==== 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 $0<r<1$, the full contracted segment
$\{r(z+te):-l_0\leq t\leq l_1\}$ is contained in $I(x)$ by
\eqref{eq:radial-interior}.

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
\[
 J_z(0)=J_z(1)=0,\qquad D_\tau J_z(0)=k.
\]
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 $k\perp z$.

\medskip\noindent
\emph{A trial-field estimate.}
The Jacobi-field trial below is related to the conjugate-cut argument of
Cordero-Erausquin, McCann and Schmuckenschl\"ager
\cite[Proposition~2.5]{CMS2001}. 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
\[
 j(\tau)=T_z(\tau)^{-1}J_z(\tau),\qquad
 J_a(\tau)=T_a(\tau)j(\tau),\qquad
 E_a(\tau)=(1-\tau)T_a(\tau)\xi.
\]
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
\[
 Q_z(E_z,J_z)
 =[\langle E_z,D_\tau J_z\rangle]_0^1
 =-\langle\xi,k\rangle_x.
\]
For $a\in I(x)$ near $z$, \cref{lem:hessian-index} can be tested
on $E_a+\lambda J_a$.  Minimizing the resulting quadratic polynomial
in $\lambda\in\R$ gives
\begin{equation}
 \begin{split}
 A_x(a)(\xi,\xi)
 &\leq Q_a(E_a,E_a)
       -\frac{Q_a(E_a,J_a)^2}{P(a)}\\
 &\leq C_0-\frac{c_0}{P(a)},
 \qquad c_0>0.
 \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
$r_0<r<1$.  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
\begin{equation}
 \begin{split}
 0<P\bigl(r(z+te)\bigr)
 &\leq P(z+te)+L(1-r)|z+te|_x\\
 &\leq Ct^2+LK(1-r)
 \leq C'\bigl(t^2+1-r\bigr).
 \end{split}
 \label{eq:quadratic-index}
\end{equation}
Combining this with \eqref{eq:trial-hessian}, and adjusting positive
constants, gives
\begin{equation}
 A_x\bigl(r(z+te)\bigr)(e,e)
 \leq C_1-\frac{c_1}{t^2+1-r},\qquad c_1>0,
 \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 $0<r<1$, define
\[
 f_r(t)=B_x\bigl(r(z+te)\bigr)(z,z),
 \qquad -l_0\leq t\leq l_1.
\]
Because $z\perp e$, \cref{lem:transverse-concavity} with test
vector $z$ and velocity direction $re$ makes $f_r$ convex.
By \eqref{eq:radial-hessian}, $f_r(0)=0$.
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
\[
 f_r(t)\leq\frac{t}{l_1}f_r(l_1)\leq C_2t
 \qquad(0<t\leq l_1)
\]
for $r$ close to $1$, with $C_2\geq0$ independent of $r$.
The radial identity also says
\begin{equation}
 B_x(a)(a,\cdot)=0,
 \qquad
 f_r(t)=t^2B_x\bigl(r(z+te)\bigr)(e,e).
 \label{eq:radial-null}
\end{equation}
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
\begin{equation}
 A_x\bigl(r(z+te)\bigr)(e,e)\geq1-\frac{C_2}{t}
 \qquad(0<t\leq l_1).
 \label{eq:singular-lower}
\end{equation}
Choose $r=1-t^2$ for sufficiently small positive $t$.  Both
\eqref{eq:singular-upper} and \eqref{eq:singular-lower} apply and
would imply
\[
 1-\frac{C_2}{t}\leq C_1-\frac{c_1}{2t^2}.
\]
Multiplying by $t^2$ and letting $t\downarrow0$ is impossible,
since $c_1>0$.  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 $0<t<1$.  Suppose that, for every
$(x,p)\in\Gamma_u$, one has $rp\in I(x)$ for $0<r<t$ and $tp$
is nonconjugate.  Put $z=\exp_x(tp)$ and let $\hat c$ be the smooth
action branch through $(x,tp)$.  Then there are $b,\rho>0$ 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 $t<s<1$.  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{equation}
 \label{geo:growth-expansion}
 \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}
\end{equation}
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 \cref{geo:slack} gives
\[
 \left.\sum_i m_iB_i\right|_{E^\perp}
 \geq\kappa_g(s-t)\Id.
\]
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 \eqref{geo:growth-expansion} 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.
\end{proof}

\subsection{The sign of a graph projection}

The next lemma is conditional: its local-minimum assumption concerns
the terminal time $t$ only.  The required short-time properties follow
separately from the semiconvexity bounds and the short-time
homeomorphism already proved in the main text.

\begin{lemma}[Positive projection degree]
\label{geo:degree}
Let $u$ be a potential and $0<t<1$.  Assume that every $rp$, with
$(x,p)\in\Gamma_u$ and $0<r\leq t$, is nonconjugate.  At each
$(x,p)$, put $z=\exp_x(tp)$ and let $\hat c$ be the smooth action
branch determined by $tp$.  Suppose that $x$ is a local minimum of
\[
 x'\longmapsto u(x')+\frac{\hat c(x',z)}t,
\]
and suppose that $F_t:\Gamma_u\to M$ is open.
Then $F_t$ is a homeomorphism.  Its pole at every endpoint is the
unique global minimizing pole of $Q_tu$, and every $tp$ lies in $I(x)$.
\end{lemma}

\begin{proof}
Choose a sufficiently small $\eps\in(0,t)$ as in
\cref{geo:short-time}.  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
\[
 q=x+ap
\]
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 $y_s=\exp_x(sp)$,
$0<s<1$, then
\begin{equation}
 \label{geo:split-support}
 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.
\end{equation}
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
\begin{equation}
 \label{geo:directional-support}
 u'(x;h)=\max_{p\in\mathcal V_u(x)}\langle p,h\rangle.
\end{equation}
The lower inequality follows from \eqref{geo:split-support} 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 \eqref{geo:directional-support}.
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 \eqref{geo:directional-support} gives \eqref{geo:active-hull}.
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.  Carath\'eodory'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.
\end{proof}

For a fixed base point $x$, write
\[
 H_s(p)=\frac1s A_x(sp).
\]
This is the source Hessian of the divided cost $c_s$ on the branch
specified by $sp$. We use it only when $sp$ is nonconjugate; at a cut
point the branch is the one defined in \cref{eq:branch-hessian}.
The base point $x$ will be clear from the velocity $p$.

\begin{lemma}[Radial divided-action identity]
\label{geo:radial-index}
Let $p\in G_x$, and let $0<t<s<1$.  Along
$\gamma(r)=\exp_x(rp)$, let $S_p(r)$ be the Jacobi tensor with
$S_p(0)=0$ and $D_rS_p(0)=\Id$, using parallel orthonormal frames.
Then
\begin{equation}
 \label{geo:radial-derivative}
 \frac{d}{dr}H_r(p)=-S_p(r)^{-1}S_p(r)^{-T},
 \qquad
 H_t(p)-H_s(p)\geq\kappa_g(s-t)\Id
\end{equation}
for a constant $\kappa_g>0$ 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
$0<t<s<1$.  Then
\begin{equation}
 \label{geo:transverse-slack}
 \left[\frac{A_x(tp)}t-
       \sum_i m_i\frac{A_x(sp_i)}s\right]\bigg|_{E^\perp}
 \geq\kappa_g(s-t)\Id.
\end{equation}
The same conclusion holds at a limiting time $t$ if the radially
contracted hulls lie in $I(x)$ and the displayed Hessians have
nonconjugate branch limits.
\end{lemma}

\begin{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~\ref{geo:radial-index} 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.
\end{proof}

\subsection{The subgradient graph and its projections}

The short-time construction is a curved version of proximal minimization
and quadratic inf-convolution; compare \cite[Sections~3--5 and~7]{Moreau1965}.
We give the local estimates explicitly, since the subsequent continuation
must use only geometric and semiconvexity bounds.
For any Lipschitz semiconvex function $f$, its ordinary subgradient
graph is compact: the subgradients are bounded by its Lipschitz constant,
and the uniform local quadratic subgradient inequalities make the graph
closed under convergence of both the base point and the vector.

\begin{lemma}[Short time for a semiconvex datum]
\label{geo:short-time}
Let $f$ be Lipschitz and semiconvex.  For sufficiently small $t>0$,
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 $0<t<1$. Suppose that
\[
 rp\in I(x)\quad\text{for every }(x,p)\in\Gamma_u
               \text{ and }0<r<t,
\]
and that every vector $tp$, $(x,p)\in\Gamma_u$, is nonconjugate.
Then $F_t:\Gamma_u\to M$ is locally injective.
\end{lemma}

\begin{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
\[
 z_j=\exp_{x_j}(tp_j)
     =\exp_{\widetilde x_j}(t\widetilde p_j)
 \longrightarrow z=\exp_x(tp).
\]

\smallskip\noindent
\emph{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.

\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 $t<s<1$. 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,
\begin{equation}
 \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.
 \label{geo:collision-expansion}
\end{equation}
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
$O(l_j^3)$.

\smallskip\noindent
\emph{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
\eqref{geo:collision-expansion} 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
\begin{equation}
 \sum_i m_{ji}d_{ji}=0.
 \label{geo:collision-cancellation}
\end{equation}
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 \cref{geo:slack} therefore gives
\begin{equation}
 \delta_*:=
 [H_t(p)-\sum_{i\in P}m_iH_s(a_i)](e,e)
 \geq\kappa_g(s-t)>0.
 \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 $0<t<1$, the projection $F_t:\Gamma_u\to M$ is a
homeomorphism; its vectors $tp$ lie in $I(x)$, and its poles are the
unique global minimizers of $Q_tu$.
\end{theorem}

\begin{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~\ref{geo:first-cut} therefore
makes its entire hull nonconjugate. Thus every scaled graph vector
at time $t_*$ is nonconjugate, and
\cref{geo:local-injectivity} makes $F_{t_*}$ locally injective.

Choose $\epsilon\in(0,t_*)$ small enough for
\cref{geo:short-time}. 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~\ref{lem:covering-homeomorphism} 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
\cref{geo:minimizer-capture}; 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, \cref{lem:interior-characterization} 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
\cref{geo:local-injectivity} 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
\cref{geo:minimizer-capture} 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 \eqref{geo:global-support}.
\end{proof}

\subsection{Convex sections and regular intermediate potentials}

\begin{corollary}[Convex injectivity domains]
\label{geo:convexity}
For every $x$, both $G_x$ and $I(x)$ are convex.
\end{corollary}

\begin{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~\ref{geo:support} 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.
\end{proof}

\begin{corollary}[Convex lifted gap sections]
\label{geo:sections}
Let $(u,v)$ be a dual pair and define
\[
 \mathcal S_x(r)=\left\{p\in G_x:
    u(x)+\frac{|p|^2}{2}+v(\exp_xp)\leq r\right\}.
\]
Every such set is compact and convex, including its ordinary and
conjugate cut endpoints.  If it is nonempty and $r\geq0$, then
\begin{equation}
 \label{geo:section-diameter}
 \operatorname{diam}\mathcal S_x(r)^2
 \leq 8\left(r+
       \osc_{p\in\mathcal S_x(r)}v(\exp_xp)\right).
\end{equation}
\end{corollary}

\begin{proof}
Write $m_p(z)=|p|^2/2-c(z,\exp_xp)$ for $p\in G_x$.
Theorem~\ref{geo:support}, 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 \eqref{geo:section-diameter}.
\end{proof}

\begin{proposition}[Intermediate regularity and pole control]
\label{geo:intermediate}
Let $(u,v)$ be a dual pair and $0<t<1$.  Then
\begin{equation}
 \label{geo:intermediate-duality}
 Q_tu=-Q_{1-t}v\in C^{1,1}(M).
\end{equation}
The $C^{1,1}$ seminorm is bounded solely in terms of $M,g,t$.
If $(x_t(z),p_t(z))=F_t^{-1}(z)$ and $\Phi_s$ denotes geodesic
flow on $TM$, then
\begin{equation}
 \label{geo:pole-flow}
 (x_t(z),p_t(z))
   =\Phi_{-t}(z,\grad Q_tu(z)),
 \qquad
 x_t(z)=\exp_z(-t\grad Q_tu(z)).
\end{equation}
In particular the pole and momentum are Lipschitz, with constants
depending only on $M,g,t$.

There are also $r_t,b_t>0$, 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')<r_t.
\end{equation}
These assertions use no density hypothesis.  On every compact
subinterval of $(0,1)$, their constants can be chosen uniformly.
\end{proposition}

\begin{proof}
Let $(x,p)=F_t^{-1}(z)$, set $\gamma(s)=\exp_x(sp)$, and put
$y=\gamma(1)$, $w=\dot\gamma(t)$.  Theorem~\ref{geo:support}
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~\eqref{geo:divided-triangle} 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 \eqref{geo:intermediate-duality}.

Equivalently, the function has the upper and lower supports
\begin{equation}
 \label{geo:two-sided-intermediate}
 -v(y)-c_{1-t}(z',y)
 \leq Q_tu(z')
 \leq u(x)+c_t(x,z'),
\end{equation}
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 \eqref{geo:two-sided-intermediate} 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
\[
 -\frac{C_g}{1-t}g\leq\Hess Q_tu\leq\frac{C_g}{t}g.
\]
The support gradients give $\grad Q_tu(z)=w$, with $|w|\leq D$.
Reversing geodesic flow proves \eqref{geo:pole-flow}.  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 $H_t(p)-H_s(p)\geq\kappa_g(s-t)\Id$ 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 \eqref{geo:uniform-growth},
uniformly in the potential and in its graph point.
\end{proof}

\begin{remark}
\label{geo:time-one}
At time one, arbitrary potentials can have many poles with the same
endpoint; $u(x)=-c(x,y_0)$ is an example.  All uses of an inverse
projection or of $C^{1,1}$ regularization in this paper concern $0<t<1$.
Only the minimizing-vector and supporting inequalities are passed
to time one.
\end{remark}
==== END Global-Support-and-Convex-Injectivity-Domains-under-Weak-MTW//geometry.tex ====

==== BEGIN Global-Support-and-Convex-Injectivity-Domains-under-Weak-MTW//introduction.tex ====
\section{Introduction}
\label{sec:introduction}

The squared geodesic distance is smooth before the cut locus, but optimal
transport and its supporting functions naturally reach that boundary.
The weak Ma--Trudinger--Wang condition controls the cost Hessian only in
directions perpendicular to a velocity segment. We prove that this
transverse condition already forces global convexity and supporting geometry
at arbitrary minimizing endpoints, including conjugate cut points.

\subsection{Weak MTW and convexity}

Let $(M,g)$ be a smooth, connected, compact Riemannian manifold without
boundary, of dimension $n\geq2$. Write $d$ for its geodesic distance,
$\exp_x:T_xM\to M$ for its exponential map, and $|\cdot|_x$ and
$\langle\cdot,\cdot\rangle_x$ for the norm and inner product of $g_x$.
The open tangent injectivity domain is
\begin{equation}
 I(x)=\left\{v\in T_xM:\ \text{there is }a>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 $0<t<1$, the map $F_t:\Gamma_u\to M$,
$F_t(x,p)=\exp_x(tp)$, is a homeomorphism; $tp\in I(x)$ and $x$
is the unique global minimizing pole of $Q_tu$ at $F_t(x,p)$.
\item For every dual pair $(u,v)$, every $x$ and every $r$, the lifted section
\[
 \mathcal S_x(r)=\left\{p\in G_x:
 u(x)+\frac{|p|^2}{2}+v(\exp_xp)\leq r\right\}
\]
is compact and convex. If it is nonempty and $r\geq0$, then
\[
 \operatorname{diam}\mathcal S_x(r)^2
 \leq8\left(r+\osc_{p\in\mathcal S_x(r)}v(\exp_xp)\right).
\]
\item For $0<t<1$, one has $Q_tu=-Q_{1-t}v\in C^{1,1}(M)$.
Its $C^{1,1}$ seminorm and the Lipschitz constants of the pole and momentum
maps $F_t^{-1}$ depend only on $M,g,t$. There are also $r_t,b_t>0$,
depending only on $M,g,t$, such that, for $z=F_t(x,p)$,
\[
 u(x')+\frac{c(x',z)}t
 \geq u(x)+\frac{c(x,z)}t+b_t d(x,x')^2
 \quad\text{if }d(x,x')<r_t.
\]
These constants can be chosen uniformly on compact subintervals of $(0,1)$.
\end{enumerate}
No density hypothesis is required.
\end{theorem}

The detailed statements and proofs are \cref{geo:support,geo:sections,geo:intermediate}.
The distinction between intermediate times and time one is essential:
only the support and minimizing-velocity assertions pass to time one.
For instance, $u(x)=-c(x,y_0)$ has many poles with the same final endpoint.

\subsection{Supporting geometry and earlier methods}

Ma, Trudinger and Wang introduced a strictly positive curvature condition
in optimal-transport regularity theory \cite{MTW2005}; Trudinger and Wang
treated its degenerate, nonnegative form, usually denoted A3w
\cite[Section~1]{TW2009}. Loeper established necessity for continuity in
the smooth-cost setting and developed the supporting-mountain principle
\cite{Loeper2009}. Kim and McCann gave its geometric formulation and a
double-mountain principle under smooth-cost and illuminated-set
hypotheses \cite[Theorem~4.10]{KM2010}. For squared distance on a compact
manifold, Figalli, Rifford and Villani obtained the global supporting
principle, including endpoint closure, from weak MTW together with convex
injectivity domains \cite[Lemma~3.1]{FRV2011}. The issue here is to derive
that convexity rather than assume it.

Several ingredients precede the MTW theory. The shortened-distance
support inequality is \cite[Claim~2.4]{CMS2001}. The endpoint-moving
Jacobi trial used at a possible conjugate cut is related to the
second-variation argument in \cite[Proposition~2.5]{CMS2001}; the
transverse MTW reduction and the joint radial--tangential growth
contradiction are proved here. Intermediate injectivity at
differentiability points and inverse estimates on compact subsets of a
positive-Hessian set appear in \cite[Lemma~5.3 and Claim~5.6]{CMS2001}.
Our conclusion concerns the entire ordinary subgradient graph and gives
bounds depending only on the geometry and the intermediate time.

The graph coordinates in the optional degree argument and the infimum
regularization also have classical
antecedents. Moreau developed quadratic inf-convolution and proximal
decomposition \cite[Sections~3--5, 7 and~12]{Moreau1965}; Minty's
monotone-graph coordinates are presented in
\cite[Proposition~1.1]{AA1999} and adapted to transport-plan
rectifiability in \cite[Section~2]{MPW2012}. We prove the local
semiconvex graph representation needed here. These constructions do not
by themselves supply continuation through cut or the global
support theorem. The resulting density-free geometry is distinct from
the additional measure estimates needed for regularity of transport
maps.

\subsection{The continuation argument}

\Cref{sec:preliminaries} develops the minimizing-branch Hessian and index
form without nonfocality. Proper radial shortening supplies smooth
distance supports, and the divided Hessian decreases strictly with
duration. These estimates also control the uniform constants in the
intermediate conclusions.

The proof works on the entire ordinary subgradient graph of an arbitrary
potential. A target is active at $x$ if it attains the maximum defining
$u(x)$; an active log is any minimizing velocity to that target.
Every ordinary subgradient is a convex combination of finitely many
active logs. All minimizing logarithms of each active target are retained.
These are finite local representations; the
potential itself need not have finitely many targets.

At a first failing scale, limits from earlier scales show that the
relevant finite convex hulls are still minimizing. The argument of
\cref{sec:conjugacy} excludes conjugacy there. Transverse concavity
confines a hypothetical conjugacy kernel to the segment direction;
the radial Hessian identity then contradicts the growth forced by a
Jacobi trial field.

To exclude local collisions of the graph projection, the finite active
representations are compared before taking limits. The first-order inequalities force the positive-weight limiting velocities
into an affine hyperplane perpendicular to the collision direction.
The linear terms are then canceled exactly at the finite stage,
before division by the squared collision distance and passage to the limit.
Weak MTW and strict radial comparison give the contradiction.
Invariance of domain now makes the projection a local homeomorphism.
Compactness and a homotopy to the short-time graph homeomorphism make
this covering one-sheeted. Capturing every minimizing logarithm
excludes ordinary cuts and continues the construction to every time
strictly below one. Closure yields global support at time one; convex
lifted sections and two-sided intermediate supports give the remaining
conclusions of \cref{thm:foundation}.

An optional alternative in \cref{sec:degree-alternative} replaces local
injectivity by local quadratic growth and openness. A Jacobian-sign
calculation, degree one, and the area formula then give the same
conditional globalization. It reuses the common setup and is not
needed for the main proof.
==== END Global-Support-and-Convex-Injectivity-Domains-under-Weak-MTW//introduction.tex ====

==== BEGIN Global-Support-and-Convex-Injectivity-Domains-under-Weak-MTW//main.tex ====
\newcommand{\DoNotLoadEpstopdf}{}
\documentclass[11pt]{article}
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\usepackage{amsmath,amssymb,amsthm,mathtools}
\usepackage{mathrsfs}
\usepackage{microtype}
\usepackage{enumitem}
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\usepackage[nameinlink,noabbrev,capitalise]{cleveref}
\hypersetup{colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black,
 pdftitle={Global Support and Convex Injectivity Domains under Weak MTW},
 pdfauthor={OpenAI}}
\setlength{\emergencystretch}{2em}
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\newcommand{\R}{\mathbb R}
\newcommand{\Id}{\mathrm{Id}}
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\title{Global Support and Convex Injectivity Domains under Weak MTW}
\author{OpenAI}
\date{September 25, 2026}
\begin{document}
\maketitle
\begin{abstract}
We prove that weak Ma--Trudinger--Wang curvature on a smooth, connected,
compact Riemannian manifold of dimension at least two without boundary implies
convexity of every tangent injectivity domain, resolving Villani's conjecture
in this setting. Conjugate cut points are allowed. More generally, every
ordinary subgradient of a squared-distance cost potential is a minimizing
velocity with a global supporting mountain. No density hypothesis is used.
\end{abstract}
\tableofcontents
\input{sections/introduction}
\input{sections/preliminaries}
\input{sections/conjugacy}
\input{sections/geometry}
\appendix
\input{appendices/degree}
\clearpage
\bibliographystyle{plain}
\bibliography{references}
\end{document}
==== END Global-Support-and-Convex-Injectivity-Domains-under-Weak-MTW//main.tex ====

==== BEGIN Global-Support-and-Convex-Injectivity-Domains-under-Weak-MTW//preliminaries.tex ====
\section{Minimizing velocities and the branch Hessian}
\label{sec:preliminaries}

Throughout the remaining sections we assume the weak MTW condition
\eqref{eq:weak-mtw}. A velocity is called \emph{nonconjugate} if the
vertical differential $d_w\exp_x:T_xM\to T_{\exp_x w}M$ is nonsingular.
We write
\begin{align*}
 L_x(q)&=\{w\in T_xM:\exp_x w=q,\ |w|_x=d(x,q)\},\\
 G_x&=\{w\in T_xM:w\in L_x(\exp_x w)\},\\
 \mathcal C&=\{(x,w)\in TM:w\in G_x\},
 \qquad
 \mathcal I=\{(x,w)\in TM:w\in I(x)\},
\end{align*}
and set $D=\diam(M)$.  Existence of minimizing geodesics shows that
$L_x(q)$ is nonempty.  Every velocity in $\mathcal C$ has norm at most
$D$.  Continuity of the distance and the exponential map shows that
$\mathcal C$ is closed; as the base manifold is compact, $\mathcal C$ is
therefore compact.

We first record the index-form facts used below. For geometric background,
see \cite[Sections~5.7--5.8]{Calegari2013}; for the unrestricted energy
second-variation formula, see \cite[Proposition~5.3.8]{Gorodski2016}.
The latter is the revised course chapter, whose formula allows tangential
as well as normal variation fields. Along a geodesic
$\gamma:[0,T]\to M$, let
\begin{equation}
 Q_\gamma(X,Y)=\int_0^T
 \bigl(\langle D_\tau X,D_\tau Y\rangle
       -\langle R(X,\dot\gamma)\dot\gamma,Y\rangle\bigr)\,d\tau .
 \label{eq:index-form}
\end{equation}
Our curvature convention gives the Jacobi equation
$D_\tau^2X+R(X,\dot\gamma)\dot\gamma=0$.
Fields in \eqref{eq:index-form} may be continuous and piecewise smooth.
A constant-speed minimizing geodesic on $[0,T]$, with $T>0$, 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 $0<r<1$, 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
\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 $0<s<s'<1$,
\begin{equation}
 A_x(sa)>\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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