Proof. Move \(x\) along a geodesic with initial velocity \(\xi\) and follow the inverse branch to the fixed endpoint \(\exp_x w\). The variational field \(Y\) is Jacobi and satisfies \(Y(0)=\xi\), \(Y(1)=0\). Differentiating the first-variation formula gives \[A_x(w)(\xi,\xi)=-\langle\xi,D_\tau Y(0)\rangle_x =Q_{\gamma_{x,w}}(Y,Y),\] where the final equality is integration by parts. There is no initial acceleration term because the moving initial point follows a geodesic. For any admissible \(X\), put \(Z=X-Y\). The endpoint conditions and the Jacobi equation give \(Q(Y,Z)=0\), while minimality of the original geodesic gives \(Q(Z,Z)\geq0\). Hence \(Q(X,X)\geq Q(Y,Y)\). Equality makes \(Z\) a Jacobi field vanishing at both endpoints; nonconjugacy forces \(Z=0\). At \(w=0\), the minimizing field is \(Y(\tau)=(1-\tau)\xi\) along the constant geodesic, which gives \(A_x(0)(\xi,\xi)=|\xi|_x^2\). ◻

Proof. Both \(sa\) and \(s'a\) belong to \(I(x)\) by 3. Time rescaling in (9) gives, along \(\gamma_{x,a}\), \[\frac1s A_x(sa)(\xi,\xi) =\min_{\substack{X(0)=\xi\\X(s)=0}}Q_{\gamma_{x,a}|_{[0,s]}}(X,X).\] Extend the unique minimizing field by zero to \([0,s']\). It is an admissible field for the corresponding minimum on \([0,s']\) and has the same index-form value. If \(\xi\ne0\), it cannot be that minimum: otherwise 4, with time rescaled, would make the extension Jacobi, although it vanishes on \((s,s')\) and is nonzero at \(0\). Thus \(A_x(sa)(\xi,\xi)/s>A_x(s'a)(\xi,\xi)/s'\), proving the claim. For \(a=0\) the same argument applies, and the difference is explicitly \((1-s/s')g_x\). ◻

Proof. For fixed \(v\in I(x)\), the second derivative at \(u=0\) of \(c(\exp_x(u\xi),\exp_x v)\) is \(A_x(v)(\xi,\xi)\), because \(u\mapsto\exp_x(u\xi)\) is a geodesic. Differentiating this identity twice in the affine velocity direction \(\eta\) and using (2) gives the equality in (11). The inequality is exactly (3). Each segment in the stated convex hull has direction perpendicular to \(\xi\), so the resulting linewise concavity gives the finite Jensen inequality by induction. Vanishing test vectors or line directions cause no exception. ◻

Proof. The divided triangle inequality is \[ c_{a+b}(z,y)\leq c_a(z,w)+c_b(w,y),\qquad a,b>0. \tag{18}\] It follows by concatenating constant-speed minimizing curves and using the energy characterization of squared distance. This shortening inequality is also used in (Cordero-Erausquin et al. 2001, Claim 2.4). Equality holds when \(w\) divides a minimizing geodesic in the time ratio \(a:b\).

In particular, if \(p\in G_x\), \(y=\exp_xp\), and \(y_s=\exp_x(sp)\), \(0<s<1\), then \[ c(z,y)\leq c_s(z,y_s)+\frac{1-s}{2}|p|^2, \qquad c_s(x,y_s)=\frac{s}{2}|p|^2. \tag{19}\] The right side is a smooth upper support near \(x\), because \(sp\in I(x)\). Taking \(s=1/2\), the relevant half-vectors form a compact subset of the open injectivity domain. The supports therefore have a common neighborhood size and a common upper Hessian bound. This proves uniform semiconcavity of the cost; symmetry handles its other variable. Also \(|c(x,y)-c(x',y)|\leq Dd(x,x')\). Suprema of the corresponding negative costs consequently give the asserted Lipschitz and semiconvexity bounds for \(u\).

For completeness, the directional derivative at \(x\) is \[ u'(x;h)=\max_{p\in\mathcal V_u(x)}\langle p,h\rangle. \tag{20}\] The lower inequality follows from (19) for every active log. To obtain the upper inequality, choose an active endpoint at \(x_r=\exp_x(rh)\) and a minimizing log there. Compactness gives, along any convergent subsequence, an active endpoint and a minimizing log at \(x\). Applying the uniform upper cost support at \(x_r\) to the point \(x\) bounds \([u(x_r)-u(x)]/r\) above by the corresponding log paired with \(h\), up to \(O(r)\). This proves (20). For a semiconvex function, the ordinary subdifferential is the set of linear forms bounded above by every directional derivative: this follows by subtracting the fixed negative quadratic term and applying the supporting-hyperplane characterization of a convex function. Thus (20) gives (17). The contact equality defines a closed subset of the compact total minimizing-vector set. Thus the total active-log graph is compact, and so is each of its fibers. Carathéodory’s Theorem gives the finite representation. Choose one with the fewest positive weights: an affine dependence would allow the weights to be varied, preserving their sum and barycenter, until one vanished. Thus its active vertices are affinely independent. ◻

Proof. The Hessian \(H_r(p)\) is the index form of the Jacobi field having initial value \(\xi\) and final value zero on the time interval \([0,r]\). Let \(C_p\) be the other fundamental Jacobi tensor, \(C_p(0)=\mathrm{Id}\), \(D_rC_p(0)=0\). The field in question is \[J(a)=C_p(a)\xi-S_p(a)S_p(r)^{-1}C_p(r)\xi.\] Integration by parts identifies its index with \(\langle S_p(r)^{-1}C_p(r)\xi,\xi\rangle\). The Wronskian identities imply \[C_p'(r)-S_p'(r)S_p(r)^{-1}C_p(r)=-S_p(r)^{-T}.\] Differentiating \(S_p^{-1}C_p\) gives (21). Jacobi evolution on the compact set \(|p|\leq D\), \(0\leq r\leq1\), bounds \(\|S_p(r)\|\) above by a geometric constant. Hence \(S_p(r)^{-1}S_p(r)^{-T}\geq\kappa_g\mathrm{Id}\) before conjugacy, which proves the second assertion by integration. Finally, for a smooth distance branch \(r\), \(\mathop{\mathrm{Hess}}(r^2/2)=r\,\mathop{\mathrm{Hess}}r+dr\otimes dr\) and \(\mathop{\mathrm{Hess}}r(\mathop{\mathrm{grad}}r,\cdot)=0\). Since its radial direction at the initial point is \(-p/|p|\), this gives (22); the identity at \(p=0\) follows by continuity. ◻

Proof. For fixed \(\xi\in E^\perp\), weak MTW says that \(q\mapsto A_x(tq)(\xi,\xi)\) is concave on every segment contained in the hull: each segment direction lies in \(E\) and is therefore perpendicular to \(\xi\). Jensen’s inequality therefore gives \(H_t(p)(\xi,\xi)\geq\sum_i m_iH_t(p_i)(\xi,\xi)\). Subtract \(\sum_i m_iH_s(p_i)\) and use Lemma 10 on each original minimizing ray. The limiting assertion follows by applying this argument to the contracted hull and passing to the specified smooth branch limits. In particular, it does not require \(sp\) to be minimizing. ◻

Proof. If \(L\) is a Lipschitz bound, a global minimizing pole satisfies \[\frac{d(x,z)^2}{2t}\leq f(z)-f(x)\leq Ld(x,z), \qquad d(x,z)\leq2Lt.\] Every graph vector has length at most \(L\), so its projection lies at distance at most \(Lt\) from its pole. For small \(t\), all these points lie in a common normal coordinate ball. On a slightly larger ball, \(D^2_{xx}c(x,z)\) is uniformly positive definite, whereas \(f\) has a fixed lower Hessian bound. Thus \(f+c_t(\cdot,z)\) is strongly convex there. Its global minimizer is unique, and every graph point projecting to \(z\) is the same minimizer by its stationarity equation. Conversely the minimizer supplies the unique log divided by \(t\) as a subgradient. This proves bijectivity; compactness proves that the inverse is continuous.

Here is also the required quantitative local assertion. In fixed coordinates write \(p_i\) as coordinate covectors. Semiconvexity gives \[\langle p_1-p_2,x_1-x_2\rangle\geq-K|x_1-x_2|^2.\] Insert \(p_i=-D_xc(x_i,z_i)/t\). On the above ball, \(D^2_{xx}c\geq a\mathrm{Id}\) and \(|D^2_{xz}c|\leq B\), uniformly. Hence \[(a-tK)|x_1-x_2|\leq B|z_1-z_2|.\] For small \(t\) this bounds the pole map in Lipschitz norm; its momentum is then Lipschitz by the smooth stationarity equation. Comparing the minimum at two nearby parameters with each other’s minimizing poles gives the displayed gradient formula. Its right side is Lipschitz, proving \(C^{1,1}\) regularity. Since \(F_t\) itself is the restriction of a smooth map on the tangent bundle, these charts are bi-Lipschitz. ◻

Proof. A minimizer exists because \(u+c_t(\cdot,z)\) is continuous on compact \(M\). Fix any \(w\in L_x(z)\) and any \(0<\lambda<1\), and put \(z_\lambda=\exp_x(\lambda w)\). Action splitting gives the upper support \[U_w(x')=\frac1{t\lambda}c(x',z_\lambda) +\frac1{t(1-\lambda)}c(z_\lambda,z) \geq c_t(x',z), \qquad U_w(x)=c_t(x,z).\] It is smooth near \(x\) because \(\lambda w\in I(x)\), and its gradient there is \(-w/t\). The function \(u+U_w\) therefore has a local minimum at \(x\). Its first-order inequality says exactly that \(w/t\in\partial u(x)\). This argument applies separately to every minimizing log. Choosing one of them gives a point \((x,w/t)\in\Gamma_u\) with \(F_t(x,w/t)=z\). ◻

Proof. Closedness of the minimizing-vector set first gives \(tp\in G_x\) for every graph point. Suppose local injectivity fails at \((x,p)\in\Gamma_u\). There are distinct graph points \[(x_j,p_j)\longrightarrow(x,p),\qquad (\widetilde x_j,\widetilde p_j)\longrightarrow(x,p)\] such that \[z_j=\exp_{x_j}(tp_j) =\exp_{\widetilde x_j}(t\widetilde p_j) \longrightarrow z=\exp_x(tp).\]

A common smooth branch. Nonconjugacy at \((x,tp)\) supplies one smooth inverse of the exponential pair map near \((x,z)\), with branch cost \(\hat c\). Both sequences of scaled velocities belong to this branch for large \(j\). Thus \(x_j\) and \(\widetilde x_j\) must be distinct: equality of their base points would force equality of their velocities in the same inverse branch. Exchange the two points if necessary so that \[\Delta_j=u(\widetilde x_j)+\frac1t\hat c(\widetilde x_j,z_j) -u(x_j)-\frac1t\hat c(x_j,z_j)\leq0.\] Write \(\ell_j=\log_{x_j}\widetilde x_j\), \(l_j=|\ell_j|>0\), and \(e_j=\ell_j/l_j\). These logs are unique for large \(j\) since \(l_j\to0\). After passing to a subsequence in a tangent-bundle chart, \(e_j\to e\in T_xM\) with \(|e|=1\). All pairs \((x',z_j)\), with \(x'\) on the short geodesic joining \(x_j\) to \(\widetilde x_j\), lie in a fixed compact product neighborhood on which \(\hat c\) is smooth.

Active supports and a uniform expansion. Choose active representations with \(n+1\) slots, \[p_j=\sum_{i=1}^{n+1}m_{ji}a_{ji},\qquad a_{ji}\in\mathcal V_u(x_j),\qquad m_{ji}\geq0,\qquad \sum_i m_{ji}=1.\] Repetitions and zero weights are allowed. Compactness of the active graph and of the weight simplex gives, after another subsequence, \[m_{ji}\to m_i,\qquad a_{ji}\to a_i\in\mathcal V_u(x), \qquad p=\sum_i m_i a_i.\] Only these finitely many active slots enter the argument; the representing family of targets need not be finite.

Fix \(s\) with \(t<s<1\). Splitting the original active rays at time \(s\) gives global lower supports for \(u\) touching at \(x_j\), with gradient \(a_{ji}\) and Hessian \(-H_s(a_{ji})\). The selected divided branch cost has gradient \(-p_j\) and Hessian \(H_t(p_j)\). Taylor expansion along \(\ell_j\) therefore yields, for every slot, \[ \Delta_j\geq l_j\langle a_{ji}-p_j,e_j\rangle +\frac{l_j^2}{2} [H_t(p_j)-H_s(a_{ji})](e_j,e_j) +\epsilon_{ji}l_j^2, \qquad \max_i|\epsilon_{ji}|\longrightarrow0. \tag{24}\] The remainder is uniform even for slots whose weights tend to zero. Indeed, the branch cost has bounded third derivatives on the fixed compact neighborhood, while the shortened active vectors \(sa_{ji}\) lie in a compact subset of the open injectivity domain. The corresponding support costs thus have one uniform third-derivative bound. Both Taylor remainders before division by \(l_j^2\) are \(O(l_j^3)\).

The limiting transverse directions. Put \[d_{ji}=\langle a_{ji}-p_j,e_j\rangle,\qquad B_{ji}=[H_t(p_j)-H_s(a_{ji})](e_j,e_j).\] Every \(B_{ji}\) is bounded and converges, because the first Hessian belongs to the fixed smooth branch and the second belongs to a uniformly shortened minimizing ray. Divide (24) by \(l_j\) and use \(\Delta_j\leq0\). The limit gives \[d_i:=\langle a_i-p,e\rangle\leq0.\] At every finite stage the active representation gives \[ \sum_i m_{ji}d_{ji}=0. \tag{25}\] Thus \(\sum_i m_i d_i=0\), and \(d_i=0\) whenever \(m_i>0\). The positive-weight limiting active velocities consequently lie in one affine hyperplane perpendicular to \(e\).

Let \(P=\{i:m_i>0\}\) and \(E=\mathop{\mathrm{span}}\{a_i-p:i\in P\}\). We have \(e\in E^\perp\) and \(p=\sum_{i\in P}m_i a_i\). For \(0<r<t\), the hull \(r\mathop{\mathrm{co}}\{a_i:i\in P\}\) lies in \(r\partial u(x)\subset I(x)\). At time \(t\) all its velocities are nonconjugate by hypothesis. The limiting transverse-slack inequality of 11 therefore gives \[ \delta_*:= [H_t(p)-\sum_{i\in P}m_iH_s(a_i)](e,e) \geq\kappa_g(s-t)>0. \tag{26}\] The later time \(s\) has been applied only to the original active velocities \(a_i\in G_x\); no assertion about \(sp\) or \(s\partial u(x)\) is used.

Cancellation before taking limits. Multiply (24) by \(m_{ji}\) and sum over all slots. Equation (25) cancels the linear terms exactly, giving \[\frac{\Delta_j}{l_j^2} \geq\frac12\sum_i m_{ji}B_{ji} +\sum_i m_{ji}\epsilon_{ji}.\] The final sum tends to zero. The first sum tends to \(\delta_*\): positive-weight slots give (26), and zero-weight slots disappear because their \(B_{ji}\) remain bounded. No rate of convergence of the weights is needed. We obtain \[\liminf_{j\to\infty}\frac{\Delta_j}{l_j^2} \geq\frac{\delta_*}{2}>0,\] contradicting \(\Delta_j\leq0\). ◻

Proof. Invariance of domain makes \(f\) a local homeomorphism (Hatcher 2002, Theorem 2B.3). Its image is open and compact, hence also closed; connectedness of \(N\) gives surjectivity. Each fiber is compact and discrete, and therefore finite. For \(f^{-1}(y)=\{q_1,\ldots,q_k\}\), choose disjoint neighborhoods \(U_i\) on which \(f\) is a homeomorphism onto a neighborhood \(V_i\) of \(y\). The compact set \(E=X\setminus\bigcup_i U_i\) has image missing \(y\), so \[W=\bigcap_i V_i\setminus f(E)\] is an evenly covered neighborhood of \(y\). Thus \(f\) is a covering.

A homotopy from \(h\) to \(f\) identifies their induced maps on fundamental groups after change of base point (Hatcher 2002, Lemma 1.19). Since \(h\) is a homeomorphism, \(f_*:\pi_1(X,q)\to\pi_1(N,f(q))\) is surjective. If \(f(q')=f(q)\), choose a path \(\gamma\) from \(q\) to \(q'\). Surjectivity of \(f_*\) supplies a loop \(\eta\) at \(q\) whose image under \(f\) is homotopic, with endpoints fixed, to \(f\circ\gamma\). Uniqueness of path lifting and the homotopy lifting property for coverings force \(\eta\) and \(\gamma\) to have the same endpoint (Hatcher 2002, Proposition 1.30). Thus \(q'=q\). A bijective local homeomorphism is a homeomorphism. ◻

Proof. The graph \(\Gamma_u\) is compact by uniform semiconvexity and Lipschitzness. Suppose that \(t_0p_0\notin I(x_0)\) for some \((x_0,p_0)\in\Gamma_u\) and some \(0<t_0<1\). The continuous map \[[0,t_0]\times\Gamma_u\longrightarrow TM,\qquad (t,(x,p))\longmapsto(x,tp)\] has a compact nonempty inverse image of the complement of the open injectivity domain. Its projection onto \([0,t_0]\) has a positive minimum \(t_*\), because graph velocities are bounded and a fixed neighborhood of the zero section lies in the injectivity domain. Consequently all earlier scaled graph vectors lie in \(I(x)\), all vectors at \(t_*\) lie in \(G_x\) by closure, and an actual failure is attained at \(t_*\).

For any \(p\in\partial u(x)\), represent \(p\) as a finite convex combination of active velocities \(a_i\in G_x\). The hull of the \(t_*a_i\) lies in \(t_*\partial u(x)\subset G_x\). Its generators are in \(I(x)\) because \(t_*<1\). Corollary 8 therefore makes its entire hull nonconjugate. Thus every scaled graph vector at time \(t_*\) is nonconjugate, and 14 makes \(F_{t_*}\) locally injective.

Choose \(\epsilon\in(0,t_*)\) small enough for 12. Then \(\Gamma_u\) is a connected compact topological \(n\)-manifold, and \[(\lambda,(x,p))\longmapsto \exp_x\bigl(((1-\lambda)\epsilon+\lambda t_*)p\bigr), \qquad 0\leq\lambda\leq1,\] is a homotopy from the homeomorphism \(F_\epsilon\) to \(F_{t_*}\). Lemma 15 shows that \(F_{t_*}\) is a homeomorphism.

Fix \((x,p)\in\Gamma_u\) and set \(z=F_{t_*}(x,p)\). A global minimizer \(x'\) of \(u+c_{t_*}(\cdot,z)\) exists. Every minimizing log \(w\in L_{x'}(z)\) gives a graph point \((x',w/t_*)\) by 13; its image under \(F_{t_*}\) is \(z\). Injectivity forces \(x'=x\) and \(w=t_*p\). Thus \(x\) is the unique global minimizing pole and \(L_x(z)=\{t_*p\}\). Together with nonconjugacy, 3 gives \(t_*p\in I(x)\) for every graph point, contradicting the attained failure. No scaled graph vector therefore reaches cut before time one.

For any fixed \(0<t<1\), all hypotheses of 14 now hold: earlier scaled vectors lie in \(I(x)\), and so does the vector at \(t\). The same short-time homotopy and covering argument make \(F_t\) a homeomorphism. Applying 13 as above proves that every graph point gives the unique global minimizing pole of \(Q_tu\).

Finally, fix \((x,p)\in\Gamma_u\) and let \(t\uparrow1\). Since \(tp\in I(x)\), \(d(x,\exp_xp)=|p|\). The global minimizing inequalities \[u(x')+\frac1t c(x',\exp_x(tp)) \geq u(x)+\frac1t c(x,\exp_x(tp)) \qquad(x'\in M)\] pass by continuity to (27). ◻

Proof. For \(p_0,p_1\in G_x\), put \(y_i=\exp_xp_i\) and consider \[w(z)=\max_{i=0,1}\{c(x,y_i)-c(z,y_i)\}.\] Both \(p_i\) belong to \(\partial w(x)\), so their segment does also. Theorem 16 puts that segment in \(G_x\). Finally \(I(x)\) is the interior of \(G_x\): it is open, while a radial cut-boundary vector has arbitrarily close outward radial vectors which are not minimizing. The interior of a convex set is convex. ◻

Proof. Write \(m_p(z)=|p|^2/2-c(z,\exp_xp)\) for \(p\in G_x\). Theorem 16, applied to \(\max\{m_{p_0},m_{p_1}\}\) at \(x\), gives \[m_{(1-\theta)p_0+\theta p_1}(z) \leq\max\{m_{p_0}(z),m_{p_1}(z)\},\qquad0\leq\theta\leq1.\] Since \[u(x)+\frac{|p|^2}{2}+v(\exp_xp) =\sup_z\{u(x)-u(z)+m_p(z)\},\] its sublevels on \(G_x\) are convex. Continuity and compactness of \(G_x\) give closedness and compactness. For \(p_0,p_1\) in a section, their midpoint \(p_m\) is also in it. If \(q(p)\) denotes the nonnegative gap, the squared-norm identity gives \[\frac{|p_0-p_1|^2}{8} =\frac{q(p_0)+q(p_1)}2-q(p_m) +v(\exp_xp_m)-\frac{v(\exp_xp_0)+v(\exp_xp_1)}2,\] which proves (28). ◻

Proof. Let \((x,p)=F_t^{-1}(z)\), set \(\gamma(s)=\exp_x(sp)\), and put \(y=\gamma(1)\), \(w=\dot\gamma(t)\). Theorem 16 makes \(y\) a supporting endpoint for \(u\) at \(x\). Since \(v\) is the dual transform, this gives \[u(x)+c(x,y)+v(y)=0.\] Formula (18) gives, for all \(x',z',y'\), \[u(x')+c_t(x',z') \geq -v(y')-c_{1-t}(z',y').\] Taking the infimum in \(x'\) and the supremum in \(y'\) gives \(Q_tu\geq-Q_{1-t}v\). At \(z'=z\), equality holds with \(x'=x\) and \(y'=y\), because the geodesic splits in the ratio \(t:(1-t)\). This proves (29).

Equivalently, the function has the upper and lower supports \[ -v(y)-c_{1-t}(z',y) \leq Q_tu(z') \leq u(x)+c_t(x,z'), \tag{32}\] both with equality at \(z\). The first is smooth there even when the full pair \((x,y)\) is conjugate: reversing the minimizing geodesic makes the segment from \(y\) to \(z\) a proper initial subsegment, and cut symmetry gives smoothness also from \(z\) to \(y\). The precise reversal identities are \[\log_zx=-tw,\qquad \log_zy=(1-t)w.\] Thus both supports in (32) have gradient \(w\).

The infimum defining \(Q_tu\) preserves the uniform semiconcavity bound of \(c_t\); its representation as the supremum of the lower supports preserves the uniform semiconvexity bound of \(-c_{1-t}\). In local coordinates it is therefore both semiconvex and semiconcave, and hence \(C^{1,1}\). More explicitly, its almost-everywhere Riemannian Hessian obeys \[-\frac{C_g}{1-t}g\leq\mathop{\mathrm{Hess}}Q_tu\leq\frac{C_g}{t}g.\] The support gradients give \(\mathop{\mathrm{grad}}Q_tu(z)=w\), with \(|w|\leq D\). Reversing geodesic flow proves (30). Smooth geodesic flow has bounded derivatives on the compact set of velocities of length at most \(D\) and times in \([-1,1]\). The Lipschitz conclusion therefore follows without inverting an exponential differential.

Finally choose \(s=(1+t)/2\). The single global support supplied by \(p\) can be split at \(y_s=\exp_x(sp)\) to give \[u(x')\geq u(x)+c_s(x,y_s)-c_s(x',y_s).\] After adding \(c_t(x',z)-c_t(x,z)\), the right side has zero gradient and Hessian \(H_t(p)-H_s(p)\geq\kappa_g(s-t)\mathrm{Id}\) at \(x\). The scaled minimizing-vector families at times \(t\) and \(s\) are compact subsets of the open injectivity domain. Their smooth action branches have uniform Taylor bounds on a common neighborhood. Shrinking that neighborhood gives (31), uniformly in the potential and in its graph point. ◻

Proof. Choose a positive finite active representation \(p=\sum_i m_ip_i\), and put \(E=\mathop{\mathrm{span}}\{p_i-p\}\). Fix \(s\) with \(t<s<1\). Each \(p_i\) is an original minimizing log, so its shortened support at time \(s\) is smooth near \(x\). In normal coordinates \(h\) at \(x\), these supports give \[u(\exp_xh)-u(x) \geq\max_i\left\{\langle p_i,h\rangle -\frac12H_s(p_i)(h,h)\right\}+o(|h|^2).\] Adding the selected branch of \(\hat c/t\) yields \[ \begin{split} &u(\exp_xh)+\frac{\hat c(\exp_xh,z)}t -u(x)-\frac{\hat c(x,z)}t\\ &\qquad\geq \max_i\left\{\langle p_i-p,h\rangle +\frac12B_i(h,h)\right\}+o(|h|^2), \qquad B_i=H_t(p)-H_s(p_i). \end{split} \tag{33}\] The contracted active hulls lie in \(I(x)\) by hypothesis, and their limits at time \(t\) are minimizing. The branch at \(tp\) is nonconjugate. Thus the limiting form of 11 gives \[\left.\sum_i m_iB_i\right|_{E^\perp} \geq\kappa_g(s-t)\mathrm{Id}.\] Only the individual active rays are evaluated at the later time \(s\).

The vectors \(p_i-p\) span \(E\), have positive weights, and have weighted mean zero. Hence, if \(E\ne\{0\}\), compactness of its unit sphere gives \[\max_i\langle p_i-p,h\rangle\geq b_E|h_E|, \qquad h_E=\operatorname{proj}_Eh, \qquad b_E>0.\] For \(E\ne\{0\}\), choose \(B\) so that \(b_E B>1+\max_i\|B_i\|\). When \(|h_E|\geq B|h|^2\), this linear term dominates all negative quadratic terms in (33) and leaves a positive multiple of \(|h|^2\). When \(|h_E|<B|h|^2\), take the weighted average inside the maximum. The linear term vanishes, the quadratic term on \(E^\perp\) is positive, and its mixed terms are \(O(|h|^3)\). The same averaged argument applies directly if \(E=\{0\}\). After shrinking the neighborhood, the claimed quadratic growth follows.

For openness, choose a small closed ball about \(x\) inside this neighborhood and inside the source domain of the branch. For an endpoint \(z'\) near \(z\), minimize \(u(\cdot)+\hat c(\cdot,z')/t\) on that ball. The positive boundary gap keeps each minimizer \(x'\) in its interior. As \(z'\to z\), every such minimizer tends to \(x\), since quadratic growth makes \(x\) the unique minimizer on the ball at \(z\). Stationarity gives \[p'=-\frac1tD_x\hat c(x',z')\in\partial u(x').\] First variation on the branch gives \(\exp_{x'}(tp')=z'\) and \((x',p')\to(x,p)\). Thus every neighborhood of \((x,p)\) in the graph projects onto a neighborhood of \(z\), as required. ◻

Proof. Choose a sufficiently small \(\varepsilon\in(0,t)\) as in 12. That result makes \(F_\varepsilon\) a homeomorphism with locally Lipschitz inverse. Since \(F_\varepsilon\) is the restriction of a smooth map on the compact graph, these are bi-Lipschitz graph charts.

We use Minty-type coordinates to compute the Jacobian sign. In a coordinate ball write \(p\) as a covector and take \(a>0\) small enough that \(h(x)=|x|^2/2+a u(x)\) is strongly convex. The coordinates \[q=x+ap\] parametrize the graph locally. To see this, restrict \(h\) to a small closed ball around the reference pole and maximize \(q\cdot x-h(x)\). Strong convexity makes the maximizer unique and Lipschitz in \(q\); for \(q\) near the reference value it is interior. Thus \(x(q)\) is the gradient of the local convex conjugate, and \(p(q)=(q-x(q))/a\). This parametrization is bi-Lipschitz. For the classical convex antecedents, see the proximal parametrization in (Moreau 1965, sec. 12) and the graph correspondence in (Alberti and Ambrosio 1999, Proposition 1.1); compare also (McCann et al. 2012, sec. 2) for related diagonal coordinates in optimal transport.

Rademacher’s theorem (Simon 2018, chap. 2, Theorem 1.4) gives, at almost every point of a graph chart, \[ X=D_qx\geq0, \qquad P=D_qp=\frac{\mathrm{Id}-X}{a}. \tag{34}\] Both matrices are symmetric and commute. Write \(\mathscr A=D^2_{xx}\hat c(x,F_t(x,p))\) for the coordinate source Hessian of the selected unit-duration action and put \(Z=D_qF_t\). Differentiating \(-D_x\hat c(x,F_t)=tp\) gives \[ Z=(-D^2_{xz}\hat c)^{-1}(tP+\mathscr A X). \tag{35}\]

Local minimality at the terminal time implies \[ X(tP+\mathscr A X)\geq0. \tag{36}\] Here is a justification requiring only first derivatives of the graph. Along a straight line in the chart, put \(x_r=x(q+r\xi)\) and \(p_r=p(q+r\xi)\). The semiconvex subgradient inequalities at two parameter values imply, almost everywhere on the line, \[\frac{d}{dr}u(x_r)=\langle p_r,x'_r\rangle.\] At a differentiability point of \((x,p)\), the expansions \(x_r=x+rX\xi+o(r)\) and \(p_r=p+rP\xi+o(r)\) give \[\begin{split} u(x_r)-u(x)-\langle p,x_r-x\rangle &=\int_0^r\langle p_s-p,x'_s\rangle\,ds\\ &=\frac{r^2}{2}\langle P\xi,X\xi\rangle+o(r^2). \end{split}\] For the last equality, integrate the term \(sP\xi\) by parts. The remaining integral is \(o(r^2)\) because \(x_s\) is Lipschitz and the remainder in \(p_s\) is \(o(s)\). No second derivative of the pole curve is used. Rewrite this as an expansion of \(u(x_r)-u(x)\), multiply by \(t\), and add the expansion of \(\hat c(x_r,z)-\hat c(x,z)\) with \(z=F_t(x,p)\) fixed. Its linear term is \(-t\langle p,x_r-x\rangle\), and its quadratic term is \(r^2\langle\mathscr A X\xi,X\xi\rangle/2\). The cancellation and the local-minimum inequality prove (36).

Split the matrices into \(\ker X\) and its orthogonal complement. Then \(tP+\mathscr A X\) is block triangular, with kernel block \((t/a)\mathrm{Id}\). On the complementary block, multiplication by the positive definite matrix \(X\) makes it symmetric positive semidefinite by (36). Consequently \[\det(tP+\mathscr A X)\geq0,\] with positive sign whenever this matrix is nonsingular. The other factor in (35) has positive determinant in compatible oriented charts. Indeed, \(-D^2_{xz}\hat c\) is the inverse exponential differential followed by the metric identification. Its sign starts positive at time zero and cannot change along the ray before a conjugate time. The nonconjugacy assumption supplies this sign through time \(t\).

Assume first that \(M\) is oriented and transfer the projections to \(M\): \[\mathcal H_r=F_r\circ F_\varepsilon^{-1}:M\to M, \qquad \varepsilon\leq r\leq t.\] These maps are Lipschitz and form a homotopy from the identity to \(\mathcal H_t\). At the small time \(\varepsilon\), semiconvexity and the positive cost Hessian make each stationary graph branch a local minimum; equivalently, \(u+c_\varepsilon(\cdot,z)\) is strongly convex on the relevant coordinate ball. The preceding sign calculation thus applies to \(F_\varepsilon\) independently of the terminal-time assumption. Its bi-Lipschitz differential is nonsingular almost everywhere, so the computed sign is positive. The chain rule transfers the nonnegative sign of \(DF_t\) to the signed Riemannian Jacobian \(J_{\mathcal H_t}\). Bi-Lipschitz changes of coordinates preserve the exceptional null sets, and hence \(J_{\mathcal H_t}\geq0\) almost everywhere.

For completeness, the signed degree identity can be justified on the smooth manifold without imposing a smooth structure on the graph. Smoothly approximate \(\mathcal H_t\) uniformly and strongly in \(W^{1,n}\), using local mollification and a smooth tubular projection after embedding the compact target. Uniformly close approximations are homotopic to \(\mathcal H_t\), hence to the identity. Stokes’ theorem gives \(\int_M h^*\omega=\int_M\omega\) for every smooth top form \(\omega\) and each such smooth approximation \(h\). Strong derivative convergence passes this identity to \(\mathcal H_t\): the pullback is multilinear of degree \(n\) in the derivative, and its coefficients converge uniformly. This identity also proves surjectivity. A missed point would give, by compactness, a missed open ball, and a top form supported in that ball with nonzero integral would contradict the identity.

Taking \(\omega=d\mathop{\mathrm{vol}}\) and applying the Lipschitz area formula (Simon 2018, chap. 2, Theorem 3.3, Remark 3.4, and (4.19)–(4.20)) in manifold charts gives \[\int_M J_{\mathcal H_t}\,d\mathop{\mathrm{vol}}=\mathop{\mathrm{vol}}(M), \qquad \int_M N(y,\mathcal H_t)\,d\mathop{\mathrm{vol}}(y) =\int_M|J_{\mathcal H_t}|\,d\mathop{\mathrm{vol}}=\mathop{\mathrm{vol}}(M),\] where \(N\) counts preimages and may be infinite. Surjectivity gives \(N\geq1\) everywhere, so \(N=1\) almost everywhere. If there were two distinct preimages, disjoint neighborhoods of them would have open images meeting in a nonempty open set, since \(F_t\) and hence \(\mathcal H_t\) are open. The multiplicity would be at least two on that set, contradicting the integral equality. Thus \(\mathcal H_t\) is injective. No discreteness of fibers is assumed in this argument.

If \(M\) is nonorientable, lift the homotopy to its orientation double cover, starting from the identity. Uniqueness of homotopy lifting shows that every lift commutes with the deck transformation. The lifted maps are Lipschitz, and the terminal lift is open because the covering maps are local diffeomorphisms. Its local Jacobian signs are the same as above: the lifted homotopy selects the target orientation transported along the geodesic used in the mixed-action calculation. The oriented argument therefore makes the terminal lift injective. Two distinct preimages downstairs could be lifted, applying a deck transformation to one of them if needed, to have the same lifted image. Deck equivariance would contradict injectivity upstairs. Thus \(F_t\) is injective in either case; surjectivity and compactness make it a homeomorphism.

Finally, take any global minimizing pole of \(Q_tu\) at an endpoint. 13 places that pole and every one of its minimizing logs divided by \(t\) in the graph fiber. Its uniqueness identifies the pole with the specified graph pole and identifies every minimizing log with \(tp\). Thus \(tp\) is the unique minimizing velocity to its endpoint. It is nonconjugate by hypothesis, so 3 places it in \(I(x)\) and proves the remaining assertions. ◻