Global Support and Convex Injectivity Domains under Weak MTW

Proof. The first inclusion in (2.2) follows from the definition. For 0<r<1, a proper initial portion of the minimizing geodesic with velocity w\in G_x remains minimizing up to time 1/r>1 when parametrized with initial velocity rw. Hence rw\in I(x). The case r=0 follows from the constant geodesic.

Suppose w\in I(x), and put \gamma(\tau)=\exp_x(\tau w). There is a T>1 for which \gamma|_{[0,T]} is minimizing. Any other minimizing geodesic from x to \gamma(1), followed by \gamma|_{[1,T]}, would be a minimizing curve of the same length from x to \gamma(T). A minimizing curve has no corner, so its velocities agree at the joining point. Uniqueness for the geodesic equation then identifies the two initial geodesics. If d_w\exp_x were singular, a nonzero kernel vector would give a nonzero Jacobi field on [0,1] vanishing at both endpoints. Extend this field by zero to [0,T]. Its index form is zero, so the preceding observation makes the extension a Jacobi field on [0,T]. Its vanishing on (1,T) contradicts uniqueness for the Jacobi equation. Thus w is nonconjugate.

Conversely, suppose that w is a unique minimizing velocity and is nonconjugate. The map

has an invertible differential at (x,w). Choose neighborhoods on which it is a diffeomorphism. After shrinking its image neighborhood, every minimizing velocity for every pair in that neighborhood belongs to this inverse branch. To see this, a sequence of contrary minimizing velocities would have, by compactness of \mathcal C, a subsequential limit minimizing from x to \exp_x w. Uniqueness makes this limit w, contradicting its exclusion from the open branch neighborhood. Every nearby pair has a minimizer, so its inverse-branch velocity is minimizing and is the unique minimizer. Thus an open neighborhood of (x,w) in TM consists of minimizing velocities. A small forward radial change of w within this neighborhood proves w\in I(x). For w=0, the same conclusion follows directly from the definition; also L_x(x)=\{0\} and d_0\exp_x=\Id.

The inverse neighborhoods just constructed show that \mathcal I is open. On their image neighborhoods the true cost is half the squared norm of the smooth inverse-branch velocity, so it is smooth. Finally, \mathcal I contains the zero section. Its openness and compactness of the zero section give a uniform \rho>0 as asserted.□

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

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 Lemma 2.1. Time rescaling in (2.6) gives, along \gamma_{x,a},

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 Lemma 2.2, 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 (1.2) gives the equality in (2.8). The inequality is exactly (1.3). 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 assertion is immediate if b_0=b_1. Otherwise, suppose there is a singular interior velocity z, and write the segment as

Here b_0=z-l_0e and b_1=z+l_1e. The velocity z is nonzero, since d_0\exp_x=\Id. For every 0<r<1, the full contracted segment \{r(z+te):-l_0\leq t\leq l_1\} is contained in I(x) by (2.2).

Choose 0\ne k\in\ker d_z\exp_x. Variation of the initial velocity in the direction k gives a Jacobi field J_z along \gamma_{x,z} on [0,1] with

The function \langle J_z,\dot\gamma_{x,z}\rangle is affine in \tau and vanishes at both endpoints. Its initial derivative is \langle k,z\rangle_x, so k\perp z.

A trial-field estimate. The Jacobi-field trial below is related to the conjugate-cut argument of Cordero-Erausquin, McCann and Schmuckenschläger [3, Proposition 2.5]. We derive the needed Hessian estimate explicitly and combine it with transverse MTW concavity and the radial identity.

Fix a vector \xi\in T_xM such that \langle\xi,k\rangle_x\ne0. For a velocity a near z, let T_a(\tau) denote parallel transport from x along \gamma_{x,a} to time \tau, and define

The geodesic and parallel-transport equations give smooth dependence on (a,\tau). Thus these are smooth families of fields along the parametrized geodesics, with J_a(0)=J_a(1)=0 and E_a(0)=\xi, E_a(1)=0. Every J_a is nonzero, since its components j(\tau) are fixed and not identically zero. The construction uses no inverse of the exponential map at z.

Write Q_a=Q_{\gamma_{x,a}} on [0,1] and set P(a)=Q_a(J_a,J_a). This is a smooth function on a full neighborhood of z, and P(z)=0 by the Jacobi equation. It is nonnegative when a\in G_x. When a\in I(x), it is strictly positive: equality would make the nonzero field J_a a Jacobi field with zero endpoints, contrary to nonconjugacy. At a=z, integration by parts gives

For a\in I(x) near z, Lemma 2.2 can be tested on E_a+\lambda J_a. Minimizing the resulting quadratic polynomial in \lambda\in\R gives

The last inequality holds in one fixed neighborhood: smoothness bounds Q_a(E_a,E_a) from above and bounds Q_a(E_a,J_a)^2 away from zero there.

Reduction to a one-dimensional kernel. For each fixed \xi\perp e, Lemma 2.4 on the contracted segment gives

The two endpoint velocities are nonconjugate. Smoothness of the branch Hessians therefore makes the right-hand side bounded from below as r\uparrow1. On the other hand, P(rz)>0 and P(rz)\to P(z)=0. Inequality (3.1) would contradict this lower bound if \langle\xi,k\rangle_x\ne0. It follows that every k\in\ker d_z\exp_x is orthogonal to e^\perp, and hence belongs to \Span\{e\}. The kernel is nontrivial, so it is exactly \Span\{e\}. In particular, e is a kernel vector and z\perp e.

A joint tangential and radial estimate. Apply the trial-field construction with k=\xi=e, and retain the notation P for this choice. Since z+te\in G_x for small t of both signs, the smooth function t\mapsto P(z+te) has a local minimum of value zero at t=0. Its first derivative vanishes, so, for some \delta>0 and C>0,

Shrink \delta and choose r_0<1 so that both z+te and r(z+te) lie in the fixed neighborhood used in (3.1) whenever |t|\leq\delta and r_0<r<1. Choose a Lipschitz constant L for P on a slightly larger ball and a bound K for |z+te|_x in this range. Since r(z+te)\in I(x), we obtain the simultaneous estimate

Combining this with (3.1), and adjusting positive constants, gives

for all these t,r. In particular, the constants do not depend on the manner in which the two parameters approach 0 and 1.

The radial identity gives a contradictory lower bound. Put B_x(a)=g_x-A_x(a) and, for 0<r<1, define

Because z\perp e, Lemma 2.4 with test vector z and velocity direction re makes f_r convex. By (2.5), f_r(0)=0. Nonconjugacy at b_1=z+l_1e bounds f_r(l_1) from above uniformly as r\uparrow1. The convexity inequality on [0,l_1] therefore yields

for r close to 1, with C_2\geq0 independent of r. The radial identity also says

Indeed, for a=r(z+te) and r>0, the first identity annihilates z+te in either slot. Inserting z=(z+te)-te in both slots gives the second identity, with no factor of r remaining. As |e|_x=1, we conclude that

Choose r=1-t^2 for sufficiently small positive t. Both (3.3) and (3.5) apply and would imply

Multiplying by t^2 and letting t\downarrow0 is impossible, since c_1>0. This contradiction proves the proposition.□

Proof. We induct on the number of positive weights in a convex representation. A representation with one positive weight gives one of the prescribed generators. For a representation

write

The vector w' is nonconjugate by induction, and a_k is nonconjugate by hypothesis. Their entire segment lies in K\subset G_x, so Proposition 3.1 makes w nonconjugate as well. Discarding zero weights gives the assertion for every point of K.□

Proof. The divided triangle inequality is

It follows by concatenating constant-speed minimizing curves and using the energy characterization of squared distance. This shortening inequality is also used in [3, 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

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

The lower inequality follows from (4.3) 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 (4.4). 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 (4.4) gives (4.1). 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)=\Id, D_rC_p(0)=0. The field in question is

Integration by parts identifies its index with \langle S_p(r)^{-1}C_p(r)\xi,\xi\rangle. The Wronskian identities imply

Differentiating S_p^{-1}C_p gives (4.5). 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\Id before conjugacy, which proves the second assertion by integration. Finally, for a smooth distance branch r, \Hess(r^2/2)=r\,\Hess r+dr\otimes dr and \Hess r(\grad r,\cdot)=0. Since its radial direction at the initial point is -p/|p|, this gives (4.6); 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 4.2 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

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

Insert p_i=-D_xc(x_i,z_i)/t. On the above ball, D^2_{xx}c\geq a\Id and |D^2_{xz}c|\leq B, uniformly. Hence

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

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

such that

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

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,

Repetitions and zero weights are allowed. Compactness of the active graph and of the weight simplex gives, after another subsequence,

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,

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

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 (4.8) by l_j and use \Delta_j\leq0. The limit gives

At every finite stage the active representation gives

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=\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\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 Lemma 4.3 therefore gives

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 (4.8) by m_{ji} and sum over all slots. Equation (4.9) cancels the linear terms exactly, giving

The final sum tends to zero. The first sum tends to \delta_*: positive-weight slots give (4.10), and zero-weight slots disappear because their B_{ji} remain bounded. No rate of convergence of the weights is needed. We obtain

contradicting \Delta_j\leq0.□

Proof. Invariance of domain makes f a local homeomorphism [8, 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

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 [8, 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 [8, 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

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 3.2 therefore makes its entire hull nonconjugate. Thus every scaled graph vector at time t_* is nonconjugate, and Lemma 4.6 makes F_{t_*} locally injective.

Choose \epsilon\in(0,t_*) small enough for Lemma 4.4. Then \Gamma_u is a connected compact topological n-manifold, and

is a homotopy from the homeomorphism F_\epsilon to F_{t_*}. Lemma 4.7 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 Lemma 4.5; 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, Lemma 2.1 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 Lemma 4.6 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 Lemma 4.5 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

pass by continuity to (4.11).□

Proof. For p_0,p_1\in G_x, put y_i=\exp_xp_i and consider

Both p_i belong to \partial w(x), so their segment does also. Theorem 4.8 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 4.8, applied to \max\{m_{p_0},m_{p_1}\} at x, gives

Since

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

which proves (4.12).□

Proof. Let (x,p)=F_t^{-1}(z), set \gamma(s)=\exp_x(sp), and put y=\gamma(1), w=\dot\gamma(t). Theorem 4.8 makes y a supporting endpoint for u at x. Since v is the dual transform, this gives

Formula (4.2) gives, for all x',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 (4.13).

Equivalently, the function has the upper and lower supports

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

Thus both supports in (4.16) 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

The support gradients give \grad Q_tu(z)=w, with |w|\leq D. Reversing geodesic flow proves (4.14). 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

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)\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 (4.15), 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=\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

Adding the selected branch of \hat c/t yields

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 Lemma 4.3 gives

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

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 (A.1) 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

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 \eps\in(0,t) as in Lemma 4.4. That result makes F_\eps a homeomorphism with locally Lipschitz inverse. Since F_\eps 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

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 [14, Section 12] and the graph correspondence in [1, Proposition 1.1]; compare also [13, Section 2] for related diagonal coordinates in optimal transport.

Rademacher’s theorem [15, Chapter 2, Theorem 1.4] gives, at almost every point of a graph chart,

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

Local minimality at the terminal time implies

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,

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

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 (A.4).

Split the matrices into \ker X and its orthogonal complement. Then tP+\mathscr A X is block triangular, with kernel block (t/a)\Id. On the complementary block, multiplication by the positive definite matrix X makes it symmetric positive semidefinite by (A.4). Consequently

with positive sign whenever this matrix is nonsingular. The other factor in (A.3) 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:

These maps are Lipschitz and form a homotopy from the identity to \mathcal H_t. At the small time \eps, semiconvexity and the positive cost Hessian make each stationary graph branch a local minimum; equivalently, u+c_\eps(\cdot,z) is strongly convex on the relevant coordinate ball. The preceding sign calculation thus applies to F_\eps 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\vol and applying the Lipschitz area formula [15, Chapter 2, Theorem 3.3, Remark 3.4, and (4.19)–(4.20)] in manifold charts gives

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. Lemma 4.5 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 Lemma 2.1 places it in I(x) and proves the remaining assertions.□