alpha\).

For continuation, write \(\Delta_h=a^{ij}\partial_i\partial_j+b^j\partial_j\) in fixed real coordinates and set \(q=e^{v_{\Omega,\alpha}}\). Directly, \[q_t-\partial_i\left(\frac{a^{ij}}q\,\partial_jq\right) =\frac{b^j-\partial_i a^{ij}}q\,\partial_jq+(q-1)R_h.\] The barriers bound \(q\) above and away from zero. The leading matrix is uniformly elliptic, and the displayed drift and forcing are bounded independently of \(\nabla q\) on the fixed cylinder. Boundary as well as interior Hölder estimates therefore apply [32]; the exterior-measure condition there holds for the smooth compact boundary. Hence \(v_{\Omega,\alpha}\) is Hölder up to the parabolic boundary. Its original equation now has Hölder coefficients and source, so [32] gives a controlled \(C^{2+\gamma,1+\gamma/2}\) norm on the closed slab for some \(0<\gamma<1\). In particular its second spatial derivatives are bounded there. Only this finite corner regularity is asserted from the first compatibility condition.

On slabs separated from the initial surface, differentiated boundary Schauder estimates give all higher bounds, since the coefficients and lateral data are smooth. At a hypothetical positive maximal time the solution thus has smooth limiting initial data satisfying the boundary equation jets. The same local construction, with those data and their first equation jet in place of \(\alpha\) and \(c\), restarts it. This proves existence up to the prescribed finite horizon. No constants near the remote boundaries need be uniform as \(\Omega\) exhausts \(M\).

Viscosity comparison with this smooth solution gives \(v\le v_{\Omega,\alpha}\). To see the only nonlinear point in that comparison, at a positive interior maximum of \(e^{-Ct}(v-v_{\Omega,\alpha})\) the test has the same spatial Hessian as \(v_{\Omega,\alpha}\). The difference of the right sides is bounded by a constant times \(v-v_{\Omega,\alpha}\), because the derivative in the value variable is \(e^{-v}(R_h-\Delta_hv_{\Omega,\alpha})\), bounded on the compact slab. Choose \(C\) larger than this bound and use a strict increasing time penalty. The initial and lateral gaps are positive, so no boundary maximum interferes. This proves the comparison directly, consistently with the usual viscosity framework [15].

Exhaust \(M\), let \(\alpha\downarrow0\), and use local uniformly parabolic estimates to extract a smooth interior limit \(w\) with \[v\le w\le\rho,\qquad w_t=e^{-w}\Delta_hw+(1-e^{-w})R_h, \qquad w(\cdot,0)=0.\] The limit at the initial surface follows from the barriers on each compact; no uniform estimates near the remote lateral boundaries are required. ◻

Proof. Fix \((x,t)\) and use Kähler-normal coordinates for \(h_t\) there. At that point \(h_t=b_t=I\), all first spatial derivatives of \(b_t\) vanish, and the Hessian of \(N\) has blocks \[\partial\bar\partial_YN=\begin{pmatrix}\mathcal R_\xi&0\\0&I\end{pmatrix}.\] The horizontal form \(\mathcal R_\xi\) is the bisectional curvature form of \(h_t\) with one pair of indices contracted against the metric dual of \(\xi\). Indeed differentiating \(b=h^{-\mathsf T}\) twice at a normal point gives the negative second derivatives of \(h\), transposed, which are precisely those curvature components. The Kähler symmetries give \[\mathop{\mathrm{tr}}\mathcal R_\xi=\xi^*\mathop{\mathrm{Ric}}(h_t)^{\mathsf T}\xi.\] On the other hand \(\partial_th=-\mathop{\mathrm{Ric}}(h)\) implies \(\partial_tb=h^{-\mathsf T}\mathop{\mathrm{Ric}}(h)^{\mathsf T}h^{-\mathsf T}\). Equation (41) now proves (42). Adding a base Hessian changes only the horizontal block. Its contribution to \(p\) is \(\mathop{\mathrm{tr}}((\partial\bar\partial_xd)b^{\mathsf T})=\mathop{\mathrm{tr}}(h_t^{-1}\partial\bar\partial_xd)\), proving (43). The calculation is tensorial, so the normal-coordinate verification proves both identities everywhere. ◻

Proof. Choose a fixed smooth function \(\chi:\mathbb R\to[0,1]\) which is zero on \((-\infty,-2]\) and one on \([-1,\infty)\), and set \[\chi_m(z)=\chi(m^{-1}\log|z|),\qquad \phi_m(z)=\frac2m\log|z|+|z|^{1/m}.\] Writing \(r=|z|\), one has \[|\bar\partial\chi_m|\le\frac{C}{mr},\qquad (\phi_m)_{z\bar z}=\frac1{4m^2}r^{1/m-2}.\] On a complex curve Griffiths and Nakano nonnegativity coincide. Apply the bundle-valued Hörmander estimate to \(E\)-valued \((1,0)\)-forms, with the metric \(ke^{-\phi_m}\) and source \(a\,\bar\partial\chi_m\wedge dz\); see [28] and [19]. One may use any complete auxiliary Kähler metric on \(\Delta^*\). In dimension one its conformal factor cancels both from the norm of the unknown top form times volume and from the curvature-inverse source pairing times volume. Dropping the nonnegative curvature of \(k\) therefore gives a solution \(v_m\) of \(\bar\partial v_m=a\,\bar\partial\chi_m\) satisfying \[\begin{align*} \int_{\Delta^*}|v_m|_k^2e^{-\phi_m}\,dA &\le C\int_{e^{-2m}<r<e^{-m}} |a|_k^2r^{-3/m}e^{-r^{1/m}}\,dA,\tag{45}\\ \int_{\Delta^*}|v_m|_k^2r^{-2/m}\,dA &\le C\int_{e^{-2m}<r<e^{-m}}|a|_k^2\,dA\mathrel{% \BeginAccSupp{method=hex,unicode,space=false,pdfliteral=direct,ActualText=27F6}% \longrightarrow% \EndAccSupp{pdfliteral=direct}% }0. \tag{46}\end{align*}\] For the second estimate we used \(r^{-3/m}\le e^6\) on the source annulus and \(e^{-1}\le e^{-r^{1/m}}\le1\). The constants are independent of \(m\). The estimate concerns smooth metrics on the punctured domain; it does not assert a curvature extension at zero.

The sections \(a_m=\chi_ma-v_m\) are holomorphic and converge to \(a\) in unweighted \(L^2\). Near zero they equal \(-v_m\), so each has a positive extra integrability exponent, namely \(2/m\). Since \(A\) is bounded, \(Aa_m\) and \(Aa\) are ordinary \(L^2\) holomorphic vector functions on the punctured disc. Their negative Laurent coefficients vanish by \(L^2\) integrability; hence they extend across zero. Their \(L^2\) convergence and the interior Cauchy estimates give convergence of every finite jet at zero.

Here is the closure argument that recovers a positive exponent for \(a\) itself. Let \(R=\mathcal O_{\mathbb C,0}\), with maximal ideal \(\mathfrak m=(z)\), and let \(S\subset R^q\) consist of all germs \(Ab\) where \(b\) is holomorphic on some punctured neighborhood and \[\int |b|_k^2r^{-\varepsilon}\,dA<\infty \quad\hbox{for some }\varepsilon>0.\] These images extend across zero by the same \(L^2\) argument. The set \(S\) is an \(R\)-submodule: for a sum take the smaller of the two positive exponents, and multiplication by a holomorphic germ is bounded after shrinking the neighborhood. The ring \(R\) is Noetherian, so \(S\) is a submodule of a finite free module and \(Q=R^q/S\) is a finite module. For each \(k\), the image of \(S\) in \(R^q/\mathfrak m^kR^q\) is a finite-dimensional vector subspace and is closed. Jet convergence of \(Aa_m\in S\) consequently gives \[Aa\in S+\mathfrak m^kR^q\qquad\hbox{for every }k.\] Krull’s intersection Theorem for the finite local module \(Q\) gives \(\bigcap_k\mathfrak m^kQ=0\) [53]. Thus \(Aa\in S\). By definition it is the image of one section \(b\) with some positive extra exponent. Pointwise injectivity of \(A\) implies \(a=b\) on a sufficiently small punctured neighborhood, proving (44). No common exponent for \(a_m\) was used. ◻

Proof. We explain the lower-anchor step rather than building it into the compactness theorem. Pass to a subsequence with \(z_j\to z_*\) and work in a coordinate ball about \(z_*\). On a slightly larger ball choose a common upper bound \(C\) and set \(w_j=C-u_j\). These functions are nonnegative and superharmonic. The supermean inequality at \(z_j\) bounds their integrals on a ball of fixed radius about \(z_j\) by its volume times \(C+A\). A fixed smaller ball about \(z_*\) is eventually contained in those balls, so the \(L^1\) norms there are bounded.

Such a bound propagates along finite chains of overlapping coordinate balls. In a fixed overlap of positive measure, the integral bound supplies a point at which \(w_j\) is bounded by its average over the overlap. Applying the supermean inequality on a larger ball centered at that point, still contained in the coordinate neighborhood, bounds the integral on the next smaller ball. On each fixed compact overlap, the Euclidean volume measures in the two coordinate charts are uniformly comparable. Increasing the local upper bound \(C\) when moving between charts only adds a fixed constant. Connectedness and a finite covering therefore give \(L^1\) bounds on every fixed compact subset of \(X\).

Families of subharmonic functions bounded in \(L^1_{\mathrm{loc}}\) are relatively compact in that topology [20]. Apply this result in coordinate balls and take a diagonal subsequence. Positivity of the complex Hessians passes to the distributional limit, so its upper semicontinuous representative is plurisubharmonic. If no subsequence has a lower anchor in any fixed compact set, then the suprema on each member of a compact exhaustion tend to \(-\infty\); this is the other alternative. This also proves the assertion for exhausting domains, since each finite chain of balls is contained in every sufficiently late \(\Omega_j\).

Finally suppose the limit is the constant \(c\). In a coordinate ball whose concentric enlargement is relatively compact, apply the submean inequality on a fixed-radius ball centered at each point of the smaller ball. It bounds \(u_j-c\) from above by a fixed constant times the \(L^1\) norm of \(u_j-c\) on the enlargement. This norm tends to zero. A finite covering proves the asserted upper bound on \(K\). ◻

Proof of Proposition 9. We first control centers of discs \(f_l\) in \(M\). The maximum principle gives uniform bounds \[ |F\circ f_l|\le C_K,\qquad |dF|_g\circ f_l\le C_K,\qquad \psi_l:=\psi\circ f_l\le C_K. \tag{47}\] The area bound controls the energy, with a fixed normalization constant. In particular, the functions \[E_l(\theta)=\int_{1/2}^1 |\partial_rf_l(re^{i\theta})|_g^2r\,dr\] have uniformly bounded angular integrals. On a set of angles of measure bounded below, \(E_l(\theta)\le C(A_0+1)\). For each such angle and every \(r\in[1/2,3/4]\), Cauchy–Schwarz gives \[\mathop{\mathrm{dist}}_g(f_l(re^{i\theta}),K) \le E_l(\theta)^{1/2} \left(\int_{1/2}^1\frac{dr}{r}\right)^{1/2}.\] Completeness and Hopf–Rinow put these images in a fixed compact. Thus on a subset of the fixed annulus of area bounded below, \(\psi_l\) has a fixed lower bound. Lemma 11, together with (47), now gives a subsequence converging in \(L^1_{\mathrm{loc}}(\Delta)\) to a subharmonic function \(\psi_\infty\); the lower bounds on the anchor subsets exclude collapse to \(-\infty\). We retain their positive-area property for the later step that avoids neighborhoods of the possible atoms. Let \(\mu_l=(2\pi)^{-1}\Delta_{\mathrm{eucl}}\psi_l\) and \(\mu=(2\pi)^{-1}\Delta_{\mathrm{eucl}}\psi_\infty\). Then \(\mu_l\to\mu\) weakly on compact subsets.

We spell out the required inverse-Jacobian estimate. Let \(Z=\{z:\mu(\{z\})\ge1\}\), a locally finite subset of \(\Delta\). Near any point outside \(Z\), choose a small disc \(D\) whose boundary has zero \(\mu\)-measure and for which \(\mu(D)<b<4/3\). For large \(l\), \(\mu_l(D)<b\). The Riesz decomposition on \(D\) [20] is \[\psi_l=h_l+\int_D\log|z-\zeta|\,d\mu_l(\zeta).\] On each smaller concentric disc the harmonic terms \(h_l\) are bounded in absolute value: their \(L^1\) norms are bounded by \(L^1\) convergence of \(\psi_l\), the bounded masses, and local integrability of the logarithmic kernel; harmonic interior estimates then apply. For a positive measure \(\nu\) of mass \(b_l\le b\), Jensen’s inequality gives, with the zero-mass case interpreted separately, \[\exp\left(-p\int_D\log|z-\zeta|\,d\nu(\zeta)\right) \le\frac1{b_l}\int_D|z-\zeta|^{-pb_l}\,d\nu(\zeta), \qquad p=\tfrac32.\] Integrating in \(z\) gives a uniform bound since \(pb<2\). Therefore \(e^{-\psi_l}\) is locally bounded in \(L^{3/2}(\Delta\setminus Z)\).

Let \(s_1\le\cdots\le s_n\) be the singular values of \(dF\) measured from \(g\) to the Euclidean metric. Cauchy–Binet and (47) imply \[e^{\psi_l}=\prod_i s_i(f_l)^2,\qquad s_1(f_l)^{-2}\le C_K e^{-\psi_l}.\] Consequently \[ |f_l'|_g^2\le C_K e^{-\psi_l}|(F\circ f_l)'|^2. \tag{48}\] The bounded holomorphic maps \(F\circ f_l\) have uniformly bounded derivatives on smaller discs. Hence \(df_l\) is locally bounded in \(L^3\) off \(Z\). For \(p\in M\) the function \(v_{l,p}(z)=\mathop{\mathrm{dist}}_g(f_l(z),p)\) has weak gradient bounded by \(|df_l|_g\). Morrey’s local gradient-only estimate in real dimension two, with exponent \(1-2/3=1/3\), therefore applies [21]. Its constant depends on the nested domain discs, not on \(p\); no bound on the values of \(v_{l,p}\) is needed. Taking \(p=f_l(w)\) gives the uniform local metric Hölder estimate.

These local estimates propagate the compact anchors. Indeed, remove small neighborhoods of the finitely many points of \(Z\) in the anchor annulus, with total area smaller than half the anchor lower bound. Choose anchor points in the remaining compact set and pass to a convergent subsequence of their domain positions. The local Hölder estimate puts their neighboring images in a fixed bounded \(g\)-neighborhood of the anchor compact. Any compact subset of the connected set \(\Delta\setminus Z\) can be reached by finitely many overlapping small discs of this kind. Completeness makes each successive bounded neighborhood compact. Arzelà–Ascoli, local holomorphic coordinates, and diagonal extraction give a holomorphic limit \(f:\Delta\setminus Z\to M\). Its area is finite by lower semicontinuity, and \(dF\) remains uniformly bounded along it.

Fix a puncture of this limit and move its domain coordinate to zero. The bundle \(E=f^*T^{1,0}M\) has Griffiths nonnegative curvature, \(A=dF:E\to\mathbb C^N\) is bounded and injective, and the holomorphic section \(a=f'\) has finite \(L^2\) norm. Lemma 10 applies. Fubini then gives an angle \(\theta\) with \[\int_0^{r_0}|a(re^{i\theta})|_g^2r^{1-\sigma}\,dr<\infty.\] Since \(\int_0^{r_0}r^{\sigma-1}\,dr<\infty\), the corresponding radial path has finite length and converges to a point \(p\in M\). The bounded map \(F\circ f\) extends holomorphically to zero and its value there is \(F(p)\). Choose a relatively compact coordinate neighborhood \(U\) of \(p\) with compact boundary. Injectivity of \(F\) gives \[\mathop{\mathrm{dist}}_{\mathrm{eucl}}(F(p),F(\partial U))>0.\] For small enough \(r\), \(F\circ f\) on \(0<|z|<r\) avoids \(F(\partial U)\), and the radial path supplies a point with image in \(U\). Connectedness traps the entire punctured image in \(U\). Bounded coordinate functions extend, so \(f\) extends holomorphically. This argument uses compactness of a local chart boundary and global injectivity of \(F\); it does not require \(F\) to be proper.

Off the puncture, \(\psi_\infty=\psi\circ f\) almost everywhere. The latter now extends smoothly across it, so the two functions define the same distribution there and \(\mu\) has no atom at the puncture. Thus every alleged point of \(Z\) is removable and has zero atomic mass. The preceding exponential-integrability and Morrey estimates apply across it as well, and the original sequence converges on all compact subsets of \(\Delta\). Since this applies to every sequence, the family of centers is relatively compact.

If complete images could escape every compact, choose an escaping interior point on each disc and precompose with a disc automorphism sending zero to that point. Both the boundary compact and the area bound are unchanged, contradicting the center conclusion. This proves the first assertion.

For the cotangent assertion take global holomorphic fields \(X_1,\ldots,X_m\) spanning \(T^{1,0}M\), as supplied by Proposition 2. The base images are compact by the first part. The functions \(\xi(X_i)\) along each disc are bounded by their boundary values. Over the resulting compact base the evaluation map \(\xi\mathrel{% \BeginAccSupp{method=hex,unicode,space=false,pdfliteral=direct,ActualText=21A6}% \mapsto% \EndAccSupp{pdfliteral=direct}% }(\xi(X_i))_i\) has a uniform positive least singular value. Its bounded values therefore bound the cotangent vectors and put the images in a compact subset of \(Y\). Local holomorphic normal-family compactness proves the stated subsequence conclusion. ◻

Proof. Equip the vector Hardy space with \(\|v\|_L^2=\langle v^*Lv\rangle\), where brackets denote normalized circle average. This norm is equivalent to the ordinary Hardy norm. The shift \(v\mathrel{% \BeginAccSupp{method=hex,unicode,space=false,pdfliteral=direct,ActualText=21A6}% \mapsto% \EndAccSupp{pdfliteral=direct}% }zv\) is an isometry, its range has codimension \(r\), and \(\bigcap_{k\ge0}z^kH^2=\{0\}\). An orthonormal basis \(p_1,\ldots,p_r\) of the orthogonal complement of the range, followed by all its successive shifts, is therefore an orthonormal basis of the weighted space. Indeed the orthogonal remainder after the first \(k\) shift levels is \(z^kH^2\), whose intersection is zero.

Let \(P\) be the holomorphic matrix with columns \(p_i\). Orthogonality of all their shifts says that every Fourier coefficient of \(P^*LP-I\) vanishes, so \(P^*LP=I\) almost everywhere on the circle. Thus \(P\) is bounded. Multiplication by \(P\) maps the ordinary Hardy space isometrically onto the weighted space by the basis property. Apply this to constant target columns to obtain a holomorphic matrix \(Q\) of Hardy functions with \(PQ=I\). Consequently \(P\) is invertible everywhere, \(Q=P^{-1}\), and the boundary identity gives \(L=Q^*Q\). It also bounds both \(P\) and \(Q\) in \(H^\infty\) by their boundary values. Taking \(H=Q\) proves (53).

For the seminorm estimate put \(L_\theta(z)=L(e^{i\theta}z)\) and \(A_\theta(z)=H(e^{i\theta}z)H(z)^{-1}\). This holomorphic matrix has \(A_\theta(0)=I\). Its mean is therefore \(I\), and \[E_\theta:=\langle\|A_\theta-I\|_F^2\rangle =\langle\mathop{\mathrm{tr}}(L_\theta L^{-1}-I)\rangle.\] Changing variables in the expression for \(E_{-\theta}\) gives \[\begin{align*} E_\theta+E_{-\theta} &=\left\langle\mathop{\mathrm{tr}}\big((L_\theta-L)L^{-1} (L_\theta-L)L_\theta^{-1}\big)\right\rangle\\ &\le m^{-2}\langle\|L_\theta-L\|_F^2\rangle. \end{align*}\] The identities \[H_\theta-H=(A_\theta-I)H,\qquad H_\theta^{-1}-H^{-1}=H_\theta^{-1}(I-A_\theta)\] bound their squared differences by \(ME_\theta\) and \(m^{-1}E_\theta\), respectively. Integrating against the symmetric kernel in (50) proves (54).

It remains to justify smooth boundary regularity. The same difference inequality, divided by \(\theta^2\), first gives one full \(L^2\) tangential derivative of \(H\) and \(H^{-1}\). With prime denoting that derivative, set \(B=H'H^{-1}\); this is the boundary value of \(izH_zH^{-1}\) and has zero constant Fourier coefficient. Differentiating \(L=H^*H\) gives \[H^{-*}L'H^{-1}=B+B^*.\] The strictly positive Fourier projection recovers \(B\). The one-dimensional Sobolev algebra property now bootstraps: if \(H,H^{-1}\in W^{k,2}\), the displayed expression is in \(W^{k,2}\), hence so is \(B\), and \(H'=BH\) gives \(H\in W^{k+1,2}\). The inverse derivative formula gives the same for \(H^{-1}\). For this one-dimensional algebra assertion, Cauchy–Schwarz on Fourier coefficients and \(\sum_k(1+k^2)^{-1}<\infty\) give \(W^{1,2}(S^1)\subset L^\infty(S^1)\). In each Leibniz term of total order at most \(k\ge1\), place a derivative of order at most \(k-1\) in \(L^\infty\) and the other factor in \(L^2\); this gives the \(W^{k,2}\) product estimate. The positive-frequency projection has norm at most one in every \(W^{k,2}\). Iterating the preceding identities therefore gives all Sobolev orders, and the same Fourier estimate for derivatives proves smoothness on the closure. ◻

Proof of Lemma 12. We first construct a preliminary frame on a neighborhood of the closed disc. The bundle \(E=f^*TY\) is embedded by \(dG\) in a trivial bundle and is generated by the pullbacks of the finite global spanning fields. Start on a disc of radius greater than one and choose a generator not identically zero. On a slightly smaller disc, divide it by the finitely many common zero factors of its ambient components, using at each zero the minimum component vanishing order. The resulting section \(e\) is holomorphic, remains in \(E\) by continuity, and is nowhere zero.

Write its ambient components as \(e_1,\ldots,e_q\) and put \(\ell_j=\overline{e_j}/|e|^2\). Thus \(\sum_j e_j\ell_j=1\). The smooth \((0,1)\)-forms \[b_{jk}=\ell_j\bar\partial\ell_k-\ell_k\bar\partial\ell_j\] are antisymmetric and satisfy \(\sum_j e_jb_{jk}=\bar\partial\ell_k\). Solve \(\bar\partial c_{jk}=b_{jk}\) with \(c_{jk}=-c_{kj}\) on a further slightly smaller disc. This scalar step follows directly by multiplying the coefficient of each \((0,1)\)-form by one common smooth compactly supported cutoff equal to one on that disc and convolving on \(\mathbb C\) with \(1/(\pi z)\); the distributional identity \(\bar\partial(1/(\pi z))=\delta_0\) gives a smooth solution for the smooth source. It does not use a frame of \(E\). The functions \[a_k=\ell_k-\sum_j e_jc_{jk}\] are holomorphic and obey \(\sum_k e_ka_k=1\). The ambient row \(a\) restricts to a holomorphic dual of \(e\) on \(E\), so \(E=\mathbb Ce\oplus\ker a\). If \(s_i\) are the current generators, then \(s_i-e\,a(s_i)\) generate \(\ker a\). Thus the kernel is again an ambiently embedded, finitely generated holomorphic bundle, now of rank one less. Repeat the construction on a finite nested sequence of discs whose radii remain greater than one. Induction gives a holomorphic frame of \(f^*TY\) on a neighborhood of \(\overline\Delta\).

Choose this preliminary frame and apply Lemma 13 to its \(dG\) Gram matrix on the circle. Multiplication of the frame by \(H^{-1}\) yields \(E_f\). The first identity in (51) follows immediately. For each constant column \(v\), the maximum principle applied to the holomorphic vector \(Uv\) gives \(|Uv|\le|v|\), proving the second.

Put \(Q=U^*U\). Since \(Q=I\) on the circle, \[\partial_r\mathop{\mathrm{tr}}Q=\mathop{\mathrm{tr}}(\partial_rQ) =\partial_r\log\det Q \quad\hbox{on }\partial\Delta.\] Green’s formula and holomorphicity give \[4\pi D(U)^2 =\int_\Delta\Delta_{\mathrm{eucl}}\mathop{\mathrm{tr}}Q\,dA =\int_\Delta\Delta_{\mathrm{eucl}}\log\det Q\,dA =2\int_\Delta(\mathfrak g\circ f)^*\omega_{\mathop{\mathrm{Gr}}},\] which proves (52).

On the boundary, \(E_f\) is orthonormal for the metric \(G^*g_{\rm eucl}\). Thus the components of \(\mathcal J\) there are bounded by its norm in that metric on \(K\). They are holomorphic functions in the frame, so the maximum principle gives the same bound inside. For the last assertion, write \(E_f=\partial S\) in the fixed chart at the center, and let \(J\) be the invertible coordinate matrix of \(\mathcal J\). Then \[J_{E_f}=S^{-1}JS^{-\mathsf T},\qquad S^{-1}=J_{E_f}S^{\mathsf T}J^{-1}.\] Let \(\gamma>0\) be the infimum on \(K_0\) of the least singular value of \(dG\) in the coordinate Euclidean metric. By (51), \(\|S(0)\|_{\mathrm{op}}\le\gamma^{-1}\). The preceding identity gives \[\|S(0)^{-1}\|_{\mathrm{op}} \le \sup_\Delta\|J_{E_f}\|_{\mathrm{op}}\, \gamma^{-1}\sup_{K_0}\|J^{-1}\|_{\mathrm{op}}.\] Every quantity on the right has the stated dependence. ◻

Proof. We give the realization, Taylor justification, and actual mixed-derivative estimate separately.

Realization of the first field. Let \(Z_1,\ldots,Z_b\) be the global holomorphic spanning fields on \(Y\). In the immersion \(G\), let \(A\) be their column matrix along \(f\). If \(r=\dim_\mathbb CY\), enumerate all \(r\)-row, \(r\)-column minors \(a_i\) of \(A\). On \(K_{j,\varepsilon}'\) the sum of their squared moduli has a common positive lower bound. Their sup norms and Dirichlet norms are bounded at the fixed parameters: the minors are smooth holomorphic functions of \(X\) on this compact, and \(D(X)\) is bounded. Set \[\sigma_i=\frac{\overline{a_i}}{\sum_k|a_k|^2},\qquad b_{ik}=\sigma_i\bar\partial\sigma_k- \sigma_k\bar\partial\sigma_i.\] These \((0,1)\) coefficients have bounded \(L^2(\Delta)\) norm. Solve \(\bar\partial Q_{ik}=b_{ik}\), skew-symmetrically, with bounded \(W^{1,2}\) norm. Explicitly, if \(b_{ik}=b_{ik,0}\,d\bar\zeta\), take \(Q_{ik}=4\partial_\zeta\Delta_D^{-1}b_{ik,0}\), with the convention \(\Delta=4\partial_\zeta\partial_{\bar\zeta}\). Here \(\Delta_D^{-1}\) is the zero-Dirichlet inverse on the unit disc. The unit disc has smooth boundary, the principal coefficients are the identity, and all lower coefficients vanish. The scalar Dirichlet estimate therefore gives \[\|\Delta_D^{-1}b_{ik,0}\|_{W^{2,2}(\Delta)} \le C\|b_{ik,0}\|_{L^2(\Delta)}\] [23], proving the asserted \(W^{1,2}\) bound. Linearity preserves skew symmetry. Smooth input on the closed disc gives smooth output there [23].

Here is the precise boundary estimate. For a smooth function \(q\) on the closed disc with boundary Fourier coefficients \(a_k\), let \(h(re^{i\theta})=\sum_k a_kr^{|k|}e^{ik\theta}\) be its harmonic extension. Integration by parts gives \[\int_\Delta|\nabla q|^2 =\int_\Delta|\nabla h|^2+\int_\Delta|\nabla(q-h)|^2 \ge2\pi\sum_k|k||a_k|^2.\] If \(q_0(r)\) is the angular average, then for \(1/2\le r\le1\) \[|q_0(1)|^2\le2|q_0(r)|^2 +2(1-r)\int_r^1|q_0'(s)|^2\,ds.\] Integrating in \(r\) and applying Jensen bounds the constant coefficient \(|a_0|^2\) by \(C\|q\|_{W^{1,2}(\Delta)}^2\). The nonconstant \(L^2\) Fourier sum is bounded by its \(|k|\)-weighted sum. Consequently \[\|q|_{\partial\Delta}\|_{L^2}^2+D(q|_{\partial\Delta})^2 \le C\|q\|_{W^{1,2}(\Delta)}^2.\] Applied entrywise to \(Q_{ik}\), this bounds its boundary \(D\) and \(L^2\) norms. The operator constants are fixed-disc constants; the input norms can still depend on \(K'_{j,\varepsilon}\). The functions \[g_i=\sigma_i+\sum_kQ_{ik}a_k\] are holomorphic and \(\sum_i g_i a_i=1\): differentiating uses \(\sum_k a_k\sigma_k=1\) and \(\sum_k a_k\bar\partial\sigma_k=0\), while skew symmetry cancels the quadratic term in the sum.

A \(W^{1,2}\) trace need not be bounded. For a large number \(P\) let \(\alpha\) be the scalar outer function with positive center value and boundary modulus \[r_P=(1+|Q|^2/P)^{-1}.\] All norms and products involving \(Q\) in the following boundary estimates refer to its trace on the circle. The maps \(Q\mathrel{% \BeginAccSupp{method=hex,unicode,space=false,pdfliteral=direct,ActualText=21A6}% \mapsto% \EndAccSupp{pdfliteral=direct}% }\log(1+|Q|^2/P)\) have Lipschitz constant \(C/\sqrt P\), and their \(L^2\) norms are at most \(C\|Q\|_2/\sqrt P\). The maps \(Q\mathrel{% \BeginAccSupp{method=hex,unicode,space=false,pdfliteral=direct,ActualText=21A6}% \mapsto% \EndAccSupp{pdfliteral=direct}% }r_PQ\) have bounded Lipschitz constant and sup norm at most \(C\sqrt P\). The phase of \(\alpha\) is the Hilbert transform of \(\log r_P\). Thus \[\|\alpha\|_\infty\le1,\quad D(\alpha)\le C D(Q)/\sqrt P,\quad D(\alpha Q)\le C D(Q),\quad \|\alpha Q\|_\infty\le C\sqrt P.\] In the third estimate the \(\sqrt P\) from the bounded product multiplies the \(P^{-1/2}\) bound for the phase. Moreover, \[\langle1-|\alpha|^2\rangle\le C\|Q\|_2^2/P, \qquad -\log\alpha(0)=\langle\log(1+|Q|^2/P)\rangle\mathrel{% \BeginAccSupp{method=hex,unicode,space=false,pdfliteral=direct,ActualText=27F6}% \longrightarrow% \EndAccSupp{pdfliteral=direct}% }0.\] Given \(\eta>0\), choose \(P\) uniformly for the sequence so that these last two quantities are at most \(\eta\) and \(D(\alpha)\le1\). The boundary estimates bound the sup norms and \(D\) norms of \(\alpha g_i\). These functions are holomorphic, so the maximum principle also bounds their interior sup norms at the fixed parameters; it is not applied to the nonholomorphic products \(\alpha Q\).

For each minor \(a_i\), Cramer’s rule supplies a vector \(c_i\), holomorphic in the disc, with \(Ac_i=a_iUl\). Namely use the adjugate on its selected rows and columns and set the other coefficient entries to zero. The identity follows by expanding the augmented determinants: every \((r+1)\)-minor of \((A,Ul)\) is zero because this matrix still has rank \(r\). This also treats selected minors that vanish identically. Consequently \(c=\sum_i\alpha g_i c_i\) has bounded sup norm and \(D\) and obeys \(Ac=\alpha Ul\). Write \(c=(c_1,\ldots,c_b)\) and compose the complex flows of \(Z_1,\ldots,Z_b\) with times \(z h c_1,\ldots,z h c_b\), starting at \(f\). For a sufficiently small common complex parameter \(z\) this gives a holomorphic family \(f_z\), smooth up to the circle, whose first field is \[ V=\left.\partial_z f_z\right|_0=\alpha hE_fl. \tag{70}\] The existence radius is uniform at the fixed parameters, since the initial images lie in \(K_{j,\varepsilon}'\) and all flow times are bounded. On the boundary the displacement is \(m\) times a uniformly bounded smooth function. Because \(m\le\delta_i^a\), decreasing this radius makes each new \(\delta_i\) comparable to its old value, with constants between \(1/2\) and \(3/2\). Thus the spatial boundary restrictions persist. The variable \(w\) and all times \(t\) are unchanged, so no uniform margin below the upper time boundary is needed. The center stays in \(\mathcal C\) by its already established margin.

Uniform Taylor remainders at the fixed parameters. We spell this out because smoothness of individual discs does not by itself imply the uniform expansion needed for an infimum. On the controlled interior compact the flow map and all its derivatives through any fixed finite order are bounded. Differentiating in the disc variable introduces one derivative of \(f\), \(h\), or \(c\). Their Dirichlet norms are bounded; for \(f\) use local metric equivalence on \(K_{j,\varepsilon}'\) and (59). Hence the family and its first three real parameter derivatives have bounded \(W^{1,2}\) norms, with bounded sup norms for their values. Differentiating the area integrands through order three yields only bounded coefficients times products of at most two such first disc derivatives. Cauchy–Schwarz supplies uniform integrable bounds. The same reasoning applies to the Gauss map, whose derivatives are bounded on the controlled compact.

At the boundary write \(\beta=h/m\). The boundary values of \(f_z\) can be written in ambient coordinates as a smooth function of \(X\), \(c\), \(\beta\), and \(z m\), on a fixed compact range. More precisely, for either cutoff, \[\delta_i(f_z)=\delta_i(f)+m R_i(z,X,c,\beta),\qquad R_i(0,\cdot)=0,\] where the real parameter derivatives through order three of \(R_i\) have bounded sup and \(D\) norms at the fixed parameters, by (64). Since \(m/\delta_i=\delta_i^{a-1}\delta_{3-i}^{a}\) is smooth on the closed boundary region, the function \[\log\delta_i(f_z)-\log\delta_i(f) =\log\bigl(1+(m/\delta_i)R_i\bigr)\] and its parameter derivatives through order three have uniformly bounded \(D\) norms. For reciprocal walls, use \[\frac1{\delta_i(f_z)}-\frac1{\delta_i(f)} =-\frac{m}{\delta_i^2}\, \frac{R_i}{1+(m/\delta_i)R_i}.\] The prefactor is smooth and bounded when \(a\ge2\). Therefore all nonzero parameter derivatives of the wall costs through order three are uniformly bounded, even when the unchanged wall cost is large. The remaining boundary integrands \(N,d_{0,j}\) have bounded derivatives on \(K_j\times[0,T]\). Finally the quadratic seminorm of the logarithmic wall has three bounded parameter derivatives by its Hilbert space pairing formula. These observations prove a uniform \(O(|z|^3)\) remainder after parameter-circle averaging of the entire functional minus the test. The constants in this paragraph are allowed to depend on the interior compact and on \(\eta\).

The actual mixed energy derivative uses boundary constants only. Set \(\widehat V=dG\,V=\alpha hUl\) on the circle. Equations (64)–(70) imply \[ \|\widehat V\|_\infty\le L, \qquad D(\widehat V)^2\le C_j(L)\bigl(1+\mathcal E+D(l)^2\bigr). \tag{71}\] Here only \(\|\alpha\|_\infty\le1\) and \(D(\alpha)\le1\) were used; the corona coefficients have disappeared.

For clarity, consider any one of the closed \((1,1)\) forms \(\omega\) in the three spatial area terms of (59). It is a smooth closed form on \(Y\) after pullback. In ambient coordinates near the boundary write its intrinsic tensor as \(i b_{p\bar q}(X)\,dX^p\wedge d\bar X^q\), extending the coefficients smoothly from \(G(\mathcal W_j)\) near \(G(K_j)\) as above. Deformed boundaries stay intrinsically in \(\mathcal W_j\) for a smaller parameter radius. Closure is used intrinsically before this extension. Cartan’s formula and Stokes give \[ \begin{split} \left.\partial_z\partial_{\bar z} \int_\Delta f_z^*\omega\right|_0 =i\int_0^{2\pi}\bigl[& b_{p\bar q}(X)\widehat V^p\partial_\theta \overline{\widehat V^q}\\ &+(\partial_{\bar r}b_{p\bar q})(X) \widehat V^p\overline{\widehat V^r}\, \partial_\theta\overline{X^q}\bigr]\,d\theta. \end{split} \tag{72}\] One can also verify the identity using local potentials: the mixed parameter derivative equals the boundary normal derivative of \(\omega(V,\bar V)\), with the corresponding Stokes normalization. In (72), the holomorphic dependence of \(Gf_z\) eliminates a mixed second parameter field. The identity is global; it does not require partitioning \(\omega\) into nonclosed forms.

The coefficient \(b\) has bounded \(C^2\) norm determined only by \(K_j\). For \(B=\|\widehat V\|_\infty\), \[\begin{split} D(b(X)\widehat V)&\le C_j\bigl(D(\widehat V)+B D(X)\bigr),\\ D((\bar\partial b)(X)\widehat V\overline{\widehat V}) &\le C_j\bigl(BD(\widehat V)+B^2D(X)\bigr). \end{split}\] Pairing each angular derivative by (64) bounds the absolute value of (72) by \(C_j(L)(D(X)^2+D(\widehat V)^2)\). This proves the required estimate for all spatial areas. The \(w\) area is unchanged. In particular for the Euclidean area the identity reduces, up to its fixed normalization, to \(D(\widehat V)^2\).

It remains to check the logarithmic seminorm; its unbounded argument cannot be handled by an unweighted smooth-composition assertion. On the boundary put \(Z=\alpha\beta Ul\), so that \(\widehat V=mZ\). Then \[\|Z\|_\infty\le L,\qquad D(Z)\le C_j(L)(1+\sqrt{\mathcal E}+D(l)).\] Regarding \(p=\delta_1\delta_2\) as a smooth ambient function, the tensors \[ \begin{split} m\partial\log p&=p^{a-1}\partial p,\\ m^2\partial\bar\partial\log p&=p^{2a-1}\partial\bar\partial p -p^{2a-2}\partial p\otimes\bar\partial p \end{split} \tag{73}\] extend smoothly to the closed boundary region. For \(q_z=\log p(f_z)\) their contractions against \(Z\) and \((Z,\bar Z)\) are exactly \(\partial_zq_z|_0\) and \(\partial_z\partial_{\bar z}q_z|_0\), respectively. Consequently each has \(D\) bounded by \(C_j(L)(1+\sqrt{\mathcal E}+D(l))\). Write \(b=\partial_zq_z|_0\) and \(c=\partial_z\partial_{\bar z}q_z|_0\), and let \(B_D\) be the sesquilinear pairing \(B_D(u,v)=\sum_k|k|u_k\overline{v_k}\). Since \(q_z\) is real, differentiating this quadratic seminorm gives exactly \[ \left.\partial_z\partial_{\bar z}D(q_z)^2\right|_0 =2D(b)^2+2\Re B_D(q,c). \tag{74}\] Since \(D(q)^2\le\mathcal E\), Cauchy–Schwarz proves \[ \left|\left.\partial_z\partial_{\bar z} \mathcal E(f_z,w)\right|_0\right| \le C_j(L)(1+\mathcal E+D(l)^2). \tag{75}\] All constants in (75) use boundary coefficients only, independently of the realization and Taylor constants.

Testing the infimum and removing the auxiliary loss. Every deformed value of (60) is at least \(\mu_{j,\varepsilon}\), while the values at \(z=0\) approach it. Average over \(|z|=r\), subtract the value at zero, and divide by \(r^2\). First take the minimizing-sequence limit with \(r\) fixed, and then \(r\downarrow0\). The uniform remainder just proved yields a nonnegative lower limit for the mixed derivative of cost minus test. Its nonpenalty terms are \[\langle|\alpha|^2 T_b(l,\bar l)\rangle -|\alpha(0)|^2h(0)^2\varphi_{E,c}(l(0),\overline{l(0)}).\] The center frames and their inverses are bounded independently of interior area by Lemma 12. Thus the test term is bounded, as is the boundary symbol by (68). Removing \(\alpha\) changes these terms by at most \(C_{j,L,\varphi}\eta\). Equation (75) has a coefficient independent of \(\eta\). Send \(\eta\downarrow0\) after the sequence and radius limits. This proves (69) with the stated uniformities. In particular, none of the constants controlling the shrinking realization radius multiplies the final \(\varepsilon\mathcal E\) error. ◻

Proof. Write \(\mathcal H=\left(\begin{smallmatrix}A&B\\B^*&D\end{smallmatrix}\right)\), \(S=A-BD^{-1}B^*\) and \(R=D^{-1}B^*\). The complex-linear change \((x,y)\mathrel{% \BeginAccSupp{method=hex,unicode,space=false,pdfliteral=direct,ActualText=21A6}% \mapsto% \EndAccSupp{pdfliteral=direct}% }(x,y+Rx)\) makes the form block diagonal with blocks \((S,D)\). It carries the canonical bivector to \[\sum_i e_i\wedge f_i+\sum_{k,i}R_{ki}f_k\wedge f_i.\] Its cross and vertical components are orthogonal for the induced exterior-square form, whence \[p(\mathcal H)=\mathop{\mathrm{tr}}(SD^{\mathsf T})+ \left|\sum_{k,i}R_{ki}f_k\wedge f_i\right|_D^2.\] The shear need not be symplectic; its additional vertical term is retained and is nonnegative. Since \(D\ge C\) and \(S>0\), the last identity gives \(p(\mathcal H)\ge\mathop{\mathrm{tr}}(SC^{\mathsf T})\). Finally, for each horizontal \(x\), \[S(x,\bar x)=\inf_y\mathcal H((x,y),\overline{(x,y)}) \ge\inf_y W((x,y),\overline{(x,y)}) =\mathop{\mathrm{Schur}}(W)(x,\bar x).\] The last infimum is finite and has the usual Schur expression because \(C>0\). This proves the Lemma. ◻

Completion of the proof of Proposition 14. Combine (78), (83), (84), (85), and (86). More explicitly, if \(r\ge0\) denotes the error tending to zero in (78), the form with frame matrix \[\mathcal H_E=(1+r)h(0)^{-2}H(0)^*H(0)\] is a positive majorant of the test Hessian at its center, after an arbitrarily small enlargement if necessary. Transform it to the fixed cotangent coordinates. Its symplectic cost is \[p(\mathcal H)=(1+r)^2h(0)^{-4} p(H(0)^*H(0),\mathcal J_E(0)).\] For fixed \(j,\delta\) first take the minimizing-sequence limits and \(\varepsilon\downarrow0\); then send \(\delta\downarrow0\); finally let \(j\to\infty\). Equations (63) and (82) give a diagonal sequence of approaching centers and positive majorants \(\mathcal H\) such that \[\limsup p(\mathcal H)\le\partial_s\varphi(y_*,s_*).\] No compactness of the complete matrices \(\mathcal H\) is needed. Their test vertical blocks \(C_c\) converge to \(C>0\), so for late centers \(C_c\ge cI\) with a fixed \(c>0\). Lemma 16 bounds their positive Schur forms \(S_c\) by \(c\mathop{\mathrm{tr}}S_c\le p(\mathcal H)\). Pass to a convergent subsequence \(S_c\to K\). The same Lemma and continuity of the test Hessian and its Schur complement give \[K\ge0,\qquad K\ge\mathop{\mathrm{Schur}}(W),\qquad \mathop{\mathrm{tr}}(C^{\mathsf T}K)\le\partial_s\varphi(y_*,s_*).\] This is (55) and proves the Proposition. ◻

Proof. The initial cometric \(b_0\) is feasible by (88). Every feasible matrix satisfies \(D\le b_t\). The constraints therefore define a compact set if positive semidefinite matrices are temporarily admitted. A determinant maximizer in that set has determinant at least \(\det b_0>0\), so is positive definite. Strict concavity of \(\log\det\) on the positive cone gives uniqueness. It also follows that \(\det b_0\le\det B\le\det b_t\), which proves (90), since \(\rho=\log\det b_t-\log\det b_0\).

Fix a feasible positive matrix \(D\) at \((x_0,t_0)\). For \(0<\varepsilon<1\), the candidate \((1-\varepsilon)D\) has a strictly positive margin on the unit fiber sphere. Lower semicontinuity of \(v\) and compactness of that sphere preserve this margin for all nearby \((x,t)\). Consequently \[\liminf_{(x,t)\to(x_0,t_0)}\mathcal L(x,t) \ge\log\det D+n\log(1-\varepsilon).\] Take \(D=B(x_0,t_0)\) and then \(\varepsilon\downarrow0\). This proves the asserted semicontinuity without asserting continuity of the optimizing matrix. Under a change of holomorphic base coordinates, both \(\mathcal L\) and \(\log\det b_t\) acquire the same time-independent pluriharmonic summand. Thus \(F_0\) is a globally defined upper semicontinuous function.

Normalize \(B(x_0,t_0)=I\), using a constant linear base-coordinate change and its induced cotangent change. On the unit sphere put \[E=\{\xi:v(x_0,\xi,t_0)=1\}.\] This is compact, because \(v\ge|\xi|^2\) at the chosen base point. Suppose that a Hermitian matrix \(H\) is strictly negative on \(E\). It is uniformly negative in a neighborhood of \(E\) on the sphere; on the complementary compact set, \(v-1\) has a positive minimum. It follows that \(I+\varepsilon H\) is feasible for all sufficiently small \(\varepsilon>0\). Optimality then implies \(\mathop{\mathrm{tr}}H\le0\).

The cone generated by \(\xi\xi^*\), \(\xi\in E\), is closed: its generators have trace one, so bounded trace bounds the total coefficient in any conic combination. If \(I\) were outside this cone, finite-dimensional separation would give \(H\) with \(\mathop{\mathrm{tr}}H>0\) and \(\xi^*H\xi\le0\) on \(E\). Subtracting a sufficiently small positive multiple of \(I\) contradicts the preceding paragraph. Hence \(I\) belongs to the cone. A finite conic representation gives (91), after discarding zero coefficients. Taking its trace gives \(\sum_i\alpha_i=n\). Replacing each pair \((\xi_i,\alpha_i)\) by \(r\) copies of \((\xi_i,\alpha_i/r)\) preserves every identity and divides the largest weight by \(r\). ◻

Completion of the proof of Proposition 18. Use the preceding construction at the lower contact, with \(B=I\). Set \(\mathcal K_\kappa=\sum_i\alpha_iK_i^\kappa\). We first show \[ \mathcal K_\kappa\ge0, \qquad \mathcal K_\kappa\ge\partial\bar\partial_x\lambda(x_\kappa,t_\kappa). \tag{106}\] For a Hermitian form with blocks \(W=\left(\begin{smallmatrix}H&D\\D^*&C\end{smallmatrix}\right)\), \(C>0\), completing the square gives \[W[(X,V),\overline{(X,V)}] =X^*(H-DC^{-1}D^*)X +(V+C^{-1}D^*X)^*C(V+C^{-1}D^*X).\] Its minimum over the complex vertical vector \(V\) is \(\mathop{\mathrm{Schur}}(W)[X,\bar X]\). The extra term on the right of (102) is nonnegative by (95). Minimizing the left side independently over every \(V_i\) therefore proves \[\sum_i\alpha_i\mathop{\mathrm{Schur}}(W_i^\kappa) \ge\partial\bar\partial_x\lambda(x_\kappa,t_\kappa).\] Equation (105) gives (106).

Write \(c_{\theta,\alpha}=(1-\theta)(1-\max_i\alpha_i)>0\). Equations (101), (103), and (105) imply \[ \partial_t\lambda(x_\kappa,t_\kappa)+(p_\kappa)_t \ge\bigl(c_{\theta,\alpha}-o(1)\bigr)\mathop{\mathrm{tr}}\mathcal K_\kappa. \tag{107}\] The positive aggregate matrices are bounded. Let \(\kappa\downarrow0\) and pass to a subsequence to obtain a matrix \(\mathcal K_{\theta,\alpha}\) such that \[\mathcal K_{\theta,\alpha}\ge0,\qquad \mathcal K_{\theta,\alpha}\ge\partial\bar\partial_x\lambda(x_0,t_0),\qquad \partial_t\lambda(x_0,t_0) \ge c_{\theta,\alpha}\mathop{\mathrm{tr}}\mathcal K_{\theta,\alpha}.\] Now let \(\theta\downarrow0\) and repeat the construction with increasingly replicated lists, so that \(\max_i\alpha_i\downarrow0\). The aggregate matrices still act on the fixed \(n\)-dimensional horizontal space and have uniformly bounded trace. Another subsequence gives \[ \mathcal K\ge0,\qquad \mathcal K\ge\partial\bar\partial_x\lambda(x_0,t_0), \qquad \partial_t\lambda(x_0,t_0)\ge\mathop{\mathrm{tr}}\mathcal K. \tag{108}\] This also specifies the parameter order: fix the list and \(\theta\); regularize and select simultaneous contacts; remove the lower-test Hessian decrease at each fixed contact; let \(\kappa\) and the slopes tend to zero; finally let \(\theta\) and the maximum weight tend to zero. No bound uniform in the number of individual contact jets is needed.

At the normalized point \(B=I\le b_{t_0}\), and \(F_0=\log\det b_{t_0}\). All eigenvalues of \(b_{t_0}\) are at least one, so each is at most their product. Consequently \[0<e^{-F_0}b_{t_0}\le I.\] Using the two matrix inequalities in (108), in the indicated order, gives \[\partial_t\lambda \ge\mathop{\mathrm{tr}}\mathcal K \ge e^{-F_0}\mathop{\mathrm{tr}}(b_{t_0}^{\mathsf T}\mathcal K) \ge e^{-F_0}\mathop{\mathrm{tr}}(b_{t_0}^{\mathsf T}\partial\bar\partial_x\lambda) =e^{-F_0}\Delta_{h_{t_0}}\lambda.\] The positivity of \(\mathcal K\) justifies the middle comparison even when \(\partial\bar\partial_x\lambda\) is indefinite. This proves (92). ◻

Proof. In any holomorphic base coordinates let \(\ell=\log\det b_t\). The flow equation and the Ricci-form identity give \[\partial_t\ell=R_{h_t},\qquad \partial\bar\partial_x\ell=\mathop{\mathrm{Ric}}(h_t),\qquad \Delta_{h_t}\ell=R_{h_t}.\] If \(\varphi\) is an upper test of \(F_0=\ell-\mathcal L\), then \(\lambda=\ell-\varphi\) is a lower test of \(\mathcal L\). Proposition 18 therefore says \[ \partial_t\varphi \le e^{-F_0}\Delta_{h_t}\varphi +(1-e^{-F_0})R_{h_t} \tag{109}\] at contact. The coefficient \(F_0\) there equals the test value \(\varphi\). By (90), \(F_0\) is upper semicontinuous, locally bounded, and between zero and \(\rho\). Since \(\rho\) is smooth and vanishes initially, extending \(F_0\) by zero at \(t=0\) preserves upper semicontinuity. Apply Proposition 6 to (109) on each interval \([0,T']\) with \(T'<T\). Since \(T'\) is arbitrary, \(F_0=0\) for every \(t<T\). Its proof does not require completeness of \(h_t\) at positive time.

We have \(B\le b_t\) and \(\det B=\det b_t\). The positive matrix \(b_t^{-1/2}Bb_t^{-1/2}\le I\) has determinant one; every one of its eigenvalues is therefore one. Hence \(B=b_t\), and feasibility and (88) imply \[N(x,\xi,t)=\xi^*B\xi\le v(x,\xi,t)\le N(x,\xi,t).\] This proves \(v=N\).

For the raw disc infimum \(v^{\rm raw}\) of (38), we have \(v\le v^{\rm raw}\), while constant discs give \(v^{\rm raw}\le N\). Thus \(v^{\rm raw}=N\) too, so \(N\) obeys the submean inequality for every admissible disc, not merely for regularized centers. Given a holomorphic disc \((f,W)\) in \(Y\times\mathbb C\), put \(t_0=e^{W(0)+\overline{W(0)}}\) and \(w=W-W(0)\). Choose \(T\) larger than the maximum of \(e^{W+\bar W}\) on the closed disc. The defining disc inequality then gives \[N(f(0),t_0) \le\frac1{2\pi}\int_0^{2\pi} N\bigl(f(e^{i\vartheta}),e^{W(e^{i\vartheta})+ \overline{W(e^{i\vartheta})}}\bigr)\,d\vartheta.\] It suffices to use discs smooth past their boundary, including all sufficiently small coordinate discs. Since the joint norm is smooth, these submean inequalities are precisely its plurisubharmonicity. The horizon \(T\) was arbitrary. ◻

Proof. By Theorem 19, the dual squared norm of the tangent metric pulled back to \((x,w)\), where \(t=e^{w+\bar w}\), is plurisubharmonic. Completing the square in the free first derivative of a cotangent section identifies this assertion with Griffiths nonnegativity of the pulled-back tangent metric. At a spatially Kähler-normal point its curvature blocks are \[\Theta_{i\bar j\,\alpha\bar\beta} =\mathop{\mathrm{Rm}}_{i\bar j\alpha\bar\beta},\qquad \Theta_{w\bar j\,\alpha\bar\beta} =t\nabla_{\bar j}A_{\alpha\bar\beta},\qquad \Theta_{w\bar w}=tA+t^2(A_t+A h_t^{-1}A).\] The spatial block gives nonnegative bisectional curvature and hence \(A\ge0\) and metric monotonicity. The last block gives (110). Trace the curvature in its fiber indices and evaluate on the spacetime direction \((X,t^{-1})\). The identities \[\mathop{\mathrm{tr}}_{h_t}A_t=R_t-|A|_{h_t}^2, \qquad \mathop{\mathrm{tr}}_{h_t}(A h_t^{-1}A)=|A|_{h_t}^2, \qquad \mathop{\mathrm{tr}}_{h_t}\nabla_X A=\partial_XR\] give (111).

At a fixed point choose a moving unitary frame satisfying \(E_t=\tfrac12 A^{\#}E\), where \(A^{\#}=h_t^{-1}A\). Differentiation shows that \[\frac{d}{dt}\bigl[tA(E,\overline E)\bigr] =t\bigl(A_t+A h_t^{-1}A+t^{-1}A\bigr)(E,\overline E)\ge0.\] Initial strict bisectional positivity and local smooth convergence give \(A>0\) at sufficiently small positive times on any prescribed compact set. The preceding inequality propagates this positivity and gives \(tA\ge a_Kh_t\) there for \(t\ge1\), after changing the compact constant. Integrating \(\partial_t h_t=-A\) proves the asserted power contraction. Applied to the trace, the same inequality gives monotonicity of \(tR(o,t)\). The linear bound for \(\rho(o,t)\) in Theorem 4 implies, for \(t\ge1\), \[tR(o,t)\log2 \le\int_t^{2t}R(o,v)\,dv =\rho(o,2t)-\rho(o,t)\le C(1+2t).\] Smoothness on compact time intervals supplies the remaining basepoint bound.

For clarity, the next integration precedes the completeness proof. Choose a piecewise smooth path from \(x\) to \(y\) whose \(h_t\) length is within \(\varepsilon\) of their distance, and run it at constant \(h_t\) speed on \([t,t']\). Such paths exist by the definition of the distance. Along this path, (111), divided by \(R>0\), and \(A\le Rh_v\) give \[\frac{d}{dv}\log R(\gamma(v),v) \ge-\frac1v-C|\dot\gamma(v)|_{h_v}^2.\] Since \(h_v\le h_t\), its energy is at most \((\mathop{\mathrm{dist}}_{h_t}(x,y)+\varepsilon)^2/(t'-t)\). Integrating and then letting \(\varepsilon\downarrow0\) proves (112) without using a minimizing geodesic or completeness of \(h_t\).

Fix \(T<\infty\). Put \(d=\mathop{\mathrm{dist}}_{h_t}(x,o)\) and apply (112) with \(y=o\) and \(t'=t+1+d^2\). The basepoint bound gives \[ R(x,t)\le\frac{C_T(1+d^2)}t, \qquad 0<t\le T. \tag{113}\] Let \(D_L=B_g(o,L)\), and let \(l_L(t)\) be the infimum of lengths of paths from \(o\) to first exit from \(D_L\), measured in \(h_t\). The initial metric is complete, so \(\overline D_L\) is compact. There is a minimizing segment realizing \(l_L(t)\), contained in \(\overline D_L\), and every initial part of it minimizes to its endpoint. On this segment the distance in (113) is at most \(l_L(t)\).

Write \(l=l_L(t)\). In the summed real index forms use perpendicular parallel fields times a cutoff which rises from zero to one on an endpoint interval of width \[a=\min\left(\frac l2,\frac{\sqrt t}{1+l}\right).\] If \(l\ge2\sqrt t/(1+l)\), the index inequality and the two endpoint curvature bounds imply \[\int_0^l\mathop{\mathrm{Ric}}_{h_t}(\dot\gamma,\dot\gamma)\,dr \le \frac C a+\frac{C_T(1+l^2)a}{t} \le\frac{C_T(1+l)}{\sqrt t}.\] Here and below fixed real-versus-complex normalization factors are included in \(C_T\). If \(l<2\sqrt t/(1+l)\), use (113) on the whole segment to obtain the same bound. The function \(l_L\) is locally Lipschitz in time, since the metrics are smoothly equivalent on the compact closure. At a differentiability time, compare a current minimizing segment with its length at a slightly earlier time. The metric variation formula therefore gives \[l_L'(t)\ge-\frac{C_T(1+l_L(t))}{\sqrt t} \quad\text{for almost every }t\in(0,T].\] Consequently \[ 1+l_L(t)\ge(1+L)e^{-2C_T\sqrt t}. \tag{114}\] As \(L\to\infty\), every divergent path has infinite \(h_t\) length. Equivalently, each finite \(h_t\) ball is trapped in a compact initial ball; its closure is compact by local equivalence of the smooth metrics. This proves completeness. ◻

Proof. For \(t\ge1\), (112), with a fixed future time increment, bounds \(R\) on every fixed-radius \(h_t\) ball about \(o\). Nonnegative bisectional curvature gives \(|\mathop{\mathrm{Rm}}|\le C_nR\). Centers in \(K\) are handled by a compact family of paths from \(o\): their \(h_t\) lengths are bounded, so balls about these centers lie in a bounded-radius ball about \(o\).

To apply local curvature derivative estimates, take a ball for the earliest time of a short time interval. It remains inside the corresponding larger later-time balls because metrics decrease. The curvature bound gives metric equivalence on this fixed region throughout the interval. A path measured in the later metric cannot first exit the earlier ball at arbitrarily small length; thus a smaller later-time ball is contained in this region. Local Ricci-flow derivative estimates [49][31] now give all curvature derivatives on smaller balls and inner time intervals. Here the estimate is applied on shifted sufficiently short cylinders, with a time buffer before each observation slice. Metric equivalence keeps the initial balls required by the local estimate compactly inside the fixed region. This construction applies verbatim after a fixed rescaling.

Here is why no injectivity-radius hypothesis enters the chart statement; compare [8]. Pull back by the exponential map on a small tangent ball. The curvature bound excludes conjugate points at this scale, although the exponential map may identify distinct points. Jacobi-field estimates and their differentiated equations give uniformly controlled pullback metrics and parallel complex structures, with a fixed Euclidean lower metric bound. In particular any sequence has a smoothly convergent subsequence on a smaller tangent ball. The identities \(J^2=-\mathop{\mathrm{Id}}\) and \(N_J=0\) pass to the smooth limit. The Newlander–Nirenberg theorem supplies local holomorphic coordinates for the limit complex structure [42] [27]. In the smooth case their components satisfy the smooth elliptic equation \(\bar\partial_J^*\bar\partial_J f=0\), so interior elliptic regularity makes these coordinates smooth. Restrict them to a still smaller ball. Their squared radius is uniformly strictly plurisubharmonic for all sufficiently close complex structures. Solve the small \(\bar\partial\) errors of these coordinate functions on a smaller strictly pseudoconvex coordinate ball. One can use the weighted \(L^2\) estimate [28, 19], viewing scalar functions as top forms with values in the anticanonical bundle. Give this bundle the metric induced by the original uniformly controlled pullback metric. A fixed large multiple of the squared radius pays its uniformly bounded curvature, and the strictly plurisubharmonic exhaustion of the ball supplies an auxiliary complete Kähler metric. The source errors tend to zero smoothly, so the correcting functions tend to zero in local \(L^2\). We first justify smoothness of each correction, then obtain uniform estimates on a fixed smaller real ball. Write the corrections as \(u_j\) and their equations as \(\bar\partial_{J_j}u_j=\eta_j\). For each index separately, the existing complex structure \(J_j\) has smooth holomorphic coordinates, with no uniform radius asserted. In those coordinates, \[\Delta_{\mathrm{eucl}}u_j =4\sum_a\partial_{z_a}\eta_{j,a}\] is an identity of distributions with smooth right side. A compactly supported smooth extension of the right side, convolved with the Euclidean fundamental solution, gives a smooth particular solution on a smaller ball. The remaining harmonic distribution is smooth: its mollifications satisfy the mean-value identity against a fixed smooth radial unit-mass kernel, and passage to the distributional limit identifies the distribution with its smooth convolution by that kernel. Thus each \(u_j\) is qualitatively smooth.

For uniform estimates return to the fixed real ball and use \(P_j=\bar\partial_{J_j}^*\bar\partial_{J_j}\) with the original controlled pullback Kähler metric, rather than the auxiliary weighted metric. The scalar Kähler identity identifies \(P_j\), up to a fixed sign and normalization, with the real Laplace–Beltrami operator. Its coefficients have uniform bounds of every fixed order and a uniform ellipticity lower bound. Moreover \(P_ju_j=\bar\partial_{J_j}^*\eta_j=:F_j\) tends smoothly to zero. Testing this equation with \(\chi^2\overline{u_j}\) and absorbing the cross term gives the Caccioppoli estimate \[\int\chi^2|\nabla u_j|^2 \le C\int_{\mathop{\mathrm{supp}}\chi}(|u_j|^2+|F_j|^2)\] for a fixed interior cutoff. The qualitative smoothness just proved legitimizes this test. Uniform metric equivalence gives a \(W^{1,2}\) bound in the fixed real coordinates. Interior elliptic estimates [23], on successively smaller fixed balls, now yield \[\|u_j\|_{H^{m+2}(B')} \le C_m\bigl(\|u_j\|_{L^2(B)}+\|F_j\|_{H^m(B)}\bigr).\] Choose \(m+2>k+n\) and use Sobolev embedding in real dimension \(2n\) [21]. This proves convergence to zero in every fixed \(C^k\) norm on smaller balls. The corrected coordinate maps are therefore \(C^1\) close to the limiting coordinate map. On a sufficiently small fixed ball their derivatives are uniformly close to its invertible center derivative. They are injective on that ball, and their images contain a common ball about their center values, by the quantitative inverse function theorem. Subtract the coordinate center values and invert on this common ball. Composing with the exponential parametrization and making a bounded linear normalization, with one further fixed reduction of radius, gives the required maps into \(M\) with unitary center derivative. If no common radius and bounds existed for the original family, a sequence witnessing their failure would have a smoothly convergent subsequence of pullbacks. The construction just given supplies common charts on that subsequence, a contradiction. This proves the uniform chart assertion.

For a flow, use the exponential parametrization at its central time as a fixed parametrization. The curvature bound controls the time derivative of the metric; the first Ricci derivative controls that of its connection. Induction using the higher covariant derivative estimates gives ordinary space-time derivative bounds for the metrics and their parallel complex structures. Arzelà–Ascoli on smaller balls gives the claimed local smooth limits. ◻

Proof. We construct a plurisubharmonic weight whose upper bound is of order \(\log(1+A)\) and whose Levi form is strict near the center. Its moving-ball definition will also give a nonnegative lower bound on the support of the weighted \(\bar\partial\) construction. These bounds make the extension estimate independent of the total distortion. We then extend a local function whose differential detects \(B\); its \(L^2\) bound gives the required polynomial bound.

Choose a constant holomorphic covector \(\alpha=df\) with \(|\alpha|_{\rm eucl}=A\) and \(|\alpha|_{\widetilde g}(0)=1\). The logarithmic dual norm \[\psi=\log|\alpha|_{\widetilde g}^2 \quad\text{satisfies}\quad \psi(0)=0,\qquad \psi\le C+2\log A.\] For a tangent \(X\), the curvature formula for this logarithmic norm is \[\partial\bar\partial\psi(X,\overline X) =\frac{\mathop{\mathrm{Rm}}(X,\overline X,v,\overline v)}{|v|^2} +\frac{|D'_X\alpha|^2|\alpha|^2 -|\langle D'_X\alpha,\alpha\rangle|^2}{|\alpha|^4},\] where \(v\) is the metric dual of \(\alpha\) and all norms in this formula use \(\widetilde g\). The second summand is nonnegative. Initial bisectional positivity therefore makes \(\psi\) plurisubharmonic. On any region whose target image lies in a prescribed compact neighborhood \(K^+\) of \(K\), it gives \[ \partial\bar\partial\psi\ge\kappa\widetilde g \tag{116}\] with \(\kappa>0\) uniform. The normalization of \(\alpha\) cancels from this estimate.

Shrink the Euclidean domain by a fixed factor. Choose \(r_0>0\) so small that any path of \(\widetilde g\) length at most \(3r_0\) starting in this smaller domain stays in \(B(r)\); the lower metric bound makes this choice uniform. The small closed metric balls used below are compact there and their distances are attained by minimizing segments. Put \[ \Psi(z)=\max_{\mathop{\mathrm{dist}}_{\widetilde g}(z,y)\le r_0}\psi(y). \tag{117}\] This function is continuous. Compactness gives upper semicontinuity. For lower semicontinuity, move a maximizing endpoint a little towards the center along a minimizing segment; the resulting endpoint is admissible for all nearby centers and its value tends to the maximum.

We verify the Levi inequality for this moving-ball maximum. Let \(\varphi\) be a smooth upper test at \(z\), let \(y\) realize the maximum, and prescribe a nonzero complex tangent \(V\) at \(z\). Choose a holomorphic center germ with this derivative. If \(\mathop{\mathrm{dist}}_{\widetilde g}(z,y)<r_0\), any holomorphic endpoint germ with derivative equal to the parallel translate of \(V\) along a minimizing segment stays admissible for a sufficiently small parameter. Suppose instead that the constraint is active. For two real parameter directions corresponding to \(V\) and \(JV\), choose trial paths whose first variation fields are the parallel translates along this segment. Their first length variations vanish. The trace of their second length variations is \[-\int_0^{r_0} \bigl[\mathop{\mathrm{Rm}}(T,V^{\parallel},V^{\parallel},T) +\mathop{\mathrm{Rm}}(T,JV^{\parallel},JV^{\parallel},T)\bigr]\,dr<0.\] Tangential components make no curvature contribution, and holomorphic endpoint germs have zero trace covariant acceleration, so there are no omitted trace boundary terms. The strict sign is precisely the initial bisectional sign.

The trace inequality alone is insufficient to preserve a two-parameter length constraint. At the far endpoint we may prescribe its holomorphic second jet freely while retaining its first derivative. Changing that jet by a real tangent \(W\) (regarded as a complex tangent by \(J\)) changes the real length Hessian by \[\begin{pmatrix} \langle W,T\rangle&\langle JW,T\rangle\\ \langle JW,T\rangle&-\langle W,T\rangle \end{pmatrix}\] at the final endpoint. Vectors in the real span of \(T,JT\) realize every trace-free symmetric matrix of size two. Choose the jet to remove the trace-free part of the length quadratic. The trace remains strictly negative; hence the full quadratic is negative definite. Smooth trial paths joining these holomorphic endpoint germs then have length less than \(r_0\) for every sufficiently small nonzero parameter. The actual distance is no greater than this trial length. This proves admissibility also at an active constraint, without differentiating a possibly nonsmooth distance function.

On either kind of admissible endpoint germ \(y(a)\), \(\varphi(z(a))\ge\Psi(z(a))\ge\psi(y(a))\), with equality at \(a=0\). Taking the parameter Levi derivative proves \(\partial\bar\partial\Psi\ge0\) in upper-test sense, and thus in the plurisubharmonic sense. If \(\mathop{\mathrm{dist}}_{\widetilde g}(0,z)<r_0/4\), all the relevant paths and endpoints map into a fixed compact \(K^+\); applying (116) to the transported tangent gives \[ \partial\bar\partial\Psi\ge\kappa\widetilde g \quad\text{on }B_{\widetilde g}(0,r_0/4), \qquad \Psi\ge0\quad\text{on }B_{\widetilde g}(0,r_0). \tag{118}\] The latter inequality uses the admissible endpoint \(0\). Throughout the fixed Euclidean domain, \(\Psi\le C+2\log A\).

We next specify the weighted extension, including its support. A fixed small initial holomorphic coordinate neighborhood \(U\) of \(p=\Phi(0)\) lifts to a single sheet through \(0\). Indeed its short radial paths and their short homotopies lift by local inversion of \(\Phi\), and their \(\widetilde g\) lengths equal their initial target lengths. The lower metric bound prevents escape from the coordinate domain. Uniqueness of local lifts and the short homotopies make this an inverse branch on \(U\). Choose \(U\) uniformly over \(p\in K\), with the whole sheet inside \(B_{\widetilde g}(0,r_0/4)\).

In these initial target coordinates \(w\), centered at \(p\), choose a linear holomorphic function \(\ell\) whose differential has target norm one and whose pullback differential at \(0\) has Euclidean norm \(B\). Let \(\chi\) be a fixed cutoff on this sheet, equal to one near \(0\) and zero near its boundary. The function \(a=\chi\ell\), extended by zero outside the sheet, is smooth and uniformly bounded. Its source \(\beta=\bar\partial a\) is supported in a fixed target coordinate annulus and satisfies \[|\beta|_{\widetilde g^{-1}}\le C, \qquad \int_{\mathop{\mathrm{supp}}\beta}dV_{\rm eucl} \le c^{-n}\int_{\mathop{\mathrm{supp}}\beta}dV_{\widetilde g} \le C.\] The last integral is the volume of one initial target annulus, since the chosen sheet is injective. These constants do not contain \(B\).

On the same sheet choose a nonpositive logarithmic-pole weight \(\lambda\), equal to \((n+1)\log(|w|^2/\delta^2)\) near \(0\) and cut off to zero within the strict region. Its negative Hessian is bounded by \(C\widetilde g\) on the cutoff annulus, uniformly also for smooth pole regularizations. Choose a fixed large \(k\) so that the weight \[\varphi_*=k\Psi+\lambda+|z|^2\] is plurisubharmonic and its Levi form dominates a positive multiple of \(\widetilde g\) on \(\mathop{\mathrm{supp}}\beta\). On that support, \(\Psi\ge0\) and \(\lambda\) is bounded below. The scalar weighted \(\bar\partial\) estimate on the fixed Euclidean ball [28][19] gives \(\bar\partial v=\beta\) with \[ \int |v|^2e^{-\varphi_*}\,dV_{\rm eucl} \le\int |\beta|^2_{(\partial\bar\partial\varphi_*)^{-1}} e^{-\varphi_*}\,dV_{\rm eucl}\le C. \tag{119}\] For a smooth-weight justification, first convolve \(\Psi\) on a slightly larger Euclidean region and regularize the pole. For each particular smooth \(\widetilde g\), the convolution scale can be chosen small enough to retain half of (118) on the compact support regions. The estimates just obtained are independent of that scale. Weak \(L^2\) limits and Fatou’s lemma give (119) for the singular weight.

The correction \(v\) is holomorphic near \(0\). Integrability with the pole \(|w|^{-2(n+1)}\) forces its constant and linear terms to vanish. Thus \(a-v\) is holomorphic on the Euclidean ball and has the prescribed differential of norm \(B\) at \(0\). Since \(\lambda\le0\) and \(\sup\Psi\le C+2\log A\), dropping the weight in (119) gives \[\|a-v\|_{L^2(dV_{\rm eucl})}^2\le C(1+A)^{2k}.\] Interior differentiation of a holomorphic function bounds its center derivative by this norm, proving (115). ◻

Proof. Proposition 20 gives monotonicity and the positive lower bound for \(q\), together with \(q\le Ce^\tau\). Hence \(s\to\infty\). If \(q(s)>2s\) for every sufficiently large \(s\), integration of \(ds/d\tau>2s\) would give \(s\ge ce^{2\tau}\), contradicting the linear-in-\(t\) bound for \(\rho(o,t)\). This proves (121).

For a compact family of initial paths from \(o\), the power contraction in Proposition 20 makes their \(h_t\) lengths tend uniformly to zero. The factor on the right of (112), with time increment one, therefore tends uniformly to one on the compact set. Apply the inequality in both spatial directions and integrate in time. More explicitly, for every \(\varepsilon>0\) and all sufficiently large \(v\), uniformly on the compact set, \[R(x,v)\le(1+\varepsilon)R(o,v+1),\qquad R(o,v)\le(1+\varepsilon)R(x,v+1).\] The first inequality also bounds \(R(x,v)\) there. Shifting either integral endpoint by one costs a bounded amount. After division by \(s\to\infty\), and then letting \(\varepsilon\downarrow0\), the two inequalities give \(\rho(x,t)/s\to1\). No integrability of the error factor is required.

Let \(\lambda_1,\ldots,\lambda_n\) be the eigenvalues of \(h_t\) relative to \(g\) at \(x\). They lie in \((0,1]\), and their product is \(e^{-\rho(x,t)}\). Thus every \(\lambda_j\ge e^{-\rho(x,t)}\), which proves the lower bound of (122). Choose the instantaneous unitary chart of Lemma 21 at \(x\). Its initial-metric singular lengths are \(\lambda_j^{-1/2}\), whose product is \(e^{\rho(x,t)/2}\). Lemma 22 bounds their largest value by a fixed power of their smallest, uniformly for \(x\) in the compact set. Since all these lengths are at least one, their product forces the smallest to be at least \(C^{-1}e^{c\rho(x,t)}\). This proves the upper bound of (122), after reducing \(c_K\) and increasing the starting time.

For each fixed nonzero covector \(\xi\in T_o^{*(1,0)}M\), the function \[L_\xi(\tau)=\log|\xi|_{h_{e^\tau}^{-1}}^2\] is convex. Indeed the logarithmic norm of a holomorphic dual section is plurisubharmonic for a Griffiths-nonnegative tangent metric, and its restriction to complexified \(\tau\) is independent of the imaginary part. Its derivative is the Rayleigh quotient of \(t\mathop{\mathrm{Ric}}\) on the dual unit vector. Normalize \(|\xi|_{g^{-1}}=1\). At \(o\), (122) gives \[L_\xi(\tau(s))-L_\xi(\tau(a_0s)) \ge(c_o-2a_0)s.\] Choose \(a_0<c_o/4\), reducing it further to be less than one if needed. Monotonicity of \(q\) gives \[\tau(s)-\tau(a_0s) =\int_{a_0s}^s\frac{dv}{q(v)} \le\frac{s}{q(a_0s)}.\] The endpoint derivative of a convex function dominates its secant slope. Taking the minimum over all covectors proves (123).

The estimates obtained so far hold at every sufficiently late clock. We now select intervals on which the lower Ricci bound is comparable to the scalar curvature. Put \(L=2/a_0>1\) and choose \(K_0>L^2\). If (124) failed for all large \(S\), then \(q(Lr)>K_0q(r)\) for all large \(r\). Iteration and monotonicity would give \(q(r)\ge cr^{\log K_0/\log L}\) for every large \(r\), contradicting (121). Thus such windows occur arbitrarily far out. On their middle halves, \[\int_{5S_i/4}^{7S_i/4}(\log q)'(s)\,ds\le\log K_0.\] Choose \(s_i\) there with \((\log q)'(s_i)\le2\log K_0/S_i\).

Write \(q_i=q(s_i)=t_iR_i\). On the rescaled clock of (125), \[ \frac{ds}{dz}=\frac{q(s)}{q_i+z},\qquad \widehat R_i(o,z)=\frac{R(o,t_i+z/R_i)}{R_i} =\frac{q(s)}{q_i+z}. \tag{126}\] Since \(q_i\ge c>0\), choose \(\delta<c/4\). As long as \(s\) lies in the good window, \(|ds/dz|\le2K_0\) for \(|z|\le\delta\). The chosen \(s_i\) lie a distance at least \(S_i/4\) from its ends, so this estimate keeps the whole fixed time interval in the window for large \(i\). Formula (126) then bounds the rescaled basepoint scalar curvature above and below by fixed positive constants. Apply the rescaled (112), using a fixed future-time buffer, to get a uniform scalar bound on every fixed-radius moving ball in a smaller interval. Bisectional nonnegativity controls \(|\mathop{\mathrm{Rm}}|\), and Lemma 21 supplies the local higher bounds. Finally (123) and (124) give at the central point \[\lambda_{\min}(\mathop{\mathrm{Ric}}(g_i)) \ge\frac{b_0q(a_0s_i)}{q_i}\ge\frac{b_0}{K_0}, \qquad \widehat R_i(o,0)=1.\] ◻

Proof. Take the rescaled flows in Proposition 23. Their pointed pullbacks on tangent balls have a smooth local subsequential limit, denoted \(g(z)\). Pass to a further subsequence so that \[k_i(z)=\frac1{q_i+z}\mathrel{% \BeginAccSupp{method=hex,unicode,space=false,pdfliteral=direct,ActualText=27F6}% \longrightarrow% \EndAccSupp{pdfliteral=direct}% }k(z),\qquad k_z=-k^2.\] This includes \(k=0\) when \(q_i\to\infty\). Denote the limit scalar curvature by \(R\) and its Ricci form by \(A\). At the central spacetime point, \(R=1\) and \(A\ge b\,g\) with \(b>0\). Formula (126) gives the exact identity \[\left[\partial_z\widehat R_i+k_i\widehat R_i\right]_{(o,0)} =(\log q)'(s_i)\mathrel{% \BeginAccSupp{method=hex,unicode,space=false,pdfliteral=direct,ActualText=27F6}% \longrightarrow% \EndAccSupp{pdfliteral=direct}% }0.\] The limit therefore has \(S:=R_z+kR=0\) at the central point.

The matrix Harnack inequality has the rescaled form \(A_z+A g^{-1}A+kA\ge0\). Its trace is \(S\), because \(R_z=\mathop{\mathrm{tr}}_g A_z+|A|_g^2\). Thus this nonnegative matrix vanishes at the point in question. Also \(A_z=\partial\bar\partial R\) under Kähler–Ricci flow. If \(H_{j\bar l}=\nabla_j\nabla_{\bar l}R\), then \[ H=-A^2-kA \tag{127}\] there, writing endomorphism products in a unitary frame. On a small spacetime neighborhood, \(A\) is positive definite. Minimizing the scalar Harnack inequality over \(X\) gives \[ Z=S-Q\ge0,\qquad Q=|\partial R|_{A^{-1}}^2. \tag{128}\] At the equality point \(S=0\), so \(\partial R=0\) and \(Z=0\).

We give the scalar Hessian calculation, since a vanishing Harnack quantity by itself does not supply real strict concavity. Use \(R_z=\Delta R+|A|^2\), \(A_z=\partial\bar\partial R\), and \(k_z=-k^2\). The derivative of the Laplacian on a scalar is \((\partial_z\Delta)R=\langle A,\partial\bar\partial R\rangle\), while \[\partial_z|A|^2=2\langle A,\partial\bar\partial R\rangle+2\mathop{\mathrm{tr}}(A^3).\] Consequently \[ (\partial_z-\Delta)S =3\langle A,\partial\bar\partial R\rangle+2\mathop{\mathrm{tr}}(A^3) +k|A|^2-k^2R. \tag{129}\] At the equality point, substitution of (127) reduces this to \(-\mathop{\mathrm{tr}}\bigl(A(A+k\mathop{\mathrm{Id}})^2\bigr)\). Diagonalize \(A\) there, with eigenvalues \(\lambda_j>0\). Since \(\partial R=0\), derivatives of the coefficients of \(A^{-1}\) make no contribution to \(\Delta Q\) or \(Q_z\), and \[Q_z=0,\qquad \Delta Q=\sum_{j,l}\lambda_j^{-1} \left(|R_{j\bar l}|^2+|R_{jl}|^2\right).\] The first square sum is exactly \(\mathop{\mathrm{tr}}\bigl(A(A+k\mathop{\mathrm{Id}})^2\bigr)\) by (127). Hence at this point \[(\partial_z-\Delta)Z =\sum_{j,l}\lambda_j^{-1}|R_{jl}|^2\ge0.\] On the other hand \(Z\ge0\) has an interior spacetime zero, so \(Z_z=0\) and \(\Delta Z\ge0\) there. The displayed sum must vanish. Thus the pure complex Hessian \(R_{jl}\) is zero, while the mixed Hessian is negative definite by (127). The real Hessian of \(R\) is therefore negative definite at the center, with a definite bound furnished by the positive lower bound for \(A\).

This conclusion holds uniformly on actual small metric balls in the original rescaled manifolds, not only on their immersed parametrizations. Smooth convergence gives the center Hessian bound and \(|\nabla\widehat R_i|(o,0)\to0\). The intrinsic third-derivative estimates from Proposition 23 allow one to parallel transport the Hessian along any minimizing radial geodesic. Shrinking a fixed radius \(r_*>0\), independently of \(i\), gives constants \(a>0\) with \[ \operatorname{Hess}_{g_i(0)}\widehat R_i \le-a g_i(0) \quad\text{on }B_{g_i(0)}(o,2r_*),\qquad |\nabla\widehat R_i|(o,0)\mathrel{% \BeginAccSupp{method=hex,unicode,space=false,pdfliteral=direct,ActualText=27F6}% \longrightarrow% \EndAccSupp{pdfliteral=direct}% }0. \tag{130}\] Indeed the change in the transported Hessian is bounded by \(Cr_*\), so one chooses \(r_*\) after the uniform center bound. This argument uses intrinsic derivative estimates along actual geodesics and is unaffected by multiple sheets of a local chart.

Hold the complete metric \(g_i(0)\) fixed and flow by the real vector field \(Y_i=\nabla\widehat R_i(\cdot,0)\). Along a minimizing radial geodesic of length \(r\le2r_*\), \[\langle Y_i,\partial_r\rangle \le |Y_i(o)|-ar.\] For large \(i\) this is negative on \(r=r_*\). At cut points, first variation along any minimizing radial segment gives the same upper bound for the upper directional derivative of distance. Thus the closed ball of radius \(r_*\) traps trajectories starting inside it. Completeness makes this closed ball compact, so these trajectories exist for all positive flow time. For a carried tangent vector \(V\), (130) gives \[\frac{d}{dv}|V|_{g_i(0)}^2 =2\operatorname{Hess}\widehat R_i(V,V) \le-2a|V|_{g_i(0)}^2.\] Every loop contained well inside the trapping ball consequently has carried length tending exponentially to zero.

Let a fixed piecewise smooth loop in \(M\) be given. Its image and a finite collection of paths from \(o\) lie in a fixed compact set. Their \(h_{t_i}\) lengths tend to zero by (122), and \(R_i\le C\) by Proposition 20. Thus this loop lies inside \(B_{g_i(0)}(o,r_*/4)\) for sufficiently large \(i\). Fix one such \(i\). The compact trapping ball has a positive minimum injectivity radius for this one smooth metric. A sufficiently short carried loop lies in a normal ball and is contractible. The gradient flow up to that finite time is a homotopy from the original loop to it. Approximation of continuous loops by piecewise smooth loops finishes the proof. ◻

Proof. Allow the constant \(-\infty\), and otherwise let \(v\) be such a function. The compact-ball maximum \(m(r)=\max_{\overline B_g(o,r)}v\) is finite and nondecreasing. It is convex as a function of \(\log r\). To see this, fix \(0<r_1<r_2\) and compare on the closed annulus with the logarithmic chord \[H(r)=m(r_1) +\frac{m(r_2)-m(r_1)}{\log r_2-\log r_1} (\log r-\log r_1).\] If its slope is zero the desired comparison is immediate from the definition of \(m(r_2)\). For positive slope, a positive maximum of \(v-H(\mathop{\mathrm{dist}}_g(o,\cdot))\) would occur in the annulus interior. At such a point choose a minimizing radial segment. The length index form with linear parallel test fields in the \(J\)-radial direction gives an upper Hessian bound for distance. More explicitly, let \(T\) be the final radial unit vector, let \(T(\ell)\) denote its parallel extension, and let \(\bar r\) be the length upper support obtained from these trial paths. At the endpoint of a segment of length \(r\) the real Hessian satisfies \[\operatorname{Hess}\log\bar r(T,T) +\operatorname{Hess}\log\bar r(JT,JT) \le-\frac1{r^3}\int_0^r \ell^2\mathop{\mathrm{Rm}}(T(\ell),JT(\ell),JT(\ell),T(\ell))\,d\ell<0.\] Indeed the radial term is \(-1/r^2\), while the linear test field in the \(JT\) direction gives \(1/r^2\) minus the displayed curvature integral. This is the strictly negative Levi derivative on the radial complex line.

This support argument applies also at cut points: smoothly vary the endpoint and use trial paths along the chosen minimizing segment. Their lengths give a smooth upper support with the same index-form bound. Since \(H\) is increasing, it produces a smooth upper test for \(v\) at the hypothetical positive maximum, with a negative Levi derivative. A plurisubharmonic function admits no such upper test, whether or not it is continuous. Hence \(v\le H\) on the annulus, proving the claimed convexity. A nondecreasing convex function of \(\log r\in\mathbb R\) which is bounded above must be constant. Upper semicontinuity implies \(\lim_{r\downarrow0}m(r)=v(o)\), so \(v\) attains its global maximum at \(o\). The plurisubharmonic maximum principle on connected \(M\) shows that \(v\) is constant. ◻

Proof. Take the charts of Lemma 21 at \(o\). We use the elementary short-path lifting property of these possibly noninjective charts; compare [9]. A path beginning at the center and having sufficiently small current-metric length lifts from \(0\) by local inversion. The lower bound in (131) bounds the Euclidean length of its lift. If the given length is smaller than the fixed distance to the chart boundary, the lift cannot escape, and hence extends over the whole path. The same reasoning applies to a homotopy all of whose paths obey this length bound. Local uniqueness of lifts supplies uniqueness throughout the homotopy.

For \(u\ge s\), the image under \(\Phi_s\) of a short radial path has \(h_{t(u)}\) length at most its \(h_{t(s)}\) length. Lift it through \(\Phi_u\) from \(0\). Short radial homotopies give a well-defined map on a fixed ball \(B(r_1)\); locally this map is \(\Phi_u^{-1}\Phi_s\), so it is holomorphic with nonsingular derivative. Its lift has a fixed Euclidean length bound, and therefore the maps are uniformly bounded on \(B(r_1)\). They compose wherever both sides are defined near \(0\) by uniqueness of local inverses. This is the local transition construction of [9] with the metric bounds already established here.

Unitary center normalization and \(h_{t(u)}\le h_{t(s)}\) give the first estimate in (132). Taking determinants of these normalizations and using \(\det h_{t(s)}(o)=e^{-s}\det g(o)\) gives the determinant identity. All singular values are at most one and their product is \(e^{-(u-s)/2}\), so the least is at least this product. This proves the inverse estimate.

We now use simple connectivity to continue the inverse germs. Fix a connected relatively compact neighborhood of a compact set and \(o\). Cover its closure by finitely many small convex initial target neighborhoods with connected pairwise intersections, enlarging the connected neighborhood if necessary to include fixed paths from \(o\) to their centers. Choose one such path to each center. On each target neighborhood it extends the inverse germ at \(o\) by path lifting. For each nonempty overlap, use one point in the overlap to form the two-path loop. By Proposition 24 this loop bounds a fixed piecewise smooth homotopy. There are only finitely many paths and homotopies; their images lie in a fixed compact set and their initial lengths have a uniform bound. Equation (122) makes every path in these homotopies sufficiently short in the current metric when \(s\) is large. All lifts stay inside the chart. Thus the two inverse germs agree at the chosen overlap point, and then throughout the connected overlap by analytic continuation. They patch to an inverse \(f_s\) on a neighborhood of the compact set.

Choose nested connected relatively compact open exhaustion sets \(D_m\), and choose nondecreasing thresholds \(T_m\ge m\) so that all shortness conditions for \(D_m\) hold whenever \(s\ge T_m\). At clock \(s\), use the construction on \(D_{\max\{m:T_m\le s\}}\). On a smaller set, the branches obtained from two larger constructions agree by continuation from \(o\). This gives the stated coherent branches with domains exhausting \(M\). The length estimate for a lift, together with (122) on the finite path compact, gives for each fixed \(K\) \[ \sup_K|f_s|\le C_Ke^{-a_Ks} \tag{136}\] with \(a_K>0\). It also gives \(f_u=F_{s,u}\circ f_s\) on any fixed compact set whenever \(s\) is sufficiently large and \(u\ge s\): both sides are the same continued inverse, and (136) keeps \(f_s\) in the transition domain.

It remains to make the exponent independent of the compact. The functions \(v_s=s^{-1}\log|f_s|\) are plurisubharmonic on their exhausting domains and locally bounded above. For any sequence \(s\to\infty\), Lemma 11 and diagonal extraction give either convergence to \(-\infty\) locally uniformly from above, or local \(L^1\) convergence to a plurisubharmonic function on \(M\). In the second case the upper-regularized limit is at most zero everywhere by (136). Lemma 25 makes it constant. On one fixed neighborhood of \(o\), (136) bounds this constant by a fixed negative number, say \(-a_*\). The constant-limit conclusion of Lemma 11 then gives \(\limsup\sup_Kv_s\le-a_*\) on every fixed compact along this subsequence. The collapsing case gives the same conclusion with \(-\infty\). A sequence violating (133), with \(\sigma=a_*/2\), would contradict these alternatives. This proves a common exponent on the full family of clocks.

For the two remaining transition estimates, transfer the common exponent from a fixed initial neighborhood back to a small ball in the chart domain. Choose a fixed compact initial neighborhood \(K_0\) of \(o\) with positive initial distance from \(o\) to its boundary. Until a path \(\Phi_s(rz)\) first exits \(K_0\), (122) and (131) bound its initial length by \(Ce^s|z|\). It follows that, for a fixed \(C_0>1\) and all large \(s\), \[\Phi_s\bigl(B(e^{-C_0s})\bigr)\subset K_0.\] For every \(u\ge s\), the map \(f_u\Phi_s\) on this ball is the same local inverse composition as \(F_{s,u}\) near zero, and hence everywhere there. Equation (133) therefore gives the bound \[ \sup_{B(e^{-C_0s})}|F_{s,u}|\le e^{-\sigma u} \qquad\text{for all }u\ge s\text{ and large }s. \tag{137}\] Cauchy’s estimates on this inner ball give the factor \(e^{dC_0s-\sigma u}\) for derivatives of order at most \(d\). Cauchy’s estimates on a fixed transition ball give a uniform bound. Combining the two proves (134).

For completeness, fix an outer radius \(R<r_1\) and a common bound \(B_*\) for the transitions there. Euclidean three-circle interpolation, applied to their scalar linear projections and then taking the supremum, gives at radius \(e^{-\gamma s}\), for fixed small \(\gamma>0\), \[\log\sup_{B(e^{-\gamma s})}|F_{s,u}| \le-\theta_s\sigma u+(1-\theta_s)\log B_*, \qquad \theta_s=\frac{\log(R/e^{-\gamma s})} {\log(R/e^{-C_0s})} \ge\frac{\gamma}{2C_0}\] for all sufficiently large \(s\). In this formula choose \(\gamma_0<C_0\), so the middle radius lies between the inner and outer radii. Since \(u\ge s\), the fixed term \(\log B_*\) is absorbed uniformly for every later endpoint, giving (135) with, for example, \(c=\sigma/(4C_0)\) and a threshold depending on \(\gamma\). This proves all the stated uniformities. ◻

Proof. Normalize the derivative to the identity. If a volume-preserving polynomial automorphism agrees with the required jet through degree \(m-1\), the first discrepancy is a homogeneous vector polynomial \(X_m\) of degree \(m\) with \(\operatorname{div}X_m=0\): this is the degree \(m-1\) part of the Jacobian determinant condition.

The space of such fields is spanned by \[ X(z)=l(z)^m v,\qquad l\in(\mathbb C^n)^*,\quad v\in\mathbb C^n,\quad l(v)=0. \tag{147}\] This is the divergence-free shear lemma of Andersén [2]; see also [22]. We include the finite-dimensional proof to track the coefficient bounds. Pair a linear functional on homogeneous vector polynomials with \(l^m v\). The result has the form \(A(l)(v)\), where \(A(l)\) is a covector-valued homogeneous polynomial of degree \(m\). If the functional annihilates all fields in (147), then \(A(l)\) is proportional to \(l\). The polynomial identities \(l_iA_j(l)=l_jA_i(l)\) show that \(A(l)=B(l)l\) for a homogeneous polynomial \(B\) of degree \(m-1\). Such functionals are exactly the image of the transpose of divergence, since \(\operatorname{div}(l^m v)=m l(v)l^{m-1}\). Taking annihilators proves the spanning assertion. In dimension one the divergence kernel in these degrees is zero, so this argument also covers \(n=1\).

Select a finite spanning list of the fields (147), once for each \(2\le m\le d\). The time-\(c\) map of such a field is the polynomial shear \(z\mathrel{% \BeginAccSupp{method=hex,unicode,space=false,pdfliteral=direct,ActualText=21A6}% \mapsto% \EndAccSupp{pdfliteral=direct}% }z+c l(z)^m v\), whose inverse has the minus sign; both have Jacobian determinant one. A fixed linear decomposition of \(X_m\) therefore gives a finite composition of shears correcting degree \(m\) without changing lower degrees. Induction through \(m=d\), followed by the original linear map, proves realization. There are a fixed number of shears, and each operation on the truncated jets is polynomial in the preceding coefficients and the inverse linear part. This proves all the coefficient and degree bounds, including those for the inverse. Restoring the original linear map gives the realizing automorphism the determinant of that linear map; the correcting shears have determinant one. ◻

Proof. Work backward, taking \(L_k^+=I\) at the last step of a finite run or horizon. Suppose the next small start adjustment is known and set \(B=L_{k+1}^-U_k\). For a real threshold \(x\), let \(A_x\) be the span of rows of \(B\) whose new weights are at most \(x\), and let \(E_x\) be the old coordinate row space with weights at most \(x\).

Projection \(P_{E_x}:A_x\to E_x\) is injective, and its inverse on its image has norm \(O(j)\). For the proof assume \(A_x\ne0\) and let \(r_x\) be the largest new rate among its defining rows. Every old rate outside \(E_x\) exceeds \(r_x+\epsilon_0\theta_j\): otherwise the passage rule would make that old weight at most the weight of a selected new row. For any unit \(a\in A_x\), (150) and the small next adjustment imply that its old rate Rayleigh quotient is at most \(r_x+o(\theta_j)\). To see the size of this error, express \(a\) as a combination of the selected rows of \(L_{k+1}^-U_k\); its coefficient norm is \(1+o(1)\), and the error is bounded by the operator norm of \(L_{k+1}^- -I\) times the largest old or new rate. The regular rate bounds make that error \(o(\theta_j)\), also at a macro junction. Nonnegativity of all old rates now gives \[(r_x+\epsilon_0\theta_j) (1-\lVert P_{E_x}a\rVert^2) \le r_x+o(\theta_j).\] As \(r_x\le C Q_j\), this implies \(\lVert P_{E_x}a\rVert^2\ge c/j^2\). Empty or full coordinate spaces give the same assertion in its evident form.

We give the graph extension explicitly. List the distinct spaces \(E_x\) increasingly. For each such space retain the largest required \(A_x\) with that \(E_x\), so that the required \(A_x\) remain nested. Starting with the full row space, descend through the list. Suppose the next larger space \(V_+\) already projects isomorphically to \(E_+\) and contains the current \(A=A_x\). Write \(T_+:E_+\to V_+\) for its inverse projection, \(E=E_x\), \(Y=P_EA\), and \(R_A:Y\to A\) for the inverse of \(P_E|_A\). Define \[ T_E(e)=R_A(P_Ye)+T_+((I-P_Y)e),\qquad e\in E. \tag{157}\] Here orthogonal projection \(P_Y\) is taken in \(E\). Both terms lie in \(V_+\), projection to \(E\) is \(e\), and the range contains \(A\). Moreover \(\lVert T_E\rVert\le\lVert R_A\rVert+\lVert T_+\rVert\). There are at most \(n\) nontrivial stages, so the resulting graph lifts have polynomial bounds in \(j\).

For each coordinate row, take its lift at its own weight threshold. These rows form \(L_k^+\). The rows up to every threshold span the corresponding nested graph: they lie there and their projections form a triangular basis of its coordinate space. Each lifted row equals its coordinate row plus terms of strictly higher weight. Hence \(L_k^+=I+N_k\) with \(N_k\) nilpotent, and \((L_k^+)^{-1}=\sum_{r=0}^{n-1}(-N_k)^r\) has polynomial bounds. The inclusion \(A_x(L_k^+)^{-1}\subset E_x\) is exactly the zero pattern in (155).

For every nonzero off-diagonal entry in row \(i\), column \(l\), the within-step passage rule gives \(\mu_l-\mu_i>\epsilon_0\theta_j\). Conjugation therefore gives \[(D_k^{-1}L_k^+D_k)_{il} =(L_k^+)_{il}\exp\bigl(- (\mu_l-\mu_i)\Delta\tau_k/2\bigr).\] By (152), \(\theta_j\Delta\tau_k\ge c\sqrt{S_j}/j^2\). This proves (156); its exponential absorbs every polynomial bound needed in the preceding backward step. After a sufficiently late start the induction thus closes uniformly in the horizon. A diagonal subsequence of finite-horizon matrices proves the assertion on infinite runs; the zero conditions and all bounds are closed under that limit. ◻

Proof. Set \[W=2N+1,\qquad D_*=D_d,\qquad B_* =\max\{0,\log(2/b_0)\},\qquad \alpha=\frac{M}{3W}.\] Choose \(M\) so large that \[ \alpha\ge\log D_*+B_*/W,\qquad \alpha>h. \tag{163}\] A flag \(f\) has expanded interval \(I_f=[f-N,f+1+N)\), of length \(W\), and growth interval \([f-N,f+1]\), on which \(\log Q\) increases by more than \(M\).

Consider a full finite bad component \([u,v)\). Choose a maximal disjoint collection of its expanded flag intervals. Their concentric triples cover the component: any unselected equal-length interval meets a selected one and is contained in its triple. Thus there are at least \((v-u)/(3W)\) selected intervals. Their growth intervals are disjoint up to endpoints and lie in \([u,v-N]\). Monotonicity gives \[ \log\frac{Q_{v-N}}{Q_u}\ge\alpha(v-u). \tag{164}\] Every full component has length \(L=v-u\ge W\), so (163) implies \[ D_*^L Q_u\le e^{-B_*L/W}Q_{v-N} \le(b_0/2)Q_{v-N}. \tag{165}\]

We now track the ordinary degree of each component of the accumulated forward polynomial. At a regular start bound these degrees by the respective weights. Allowed nonlinear terms preserve that bound under composition, and (155) preserves it at a regular jump. If the next macro is bad, retain the last end weights; at entry to \([u,v)\) their maximum is at most \(\lambda_{u-1}Q_u\). Each bad map multiplies maximum degree by at most \(D_*\). At the next regular macro \(v\), its flag test fails, and hence \(Q_v\le e^M Q_{v-N}\). For sufficiently large \(v\) every new start weight is at least \[\lambda_v\bigl(b_0Q_{v-N}-Q_v/v^2\bigr) \ge\lambda_v(b_0/2)Q_{v-N}.\] As \(\lambda_v\ge\lambda_{u-1}\), (165) resets all degree bounds exactly to the new weights. There is no multiplicative loss for a completed bad component.

Prefixes require a separate estimate. For a bad prefix \([u,j)\), the expanded flag intervals wholly contained in the prefix cover \([u,j-W)\), when this interval is nonempty. Indeed an expanded interval covering a point before \(j-W\) ends before \(j\), and none crosses the left endpoint \(u\) of the component. The same disjoint selection gives \[ \log(Q_j/Q_u)\ge\alpha(j-u-W)_+, \qquad D_*^{j-u}Q_u\le D_*^W Q_j. \tag{166}\] This proof also applies to a prefix of an infinite bad component. Such a component would give \(\log(Q_j/S_j)\ge(\alpha-h)j-O(1)\to\infty\). It is impossible: choose continuous-clock points \(x_r\to\infty\) with \(q(x_r)\le3x_r\), and let \(S_{j_r}\le x_r<S_{j_r+1}\). Then \(Q_{j_r}\le3e^h S_{j_r}\), contradicting that growth. Thus there are regular macro starts arbitrarily far out.

Start at one of them, after every fixed-parameter threshold above, and label this grid point \(k=0\), retaining the original large macro indices \(j\). The accumulated map is initially the identity and its component degrees are one, so the initial weight bound holds after the overall scaling already allowed for \(\lambda\). Regular propagation, the exact full-component reset, and the single prefix loss in (166) give, at all macro endpoints, \[\deg G_{0,k(j)}\le C\max\{e^M,D_*^W\}Q_j,\] where \(C\) can be taken to depend on the bounded supremum of the multipliers. On a regular macro one uses \(Q_{j+1}\le e^M Q_j\); inside a bad component one uses the prefix estimate. Applying the bound at the endpoints \(j_r\) just selected proves (143). This uses only a subsequence with \(Q_j=O(S_j)\), never an all-time bound on \(q(s)/s\). ◻

Proof. For a terminal index \(l\) put \(y_l^{(l)}=\zeta_l\) and \(y_i^{(l)}=G_i^{-1}y_{i+1}^{(l)}\) for \(i<l\). Simultaneously establish, by backward induction, that the comparison segments at step \(i\) lie in \(B(e^{-\nu s_{i+1}})\) and that \[ \lvert y_k^{(l)}-\zeta_k\rvert \le \sum_{i=k}^{l-1} e^{-10s_i}\exp(s_{i+1}-s_k). \tag{175}\] The terminal assertion is immediate. At a backward step compare \(G_i^{-1}y_{i+1}^{(l)}\) with \(G_i^{-1}G_i\zeta_i=\zeta_i\). Both arguments are close to \(\zeta_{i+1}\): the first by the inductive bound and the second by (172). Since \(\lvert\zeta_{i+1}\rvert\le e^{-2\nu s_{i+1}}\), they and their joining segment lie in the required larger tube whenever the error sum has the bound asserted below. Applying (173) then gives the next error sum.

Indeed \(s_{i+1}\le(1+\delta)s_i\) and the grid eventually has spacing at least one, so \[\sum_{i\ge k}e^{-10s_i+s_{i+1}-s_k} \le e^{-s_k}\sum_{i\ge k}e^{-(9-\delta)s_i} \le C e^{-(10-\delta)s_k}\le e^{-6s_k}.\] As \(\nu<1/10\), this is much smaller than the difference between the two tube radii. The estimates therefore close the simultaneous induction without evaluating an inverse outside its proved domain.

For completeness, consecutive terminal approximations are Cauchy by the same reasoning. At level \(l\) their difference is at most \(e^{\Delta s_l}e^{-10s_l}\), and propagation down to \(k\) bounds it by \(e^{-10s_l+s_{l+1}-s_k}\). These bounds are summable in \(l\). They are uniform on the compact sets in either regime, so the limits are holomorphic. The finite identities give \(\xi_{k+1}=G_k\xi_k\), and the infinite error sum proves (174). ◻

Proof. On a fixed exhaustion compact choose an index \(k\) after which regime (i) is available. Lemma 31 gives \(\xi_k\) there, and composing it with the single fixed polynomial automorphism \(G_{0,k}^{-1}\) gives the limit (176). On overlaps the inverse charts agree by continuation, and (174) makes the limits agree. Thus these maps define a holomorphic \(\Psi\) on all of \(M\). All maps fix zero in charts, so \(\Psi(o)=0\).

Suppose \(\Psi(x)=\Psi(y)\) for two fixed points. For every large \(k\), \(G_{0,k}\Psi=\xi_k\) on a compact containing them, and hence (174) implies \[\lvert H_kf_{s_k}(x)-H_kf_{s_k}(y)\rvert\le2e^{-6s_k}.\] On the tiny raw-coordinate ball containing these points, write \(H_k(z)=A_kz+E_k(z)\), where \(A_k=H_k'(0)\) has conorm at least \(s_k^{-C}\) and \(\lVert DE_k\rVert\) is bounded by \(\exp(o(s_k))\) times the radius. Integrating only \(DE_k\) along the segment and retaining the fixed linear term \(A_k\) gives \[|H_k(z)-H_k(w)|\ge \bigl(s_k^{-C}-\sup\lVert DE_k\rVert\bigr)|z-w|.\] The nonlinear error is eventually less than \(s_k^{-C}/2\), so \[ \lvert f_{s_k}(x)-f_{s_k}(y)\rvert\le e^{-5s_k} \tag{179}\] eventually.

If \(x\ne y\), choose a small relatively compact initial metric ball about \(x\) whose closure excludes \(y\). Map the straight segment between the two raw coordinates by \(\Phi_{s_k}\), obtaining a path from \(x\) to \(y\). The segment stays in the uniform chart ball. Up to its first exit from the fixed ball about \(x\), the compact metric comparison \(e^{-2s_k}g\le h_{t(s_k)}\) from Proposition 23, and the uniform chart upper bound \(\Phi_{s_k}^*h_{t(s_k)}\le Cg_{\mathrm{Eucl}}\), bound its initial-metric length by \(Ce^{s_k}e^{-5s_k}\). This tends to zero and cannot reach the boundary of that fixed ball. The contradiction proves injectivity. This argument uses immersed charts only on their indicated sheet; it requires no global injectivity of \(\Phi_{s_k}\).

Nonsingularity follows directly from the same quantitative estimates. Fix a point and two coordinate neighborhoods \(U\Subset V\) whose closures lie in one compact exhaustion set. The estimates of regime (i) hold uniformly on \(\overline V\) for every sufficiently large \(k\). Differentiating \(\Phi_{s_k}f_{s_k}=\mathop{\mathrm{Id}}\), and using (122) and (131), gives \[|df_{s_k}(v)|\ge c e^{-s_k}|v|_g \quad\text{on }\overline V.\] The coefficient bounds on \(H_k\), its polynomially bounded inverse linear part, and \(|f_{s_k}|\le e^{-\sigma s_k}\) therefore imply \[|d(H_kf_{s_k})(v)|\ge c s_k^{-C}e^{-s_k}|v|_g.\] On the other hand, the shadow bound \(|\xi_k-H_kf_{s_k}|\le e^{-6s_k}\) on \(\overline V\) and Cauchy’s estimate on the fixed pair \(U\Subset V\) bound its differential on \(U\) by \(C e^{-6s_k}|v|_g\). Thus \[|d\xi_k(v)|\ge \bigl(c s_k^{-C}e^{-s_k}-C e^{-6s_k}\bigr)|v|_g>0 \qquad(v\ne0)\] for all sufficiently large \(k\). Since \(\xi_k=G_{0,k}\Psi\) and \(G_{0,k}\) is a polynomial automorphism, \(d\Psi\) is nonsingular. The holomorphic inverse theorem now makes the image \(\Omega\) open; the local inverses agree by the proved injectivity, giving a holomorphic inverse on \(\Omega\). Consequently \(K'=\Psi^{-1}(K)\) is compact whenever \(K\subset\Omega\) is compact. On \(K'\), the common chart bound \(\lvert f_{s_k}\rvert\le e^{-\sigma s_k}\), the coefficient bound on \(H_k\), and (174) give \[\sup_K\lvert G_{0,k}\rvert \le \exp\bigl(-\sigma s_k+o(s_k)\bigr)+e^{-6s_k},\] which proves (177).

It remains to prove the ball inclusion. Fix a sufficiently large \(k\) and use regime (ii) with \(a=k\). For large \(l\), the inverse chart \(f_{s_l}\) is defined on the compact image under \(\Phi_{s_k}\) of the closed ball \(\overline B(e^{-\gamma s_k})\). The identity \(f_{s_l}\Phi_{s_k}=F_{s_k,s_l}\) holds first as a germ at zero and then throughout that ball by holomorphic continuation. Therefore Lemma 31 gives \[ \sup_{\lvert z\rvert\le e^{-\gamma s_k}} \lvert G_{0,k}\Psi\Phi_{s_k}(z)-H_k(z)\rvert \le e^{-6s_k}. \tag{180}\] Write \(A=H_k'(0)\), \(r=e^{-\gamma s_k}\), and \[T_k=G_{0,k}\Psi\Phi_{s_k},\qquad E_k(z)=T_k(z)-Az.\] The map \(T_k\) is holomorphic on the fixed chart ball, and fixes zero. Its shadow estimate and the coefficient bound on \(H_k\) give \[E_k(0)=0,\qquad \lVert A^{-1}\rVert\le s_k^C,\qquad \sup_{\lvert z\rvert\le r}|E_k(z)| \le \exp(o(s_k))r^2+e^{-6s_k}.\] Cauchy’s estimate on the smaller ball then gives \[\sup_{\lvert z\rvert\le r/2}\lVert A^{-1}DE_k(z)\rVert \le C_n s_k^C\bigl(\exp(o(s_k))r+e^{-6s_k}/r\bigr) \mathrel{% \BeginAccSupp{method=hex,unicode,space=false,pdfliteral=direct,ActualText=27F6}% \longrightarrow% \EndAccSupp{pdfliteral=direct}% }0.\] For large \(k\) this supremum is at most \(1/4\). Moreover, \(\gamma'>\gamma\) implies \(|A^{-1}w|\le r/4\) whenever \(|w|\le e^{-\gamma's_k}\), after increasing the starting index once more. For such a target \(w\), the map \[z\mathrel{% \BeginAccSupp{method=hex,unicode,space=false,pdfliteral=direct,ActualText=27FC}% {\let\mathrel{% \BeginAccSupp{method=hex,unicode,space=false,pdfliteral=direct,ActualText=27F6}% \longrightarrow% \EndAccSupp{pdfliteral=direct}% }\longrightarrow\longmapsto}% \EndAccSupp{pdfliteral=direct}% }A^{-1}w-A^{-1}E_k(z)\] is \(1/4\)-Lipschitz on \(\overline B(r/2)\) and maps this closed ball into \(B(3r/8)\), because \(E_k(0)=0\). Its successive iterates are Cauchy, stay in the closed ball, and converge to a fixed point \(z\). The fixed-point identity is \(T_k(z)=w\). Therefore \(B(e^{-\gamma's_k})\subset G_{0,k}(\Omega)\), which is precisely (178). ◻

Proof of Theorem 1. The preceding analytic and chart constructions, followed by Propositions 27 and 32, give a biholomorphism \(\Psi:M\to\Omega\) onto an open set containing zero, with the forward smallness and inverse-ball inclusion proved above. It remains to show that \(\Omega=\mathbb C^n\).

Suppose \(R_*=\sup\{R>0:B(R)\subset\Omega\}<\infty\). It is positive because \(0\in\Omega\) and \(\Omega\) is open. By (143) there is an unbounded subsequence with \(\deg G_{0,k}\le Ds_k\) for some finite \(D\). Choose \(0<R_1<R_*<R_2\) so close to \(R_*\) that \[ D\log(R_2/R_1)<\sigma/2-\gamma'. \tag{182}\] The closed smaller ball lies compactly in \(\Omega\): it lies inside some still larger centered ball of radius below \(R_*\). By (177), (181), and (182), the chosen subsequence satisfies \[\sup_{B(R_2)}\lvert G_{0,k}\rvert \le \exp\left[ -\frac{\sigma}{2}s_k+Ds_k\log(R_2/R_1)\right] < e^{-\gamma's_k}\] for all sufficiently large indices. Equation (178) then implies \(B(R_2)\subset\Omega\), contradicting the definition of \(R_*\). Therefore \(R_*=\infty\), and \(\Omega=\mathbb C^n\).

The map \(\Psi\) in (176) is consequently a biholomorphism \(M\to\mathbb C^n\). ◻