==== TABLE OF CONTENTS ====
Global-Support-and-Convex-Injectivity-Domains-under-Weak-MTW//conjugacy.tex
Global-Support-and-Convex-Injectivity-Domains-under-Weak-MTW//degree.tex
Global-Support-and-Convex-Injectivity-Domains-under-Weak-MTW//geometry.tex
Global-Support-and-Convex-Injectivity-Domains-under-Weak-MTW//introduction.tex
Global-Support-and-Convex-Injectivity-Domains-under-Weak-MTW//main.tex
Global-Support-and-Convex-Injectivity-Domains-under-Weak-MTW//preliminaries.tex
Global-Support-and-Convex-Injectivity-Domains-under-Weak-MTW//references.bib
===========================
==== BEGIN Global-Support-and-Convex-Injectivity-Domains-under-Weak-MTW//conjugacy.tex ====
\section{Excluding conjugacy in minimizing convex hulls}
\label{sec:conjugacy}
The continuation argument will encounter finite convex hulls whose every
velocity is known to be minimizing. Their generators will be nonconjugate,
but their other points may lie at cut. We show here that none of those
points can be conjugate. This removes the focal obstruction before the
next section addresses uniqueness of minimizing geodesics.
The essential step concerns a segment with nonconjugate endpoints.
Transverse concavity first confines a
hypothetical conjugacy kernel to the segment direction. The radial
Hessian identity then controls that remaining direction and contradicts
the singular growth forced by an index-form trial field. All uses of
weak MTW occur at proper radial contractions, where the true cost is
smooth.
\begin{proposition}
\label{prop:segment-nonconjugacy}
Suppose that $[b_0,b_1]\subset G_x$ and that $b_0,b_1$ are
nonconjugate. Then every velocity in $[b_0,b_1]$ is nonconjugate.
\end{proposition}
\begin{proof}
The assertion is immediate if $b_0=b_1$. Otherwise, suppose there
is a singular interior velocity $z$, and write the segment as
\[
z+te,\qquad -l_0\leq t\leq l_1,
\qquad |e|_x=1,\quad l_0,l_1>0.
\]
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 $00.
\end{split}
\label{eq:trial-hessian}
\end{equation}
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.
\medskip\noindent
\emph{Reduction to a one-dimensional kernel.}
For each fixed $\xi\perp e$, \cref{lem:transverse-concavity} on the
contracted segment gives
\[
A_x(rz)(\xi,\xi)
\geq \frac{l_1}{l_0+l_1}A_x(rb_0)(\xi,\xi)
+\frac{l_0}{l_0+l_1}A_x(rb_1)(\xi,\xi).
\]
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 \eqref{eq:trial-hessian} 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 $z\perp e$.
\medskip\noindent
\emph{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 $C>0$,
\[
0\leq P(z+te)\leq Ct^2\qquad(|t|\leq\delta).
\]
Shrink $\delta$ and choose $r_0<1$ so that both $z+te$ and
$r(z+te)$ lie in the fixed neighborhood used in
\eqref{eq:trial-hessian} whenever $|t|\leq\delta$ and
$r_00,
\label{eq:singular-upper}
\end{equation}
for all these $t,r$. In particular, the constants do not depend
on the manner in which the two parameters approach $0$ and $1$.
\medskip\noindent
\emph{The radial identity gives a contradictory lower bound.}
Put $B_x(a)=g_x-A_x(a)$ and, for $00$, 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
\begin{equation}
A_x\bigl(r(z+te)\bigr)(e,e)\geq1-\frac{C_2}{t}
\qquad(00$. This contradiction proves the proposition.
\end{proof}
\begin{corollary}[Nonconjugacy of a minimizing convex hull]
\label{geo:first-cut}
Let $a_1,\ldots,a_m\in T_xM$ be nonconjugate velocities, with $m\geq1$,
and suppose
\[
K=\operatorname{co}\{a_1,\ldots,a_m\}\subset G_x.
\]
Then every velocity in $K$ is nonconjugate.
\end{corollary}
\begin{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
\[
w=\sum_{i=1}^k\theta_i a_i,\qquad
\theta_i>0,\qquad \sum_{i=1}^k\theta_i=1,\qquad k\geq2,
\]
write
\[
w=(1-\theta_k)w'+\theta_k a_k,\qquad
w'=\sum_{i=1}^{k-1}\frac{\theta_i}{1-\theta_k}a_i.
\]
The vector $w'$ is nonconjugate by induction, and $a_k$ is
nonconjugate by hypothesis. Their entire segment lies in $K\subset G_x$,
so \cref{prop:segment-nonconjugacy} makes $w$ nonconjugate as well.
Discarding zero weights gives the assertion for every point of $K$.
\end{proof}
In the continuation argument, membership of the entire hull in $G_x$
will follow by taking limits from smaller scales. The corollary then
removes conjugacy without assuming convexity of the full minimizing
domain.
==== END Global-Support-and-Convex-Injectivity-Domains-under-Weak-MTW//conjugacy.tex ====
==== BEGIN Global-Support-and-Convex-Injectivity-Domains-under-Weak-MTW//degree.tex ====
\section{An alternative globalization by positive degree}
\label{sec:degree-alternative}
The covering argument in the main text uses local injectivity of the
graph projection. Here we give a different sufficient condition:
openness, together with local minimality of its action branches, also
forces global injectivity. The reason is that local minimality makes
the projection Jacobian nonnegative in coordinates on the subgradient
graph, while the homotopy from a short-time projection has degree one.
The area formula then rules out two overlapping open images.
This argument uses the same active logs and divided-action estimates
as the main proof; it does not require the global supporting theorem.
\subsection{Local minima and openness}
\begin{lemma}[Local quadratic growth]
\label{geo:local-growth}
Let $u$ be a potential and $00$ such that
\[
u(x')+\frac{\hat c(x',z)}t
\geq u(x)+\frac{\hat c(x,z)}t+b\,d(x,x')^2
\qquad\text{if }d(x,x')<\rho.
\]
In particular, $F_t:\Gamma_u\to M$ is open.
\end{lemma}
\begin{proof}
Choose a positive finite active representation
$p=\sum_i m_ip_i$, and put $E=\Span\{p_i-p\}$.
Fix $s$ with $t0.
\]
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 \eqref{geo:growth-expansion} and leaves a positive
multiple of $|h|^2$. When $|h_E|0$ 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 \cite[Section~12]{Moreau1965} and the graph correspondence in
\cite[Proposition~1.1]{AA1999}; compare also
\cite[Section~2]{MPW2012} for related diagonal coordinates in optimal
transport.
Rademacher's theorem \cite[Chapter~2, Theorem~1.4]{Simon2018} gives,
at almost every point of a graph chart,
\begin{equation}
\label{geo:minty-matrices}
X=D_qx\geq0,
\qquad P=D_qp=\frac{\Id-X}{a}.
\end{equation}
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
\begin{equation}
\label{geo:projection-matrix}
Z=(-D^2_{xz}\hat c)^{-1}(tP+\mathscr A X).
\end{equation}
Local minimality at the terminal time implies
\begin{equation}
\label{geo:sign-form}
X(tP+\mathscr A X)\geq0.
\end{equation}
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
\eqref{geo:sign-form}.
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 \eqref{geo:sign-form}. Consequently
\[
\det(tP+\mathscr A X)\geq0,
\]
with positive sign whenever this matrix is nonsingular.
The other factor in \eqref{geo:projection-matrix} 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 $M$:
\[
\mathcal H_r=F_r\circ F_\eps^{-1}:M\to M,
\qquad \eps\leq r\leq t.
\]
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
\cite[Chapter~2, Theorem~3.3, Remark~3.4, and (4.19)--(4.20)]{Simon2018} in manifold
charts gives
\[
\int_M J_{\mathcal H_t}\,d\vol=\vol(M),
\qquad
\int_M N(y,\mathcal H_t)\,d\vol(y)
=\int_M|J_{\mathcal H_t}|\,d\vol=\vol(M),
\]
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.
\Cref{geo:minimizer-capture} 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 \cref{lem:interior-characterization} places it in
$I(x)$ and proves the remaining assertions.
\end{proof}
==== END Global-Support-and-Convex-Injectivity-Domains-under-Weak-MTW//degree.tex ====
==== BEGIN Global-Support-and-Convex-Injectivity-Domains-under-Weak-MTW//geometry.tex ====
\section{Global supports and intermediate transport}
\label{sec:geometry}
We prove the global supporting property by continuing the ordinary
subgradient graph from short times to every time below one.
The nonconjugacy result of \cref{sec:conjugacy} handles a possible
conjugate first cut. A local injectivity argument and covering theory
then exclude a second minimizing geodesic at that scale.
We retain the branch Hessian $A_x(w)$ and minimizing domain $G_x$
from \cref{sec:preliminaries}, and identify tangent vectors and covectors
by the metric except when explicitly using coordinate covectors.
Put $D=\operatorname{diam}M$ and
\[
G_x=\{p\in T_xM:d(x,\exp_xp)=|p|\}=\overline{I(x)},
\qquad c_t=\frac ct,
\qquad Q_tf(z)=\min_x\{f(x)+c_t(x,z)\}.
\]
The total set of minimizing vectors $\{(x,p):p\in G_x\}$ is compact.
The total open injectivity domain is open, and it contains a fixed
neighborhood of the zero section. We use the usual cut-point alternative:
a minimizing vector on its boundary either has a conjugate endpoint or
has a second minimizing geodesic to the same endpoint.
A \emph{potential} means a function represented by a continuous datum $v$ as
\[
u(x)=\max_y\{-c(x,y)-v(y)\}.
\]
In \cref{geo:prelim}, the active logs may be defined using any such
representing datum; duality of that datum is not required. When a dual
pair is needed, we may replace it by
$v(y)=\max_x\{-c(x,y)-u(x)\}$; then $(u,v)$ is a dual pair and
$u(x)+c(x,y)+v(y)\geq0$. Additive constants play no role below.
A finite maximum $u(x)=\max_i\{h_i-c(x,y_i)\}$ is included in this
definition after dual completion. Indeed, if
$v(y)=\max_x\{-c(x,y)-u(x)\}$, then $v(y_i)\leq-h_i$ and
$\max_y\{-c(x,y)-v(y)\}\leq u(x)$. An index active at $x$ gives the
opposite inequality. This identifies the functions, without asserting
that dual completion leaves the set of active targets unchanged.
\subsection{Supports, divided action, and subgradients}
For a semiconvex function, $\partial u(x)$ denotes its ordinary
subdifferential, with vectors identified by the metric. Equivalently,
$p\in\partial u(x)$ if, in normal coordinates at $x$,
\[
u(\exp_xh)\geq u(x)+\langle p,h\rangle+o(|h|).
\]
For a fixed uniform semiconvexity constant, the remainder in this
inequality can instead be bounded below by a negative quadratic term.
\begin{lemma}[Uniform supports and active logs]
\label{geo:prelim}
There is a geometric constant $C_g$ such that $c(\cdot,y)$ and
$c(x,\cdot)$ are uniformly semiconcave, with constant $C_g$, including
at cut points. Every potential is $D$-Lipschitz and uniformly
semiconvex. If
\[
\mathcal V_u(x)=\{p\in G_x:
u(x)+c(x,\exp_xp)+v(\exp_xp)=0\},
\]
then
\begin{equation}
\label{geo:active-hull}
\partial u(x)=\operatorname{conv}\mathcal V_u(x).
\end{equation}
The total active-log graph
$\{(x,p):p\in\mathcal V_u(x)\}$ is compact. Every member of the convex
hull in \eqref{geo:active-hull} has a representation with positive weights using at most $n+1$
affinely independent members of $\mathcal V_u(x)$.
\end{lemma}
\begin{proof}
The divided triangle inequality is
\begin{equation}
\label{geo:divided-triangle}
c_{a+b}(z,y)\leq c_a(z,w)+c_b(w,y),\qquad a,b>0.
\end{equation}
It follows by concatenating constant-speed minimizing curves and using
the energy characterization of squared distance. This shortening
inequality is also used in \cite[Claim~2.4]{CMS2001}.
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 $y_s=\exp_x(sp)$,
$00$ depending only on the manifold.
Furthermore,
\begin{equation}
\label{geo:radial-identity}
A_x(w)w=w
\end{equation}
on every smooth minimizing branch.
\end{lemma}
\begin{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
\[
J(a)=C_p(a)\xi-S_p(a)S_p(r)^{-1}C_p(r)\xi.
\]
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
\eqref{geo:radial-derivative}. 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
\eqref{geo:radial-identity}; the identity at $p=0$ follows by continuity.
\end{proof}
\begin{lemma}[Transverse slack]
\label{geo:slack}
Let $p_i\in G_x$, $m_i>0$, $\sum_i m_i=1$, and
\[
p=\sum_i m_ip_i,\qquad E=\Span\{p_i-p\}.
\]
Suppose $t\operatorname{conv}\{p_i\}\subset I(x)$ and
$00$,
depending only on its Lipschitz and semiconvexity bounds and on the
manifold, the map
\[
F_t:\Gamma_f\longrightarrow M,\qquad
\Gamma_f=\{(x,p):p\in\partial f(x)\},\qquad
F_t(x,p)=\exp_x(tp),
\]
is a homeomorphism. Its inverse is locally Lipschitz, as a map into
the tangent bundle. The minimizing pole of $Q_tf$ is unique,
$Q_tf\in C^{1,1}$, and
\[
\grad Q_tf(z)=-\frac1t\log_zx_t(z).
\]
The compact graph $\Gamma_f$ is an $n$-dimensional Lipschitz manifold.
\end{lemma}
\begin{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,
$D^2_{xx}c\geq a\Id$ 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.
\end{proof}
\begin{lemma}[Capture of every minimizing velocity]
\label{geo:minimizer-capture}
Let $u$ be a potential, let $t>0$, and let $z\in M$.
If $x$ is a global minimizer of $u+c_t(\cdot,z)$, then
\[
\frac wt\in\partial u(x)\qquad\text{for every }w\in L_x(z).
\]
Consequently, $F_t:\Gamma_u\to M$ is surjective for every $t>0$.
\end{lemma}
\begin{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 $F_t(x,w/t)=z$.
\end{proof}
\subsection{Local injectivity and continuation}
The short-time lemma makes $\Gamma_u$ a compact topological manifold
homeomorphic to $M$. To continue its projection, we will prove local
injectivity at a scale $t$ whenever the scaled graph has reached no
conjugate point and all earlier scaled graphs lie in the injectivity
domain. The following argument does not assume that a cut has actually
occurred at $t$.
\begin{lemma}[Local injectivity of the projection]
\label{geo:local-injectivity}
Let $u$ be a potential and $00$,
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.
\smallskip\noindent
\emph{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 $t0$.
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 $00.
\label{geo:collision-gap}
\end{equation}
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.
\smallskip\noindent
\emph{Cancellation before taking limits.}
Multiply \eqref{geo:collision-expansion} by $m_{ji}$ and sum over
all slots. Equation \eqref{geo:collision-cancellation} 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 \eqref{geo:collision-gap}, 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 $\Delta_j\leq0$.
\end{proof}
Local injectivity has a global consequence because the projection is
homotopic to its short-time homeomorphism. We record the topological
fact in a form that does not require an orientation.
\begin{lemma}[A covering homotopic to a homeomorphism]
\label{lem:covering-homeomorphism}
Let $X$ and $N$ be nonempty connected compact topological
$n$-manifolds without boundary. A continuous locally injective map
$f:X\to N$ that is homotopic to a homeomorphism $h:X\to N$
is itself a homeomorphism.
\end{lemma}
\begin{proof}
Invariance of domain makes $f$ a local homeomorphism
\cite[Theorem~2B.3]{Hatcher2002}. 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 \cite[Lemma~1.19]{Hatcher2002}.
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
\cite[Proposition~1.30]{Hatcher2002}. Thus $q'=q$.
A bijective local homeomorphism is a homeomorphism.
\end{proof}
\begin{theorem}[Global supporting property]
\label{geo:support}
For every potential $u$, every $x\in M$, and every
$p\in\partial u(x)$, one has $p\in G_x$ and
\begin{equation}
\label{geo:global-support}
u(x')\geq u(x)+c(x,\exp_xp)-c(x',\exp_xp)
\qquad\text{for all }x'\in M.
\end{equation}
For each $00$, depending only on $M,g,t$, such that
every projected pair $z=\exp_x(tp)$ satisfies
\begin{equation}
\label{geo:uniform-growth}
u(x')+c_t(x',z)
\geq u(x)+c_t(x,z)+b_t d(x,x')^2
\qquad\text{if }d(x,x')1\text{ such that }
d(x,\exp_x(av))=a|v|_x\right\}.
\label{eq:injectivity-domain}
\end{equation}
Its geodesics remain minimizing beyond time one. Put $c(x,y)=d(x,y)^2/2$.
Our MTW convention is
\begin{equation}
\mathfrak S_{(x,v)}(\xi,\eta)
=-\frac32\left.
\frac{\partial^4}{\partial s^2\partial t^2}
c\bigl(\exp_x(t\xi),\exp_x(v+s\eta)\bigr)
\right|_{s=t=0},\qquad v\in I(x).
\label{eq:mtw-definition}
\end{equation}
The cost is smooth near the pairs used here, as recalled in
\cref{sec:preliminaries}. Throughout the paper we assume precisely
\begin{equation}
\langle\xi,\eta\rangle_x=0
\quad\Longrightarrow\quad
\mathfrak S_{(x,v)}(\xi,\eta)\geq0
\qquad(x\in M,\ v\in I(x),\ \xi,\eta\in T_xM).
\label{eq:weak-mtw}
\end{equation}
No condition on nonorthogonal pairs, strict lower bound, nonfocality,
or prior convexity of $I(x)$ is assumed.
Write $G_x=\{p\in T_xM:d(x,\exp_xp)=|p|_x\}$ for the closed
minimizing domain.
\begin{theorem}[Convexity of injectivity domains]\label{thm:main}
If $(M,g)$ satisfies \eqref{eq:weak-mtw}, then $I(x)$ is convex for
every $x\in M$. More precisely, for every $v_0,v_1\in I(x)$ and
$\theta\in[0,1]$,
\[
(1-\theta)v_0+\theta v_1\in I(x).
\]
\end{theorem}
In fact the closed minimizing domain $G_x$ is convex as well
(\cref{geo:convexity}). The conclusion is ordinary linear convexity;
no strict convexity or regularity of the boundary is asserted.
Villani proposed that weak MTW should force convexity of every tangent
injectivity domain \cite[Section~4.2]{Villani2011}. In the compact setting
of \cref{thm:main}, the conjecture imposes neither nonfocality nor prior
convexity. Figalli, Gallou\"et and Rifford proved the implication for
nonfocal manifolds \cite[Theorem~1.7]{FGR2014}, where minimizing geodesics
reach cut strictly before their first conjugate point. Here and below the
numbering in that citation refers to the published article.
\Cref{thm:main} removes this restriction: conjugate cut endpoints are
allowed in every dimension $n\geq2$.
Loeper and Villani connected MTW to cut geometry, obtaining convexity
under strong MTW and nonfocality and treating some focal configurations
under stronger geometric conditions \cite{LoeperVillani2010}; see also
\cite[Section~3.5.2]{Figalli2010}. Their work and the nonfocal theorem of
\cite{FGR2014} distinguish the two obstructions in the present problem.
The cost may cease to be smooth along a proposed velocity segment, and
weak MTW controls only directions perpendicular to that segment.
Neither difficulty can be removed by applying smooth-cost concavity
beyond its domain of definition.
\subsection{Global support and intermediate geometry}
A \emph{potential} is a function
$u(x)=\max_y\{-c(x,y)-v(y)\}$ for a continuous datum $v:M\to\mathbb R$;
replacing $v$ by the dual transform
$v(y)=\max_x\{-c(x,y)-u(x)\}$ gives a dual pair with nonnegative gap
$u(x)+c(x,y)+v(y)$. Such functions are Lipschitz and semiconvex.
We use their ordinary semiconvex subdifferentials $\partial u(x)$,
identifying vectors and covectors by the metric. Define
\[
\Gamma_u=\{(x,p):p\in\partial u(x)\},\qquad
Q_tf(z)=\min_x\left\{f(x)+\frac{c(x,z)}t\right\}.
\]
\begin{theorem}[Global support and intermediate geometry]
\label{thm:foundation}
Under \eqref{eq:weak-mtw}, every potential $u$, with its dual potential $v$,
has the following properties.
\begin{enumerate}[label=\textup{(\roman*)}]
\item Every $p\in\partial u(x)$ belongs to $G_x$ and satisfies
\[
u(x')\geq u(x)+c(x,\exp_xp)-c(x',\exp_xp)
\qquad(x'\in M).
\]
For every $00$,
depending only on $M,g,t$, such that, for $z=F_t(x,p)$,
\[
u(x')+\frac{c(x',z)}t
\geq u(x)+\frac{c(x,z)}t+b_t d(x,x')^2
\quad\text{if }d(x,x')0$, also
minimizes the fixed-endpoint energy
$E(\alpha)=\tfrac12\int_0^T|\dot\alpha|^2\,d\tau$. Indeed,
Cauchy--Schwarz gives
\[
E(\alpha)\geq\frac{\operatorname{length}(\alpha)^2}{2T}
\geq\frac{d(\gamma(0),\gamma(T))^2}{2T}=E(\gamma).
\]
Energy second variation therefore gives $Q_\gamma(Z,Z)\geq0$ for every
field with $Z(0)=Z(T)=0$, without a normality restriction.
Approximation gives the same assertion for piecewise smooth fields. Moreover, equality forces
$Z$ to be a Jacobi field. Indeed, nonnegativity applied to $Z+tW$
implies $Q_\gamma(Z,W)=0$ for every endpoint-zero test field $W$.
Integration by parts first gives the Jacobi equation on each smooth
piece and then continuity of $D_\tau Z$ at the break points. Thus $Z$
is a Jacobi field on the entire interval.
\begin{lemma}[Interior characterization]
\label{lem:interior-characterization}
For every $x\in M$,
\begin{equation}
I(x)\subset G_x,\qquad rG_x\subset I(x)\quad(0\leq r<1).
\label{eq:radial-interior}
\end{equation}
A velocity $w\in T_xM$ belongs to $I(x)$ if and only if it is the
unique minimizing velocity from $x$ to $\exp_x w$ and is nonconjugate.
The set $\mathcal I$ is open, and $c$ is smooth in a neighborhood of
each pair $(x,\exp_x w)$ with $w\in I(x)$. There is a number
$\rho>0$ such that $|w|_x<\rho$ implies $w\in I(x)$ for every $x$.
\end{lemma}
\begin{proof}
The first inclusion in \eqref{eq:radial-interior} follows from the
definition. For $01$ 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
\begin{equation}
\Psi:TM\longrightarrow M\times M,
\qquad \Psi(z,u)=(z,\exp_z u)
\label{eq:exponential-pair-map}
\end{equation}
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.
\end{proof}
\subsection{Smooth geodesic branches}
At any nonconjugate $(x,w)$, the local inverse of
\eqref{eq:exponential-pair-map} defines a smooth branch cost
$\hat c(z,q)$: it is half the squared norm of the inverse-branch
velocity. Define the symmetric bilinear form
\begin{equation}
A_x(w)=\Hess_{xx}\hat c(x,\exp_x w),
\label{eq:branch-hessian}
\end{equation}
where the second endpoint is held fixed in taking the covariant
Hessian. Local inverse branches agree near a prescribed velocity,
so $A$ depends smoothly on $(x,w)$ throughout the nonconjugate set.
When $w\in I(x)$, the branch cost agrees locally with $c$ by
\cref{lem:interior-characterization}. At a nonconjugate minimizing
cut velocity its central value is still $c(x,\exp_x w)$, and
\eqref{eq:branch-hessian} defines its smooth geodesic-branch Hessian.
In particular, as velocities in $\mathcal I$ approach a nonconjugate
minimizing velocity, their true-cost Hessians converge to this branch
Hessian. This assertion also holds with the base point varying, using
local tangent-bundle coordinates.
First variation gives $\nabla_x\hat c(x,\exp_x w)=-w$.
For small $\tau$, with $\exp_x w$ fixed, the branch velocity at
$\gamma_{x,w}(\tau)=\exp_x(\tau w)$ is
$(1-\tau)\dot\gamma_{x,w}(\tau)$. Its negative has covariant
derivative $w$ at $\tau=0$. Consequently,
\begin{equation}
A_x(w)(w,\cdot)=\langle w,\cdot\rangle_x.
\label{eq:radial-hessian}
\end{equation}
This identity is valid at every nonconjugate velocity.
\begin{lemma}[Index-form representation]
\label{lem:hessian-index}
If $w\in G_x$ is nonconjugate, then, along
$\gamma_{x,w}(\tau)=\exp_x(\tau w)$ on $[0,1]$,
\begin{equation}
A_x(w)(\xi,\xi)
=\min_{\substack{X(0)=\xi\\X(1)=0}} Q_{\gamma_{x,w}}(X,X).
\label{eq:hessian-index-minimum}
\end{equation}
The minimum is attained uniquely by the Jacobi field with these
endpoints. In particular, $A_x(0)=g_x$.
\end{lemma}
\begin{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
$A_x(0)(\xi,\xi)=|\xi|_x^2$.
\end{proof}
\begin{lemma}[Strict radial comparison]
\label{lem:radial-comparison}
For every $a\in G_x$ and $0\frac{s}{s'}A_x(s'a)
\label{eq:radial-comparison}
\end{equation}
as quadratic forms; that is, the difference is positive on every
nonzero vector.
\end{lemma}
\begin{proof}
Both $sa$ and $s'a$ belong to $I(x)$ by
\cref{lem:interior-characterization}. Time rescaling in
\eqref{eq:hessian-index-minimum} 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 \cref{lem:hessian-index}, 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
$(1-s/s')g_x$.
\end{proof}
\begin{lemma}[Transverse concavity]
\label{lem:transverse-concavity}
Let $b_t=b_0+t\eta$ be an affine velocity line in $T_xM$, and let
$\xi\perp\eta$. On every interval on which $b_t\in I(x)$,
\begin{equation}
\frac{d^2}{dt^2}A_x(b_t)(\xi,\xi)
=-\frac23\mathfrak S_{(x,b_t)}(\xi,\eta)\leq0.
\label{eq:transverse-concavity}
\end{equation}
Consequently, if $b=\sum_{i=1}^m\theta_i b_i$, with
$\theta_i\geq0$ and $\sum_i\theta_i=1$, the convex hull of the
$b_i$ is contained in $I(x)$, and $\langle b_i,\xi\rangle_x$
is independent of $i$, then
\[
A_x(b)(\xi,\xi)\geq\sum_{i=1}^m\theta_iA_x(b_i)(\xi,\xi).
\]
\end{lemma}
\begin{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
\eqref{eq:mtw-definition} gives the equality in
\eqref{eq:transverse-concavity}. The inequality is exactly
\eqref{eq:weak-mtw}. 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.
\end{proof}
==== END Global-Support-and-Convex-Injectivity-Domains-under-Weak-MTW//preliminaries.tex ====
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pages = {219--257},
year = {2001},
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note = {\href{https://doi.org/10.1007/s002220100160}{doi:10.1007/s002220100160}.
\href{https://webusers.imj-prg.fr/~dario.cordero/Docs/articles/CMSinvent.pdf}{Author's copy of the published article}}
}
@article{AA1999,
author = {Alberti, Giovanni and Ambrosio, Luigi},
title = {A geometrical approach to monotone functions in {$\mathbb R^n$}},
journal = {Mathematische Zeitschrift},
volume = {230},
pages = {259--316},
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doi = {10.1007/PL00004691},
note = {\href{https://doi.org/10.1007/PL00004691}{doi:10.1007/PL00004691}.
\href{https://people.dm.unipi.it/alberti/ricerca/1997-99/monotone-rdc.pdf}{Author's manuscript}}
}
==== END Global-Support-and-Convex-Injectivity-Domains-under-Weak-MTW//references.bib ====