The extension problem and its geometric obstruction

Hamilton introduced Ricci flow and established its short-time existence on closed manifolds (Hamilton 1982). A compact flow continues while its full curvature remains bounded. Šešum showed that a uniform bound for the Ricci tensor also suffices (Šešum 2005). A scalar bound is weaker: in a possible curvature blowup, the scalar curvature may vanish after rescaling while the full curvature stays nonzero. The limiting geometry can therefore be Ricci-flat without being flat.

Bamler and Zhang developed heat-kernel and curvature estimates under scalar bounds, including the four-dimensional energy and compactness theory (Bamler and Zhang 2017, 2019). Their work also gives smooth extension when \(H_2(M;\mathbb F)=0\) for every field \(\mathbb F\) (Bamler and Zhang 2017, arXiv version, Corollary 1.10). Simon constructed continuation from the limiting space by an orbifold Ricci flow (Simon 2020a). Bamler’s convergence theory gives smooth convergence away from a set of codimension at least four (Bamler 2018); four-dimensional convergence under spatial scalar-integrability hypotheses is also studied by Liu and Simon (Liu and Simon 2026). These compactness conclusions permit concentrated curvature and orbifold points. The assertion here is smooth continuation of the given flow on its original manifold. The scalar extension problem and its known special cases are discussed in (Li and Li 2026, sec. 2). Finite-time scalar blowup was established for closed Kähler–Ricci flows by Zhang (Zhang 2010) and for compact Ricci flows with a global Type I bound, \(\sup_M|\mathop{\mathrm{Rm}}(g(t))|\le C/(T-t)\), by Enders, Müller and Topping (Enders et al. 2011). More recently, Buzano and Di Matteo proved scalar blowup at local Type I singular points without a global Type I assumption (Buzano and Di Matteo 2026, Definition 1.4 and Theorem 1.1). Here “local Type I” means \(0<\limsup_{t\uparrow T}(T-t)r_{\mathop{\mathrm{Rm}}}(p,t)^{-2}<\infty\), where \(r_{\mathop{\mathrm{Rm}}}\) is their parabolic curvature scale at the point \(p\). That result addresses a specified blowup regime; the argument below allows unrestricted rates at the hypothetical concentration points.

The spatial degeneration analysis has its origins in the theory of Einstein metrics. Anderson’s compactness theorem produces orbifold limits with smooth convergence on their regular parts (Anderson 1989, Theorem C). Bando’s bubbling analysis, including its correction, organizes Ricci-flat bubbles and quantitative necks (Bando 1990a, 1990b); the work of Bando, Kasue and Nakajima develops coordinates at infinity under curvature decay and volume-growth hypotheses (Bando et al. 1989). Cheeger and Tian’s curvature-energy estimates provide a further important part of this Einstein four-manifold theory (Cheeger and Tian 2006). Our slices need not be Einstein. We prove the required geometric statements below using the scalar-bound estimates and retaining the Ricci forcing in the neck argument.

At a hypothetical concentration point, successive spatial rescalings produce a finite tree of Ricci-flat pieces. A vertex represents a nonflat limit; its singular points may contain smaller vertices. The outermost nonflat vertex has a distinguished length scale \(a(t)\). The central problem is to rule out contraction of this outermost scale while allowing arbitrarily smaller descendants.

The static functional and the flow estimates

Retain the true metric from the cluster through an annular collar of radius \(b=Na\), with \(N\) large and fixed, and attach an asymptotically locally Euclidean end. On that end write the metric in Euclidean quotient coordinates. The renormalized Einstein–Hilbert functional is \[E(g)=\lim_{D\to\infty}\left[ \int_{\{r<D\}}R(g)\,d\mu_g- \int_{\{r=D\}}(\partial_jg_{ij}-\partial_ig_{jj})n_i\,dS_\delta \right].\] The boundary subtraction cancels the scalar curvature’s linear divergence term. The functional scales by the square of the length unit, and its first variation pairs the metric velocity with \(\tfrac12 Rg-\mathop{\mathrm{Ric}}\).

The companion article (OpenAI 2026) proves a sequential inequality \(E=o(M)\) on each fixed limiting tree, where \(M\) is its weighted, dimensionless Ricci error. Section 2 records the complete input, including the end weights, derivative orders and matching quotient groups. Our task is both to verify those hypotheses for metrics obtained from the flow and to compare their functional variation with actual spacetime Ricci energy.

Two flow estimates are decisive. The divergence constraint on Ricci removes the critical radial harmonic mode on a nearly flat annulus. This improves annular decay enough that the loss outside the collar is small compared with the energy inside the root. The exterior is then constructed as an analytic function of the metric on one fixed material collar. Longer exterior coordinates enter only an error estimate. Consequently the extension can be differentiated on time patches without choosing a smoothly varying bubble tree. The collar construction uses the classical compatibility between metric strains and linearized curvature, together with Korn control modulo rigid motions (Khavkine 2017; Ciarlet and Ciarlet 2005). The scale-specific estimates, filled-space inverse and dependence on the fixed collar are proved in Section 5.

Proof structure

Section 3 establishes the scale estimates, genuine orbifold limits and finite trees needed here. It defines a continuous outermost scale and selects buffered time intervals on which that scale changes by a fixed factor. After parabolic normalization, the interval has fixed duration and its root length \(a\) tends to zero.

Section 4 controls Ricci on exterior annuli and on the root’s compact and joining regions. A time maximal function converts root spacetime energy into the single-time weighted error required by the static theorem. Section 5 constructs the collar-dependent exterior and proves an error bound independent of its length.

Section 6 applies the static theorem separately to arbitrary sequences of active time patches. If \(K^2\) denotes the root spacetime Ricci energy divided by \(a^4\), the positive functional variation and its static smallness imply \[cK^2\le o(1)(K+1)^2.\] Hence \(K\to0\), without assuming it was bounded. Dense sets of material points then preserve distances at scale \(a\) between the two endpoints. The resulting pointed isometry preserves the curvature-radius maximum defining the outermost scale. This contradicts its prescribed factor change. The absence of concentration gives a smooth terminal metric and continuation on \(M\).

Conventions

We use \(\Delta=\mathop{\mathrm{tr}}\nabla^2\). A tensor norm at scale \(r\) includes \(r^j\) on its \(j\)th spatial derivative, while leaving its pointwise magnitude unchanged. Tensor magnitudes use the current metric unless specified otherwise. Quotient-chart components are Cartesian components on the Euclidean cover; all constructions are equivariant and descend to the quotient.

Constants may depend on a fixed finite differentiation order. After extraction they may also depend on a fixed finite tree. Uniformity in a sequence index, a length, or the collar factor \(N\) is stated where needed. A sequential little-oh is taken after fixing the parameters in its statement.

The geometric data and the functional

Let \(\mathcal T\) be a finite rooted tree of four-dimensional Ricci-flat spaces \((V_v,h_v)\). A vertex has one exterior end and one punctured end for each child. Every end has Cartesian coordinates on a Euclidean annulus modulo a finite subgroup of \(O(4)\) acting freely on \(S^3\). The group on a joined end is nontrivial. The two groups on an edge are identified; a fixed orthogonal identification is included in the charts. Put \(\tau=\log r\) on an exterior end and \(\tau=-\log r\) at a puncture. In these charts, in vertex units, \[ h_v=\delta+O(e^{-\mu\tau}) \tag{1}\] on joined ends, with derivatives at scale, for some \(\mu>0\). On the unjoined exterior end of the root we require instead \(h_v-\delta=O(r^{-q})\), where \(q>1\). All remaining parts are smooth compact manifolds with the indicated boundary collars. Bounds through order \(20\) suffice in the data below.

For an edge \(e\) from \(p\) to \(c\), truncate both end coordinates at \(\tau=L_e\) and identify the boundary collars by \[ x_p=e^{-2L_e}x_c,\qquad a_c=a_p e^{-2L_e},\qquad a_{\rm root}=1. \tag{2}\] Thus the metric tensor in vertex units is multiplied by \(a_v^2\) in physical units. The physical radius \(\rho=a_vr\) agrees in both charts, and \(\tau_p+\tau_c=2L_e\). Gluing just the boundary faces, with their compatible smooth collars, gives the same manifold.

Consider a sequence of metrics \(g\) for which every \(L_e\to\infty\), which converges to \(h_v\) on each compact subset of every open vertex. Assume the same end estimates as in Equation (1) on each joining half, and the stated root estimate, with uniform finite derivative bounds. On a compact core put \(\rho=a_v\). Suppose \(M\to0\) and \[ |\rho^2\mathop{\mathrm{Ric}}(g)|_j\le CM \begin{cases} e^{-\mu\tau},&\text{on joined half-ends},\\ r^{-q},&\text{on the root exterior end},\\ 1,&\text{on vertex cores}, \end{cases} \qquad j=10. \tag{3}\] Here the tensor magnitude is physical, and the subscript includes covariant derivatives multiplied by the corresponding powers of \(\rho\). The metric bounds convert these to coordinate estimates. Decreasing \(\mu\) or \(q\), while retaining \(\mu>0\) and \(q>1\), is allowed.

Define, in physical root units, \[ E(g)=\lim_{D\to\infty}\left\{ \int_{r<D}R(g)\,d\mu_g- \int_{r=D}(\partial_jg_{ij}-\partial_ig_{jj})n_i\,dS_\delta \right\}. \tag{4}\] All chart integrals are quotient integrals. The two terms inside the braces need not converge separately.

The boundary subtraction is the mass term in the renormalized Einstein–Hilbert expression discussed by Deruelle and Ozuch (Deruelle and Ozuch 2025, Definition 4.2, Remark 4.3, and Proposition 4.4). Their ALE version of Perelman’s functional also minimizes over an auxiliary potential; the functional \(E\) here is the unminimized scalar-curvature-minus-flux expression.

The companion proof constructs a finite-dimensional analytic family with complex length parameters. It does not assume that Einstein deformations are unobstructed, or that the given metrics or the coefficients of a fixed core metric are spatially analytic.

The theorem is applied in Proposition 28. There we verify every hypothesis after extracting a fixed tree from an arbitrary sequence of active times and material collars. The root exterior has decay exponent greater than one because it is replaced by the extension of Section 5; the true joining necks retain their own positive decay exponent.

Heat localization and two-sided persistence of the spatial radius

The spatial curvature radius does not by itself control a time cylinder. Our first goal is to obtain that control in both time directions, and then to improve the Ricci bound by one power of the spatial radius. We give the heat estimates first so that the curvature-radius argument does not assume curvature-dependent cutoff bounds. Set \(n=4\) for the applications, retaining \(n\) temporarily in the heat formulas. If \(u=(4\pi\tau)^{-n/2}e^{-f}\) is a positive unit-mass conjugate heat density, \(\partial_\tau=-\partial_t\) and \(\partial_\tau u=\Delta u-Ru\), put \[v=\{\tau(2\Delta f-|\nabla f|^2+R)+f-n\}u.\] A direct computation gives \[ (\partial_\tau-\Delta+R)v =-2\tau|\mathop{\mathrm{Ric}}+\nabla^2f-g/(2\tau)|^2u. \tag{9}\] For completeness, after factoring out \(u\) the operator is \(D=\partial_\tau-\Delta+2\nabla f\cdot\nabla\). The identities used are \[\begin{align*} f_\tau&=\Delta f-|\nabla f|^2+R-n/(2\tau),\\ D\Delta f&=-2|\nabla^2 f|^2-2\mathop{\mathrm{Ric}}(\nabla f,\nabla f) +\Delta R-2\mathop{\mathrm{Ric}}:\nabla^2f,\\ D|\nabla f|^2&=2\nabla f\cdot\nabla R -2|\nabla^2 f|^2-4\mathop{\mathrm{Ric}}(\nabla f,\nabla f),\\ DR&=-2\Delta R-2|\mathop{\mathrm{Ric}}|^2+2\nabla f\cdot\nabla R. \end{align*}\] The metric term in \(\partial_\tau\Delta f\) is \(-2\mathop{\mathrm{Ric}}:\nabla^2f\); the contracted Bianchi identity cancels the first-derivative terms. Substitution gives Equation (9), including its time direction. This is Perelman’s entropy calculation (Perelman 2002, Proposition 9.1).

Consequently \(\int v\,d\mu_t\) is nondecreasing in forward time. At time zero it has a lower bound uniform for \(0<\tau\leq T+1\): writing \(\phi=\sqrt u\) and \(\int\phi^2=1\), its expression is \[\tau\int(4|\nabla\phi|^2+R\phi^2) -\int\phi^2\log\phi^2-\frac n2\log(4\pi\tau)-n.\] Jensen’s inequality followed by the Sobolev inequality of the fixed initial metric gives this bound, including as \(\tau\downarrow0\) by optimizing the coefficient of \(\|\nabla\phi\|_2^2\). Solve the conjugate equation backwards from an arbitrary smooth positive terminal density. Monotonicity then gives the same logarithmic Sobolev inequality at every time for \(0<\tau\leq1\). Positive approximation allows Lipschitz or nonnegative test functions.

Here are two consequences with their normalizations. If a small ball has arbitrarily small ratio \(\mathop{\mathrm{Vol}}(B_r)/r^n\), choose a dyadic descendant whose inner half has a definite fraction of its volume and whose ratio is still small. Such a descendant exists: otherwise successive ratios decrease geometrically to zero, contradicting the Euclidean ratio at a smooth center. A tent function supported in this descendant, equal to one on its inner half, has normalized Dirichlet energy \(O(r^{-2})\). Jensen gives \(\int\phi^2\log\phi^2\geq-\log\mathop{\mathrm{Vol}}(B_r)\). The logarithmic Sobolev inequality with \(\tau=r^2\) therefore gives a uniform positive lower volume ratio. Also \(\int\phi^2\log\phi^2\geq-2\log\|\phi\|_1\) for \(\|\phi\|_2=1\). Optimization in \(\tau\), using \(\tau=1\) when necessary, gives \[ \|\phi\|_2^{2+4/n} \leq C(\|\nabla\phi\|_2^2+\|\phi\|_2^2)\|\phi\|_1^{4/n}. \tag{10}\] For both forward ordinary heat and backwards conjugate heat, the \(L^1\) norm is bounded by its initial norm times \(e^{Ch}\), and the derivative of the squared \(L^2\) norm in the diffusion direction is at most \(-2\|\nabla u\|_2^2+C\|u\|_2^2\). This uses only \(\partial_t d\mu_t=-R\,d\mu_t\). Equation (10) and integration of the resulting scalar differential inequality give \(\|P_h\|_{1\to2}\leq Ch^{-n/4}\) for \(h\leq1\). Duality with the conjugate equation and composition at the midpoint give the heat kernel bound \(K\leq Ch^{-n/2}\).

For a conjugate kernel with terminal pole \(p\), Equation (9) also gives \(v\leq0\); compare (Perelman 2002, Corollaries 9.3–9.4). One may verify the terminal sign without a uniform curvature bound near \(T\). Fix the terminal time first, when the flow is smooth, and a positive forward heat test \(\zeta\). Integration by parts gives \[\int v\zeta =\partial_\tau\left(\tau\int(f-n/2)\zeta u\right) -2\tau\int u\Delta\zeta.\] The kernel upper bound bounds the integral in parentheses from below after division by \(\tau\). For its upper bound, apply Jensen relative to \((4\pi\tau)^{-n/2}e^{-d_*^2/(4\tau)}\), where \(d_*^2\) is a smooth function agreeing with squared terminal distance near \(p\) and has \(p\) as its unique zero. Its mass tends to \(1\), and the heat equation, tested against \(d_*^2\zeta\), gives \(\tau^{-1}\int d_*^2\zeta u\to2n\zeta(p)\). Hence \(\limsup\int(f-n/2)\zeta u\leq0\). Along a sequence approaching zero the displayed derivative has nonpositive liminf. At this fixed smooth terminal time, \(2\tau\int u\Delta\zeta\to0\), so the same is true of \(\int v\zeta\). Equation (9) makes \(\int v\zeta\) nonincreasing in \(\tau\); hence it is nonpositive at every earlier time. Arbitrary positive forward heat tests now give \(v\leq0\). Equivalently, \[2\tau f_\tau+\tau|\nabla f|^2-\tau R+f\leq0.\] Integration along the constant material path at \(p\) gives \(f(p,\tau)\leq C\tau\). The initial term in this integration can also be obtained from the preceding second moment: some point within \(O(\sqrt\tau)\) of \(p\) has bounded \(f\); moving from that point to \(p\) over a time interval of size \(\tau\) costs \(O(\sqrt\tau)\) in the inequality for \(\sqrt\tau f\), by completing the gradient square. Letting the starting time tend to zero removes this cost. Thus \[ K(p,t;p,s)\geq c(t-s)^{-n/2}. \tag{11}\]

For a forward heat solution \(|U|\leq1\) over elapsed time \(h\leq1\), the cancellation in the Bochner formula gives \[(\partial_t-\Delta)|\nabla U|^2=-2|\nabla^2U|^2, \qquad |\nabla U|\leq Ch^{-1/2}.\] The gradient bound follows by applying the maximum principle to \(h|\nabla U|^2+CU^2\). To obtain a time derivative bound observe \((\partial_t-\Delta)\Delta U=2\mathop{\mathrm{Ric}}:\nabla^2U\) and \((\partial_t-\Delta)R=2|\mathop{\mathrm{Ric}}|^2\). For either sign, apply the maximum principle to \(Q=h^2(\pm\Delta U+A|\nabla U|^2-AR)\), with fixed large \(A\). The negative terms \(-2Ah^2(|\nabla^2U|^2+|\mathop{\mathrm{Ric}}|^2)\) absorb \(2h^2\mathop{\mathrm{Ric}}:\nabla^2U\). The remaining negative Hessian square absorbs \(2h|\Delta U|\); the already proved gradient bound controls the other elapsed-time terms. At a positive maximum with \(Q\geq C h\) the resulting upper bound is \(C-cQ^2/h^2\), which rules out a first crossing of a sufficiently large multiple of \(h\). Therefore \[ |\nabla U|\leq Ch^{-1/2},\qquad |\partial_tU|=|\Delta U|\leq C/h. \tag{12}\]

Take a forward kernel from \((x,t-r^2)\) and multiply it by \(r^n\). On a short slab about \(t\) it is bounded, has value at least \(c\) at \((x,t)\), has spatial gradient \(C/r\), time derivative \(C/r^2\), and integral at most \(Cr^n\). It is therefore bounded below on a ball of radius \(cr\) at time \(t\), proving the upper volume bound. Each connected component of a fixed positive superlevel has diameter \(O(r)\): a connecting path of greater diameter contains arbitrarily many disjoint balls of radius \(c'r\), where the function is bounded below; lower volume bounds and the integral estimate bound their number. The same function, using its time derivative bound and slightly nested levels, proves \[ B_{t_1}(x,cr)\subset B_{t_2}(x,Cr) \quad\text{if }|t_1-t_2|\leq c'r^2, \tag{13}\] in either time direction. Indeed every initial radial path in the smaller ball remains in a positive superlevel at the second time, so its endpoint remains in the same component as \(x\).

In particular, for small \(l\) there is a nonnegative Lipschitz cutoff on \([t-cl^2,t]\), bounded by a fixed constant, with \[ |\nabla\chi|\leq C/l,\quad c/l^2\leq\partial_s\chi\leq C/l^2\ \text{ on }\{\chi>0\}, \quad\chi(x,s)\geq c,\quad\mathop{\mathrm{supp}}\chi\subset B_t(x,l). \tag{14}\] Choose the kernel radius to be a small fixed fraction of \(l\), subtract a suitable level, add \(C'(s-t)/r^2\), and take the positive part. Keep the final superlevel component containing \(x\). Choosing \(C'\) larger than the kernel time-derivative constant makes the positive domains monotone in \(s\), so extension by zero outside this component is legitimate. The component diameter bound puts its support in \(B_t(x,l)\). All constants are independent of the cap \(l_0\).

We now bootstrap, rather than assume, the time control needed to use a spatial curvature radius. Fix a tensor-index convention and bootstrap \[ |\partial_t\mathop{\mathrm{Rm}}|+C_n|\mathop{\mathrm{Ric}}||\mathop{\mathrm{Rm}}|\leq A l^{-3}. \tag{15}\] The left side controls the time derivative of the curvature norm. Assume Equation (15) at preceding times, set \(l=l(x,t)\), and use Equation (14). At time \(t\), \(\max\chi^2|\mathop{\mathrm{Rm}}|\leq C l^{-2}\). Choose \(B>C\). If this maximum had been larger than \(Bl^{-2}\), take its last crossing down to that value, and let \(y\) be a maximizing point at the crossing. Put \(\chi_y=\chi(y,s)>0\). The maximum and the cutoff gradient bound give, for \(d_s(y,z)\leq c\chi_y l\), \[\chi(z,s)\geq\chi_y/2,\qquad |\mathop{\mathrm{Rm}}|(z,s)\leq4B/(\chi_y^2 l^2).\] Thus, taking \(c_B\leq\min(c,(4B)^{-1/2})\), \[ l(y,s)\geq c_B\chi_y l. \tag{16}\] This inference uses no curvature derivative estimate. At the crossing the positive cutoff contribution to \(\partial_s(\chi^2|\mathop{\mathrm{Rm}}|)\) is at least \(cB/(\chi_y l^4)\), whereas its possible negative curvature contribution is at most \(C_BA/(\chi_y l^3)\) by the bootstrap. Choosing \(A l_0\) sufficiently small makes the derivative strictly positive. The upper one-sided derivative of the spatial maximum cannot then permit the crossing. Approximation of the positive-part cutoff, or the corresponding one-sided maximum argument on its positive set, gives the same conclusion. Consequently curvature is bounded by \(Cl^{-2}\) on a definite smaller backward parabolic ball at \((x,t)\). Its moving spatial balls are contained in the positive cutoff region by \(\chi(x,s)\geq c\) and the gradient bound.

On this parabolic ball the usual differentiated curvature argument can now be applied without circularity. The connection variation is \[\partial_t\Gamma^k_{ij} =-g^{km}(\nabla_i\mathop{\mathrm{Ric}}_{jm}+\nabla_j\mathop{\mathrm{Ric}}_{im}-\nabla_m\mathop{\mathrm{Ric}}_{ij}).\] Skew differentiation, the second Bianchi identity, and commuting derivatives give the heat equation for \(\mathop{\mathrm{Rm}}\) and its differentiated equations, with products \(\nabla^i\mathop{\mathrm{Rm}}*\nabla^{k-i}\mathop{\mathrm{Rm}}\). At unit scale, after bounds through order \(k-1\), the quantity \((D+|\nabla^{k-1}\mathop{\mathrm{Rm}}|^2)|\nabla^k\mathop{\mathrm{Rm}}|^2\), with large fixed \(D\), satisfies a differential inequality bounded above by a constant minus a positive multiple of its square. Cauchy’s inequality absorbs the cross-gradient terms. Cutoffs on smaller parabolic balls, whose distance and Laplacian bounds now follow from the curvature bound, prove the derivative estimate inductively; distance cutoffs are interpreted by shortened radial barriers at cut points. This is the local derivative argument of (Shi 1989).

The coordinate estimates used here have a uniform radius. Normalize a smaller controlled curvature ball by the length \(l\). On a fixed ball in these units, sectional curvature is bounded above and below, and every sufficiently small contained ball has volume at least \(c r^4\). The local injectivity estimate of Cheeger–Gromov–Taylor (Cheeger et al. 1982, Theorem 4.3) therefore gives a positive uniform injectivity radius on a further smaller ball. To check its local hypotheses, reserve a fixed outer ball within the curvature bound and choose all comparison radii below its conjugate-radius bound. The upper sectional bound supplies that bound; the lower sectional bound controls the volume of the pulled-back tangent ball in the theorem’s loop estimate, while the lower ball-volume bound controls its denominator. Closedness supplies compactness of the smaller closed balls. No curvature bound outside this controlled region is used. Returning to original units gives normal radius \(c l\). Differentiating radial Jacobi equations then gives the higher coordinate bounds from the covariant curvature bounds.

At this stage the crude estimates \(|\nabla^k\mathop{\mathrm{Ric}}|\leq C_k l^{-2-k}\) follow by contraction from the curvature estimates. These already suffice, after dilation to unit scale, to integrate the metric and connection equations in a fixed initial normal chart for a time of order \(l^2\). Consequently the chart coefficients and all needed derivatives stay uniformly controlled, and smooth spacetime cutoffs with the usual scaled bounds are available before the improved Ricci estimate is proved.

The lower-index Ricci tensor satisfies \[ \partial_t\mathop{\mathrm{Ric}}=\Delta\mathop{\mathrm{Ric}}+2\mathop{\mathrm{Rm}}(\mathop{\mathrm{Ric}})-2\mathop{\mathrm{Ric}}^2, \qquad \mathop{\mathrm{Rm}}(g)=\mathop{\mathrm{Ric}}. \tag{17}\] Test the scalar equation against a scaled smooth spacetime cutoff on the smaller parabolic ball. Such cutoffs are now available in the controlled coordinates. Integration by parts, \(|R|\leq C_R\), and volume \(O(l^4)\) over time \(O(l^2)\) give \[ \int_{t-cl^2}^{t}\int_{B_s(x,cl)}|\mathop{\mathrm{Ric}}|^2\,d\mu_s\,ds \leq Cl^4. \tag{18}\] For example, derivatives of the cutoff cost \(l^{-2}\), so its scalar terms cost \(l^{-2}l^4l^2= l^4\); time integration also includes the bounded \(R^2\) measure term. Rescaling lengths by \(l^{-1}\) makes the squared spacetime Ricci norm \(O(l^2)\). Equation (17) is a homogeneous linear tensor heat equation once its bounded coefficients are fixed. Interior energy estimates followed by local Sobolev iteration and differentiated interior estimates give \(|\nabla^k\mathop{\mathrm{Ric}}|\leq C_k l^{-1-k}\) in original units. This is the local scalar-to-Ricci improvement of (Bamler and Zhang 2017, arXiv version, Lemma 6.1); the preceding argument has verified its bounded-curvature cylinder and cutoff inputs from the spatial radius. The uniform coefficients used in these interior estimates are precisely those obtained from the crude curvature bounds above; the improved Ricci estimates are their conclusion.

Curvature variation now bounds the left side of Equation (15) by \(A_0 l^{-3}\), with \(A_0\) independent of \(A,l_0\). Choose \(A>A_0\), and then \(l_0\) small enough for the crossing argument and for the bootstrap to hold on a fixed initial compact time interval. A first-failure argument proves it for all later times. This proves Equation (8). For forward persistence, use the same heat function with a negative drift, so that \(-C/l^2\leq\partial_s\chi\leq-c/l^2\) on its positive set and its support stays in the initial radius-\(l\) ball. At the first upward crossing of \(\max\chi^2|\mathop{\mathrm{Rm}}|=Bl^{-2}\), the cutoff gradient again gives \(l(y,s)\geq c_B\chi_y l\). The negative cutoff contribution is at most \(-cB/(\chi_y l^4)\) while the curvature contribution is at most \(C_BA/(\chi_y l^3)\). Their sum is negative for the chosen cap, contradicting the crossing. The cutoff is positive near the initial center throughout a fixed shorter time slab, proving the forward bounded-curvature neighborhood. We have proved both directions of spatial-scale persistence and the derivative bounds in Equation (8); the two-sided volume bounds were established by heat localization. The remaining global input for compactness is the slice energy bound.

Energy, small bad sets, and volume comparison

The local estimates now give definite curvature energy at each concentration scale. To bound how many such regions can occur, we first control the total slice energy and then derive a summable volume-comparison error. The latter will distinguish genuine quotient singularities from smooth points in a limit. Choose \(C_0\) so \(1\leq R+C_0\leq C\). Completing the gradient square in the quotient evolution, as in Simon’s integral-curvature argument (Simon 2020b, sec. 3, equation (3.2) and Theorems 3.5–3.6), gives \[(\partial_t-\Delta)\frac{|\mathop{\mathrm{Ric}}|^2}{R+C_0} \leq C|\mathop{\mathrm{Rm}}||\mathop{\mathrm{Ric}}|^2-c|\mathop{\mathrm{Ric}}|^4.\] If \(F=\int|\mathop{\mathrm{Ric}}|^2/(R+C_0)\), the volume derivative and Young’s inequality imply \[F'\leq C\int|\mathop{\mathrm{Rm}}|^2-c'\int|\mathop{\mathrm{Ric}}|^4+CF.\] In four dimensions the Chern–Gauss–Bonnet formula (Chern 1944) in the full-tensor normalization of (Simon 2020b, Equation (3.6)) gives \(\int|\mathop{\mathrm{Rm}}|^2\leq C+4\int|\mathop{\mathrm{Ric}}|^2\leq C+CF\). No orientability assumption is needed: for a nonorientable \(M\) apply the identity to its orientation double cover with the pulled-back flow and divide the integral identity by two. Dropping the negative quartic term and applying Gronwall bounds \(F\) and then the slice curvature energy, proving Equation (7).

There is a fixed energy \(e_0>0\) associated to a point-picked curvature ball. Indeed maximize curvature times squared distance to the boundary of a slightly larger ball. At the maximizing point, curvature \(Q\) is bounded by \(CQ\) on a ball of radius \(cQ^{-1/2}\). The spatial derivative estimate makes curvature at least \(Q/2\) on a smaller ball; lower volume then gives \(\int|\mathop{\mathrm{Rm}}|^2\geq e_0\). If \(l(x)<r\), curvature reaches at least \(r^{-2}\) within distance \(r\) of \(x\) by the definition of \(l\). The preceding point selection therefore produces an energy ball inside \(B(x,Cr)\). A maximal disjoint collection of these larger radius-\(Cr\) balls has at most \(C/e_0\) members by the slice energy bound; enlarging them covers \(\{l<r\}\). Thus \[ \mathop{\mathrm{Vol}}_t\{l<r\}\leq Cr^4. \tag{19}\] On \(B(x,h)\cap\{l\geq h\}\), the cubic Ricci integral is at most \(Ch^{-3}h^4\). On \(2^{-j-1}h\leq l<2^{-j}h\) it is at most \(C(2^{-j}h)^{-3}(2^{-j}h)^4\). Summing gives \[ \int_{B_t(x,h)}|\mathop{\mathrm{Ric}}|^3\,d\mu_t\leq Ch. \tag{20}\]

Distances have a uniform terminal limit. For \(|t-s|\leq cr^2\), cover the bad set at the first time by the bounded family of radius-\(Cr\) balls, omitting from a minimizing path its visits to slightly enlarged balls. Replace first-to-last visits when a ball is revisited. There are uniformly many replacement endpoints, with total replacement cost \(O(r)\) at the other time by Equation (13). On the remaining path the curvature scale is \(\geq cr\) throughout the interval by forward and backward persistence. Its length changes by a factor at most \(1+Cr\) since \(|\mathop{\mathrm{Ric}}|\leq C/r\). Comparing also in reverse gives \[|d_t(x,y)-d_s(x,y)|\leq Cr(1+d_s(x,y)+d_t(x,y)).\] First compare with a fixed late time to bound diameter, then take \(r\) a sufficiently large constant times \(\sqrt{|t-s|}\). This proves uniform convergence to a pseudodistance \(d_T\) on \(M\) and defines its compact metric quotient \(Z\). For scalar-bound distance distortion and terminal metric convergence, see also (Bamler and Zhang 2019, Theorem 1.1 and Corollary 1.2). Volume forms at two times differ by factors \(e^{\pm C_R|t-s|}\).

We will need a comparison estimate stronger than mere volume bounds. In polar coordinates about a smooth point, before the cut radius, let \(P\) be the polar Jacobian and \(m=(\partial_r\log P-3/r)_+\). The Riccati inequality gives, on \(\{m>0\}\), \[m'+2m/r+m^2/3\leq|\mathop{\mathrm{Ric}}|.\] Since \(P'=(m+3/r)P\) there, \[(m^5P)'\leq5|\mathop{\mathrm{Ric}}|m^4P-7m^5P/r-(2/3)m^6P.\] Integrate to cuts and use Young’s inequality on the first term. The boundary contributions have the correct sign, with \(m=0\) at entrance into a positivity interval. Setting \(P=0\) after cuts gives \[ \int_{B(x,h)}m^6\leq C\int_{B(x,h)}|\mathop{\mathrm{Ric}}|^3\leq Ch. \tag{21}\] Furthermore \((P/r^3)'\leq mP/r^3\). Integrating between comparable radii yields \[ \frac{\mathop{\mathrm{Vol}}B(x,h)}{h^4} \leq\frac{\mathop{\mathrm{Vol}}B(x,h/2)}{(h/2)^4}+Ch^{1/2}. \tag{22}\] To see the exponent, the error is bounded by \(Ch^{-3}\int m\leq Ch^{-3}(Ch)^{1/6}(Ch^4)^{5/6}=Ch^{1/2}\). The same argument holds for any fixed comparable pair of radii. Errors sum to \(Ch^{1/2}\) over all dyadic descendants. Polar integration including the nonpositive cut measure also gives weakly \[ \Delta(d^2/2)\leq4+dm. \tag{23}\]

The scale, energy, and bad-set bounds of Theorem 6 are now established, as is uniform convergence of distances. We next identify the geometry on every shrinking spatial scale; smooth terminal convergence away from a fixed finite set will then follow by persistence.

Slice limits, quotient cones, and quantitative necks

We next turn these estimates into a description of every small-scale slice limit. The essential conclusions are smooth Ricci-flat convergence off a finite set and quantitative quotient coordinates on the intervening annuli. Puncture removal and the nontriviality of the quotient groups will be established after those coordinates are available. This is the part of the argument corresponding to classical Einstein orbifold compactness and bubbling (Anderson 1989; Bando 1990a, 1990b); the estimates below include the Ricci error present in the flowing slices. Consider \(h_i^{-2}g(t_i)\), based at arbitrary \(x_i\), where \(h_i\to0\) and \(t_i\uparrow T\). Two-sided volume bounds uniformly bound the number of disjoint radius-\(\rho\) balls in any bounded ball, giving pointed subsequential compactness as length spaces. Completeness follows by exhausting with successively larger balls. The cover of \(\{l<\rho h_i\}\) has uniformly bounded cardinality. Letting \(\rho\downarrow0\) diagonally identifies a finite exceptional set \(S\), with \(l/h_i\) bounded below on its regular compact complement. The coordinate and derivative bounds give smooth convergence there. In the rescaled metrics, \[|\nabla^k\mathop{\mathrm{Ric}}|\leq C_kh_i(l/h_i)^{-1-k}\longrightarrow0\] on these compacts, so the limit is Ricci-flat.

Smooth convergence includes embeddings of relatively compact regular subregions. Take finitely many normal charts with margins and extract their transition maps together with their metrics. The local inverse identifications differ smoothly by quantities tending to zero. In convex limiting balls combine them by a smooth partition and the unique local squared-distance barycenter. The resulting map is locally invertible on smaller charts, covers the desired region by the chart margins, and is injective: two points whose images converge to the same point have distance tending to zero, hence lie in one chart where injectivity holds. Construct on a larger compact region before restricting. This supplies actual embeddings for the later gluing.

There is no volume mass at \(S\), since its \(\rho\)-neighborhood has volume \(O(\rho^4)\). Volume is therefore regular Riemannian volume, and balls have converging volumes. Indeed at almost every regular point away from the center, the last regular segment of a minimizing path gives \(|\nabla d|=1\). Distance levels consequently have zero four-dimensional measure. The regular limit has finite curvature energy by lower semicontinuity on compact exhaustions. In \(h_i^{-1}\) length units, Equation (20) gives a cubic Ricci integral \(O(h_i^2)\) on each fixed ball. Thus the normalized polar defect vanishes, and Equations (22) and (23) give exact Bishop monotonicity and \(\Delta(d^2/2)\leq4\) weakly on the regular part of the limit.

The finite exceptional set cannot separate positive-volume regular components. Fix a regular initial point and a small regular endpoint ball \(U\) in a supposed other component, both a positive distance from \(S\). All approximating minimizing segments between them must enter \(N_\rho(S)\); otherwise a subsequential limit of such bounded segments would connect the two components in the regular complement. Their passage through \(N_{C\rho}(S)\) has length at least \(c\rho\), at radii in a fixed compact subinterval of \((0,\infty)\). The polar inequality gives, for a passage radius \(s\) and endpoint radius \(r\), \[P(r)\leq C P(s)+C\int_s^r m(u)P(u)\,du.\] Integrating over passage times, endpoint radii, and angles, and using Fubini on these bounded ranges, gives \[ c\rho\mathop{\mathrm{Vol}}(U)\leq C\left(\mathop{\mathrm{Vol}}(N_{C\rho}(S))+\int m\right). \tag{24}\] Cuts only decrease the polar integrals. First take the sequence limit, eliminating \(\int m\), and then let \(\rho\downarrow0\). The right side is \(O(\rho^4)\), a contradiction. This argument also applies to iterated tangent and end limits: a diagonal choice ensures that the original physical radii still tend to zero.

Write \(d\) for distance from a puncture, or from a fixed point when studying infinity. At a puncture, finite energy makes the curvature energy on annuli \(\{r/4<d<4r\}\) tend to zero as \(r\downarrow0\); at infinity the same holds as \(r\to\infty\). First, point-picking and the fixed energy \(e_0\) rule out unbounded \(r^2|\mathop{\mathrm{Rm}}|\) on a smaller annulus. The inherited local estimates then give \(r^3|\nabla\mathop{\mathrm{Rm}}|\le C\) with further fixed boundary margins. If \(r^2|\mathop{\mathrm{Rm}}|\ge\varepsilon>0\) at a point of this middle annulus, curvature remains at least \(\varepsilon/(2r^2)\) on a ball of radius \(c\varepsilon r\) contained in the wider annulus. For each fixed \(\varepsilon\), lower volume bounds give energy at least \(c\varepsilon^6\) on that ball, contradicting the vanishing annular energy. Thus the scaled curvature tends to zero. Interior stationary estimates, or interpolation with the inherited higher bounds, give the same conclusion for every fixed number of scaled derivatives. Tangent and end limits have flat regular annuli and only the dilation center as a possible incomplete point. Their ball volumes are exactly \(vr^4\): the monotone ratio has a limit at the relevant end, and dilation freezes that limit at every fixed radius.

Put \(u=d^2/2\). The nonnegative distribution \(4-\Delta u\) pairs to zero with every compactly supported radial test by this volume formula and \(|\nabla d|=1\). Dominate an arbitrary nonnegative test by a radial test on a larger compact annulus to see that the defect vanishes. Elliptic regularity makes \(u\) smooth and \(\Delta u=4\). Bochner’s formula and \(|\nabla u|^2=2u\) give \(|\nabla^2u|^2=4\); since its trace is \(4\), equality in Cauchy’s inequality gives \(\nabla^2u=g\). Its gradient curves produce the cone metric \(dr^2+r^2g_L\). Flatness makes \(L\) a compact spherical manifold and nonseparation makes it connected. Thus \(L=S^3/\Gamma\) for a finite orthogonal group acting freely. Two-sided volumes bound \(|\Gamma|\). All fixed-ratio annuli sufficiently far along the original end are therefore arbitrarily close, with derivatives and radial distances, to round quotient annuli. Enlarged overlapping annuli identify the groups.

Removal of punctures and intrinsic singular points

The neck estimate supplies coordinates in which a puncture has a continuous metric completion. We now improve that completion to a smooth metric on its local cover, and then prove that every actual concentration point has a nontrivial quotient group. This second assertion is needed both for organizing trees and for tracking them by their distances. At a limit puncture Lemma 7, with zero Ricci forcing, gives on the cover \[ |D^k(g-\delta)|\leq C_kr^{\mu-k},\qquad \mu>0. \tag{27}\] Choose \(0<\alpha<\min(\mu,1)\) and fill \(g(0)=\delta\). Points separated by a distance comparable to their radii are controlled by the zeroth-order bound; closer points by the first-derivative bound along their segment. Thus \(g\) and \(A^{ij}=\sqrt{\det g}\,g^{ij}\) are \(C^{0,\alpha}\), and after scaling \(B_R\) to \(B_1\), \(\|A(R\cdot)-I\|_{C^{0,\alpha}}=O(R^\mu)\).

Solve \[\partial_i(A^{ij}\partial_jy^a)=0\quad\text{in }B_R,\qquad y^a=x^a\quad\text{on }\partial B_R.\] Coercivity gives the unique variational solution. For \(w=y-x\), \[\partial_i(A^{ij}\partial_jw^a) =-\partial_i(A^{ia}-\delta^{ia}),\qquad w^a|_{\partial B_R}=0.\] After rescaling to \(B_1\), the forcing vector has \(C^{0,\alpha}\) norm \(O(R^\mu)\). The weak-solution gradient estimate (Gilbarg and Trudinger 2001, Theorem 8.16 and Corollary 8.35) gives \(\|Dy-I\|_\infty=O(R^\mu)\) and \(y\in C^{1,\alpha}\). This estimate also follows by absorbing the small coefficient perturbation in the constant-coefficient divergence estimate. Thus \(y\) is a coordinate diffeomorphism for small \(R\). Uniqueness makes it equivariant, and \(y(0)\) is \(\Gamma\)-fixed, so recentering is permitted. Put \(B=Dy(0)\), and then make a constant linear normalization. Since \(y(x)-y(0)-Bx=O(r^{1+\alpha})\), rescaled interior estimates after subtracting the affine part give \(D^ky=O(r^{1+\alpha-k})\) for \(k\geq2\). The forcing from that affine part has the stronger coefficient exponent \(\mu>\alpha\). The inverse coordinates obey the same estimates. Consequently the harmonic-coordinate metric satisfies \[ G-G(0)=O(r^\alpha),\quad DG=O(r^{\alpha-1}),\quad D^2G=O(r^{\alpha-2}). \tag{28}\]

Thus \(G\in W^{2,p}\) across zero for all \(p<4/(2-\alpha)\); take \(p_0=2+\alpha/2>2\). There is no distributional point defect: integration by parts across \(\partial B_\rho\) has first-derivative boundary terms \(O(\rho^3)\) and second-derivative terms involving \(DG\) of size \(O(\rho^{2+\alpha})\). Both tend to zero. The harmonic-coordinate Einstein equation holds almost everywhere: \[ G^{ab}\partial_a\partial_bG_{ij}=Q_{ij}(G^{-1},DG,DG). \tag{29}\] Here \(Q\) is quadratic in \(DG\) with smooth coefficients in \(G^{-1}\); this is the Lanczos harmonic-coordinate formula, as recorded in (DeTurck and Kazdan 1981, Lemma 4.1, equation (4.3)). For \(2<p<4\), Sobolev gives \(DG\in L^{4p/(4-p)}\), so elliptic regularity improves the exponent by \[ p_{k+1}=\frac{2p_k}{4-p_k}>p_k,\qquad \frac1{p_k}=\frac12-2^k\left(\frac12-\frac1{p_0}\right). \tag{30}\] After finitely many steps one can take \(p>4\). If a step reaches exactly \(4\), the local embedding \(W^{1,4}\subset L^q\) for every finite \(q\) permits the next exponent above \(4\).

This gain does not assume the stronger regularity in advance. On a sufficiently small ball the continuous leading coefficients are uniformly close to a constant positive matrix. Localize \(G\); its equation has right side in the higher \(L^q\) since the cutoff terms contain only \(G,DG\). The constant-coefficient \(W^{2,q}\) inverse and a Neumann series for the small coefficient perturbation give a \(W^{2,q}\) solution. Uniqueness in the previously available \(W^{2,p}\) class identifies it with the localized metric. Alternatively, the global Dirichlet regularity theorem (Chiarenza et al. 1993, Theorem 4.2) gives this upgrade for the localized metric components, which have zero boundary trace. Keep the continuous leading coefficients equal to \(G^{-1}\) near the support of the cutoff and extend them to a constant positive matrix outside a larger compact set. A convex interpolation preserves ellipticity and gives bounded uniformly continuous, hence vanishing-mean-oscillation, coefficients. Once \(p>4\), \(G\in C^{1,\beta}\). Equation (29), Schauder estimates, and differentiation give smoothness. This proves the orbifold filling in changed equivariant coordinates without an assumption about Einstein deformation integrability.

A concentration point cannot have trivial group. If it did, its smooth filling would have almost-Euclidean volume ratios on sufficiently small balls. Smooth convergence on surrounding annuli, negligible tip volume, and Equation (22) propagate these lower ratios to every smaller scale in the approximations: the total error below original radius \(h\) is \(O(h^{1/2})\). Concentration and point-picking give, at a much smaller scale, a complete nonflat Ricci-flat limit with a regular nonflat region. Point-picked centers are \(o(h)\) from the smoothly filled center. Inclusions between their radius-\(h\) balls and the filled-center balls of radii \(h\pm o(h)\) transfer the same lower starting ratios. Let the initial ratio approach the Euclidean one, choosing the approximating index after each outer radius; the summable comparison error tends to zero in physical units. Every fixed ball of the smaller limit has at least Euclidean volume ratio. At a regular center Bishop comparison gives the opposite bound. Equality, by the radial Laplacian and Hessian argument, forces a smooth flat cone from that center, a contradiction. Thus each exceptional point is a genuine intrinsic singularity with nontrivial group; in particular a metrically regular point of a limit cannot conceal a further curvature concentration. The same argument shows the end group of a nonflat complete limit is nontrivial: a trivial end gives Euclidean asymptotic volume ratio, which is independent of the chosen center.

A complete flat limit here is a global Euclidean quotient with at most one tip. Develop the universal cover of its regular locus. The faithful free linear local holonomy at a puncture ensures that its local fundamental group injects into that of the regular locus: the composition of the inclusion with flat holonomy is the faithful linear action of its quotient group. Hence its lifts contain full punctured Euclidean balls. Fill their centers. The resulting flat manifold is complete: a finite-length path projects to a convergent path downstairs and eventually remains in one lifted local ball. Adding isolated points preserves simple connectivity in dimension four. Its development is therefore all of Euclidean space. The deck group is finite. For any prescribed finite number of distinct deck transformations, lifts of a regular expanding ball and their transforms occupy that many sheets in a comparably enlarged Euclidean ball. Every downstairs point has a lift within comparable radius by path lifting; paths detour isolated tips with arbitrarily small excess using their connected links. Arbitrarily many sheets would force the downstairs Euclidean growth ratio to zero, contrary to noncollapse. A finite Euclidean isometry group fixes its barycenter and is orthogonal after translation. Since singularities are isolated, its only possible singular point is the origin. This finishes the small-scale orbifold, genuine-tip, and end assertions of Theorem 6. We can now use the intrinsic singular points to identify the vertices and edges of a finite tree.

Finite bubble trees and the definition of the root scale

We have shown that a flat limit has at most one genuine tip. This excludes two separated concentrations without a larger nonflat ancestor. Together with the slice energy bound it will produce a finite tree and identify its outermost member by a continuous maximum. The terminal space \(Z\) has only finitely many concentration points. Fix a sequence \(t_i\uparrow T\) and write \(\pi:M\to Z\) for the terminal quotient. For each sufficiently large integer \(m\), let \(E_{i,m}=\pi(\{l(\cdot,t_i)\leq2^{-m}\})\). Extract their Hausdorff limits \(F_m\) diagonally in the compact space \(Z\), allowing an empty limit if the sets eventually become empty. They are nested compact sets. The bad-set cover and uniform convergence of distances show that each \(F_m\) is covered by at most \(N\) balls of radius \(C2^{-m}\), where \(N,C\) are independent of \(m\). Consequently \(S=\bigcap_mF_m\) has at most \(N\) points: \(N+1\) distinct points would violate such a cover for large \(m\).

Let \(K\subset Z\setminus S\) be compact. Nested compactness gives an \(m\) with \(K\cap F_m=\varnothing\) and a fixed neighborhood of \(K\) disjoint from \(F_m\). For all sufficiently large \(i\), every material point mapping to a slightly smaller neighborhood has \(l(\cdot,t_i)>r=2^{-m}\). Crucially, \(r\) is now fixed. Choose one such \(i\) with \(T-t_i<c r^2\). Forward spatial persistence on these radius-\(r\) balls controls the entire remaining interval \([t_i,T)\), and a finite ball cover gives fixed material charts, bounded derivatives, and smooth terminal convergence over \(K\). Local injectivity gives a single sheet. Thus the possible exceptional set is fixed and finite throughout late time; this conclusion does not infer uniformity from shrinking time windows. It completes the terminal-convergence assertion of Theorem 6.

If the exceptional set is empty, finitely many such charts bound curvature globally. Their derivative estimates, metric evolution, and compactness give a smooth limiting metric on the original manifold at \(T\). The DeTurck formulation, with \(W^a=g^{ij}(\Gamma^a_{ij}(g)-\Gamma^a_{ij}(g_{\rm fixed}))\), has principal part \(g^{ij}\partial_i\partial_jg\) after adding \(\mathcal L_Wg\). Short-time quasilinear parabolic existence and pullback by \(-W\) continue the flow on the same manifold. The smooth limit and evolution equation give matching derivatives at \(T\); compare (Hamilton 1982, Theorem 14.1) for the curvature continuation criterion. Hence only the case of a genuine concentration point needs further work. Fix a small neighborhood \(\mathcal U\) in the material manifold, pulled back from a \(d_T\)-ball, whose regular boundary isolates one such point. Its punctured regular annuli have finite limiting curvature energy.

For a sequence \(t_i\uparrow T\), call two nonflat small-scale limits in this cluster equivalent if their scales have ratio bounded above and below and their centers remain a bounded distance apart in either scale. Pass to subsequences so all scale and separation comparisons converge in the extended sense. Each nonflat limit contains a regular ball with a uniformly positive energy. Indeed its orbifold filling and end decay make its maximum curvature finite and attained. Dilate so that maximum equals \(1\). The complete stationary curvature equation gives a universal derivative bound: the differentiated-curvature product used above, now stationary, satisfies \(-\Delta Q\leq C-cQ^2\) for \(Q=(D+|\mathop{\mathrm{Rm}}|^2)|\nabla\mathop{\mathrm{Rm}}|^2\). Its maximum is controlled, using decay at infinity and lifted maximum tests at quotient points. Thus a definite ball around a curvature maximum has curvature bounded below. Noncollapse, upper volume bounds, and the uniformly finite number of tips allow removal of small tip neighborhoods without exhausting that ball; some smaller regular ball remains with definite curvature and volume. Its energy is at least a fixed \(e_1>0\).

Such regular energy balls belonging to inequivalent entries are disjoint in the original slices for large \(i\). Separated centers give disjoint balls directly. If one scale is much smaller and its center remains bounded on a larger scale, it cannot meet a fixed regular ball there: smooth regular convergence bounds its curvature radius from below proportionally to the larger scale. The smaller entry must approach a tip. Therefore the total energy bounds the number of inequivalent entries by \(C/e_1\).

Construct a maximal list by adjoining, whenever possible on a further subsequence, a nonflat limit inequivalent to those already listed. Previous convergences and comparisons persist after subsequence extraction. The energy bound forces this process to terminate, including against tests on further subsequences. The list is nonempty by high-curvature point-picking. If two entries are separated by a distance much greater than both scales, their separation-scale limit has two genuine tips. It cannot be flat, since a flat limit has at most one tip. It is consequently a new nonflat ancestor unless it is already in the list. Comparable scales at bounded distance are equivalent; at separated scales and bounded larger-scale distance the smaller entry approaches a tip of the larger one. These observations define an ancestor order on the finite list and show that it has a unique maximal entry. Applying the same argument to entries at each tip gives a unique outermost child in that branch, and every tip has a child by concentration and point-picking. This is the finite rooted tree.

Every intermediate limit between consecutive entries is flat, since a nonflat one would enlarge the maximal list. The child becomes its cone tip, and the same conclusion holds between the root and arbitrarily slowly vanishing outer radii. On these flat annuli \(l\geq cr\), so Equation (8) gives \(r^2|\mathop{\mathrm{Ric}}|_{k,r}\leq Cr\) in physical units. Lemma 7 therefore supplies the finite necks and positive decay exponents. This completes the tree assertion of Theorem 6.

These convergences identify actual slice regions with the core/neck domains of Section 2. Use compact regular embeddings on truncated vertices and the single annular coordinates on each joining region. Dividing physical coordinates by the corresponding vertex scale gives its end coordinates. If the parent and child scales are \(A\) and \(Ae^{-2L}\), their end variables satisfy \[\tau_p=\log(A/r),\qquad \tau_c=\log(r/(Ae^{-2L})),\qquad \tau_p+\tau_c=2L.\] On the parent half \(\tau_p\leq L\), \(e^{-\mu\tau_p}+e^{-\mu\tau_c}\leq2e^{-\mu\tau_p}\); the child half has the analogous bound. Thus the two-sided neck estimate supplies exactly the one-sided half-end weights used there. Fixed initial end collars are absorbed by enlarging constants.

On fixed overlaps the annular charts converge, after a subsequence, with all required derivatives; they prescribe end charts for the limiting vertices by diagonal extraction. Adjust the regular embeddings by the small resulting overlap differences. Regular convergence covers each vertex except holes about its tips; annular convergence covers the intervening distances to its children, and at a leaf a bounded rescaled core is entirely regular. The same construction passes through a fixed compact root collar if the exterior is later replaced. It is a sequential construction: there is no assertion that one tree or one family of identifications is smooth in time. It gives only a positive joining exponent. The stronger exponent greater than one on the unjoined exterior will be supplied by Theorem 21 before Theorem 4 is applied.

Motion on short intervals

The scale just defined must be compared at different times on the same material manifold. We first use the genuine quotient tips to control curvature radii along minimizing segments, then construct dense sets of material points whose distance changes are small.

Geometric inputs and local estimates

We specify the consequences of Section 3 that enter the argument. They are sequential statements: they apply to arbitrary choices of times, radii, and windows after passage to a subsequence.

1. If \(h\) is bounded above and \(\epsilon\Delta t/h\to0\), the geometry at length \(h\) moves negligibly on a window of duration \(\Delta t\). On regular charts with \(l\ge c h\), its coefficients in fixed material coordinates change by \(O(\epsilon\Delta t/h)\), with any fixed number of spatial derivatives measured at scale \(h\).

2. For \(h\asymp a\) on these windows, the old and new centers have distance \(O(a)\). For \(a/h\to0\), they have distance \(o(h)\). For the latter assertion the old cone tip persists as an intrinsic singularity. If a concentration representing it were separated from the current center by a definite fraction of \(h\), the limit at their separation scale would have two distinct genuine exceptional points. The tree construction would then give a nonflat ancestor larger than \(a\), contrary to Proposition 8. The cluster itself is fixed by convergence of the original distances to the isolated point of \(Z\).

3. For sufficiently large fixed \(C\) and every fixed \(C'>0\), distance annuli with \(Ca\le r\le C'\) admit the quotient charts of Equation (33), with error, through any prescribed finite order, bounded by a constant times \[ \Theta(r)=(a/r)^\mu+\delta_i r^\mu, \qquad \delta_i\longrightarrow0, \qquad \epsilon=o(\delta_i). \tag{35}\] We decrease \(\mu\) when necessary and assume \(0<\mu<1\). The charts cover whole shortened distance annuli, their radial positioning has arbitrarily small relative error when \(C\) is sufficiently large, and their finite deck groups are nontrivial.

The quantitative estimate in Item (iii) is Equation (33), with the prescribed finite derivative order fixed in advance and a fixed positive exterior exponent and constants uniform in time and radius. It covers every fixed upper radius \(C'\) for late sequence indices, since \(R_i^{\mathrm{out}}\to\infty\). Whole-band coverage and radial positioning are part of the quotient-chart construction there. Positioning is only required with small relative error, not at the rate \(\Theta\).

For a tensor \(U\), write \[|U|_{j,r}=\sum_{k=0}^j r^k|\nabla^kU|.\] In material coordinates the lower-index Ricci equation and its constraint are \[\begin{align*} (\partial_t-\Delta_g)\mathop{\mathrm{Ric}}_{ij} &=2R_{ikjl}\mathop{\mathrm{Ric}}^{kl}-2\mathop{\mathrm{Ric}}_i{}^k\mathop{\mathrm{Ric}}_{kj}, \tag{36}\\ (\partial_t-\Delta_g)G_{ij} &=2R_{ikjl}G^{kl}-2\mathop{\mathrm{Ric}}_i{}^kG_{kj}+\eta'\mathop{\mathrm{Ric}}_{ij}, \tag{37}\\ \nabla^i\bigl(G_{ij}-\tfrac12 g_{ij}\mathop{\mathrm{tr}}_gG\bigr)&=0. \tag{38}\end{align*}\] The last identity uses that \(\eta\) is spatially constant. On a regular patch of radius \(r\), use time \(\tau=(t-t_0)/r^2\) and tensor components whose size records the original \(I\)-unit magnitude. More precisely, for the spatial chart \(\Phi_r\) set \[g_r=r^{-2}\Phi_r^*g,\qquad G_r=r^{-2}\Phi_r^*G,\qquad \mathop{\mathrm{Ric}}_r=r^{-2}\Phi_r^*\mathop{\mathrm{Ric}}.\] Thus \(|G_r|_{g_r}=|G|_g\circ\Phi_r\); the symbol \(\mathop{\mathrm{Ric}}_r\) records the original magnitude and is \(r^{-2}\mathop{\mathrm{Ric}}(g_r)\). We suppress these pullbacks in norm estimates. Equation (8) gives \[ |\mathop{\mathrm{Ric}}|_{j,r}\le C_j\epsilon/r, \qquad |r^2\eta'\mathop{\mathrm{Ric}}|_{j,r}\le C_j\epsilon r. \tag{39}\] In particular the cutoff source requires no division by \(\eta\).

The spherical projection and the missing radial mode

Lift a quotient annulus to its Euclidean cover and record tensor components in a Cartesian frame. Let \(P\) subtract the componentwise spherical average at every radius. This operation preserves equivariance and commutes with the componentwise flat heat operator.

The two Lemmas explain why a degree-zero critical mode cannot obstruct the estimates. Although \(s^{-2}A\) is componentwise harmonic for a constant symmetric matrix \(A\), its flat trace-reversed divergence is \[-2s^{-3}\bigl(A-\tfrac12\delta\mathop{\mathrm{tr}}_\delta A\bigr)n.\] It vanishes for every \(n\) only if \(A=0\). Spatially constant tensors can survive, but have no positive spatial derivatives; their values will be fixed by an outer boundary estimate. The trace-reversal calculation is closely related to the improvement of ALE metric decay in the proof of Deruelle–Kröncke’s Theorem 2.7 in the arXiv version (Deruelle and Kröncke 2021). Their field is a stationary metric perturbation in a chosen gauge. Here the contracted Bianchi identity constrains the evolving Ricci tensor itself, and the two lemmas provide the quantitative derivative bounds needed below.

The exterior estimate

We apply the missing-mode argument on successive annuli. The goal is decay faster than the dimensional \(r^{-2}\) rate for the part of Ricci controlled by the root energy. The spatially constant part is instead controlled from a fixed outer annulus; separating these two contributions prevents a loss proportional to the number of shells. Take dyadic radii based on \(a\), and define \[\mathcal A_t(r)=\{y:r/2\le d_t(x_t,y)\le2r\},\quad A(r)=\left\|\sup_{\mathcal A_t(r)}|G|\right\|_{L^2(I)},\quad V(r)=\left\|\sup_{\mathcal A_t(r)} \sum_{j=1}^m r^j|\nabla^jG|\right\|_{L^2(I)}.\] Here \(m\) is a fixed sufficiently high order, for example \(60\). We may replace widths by fixed neighboring dyadic widths; a star will indicate a maximum over a fixed number of such neighbors.

The maximal function controlling the root

Fix a sufficiently large \(D_0\) and consider the tree pieces inside \(B_t(x_t,D_0a)\). Choose an ample fixed \(D_1>D_0\), enlarging it to include the short-time comparison patches. Extend functions of time by zero outside \(I\), fix \(1<p<2\), and set \[ f(t)=a^{-2}\|G(\cdot,t)\|_{L^2(B_t(x_t,D_1a))},\qquad H_*(t)=\bigl(\mathcal M(f^p)(t)\bigr)^{1/p}, \tag{52}\] where \(\mathcal M\) is the centered Hardy–Littlewood maximal operator on the time line (Hardy and Littlewood 1930). We include its exact boundedness argument at the exponent needed below.

A weighted estimate at a single time

The exterior estimate controls the collar error in time-integrated norms. The static input also needs a bound at a single time on every smaller descendant and joining neck. The maximal function just defined provides the time control; we now obtain the additional exponential weight along a deep neck.

There is also an uncut qualitative estimate on these true pieces: \[ r^2|\mathop{\mathrm{Ric}}|_{j,r}\le C\epsilon r \le o_i(1)e^{-\nu D}, \tag{62}\] with \(\nu\) decreased below one, using \(r\le Ca e^{-D}\) on deep necks and \(r\le Ca\) on compact pieces. Equation (54) supplies the stronger estimate relative to the energy and cutoff. In the root units used in Section 2, its precise form is \[\eta\,r^2|\mathop{\mathrm{Ric}}|_{j,r} \le Ca^2(H_*+\epsilon)e^{-\nu D}.\] Together with Equation (53), this is the interface used for the functional estimate in Section 6.

Norms and the extension statement

All lengths in this Section are in collar units. Let \(\Gamma\subset O(4)\) be a nontrivial finite group acting freely on \(S^3\), and put \[C_1=\{1\le |x|\le2\}/\Gamma,\qquad \mathcal E=\{|x|\ge1\}/\Gamma.\] Calculations are made with equivariant Cartesian components on the Euclidean cover. For a fixed-ratio shell \(\Omega_r\), use the norm \[ \|v\|_{H^s(r)}^2 =\sum_{j=0}^s r^{2j-4}\int_{\widetilde\Omega_r}|\partial^jv|^2\,dx, \qquad |v|_{C^s,r}=\sum_{j=0}^s r^j\sup_{\widetilde\Omega_r}|\partial^jv|. \tag{63}\] Here \(\widetilde\Omega_r\) is the lifted shell. Thus derivatives, but not the sizes of tensor components, are rescaled. In particular, \(\|\mathcal K(v)\|_{H^{s-2}(r)}\) is bounded by \(Cr^{-2}\|v\|_{H^s(r)}\). Equivalent fixed enlargements of the shells will be used without changing notation. At a boundary they are one-sided annuli of a fixed positive relative width. Define \[\|v\|_{H^s_\beta(\mathcal E)} =\sup_{r\in\{1,2,4,\ldots\}}r^\beta\|v\|_{H^s(r)}.\] On filled Euclidean space the corresponding norm also includes an ordinary Sobolev norm on the ball of radius two. These conventions give constants independent of \(\Gamma\): all estimates are on the fixed cover and finite-group averaging has norm at most one. Every relation \(\rho\asymp r\) below ranges over a fixed finite number of neighboring dyadic shells. Its comparison width is chosen once to include the initial shells affected by the collar correction and is independent of the outer radius \(R\).

Fix \(k=30\) and \(1<q<2\). Let \(R\) be a large dyadic radius; \(R\) in this Section does not denote scalar curvature. Suppose a metric \(g\), expressed in these coordinates on \(1\le |x|\le4R\), satisfies \[ |g-\delta|_{C^{k+2},r}\le\theta_r, \qquad \sum_{1\le r\le R}\theta_r\le\sigma, \qquad \theta_r\ge \|g-\delta\|_{H^k(C_1)}r^{-q}. \tag{64}\] The weights are positive and comparable on neighboring shells, with fixed comparison constants; the hypotheses include the fixed enlargements needed below. A fixed multiplicative constant in the last inequality can be incorporated into the weights. Set \[ \begin{split} m_1={}&\sup_{C_1}|\mathop{\mathrm{Ric}}(g)| +\sum_{1\le r\le R}\sup_{\Omega_r^+} \left(\theta_r|\mathop{\mathrm{Ric}}(g)|+ \sum_{j=1}^{k-2}r^j|\nabla_g^j\mathop{\mathrm{Ric}}(g)|\right) +\frac{\theta_R}{R^2}, \end{split} \tag{65}\] where \(\Omega_r^+\) denotes the specified ample enlargement. Curvature norms on the right are intrinsic; since the metrics are uniformly close to \(\delta\), changing to Cartesian component magnitudes costs only a fixed factor.

The last term in Equation (65) cannot simply be discarded. A regular vacuum curvature field on a finite annulus is detected at its outer boundary even when its Ricci tensor vanishes. The term \(\theta_R/R^2\) retains this information.

Two local linear estimates

The flat compatibility used here is the Saint-Venant relation between metric strain and linearized curvature. Its place in the Calabi complex is described in (Khavkine 2017, sec. 2.2). The relation between compatibility, potentials and Korn’s inequality is also developed in (Ciarlet and Ciarlet 2005) in dimension three. We give the required four-dimensional, equivariant estimates with their actual derivative orders and kernel below.

Write \(Sv\) and \(\mathcal K(v)\) for the flat linearizations of Ricci and Riemann curvature. With plus-contraction divergence, set \(Bv=\partial^iv_{ij}-\tfrac12\partial_j\mathop{\mathrm{tr}}_\delta v\). Directly differentiating the coordinate formulas for curvature gives \[ Sv=-\tfrac12\Delta v+\tfrac12\mathcal L_{(Bv)^\sharp}\delta, \qquad BSv=0, \qquad \Delta\mathcal K(v)=\mathcal T(\partial^2Sv). \tag{67}\] Here \(\mathcal T\) is a fixed linear combination of components, independent of the annulus. These identities hold distributionally at the Sobolev orders used below. Also \(S\mathcal L_V\delta=\mathcal K(\mathcal L_V\delta)=0\).

A filled-space generalized inverse and the analytic map

Let \[\mathcal X=H^k_q(\mathcal E;\operatorname{Sym}^2),\qquad \mathcal Y=H^k(C_1;\operatorname{Sym}^2) \mathbin{\times}H^{k-2}_{q+2}(\mathcal E;\operatorname{Sym}^2), \qquad Av=(v|_{C_1},Sv).\] The pair \((g|_{C_1}-\delta,0)\) need not lie in the range of \(A\): prescribed collar data need not admit a linearized Ricci-flat extension. We first solve a projected nonlinear equation. The supplied metric on the finite annulus will then control the residual left by the projection.

We construct a bounded linear \(L:\mathcal Y\to\mathcal X\) such that \[ ALA=A. \tag{72}\] Fix a bounded linear extension \(E\) from collar \(H^k\) to \(H^k(B_2)\), equal to its input on the entire collar. Such an extension is obtained by reflection across the inner smooth boundary in finitely many boundary charts, with the usual finite linear combination of reflections matching derivatives through order \(k\), followed by a partition of unity and an interior cutoff. Average it over \(\Gamma\). Fix also a radial cutoff \(\chi\) equal to one near radius one and zero before radius two. For \(d=(v_1,f)\), define \[ f^\#= \begin{cases} S(Ev_1),& |x|<1,\\ \chi Sv_1+(1-\chi)f,&1\le |x|\le2,\\ f,&|x|>2. \end{cases} \qquad u=\Delta^{-1}(-2f^\#). \tag{73}\] The source lies in filled \(H^{k-2}_{q+2}\), with norm at most \(C\|d\|_{\mathcal Y}\); Lemma 23 gives \(\|u\|_{H^k_q}\le C\|d\|_{\mathcal Y}\). Apply Lemma 22 to \(v_1-u|_{C_1}\), extend its extracted vector linearly to the exterior, and cut that vector off outside a fixed annulus. Write the resulting vector as \(V_d\), and set \[Ld=u|_{\mathcal E}+\mathcal L_{V_d}\delta.\] This operator is bounded. Its collar residual satisfies \[ \|v_1-(Ld)|_{C_1}\|_{H^k} \le C\|\mathcal K(v_1-u)\|_{H^{k-2}(C_1)}, \qquad S(Ld)=Su. \tag{74}\]

To prove Equation (72), take \(d=Av\) and fill \(v\) itself by the same extension \(E(v|_{C_1})\), obtaining \(V\) on all of \(\mathbb{R}^4\). The construction gives \(f^\#=SV\) everywhere, including the interpolation region. Equation (67) gives \(Bf^\#=0\). Thus \(Bu=0\): either commute the constant coefficient derivative with the Newton integral in distributions, or observe that \(Bu\) is an entire harmonic field decaying at infinity and apply the mean-value property. Consequently \(Su=f^\#\). The last identity in Equation (67) shows that \(\mathcal K(V-u)\) is harmonic on the whole space. It decays at infinity and hence vanishes by the same mean-value argument. The local extraction in Equation (74) is now exact. We obtain \((Ld)|_{C_1}=v|_{C_1}\) and \(S(Ld)=Sv\), which proves Equation (72). Filling the hole is essential here: a decaying harmonic field on the punctured exterior need not vanish.

Put \(Q=LA\). Equation (72) implies \[Q^2=Q,\qquad AQ=A.\] In particular \(\mathcal X_0=\operatorname{im}Q\) is a closed Banach subspace. For small \(h\in\mathcal X_0\), define \[\Phi(h)=QL\bigl(h|_{C_1},\mathop{\mathrm{Ric}}(\delta+h)\bigr).\] This map is analytic. Indeed the inverse metric is given by its convergent Neumann series, and the coordinate expression for Ricci is a sum of inverse-metric multiples of \(D^2h\) and products of \(Dh\). On every rescaled shell, \(H^{k-2}\) is a multiplication algebra; inserting weights gives an analytic map \(H^k_q\to H^{k-2}_{q+2}\). Its differential at zero is \(S\). For \(h\in\mathcal X_0\), \[D\Phi(0)h=QLAh=Q^2h=h.\] The analytic inverse function theorem on this Banach space therefore solves \[ \Phi(\bar h-\delta)=QL(g|_{C_1}-\delta,0) \tag{75}\] in a fixed neighborhood. It yields the first estimate in Equation (66) and the asserted bounded first derivative. Everything in this definition is fixed by the collar, \(q\), and the choices of linear operators.

It remains to estimate the error. Set \[ d_1=(p_1,f) =(g|_{C_1}-\bar h|_{C_1},-\mathop{\mathrm{Ric}}(\bar h)), \qquad e=\|d_1\|_{\mathcal Y}. \tag{76}\] Equation (75) says \(QLd_1=0\). Applying \(A\), and using \(AQ=A\), gives \[ ALd_1=0. \tag{77}\] No identity \(LAL=L\) is needed. We will show that \(d_1-ALd_1\) is small, and then use Equation (77).

A comparison gauge on the long finite annulus

The analytic extension has already been defined from the collar alone. To estimate its error, we are free to compare it with the supplied metric in a separate gauge on the longer annulus. The construction below must retain constants independent of that annulus’s length, even when its coordinate rotations accumulate. We first construct a linear projection which removes coordinate strain without a constant depending on \(R\). For tensors on \(1\le |x|\le R\), there are linear operators \(P_0\) and \(\mathcal D z=\mathcal L_{P_0z}\delta\), with \(P_0z=0\) on all of \(C_1\), such that \[ \|z-\mathcal Dz\|_{H^k(r)} \le C\left(r^2\max_{\rho\asymp r} \|\mathcal K(z)\|_{H^{k-2}(\rho)} +\mathbf1_{r\le C_0}\|z\|_{H^k(C_1)}\right). \tag{78}\] Here and below \(C_0\) is a fixed width, not a growing end length.

Here is the construction. Take a sequence of roomy dyadic annuli \(\Omega_j\) and smaller neighboring overlaps with fixed relative widths. Lemma 22 supplies local vectors \(v_j\) with \[r_j^{-1}\|v_j\|_{H^{k+1}(r_j)}\le C\|z\|_{H^k(r_j)}, \quad \|z-\mathcal L_{v_j}\delta\|_{H^k(r_j)}\le e_j, \quad e_j=Cr_j^2\|\mathcal K(z)\|_{H^{k-2}(\Omega_j)}.\] On each overlap, Equation (69) writes \[v_{j+1}-v_j=A_jx+w_j, \qquad r_j^{-1}\|w_j\|_{H^{k+1}} \le C(e_j+e_{j+1}),\] where \(A_j\) is skew and commutes with \(\Gamma\). There is no equivariant translation: a nonzero invariant vector would be fixed by each nonidentity group element, contrary to its free action on the sphere. The rigid projection can be chosen orthogonal, so its coefficient also obeys \[|A_j|\le C\bigl(\|z\|_{H^k(\Omega_j)} +\|z\|_{H^k(\Omega_{j+1})}\bigr).\] Choose \(B_{j+1}=B_j-A_j\), and write \(W_j=v_j+B_jx\). Then \(W_{j+1}-W_j=w_j\) on the overlap. With a radial partition \(\sum_j\psi_j=1\), let \(W=\sum_j\psi_jW_j\). The part of its strain coming from partition derivatives is \[\sum_j\bigl(d\psi_j\otimes W_j^\flat +W_j^\flat\otimes d\psi_j\bigr).\] On a shell subtract any one active \(W_i\) inside this sum; \(\sum d\psi_j=0\) cancels it exactly. Only differences of neighboring \(W_j\)’s remain, whose norms are controlled by the nearby \(e_j\)’s. Thus the residual estimate in Equation (78) holds for \(W\) without its collar term. In particular a cumulative rotation \(B_j\) never appears in a residual bound.

Normalize the global rigid ambiguity by projecting \(W|_{C_1}\) off the rigid space. By Equation (69), its remaining collar norm is bounded by \(C\|z\|_{H^k(C_1)}\) plus the nearby curvature residuals. Subtract a fixed bounded extension of this entire collar vector, supported in \(r\le C_0\) and equal to it on all of \(C_1\). The resulting vector is \(P_0z\); this proves Equation (78). If \(z=\mathcal L_V\delta\) with \(V=0\) on \(C_1\), both terms on the right vanish. Thus \(\mathcal Dz=z\) on such strains. Moreover \(P_0z-V\) is a Euclidean Killing field vanishing on a whole collar, so it is zero. Since every \(\mathcal Dz\) is itself such a strain, \(\mathcal D^2=\mathcal D\).

For the nonlinear step, define \[n(z)=\sup_{r\le R}\theta_r^{-1}\|z\|_{H^k(r)}.\] The preceding construction gives more information than Equation (78). Namely the local potentials have size \(Cr\theta_r n(z)\), successive correcting rotations differ by \(C\theta_r n(z)\), and the first correcting rotation has size \(C\theta_1n(z)\). Consequently \(\mathcal D\) is bounded for \(n\), independently of \(R\). Radially interpolate the correcting rotations, with value zero on \(C_1\), to write linearly \[ P_0z=J(x)x+U(x),\qquad J(x)^t=-J(x),\qquad J\gamma=\gamma J\quad(\gamma\in\Gamma), \tag{79}\] where \(J=U=0\) on \(C_1\), and \[ \sup|J|\le C\sigma n(z),\qquad \sum_{d=1}^{k+1}\|r^d\partial^dJ\|_{L^\infty(\Omega_r)} +r^{-1}\|U\|_{H^{k+1}(r)}\le C\theta_r n(z). \tag{80}\] To see these bounds explicitly, before the collar correction use \(J=\sum\psi_jB_j\) and \(U=\sum\psi_jv_j\). For a derivative of \(J\), subtract a neighboring \(B_i\) using the differentiated partition identity. The resulting coefficient is bounded by a fixed sum of \(|B_j-B_i|\), hence by \(C\theta_rn(z)\). The undifferentiated bound is the telescoping sum \(\sum\theta_rn(z)\). The collar subtraction and a cutoff of \(J\) there change these estimates only on finitely many comparable shells. Bounds for arbitrarily many fixed derivatives of \(J\) follow from the smooth fixed partition. This justifies the supremum version used in Equation (80).

For \(z\in\operatorname{im}\mathcal D\), define \[ F_z(x)=e^{J(x)}(x+U(x)). \tag{81}\] Fix a sufficiently large but fixed ball \(n(z)\le M_0\). If \(\sigma\) is small in terms of \(M_0\), the map is the identity on \(C_1\), is equivariant, and is a coordinate map into \(1\le|x|\le4R\). Indeed Sobolev embedding and Equation (80) give \(\|DF_z-\mathop{\mathrm{Id}}\|_\infty\le C M_0\sigma\) and \(|F_z(x)-x|\le C M_0\sigma|x|\). Extend it by the identity to the inner ball. The derivative bound on this convex ball of radius \(R\) proves injectivity by integrating along segments. Its image cannot enter the unit ball from the exterior, since it already fixes that ball; the upper radius is less than \(4R\). This also verifies the boundary and domain margins needed for composition.

The use of an orthogonal exponential in Equation (81) gives \[ F_z^*g-\bar h=g-\bar h+z+\mathcal E(z),\qquad \mathcal E(0)=0, \qquad n(\mathcal E(z)-\mathcal E(z'))\le C\sigma n(z-z') \tag{82}\] on this fixed ball. We verify the relative error bound. Write \(O=e^J\) and \(\Omega_i=O^{-1}\partial_iO\). Then \[O^{-1}\partial_iF_z=e_i+\partial_iU+\Omega_i(x+U), \qquad \Omega_i=\partial_iJ+O(|J|\,|\partial_iJ|).\] The undifferentiated \(O\) cancels exactly from the Euclidean pullback. Its linear metric term is the symmetric derivative of \(U+Jx\): the extra undifferentiated \(J\) in \(D(Jx)\) has zero symmetric part. The remaining products have a shell factor \(\theta_r\) and a second factor bounded by \(C\sigma\). The same statement for differences follows from linearity of \(J,U\), the integral formula for a difference of matrix exponentials, and Sobolev multiplication on rescaled shells. For the part involving \(g-\delta\), the maps and their differences have relative \(H^{k+1}\) size at most \(C\sigma\) and \(C\sigma n(z-z')\). The mean-value formula for \((g-\delta)\circ F_z-(g-\delta)\circ F_{z'}\), together with the \(C^{k+2}\) bound in Equation (64), gives the factor \(C\theta_r\sigma n(z-z')\). Differentiating the pullback formula through order \(k\) produces products with the same bound; the highest derivative of a map occurs linearly and is controlled in \(L^2\), while the other factors are controlled by Sobolev multiplication. Neighboring-shell comparability covers the displaced points. This proves Equation (82). In particular an undifferentiated accumulated rotation produces no Euclidean metric error of size \(|J|^2\) on a remote shell.

Equation (64) and the first estimate for \(\bar h\) imply \(n(g-\bar h)\le C\). The equation \[z=-\mathcal D(g-\bar h)-\mathcal D\mathcal E(z)\] is a contraction on a sufficiently large fixed \(n\)-ball in \(\operatorname{im}\mathcal D\), once \(\sigma\) is small. For its solution set \(F=F_z\) and \(p=F^*g-\bar h\). Then \(\mathcal Dp=0\), \(p|_{C_1}=p_1\), and Equation (78) gives \[ \|F^*g-\delta\|_{H^k(r)}\le C\theta_r, \qquad Y:=\sup_{r\le R}r^{-2}\|p\|_{H^k(r)}\le C(e+X), \quad X:=\sup_{r\le R}\|\mathcal K(p)\|_{H^{k-2}(r)}. \tag{83}\] This gauge is only a comparison device for the present estimate. It is not used to define the analytic map in Equation (75).

Curvature control and the compatibility residual

The comparison gauge controls metric differences by linearized curvature. We now estimate that curvature and use the filled-space inverse again to bound the incompatibility of the collar and Ricci data. Filling the inner ball is what removes an otherwise uncontrolled inner boundary term. Put \(T=\mathop{\mathrm{Ric}}(F^*g)\). The coordinate Ricci formula and Equation (83) imply \[ Sp=T-\mathop{\mathrm{Ric}}(\bar h)+E_2=T+f+E_2, \qquad \|E_2\|_{H^{k-2}(r)}\le C\theta_rY. \tag{84}\] Indeed, writing \(\mathop{\mathrm{Ric}}(\delta+v)=Sv+N(v)\), its nonlinear terms satisfy on a shell \[\|N(v)-N(w)\|_{H^{k-2}(r)} \le Cr^{-2}(\|v\|_{H^k(r)}+\|w\|_{H^k(r)}) \|v-w\|_{H^k(r)}.\] Both first factors are \(O(\theta_r)\); the last factor is at most \(r^2Y\). On the collar the sharper bound \(\|E_2\|_{H^{k-2}(C_1)}\le C\sigma e\) holds because \(p=p_1\) there.

We will repeatedly use the consequence of Equation (65) \[ \|T\|_{H^{k-2}(C_1)} +\sum_{r\le R}\sum_{j=1}^{k-2} \|r^j\partial^jT\|_{H^0(r)} \le Cm_1. \tag{85}\] Here no sum of unweighted zeroth-order exterior values is being claimed. For completeness, covariant derivatives commute with pullback. The first derivative of \(F\) is bounded, so the intrinsic terms \(F^*(\nabla_g^j\mathop{\mathrm{Ric}}(g))\) are bounded by the suprema in Equation (65) on the enlarged shells. To pass from these tensors to Cartesian derivatives, expand iteratively \(\partial T=\nabla_{F^*g}T+\Gamma(F^*g)*T\). A term involving no positive covariant derivative of \(T\) contains a metric derivative and hence a factor \(\theta_r\); the other terms are bounded by the positive covariant derivatives already present in the data. Products through order \(k-2\) are controlled in scaled \(L^2\) by the \(H^k\) bound for \(F^*g\). This proves the shell sum. On \(C_1\), the unweighted zeroth-order term is supplied separately in Equation (65). Smooth approximation of \(F\) on the roomy shells proves the identities first for smooth maps and then in these finite Sobolev orders; their bounds require no uncontrolled higher derivatives of \(F\).

Apply the last identity in Equation (67) to \(p\). Its forcing satisfies \[ \sum_{r\le R}r^2 \|\mathcal T(\partial^2Sp)\|_{H^{k-4}(r)} \le C(m_1+e+\sigma Y). \tag{86}\] The contribution of \(T\) is controlled by Equation (85); the contribution of \(f\) is at most \(Ce\sum r^{-q-2}\); and that of \(E_2\) is at most \(CY\sum\theta_r\). The inner boundary strips have \(\mathcal K(p)\)-norm at most \(Ce\), because \(p=p_1\) on the collar. Outer boundary strips have norm at most \(C\theta_R/R^2\) by Equation (83) and the lower bound for \(\theta_R\) in Equation (64). Lemma 23 therefore yields \[X\le C(m_1+e+\sigma Y).\] Together with Equation (83), and then absorption of \(C\sigma(X+Y)\), this gives \[ X+Y\le C(m_1+e). \tag{87}\] The two derivatives of \(Sp\) are essential: a constant Ricci component is killed, so it does not accumulate once per shell.

We now run the fixed linear construction \(L\) on the data \(d_1=(p_1,f)\) from Equation (76). In particular \(f^\#\) and \(u\) always mean precisely the filled source and Newton potential in Equation (73). Let \(D_0=m_1+\sigma e\). We claim \[ \|Bf^\#\|_{H^{k-3}_{q+3}(\mathbb{R}^4)}\le CD_0, \qquad \int_{\mathbb{R}^4}Bf^\#\,dx=0. \tag{88}\] Inside radius one, \(Bf^\#=BS(Ep_1)=0\). On \(C_1\), Equation (84) and Equation (85) give \[\|Sp_1-f\|_{H^{k-2}(C_1)} \le C(m_1+\sigma e)=CD_0.\] Thus derivatives of the fixed blending cutoff cost at most \(CD_0\). On the exterior, the contracted Bianchi identity for \(\bar h\) gives \(B_{\bar h}f=0\). Comparing it with flat \(B\) yields terms of the form \((\bar h^{-1}-\delta^{-1})\partial f\) and \(\partial\bar h*f\), including the corresponding trace terms. Their weighted \(H^{k-3}_{q+3}\) norm is at most \(C\sigma e\) by Sobolev multiplication. This proves the first assertion. For the second, integrate the divergence expression for each component of \(Bf^\#\) over a ball. Its boundary flux is at most \(Ce\rho^3\rho^{-q-2}=Ce\rho^{1-q}\), which tends to zero because \(q>1\). Integrability follows from the first assertion. There is no inner boundary in this computation.

Commutation with the Newton operator and Equation (70) now give \[Bu=\Delta^{-1}(-2Bf^\#),\qquad \|Bu\|_{H^{k-1}_{q+1}}\le CD_0.\] If desired, the commutation identity follows by comparing the two distributional solutions: their difference is entire harmonic and decays, hence is zero. By Equation (67), \(Su-f^\#=\tfrac12\mathcal L_{(Bu)^\sharp}\delta\). On the collar \(f^\#-f=\chi(Sp_1-f)\), while outside it \(f^\#=f\). Hence \[ \|Su-f\|_{H^{k-2}_{q+2}(\mathcal E)}\le CD_0. \tag{89}\]

To control the other component of the residual, fill \(p\) by \(Ep_1\) in the unit ball, obtaining \(P\) on the full ball of radius \(R\). Consider \(W=\mathcal K(P-u)\). On the filled inner ball, \(SP-f^\#=0\); on the collar, \(SP-f^\#=(1-\chi)(Sp_1-f)\). Therefore \(S(P-u)\) on this whole fixed ball is bounded in \(H^{k-2}\) by \(CD_0\). Farther out it is \[S(P-u)=T+E_2+(f-Su).\] The shell sum of the forcing \(\Delta W=\mathcal T(\partial^2S(P-u))\) is consequently bounded by \[C\bigl(m_1+\sigma Y+D_0\bigr) \le C(m_1+\sigma e),\] where Equation (87) was used in the last step. The filled ball contributes one ordinary fixed-scale term to this sum. At its only boundary, near radius \(R\), \[ \|W\|_{H^{k-2}(R)} \le C\left(\frac{\theta_R}{R^2}+eR^{-q-2}\right) \le C(m_1+eR^{-q-2}). \tag{90}\] Here \(\|u\|_{H^k_q}\le Ce\) follows from the bounded linear construction of \(L\), and \(\mathcal K(p)\) is controlled on the outer strip by Equation (83). The filled-ball version of Equation (71), followed by the fixed-scale interior estimate around \(C_1\), gives \[ \|\mathcal K(p_1-u)\|_{H^{k-2}(C_1)} \le C\bigl(m_1+(\sigma+R^{-q-2})e\bigr). \tag{91}\] There is no inner boundary term proportional to \(e\); that is the purpose of using the same inward extension a second time.

Equations (74), (89), and (91) imply \[\|d_1-ALd_1\|_{\mathcal Y} \le C\bigl(m_1+(\sigma+R^{-q-2})e\bigr).\] Since \(ALd_1=0\), first choosing \(\sigma_*\) small and then \(R_*\) large makes the coefficient of \(e\) at most one half. Thus \(e\le Cm_1\), proving the second estimate in Equation (66). All absorptions have constants independent of the exterior length.

Finally choose a fixed smooth radial \(\zeta\), equal to one near radius one and zero near radius two, and define \[\hat h=\bar h+\zeta(g-\bar h)\quad\hbox{on }C_1, \qquad \hat h=\bar h\quad\hbox{for }|x|\ge2.\] It is a positive metric for the same smallness choice. The coordinate Ricci formula and bounded cutoff derivatives give \[\|\mathop{\mathrm{Ric}}(\hat h)\|_{H^{k-2}(C_1)} \le C\bigl(\|\mathop{\mathrm{Ric}}(\bar h)\|_{H^{k-2}(C_1)} +\|g-\bar h\|_{H^k(C_1)}\bigr)\le Cm_1.\] Outside the collar the estimate is already proved. The interpolation is linear in the collar input and \(\bar h\), so analyticity and the bounded derivative persist. This completes the proof of Theorem 21.

Material collars and time patches

At any individual time in such a patch, complete exterior coordinates can be chosen to agree with the fixed chart on \(C_1=\{1\le|x|\le2\}/\Gamma\). Here and below an outward fixed rescaling of the chosen middle collar is harmless. To see the agreement, take the single neck coordinates from Equation (25) out to a sufficiently small fixed radius \(r_*<r_0\) in interval units. On the roomy overlap their transition with the fixed chart has metric distortion \(O(\sigma_0+N^{-\mu})\) and controlled connection difference. Lifting the overlap, integration of the connection transformation makes this transition close to a rigid equivariant map, through all needed orders. Its linear part can be adjusted to be orthogonal by polar factorization. The group and whole-band identifications are those proved in Lemma 25. After that rigid alignment the maps are close to identity, so a radial cutoff interpolation on a shorter overlap has invertible derivative and patches the coordinate systems. This is precisely the overlap construction of the neck charts, now performed only in a regular fixed-width band. Decreasing \(\mu>0\) if needed, the shell weights in Theorem 21 can consequently be chosen as \[ \theta_r\le C\bigl[(\sigma_0+N^{-\mu})r^{-\mu} +\delta_i(rb)^\mu\bigr], \qquad 1\le r\le4R,\qquad Rb\asymp r_*. \tag{93}\] They include the collar norm times \(r^{-q}\) after decreasing \(\mu<q\). The dyadic sum of their first term is small with the fixed choices; the second sum tends to zero because \(Rb\asymp r_*\). These coordinates outside \(C_1\) are used only to estimate the error. They are not differentiated in time.

Let \(B_1(t)\) be an upper bound for \(|\mathop{\mathrm{Ric}}|_{k+4,b}\) on a fixed ample band of true distance radii \([c'b,C'b]\) around \(x_t\), in interval units. Suprema over all admissible center choices can be used to make this bound independent of a choice of chart. It is measurable and locally bounded for every \(i\). Define an increasing absolutely continuous time coordinate by \[ u(t)=\int_{t_*}^{t}(1+B_1(s))\,ds . \tag{94}\] Start each chart with a tolerance strictly better than \(\sigma_0\), as is possible by first taking \(N\) large. As long as the chart remains within \(\sigma_0\), Lemma 25 places its entire closed domain in the band defining \(B_1\). The flow equation \(\partial_tg=-2\mathop{\mathrm{Ric}}\), together with conversion between covariant and coordinate derivatives in that controlled chart, then gives \[\|g(t)-g(s)\|_{C^{k+3}(\text{fixed collar})} \le C\int_s^t B_1(v)\,dv .\] A first-exit argument proves validity for a fixed sufficiently small \(u\)-width. This holds in both time directions, using the absolute value of the integral.

On an interval with room around \(\mathop{\mathrm{supp}}\eta\), choose regularly spaced centers in the \(u\) coordinate, subordinate smooth nonnegative bumps of that fixed width, and normalize their sum. Pulling them back gives an absolutely continuous partition \(\{\chi_j\}\) with \[ \sum_j\chi_j=1\quad\hbox{near }\mathop{\mathrm{supp}}\eta,\qquad \sum_j|\chi_j'(t)|\le C(1+B_1(t))\quad\hbox{a.e.} \tag{95}\] Its overlap is bounded by a fixed number. For each \(i\) it is finite on the compact interval in use; no bound on the number of members uniform in \(i\) is asserted or needed.

On patch \(j\), retain the true metric on the material inner component and apply Theorem 21 in its fixed collar. Interpolate as in that theorem. Denote the resulting metric on the fixed manifold with one end by \(\hat g_j(t)\), and its functional in interval length units by \(E_j(t)\), using Equation (4). Different patches are allowed to have different extensions and different functional values.

The lower bound for functional variation

Applying the static inequality to every time patch

The lower variation bound is expressed in terms of the true root energy. We now obtain the opposing upper bound. The only required uniformity is over the ratios of functional value to weighted Ricci error: a new fixed tree may be extracted from each proposed failure of that uniformity.

Small energy fixes the pointed root geometry

The vanishing energy is an integral statement. To contradict the prescribed change of scale, it remains to translate it into distance preservation on a single dense set of material points and then into preservation of the curvature-radius maximum.

Smooth continuation on the original manifold

Proposition 32 contradicts the definite factor change of the comparable positive root scales at the endpoints of the intervals in Proposition 12. Hence a scalar-bounded flow cannot have a concentration point. Theorem 6 then gives a uniform full-curvature bound up to the original finite time \(T\).

For completeness, this yields the precise extension required in Theorem 1. Curvature differentiation on a closed manifold gives bounds for every covariant derivative of curvature on any interval \([t_0,T)\) with \(t_0>0\). Bounded Ricci curvature and \(\partial_tg=-2\mathop{\mathrm{Ric}}\) first make \(g(t)\) uniformly equivalent to \(g(t_0)\). Integrating the connection variation and its successive derivatives, or working in the resulting fixed controlled charts, then gives uniform bounds and convergence for all coordinate derivatives of \(g(t)\). Thus \(g(t)\) converges smoothly to a positive definite smooth metric \(g_T\) on the original manifold \(M\). Short-time existence from \(g_T\) follows from the Ricci–DeTurck equation on this same closed smooth manifold, followed by its diffeomorphism pullback. The flow equation determines all time derivatives from the spatial derivatives, so the old and new solutions join smoothly at \(T\). Retaining the given flow for \(t<T\) and this solution for \(t\ge T\) gives a smooth Ricci flow \(\widetilde g\) on \(M\times[0,T+\varepsilon)\), for some \(\varepsilon>0\), with \(\widetilde g(t)=g(t)\) for every \(t<T\). The argument uses arbitrary smooth initial data in real dimension four; no change of manifold or orbifold continuation is involved.