Weak MTW curvature: convex injectivity domains and regular optimal transport

Source and verification

This article explains family 360 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

🏷️ Optimal transport meets the cut locus

Moving mass on a curved manifold involves two different questions. Does the geometry admit the supporting functions needed for transport? If it does, are the resulting transport maps continuous, with estimates uniform over rough densities? Family 360 treats these questions in two dependent manuscripts. The first establishes supporting geometry from weak Ma–Trudinger–Wang curvature, abbreviated weak MTW. The companion uses that geometry to obtain uniform Hölder estimates for optimal maps and their inverses.

Let $(M,g)$ be a smooth, connected, compact Riemannian manifold without boundary, of dimension $n\ge 2$. The cost of moving $x$ to $y$ is

$$c(x,y)=\frac{1}{2}d(x,y{)}^2.$$

For probability densities ${\rho }_0,{\rho }_1$, an optimal map $T$ pushes ${\rho }_{0}d\mathrm{vol}$ to ${\rho }_{1}d\mathrm{vol}$ and minimizes the integral of $c(x,T(x))$. Pushforward means that each target region receives exactly its prescribed mass. Existence and almost-everywhere uniqueness are inputs from classical optimal transport; the new issue in the companion is regularity uniformly over a fixed density class.

The exponential map ${\exp }_x(p)$ follows the geodesic with initial velocity $p$ for time one. The closed minimizing domain $G_x$ consists of velocities whose geodesics minimize distance through that time. The open tangent injectivity domain $I(x)$ consists of velocities that remain minimizing for a little longer:

$$\begin{array}{rl}{G}_x& =\{p:d(x,{\exp }_{x}p)=|p|\},\\ I(x)& =\bigcup _{a>1}{a}^{-1}{G}_x.\end{array}$$

A cut endpoint is where this minimizing behavior first stops. Two phenomena can cause it: another minimizing geodesic reaches the same endpoint, or the differential of the exponential map becomes singular. The latter is conjugacy. Before cut, the minimizing velocity is unique and nonconjugate, and the squared-distance cost is smooth. Neither manuscript assumes that cut always occurs strictly before conjugacy.

🏷️ The curvature condition and the two conclusions

For $p\in I(x)$, define the cost Hessian

$$A_x(p)={\mathrm{Hess}}_{xx}c(x,{\exp }_{x}p),$$

holding the endpoint fixed when taking the Hessian. Weak MTW has a useful operational form: for every affine velocity segment $p+s\eta$ contained in $I(x)$,

$$\begin{array}{c}\xi \perp \eta ⟹\\ \frac{d^2}{d{s}^2}{A}_x(p+s\eta )(\xi ,\xi )\le 0.\end{array}$$

In the manuscripts’ normalization, the negative of this second derivative, multiplied by $3/2$, is the MTW tensor evaluated on $(\xi ,\eta )$. Thus the assumption controls a fourth derivative of the cost, but only in transverse directions. It is not concavity on arbitrary vector pairs, and it is imposed only where the cost is smooth.

For the Euclidean quadratic cost, $A_x(p)$ is the identity independently of $p$, so this tensor vanishes. On a manifold, variation of the endpoint changes the Hessian and makes the condition substantive.

Convexity asserted by the foundation manuscript

Under weak MTW, every $I(x)$ and every $G_x$ is linearly convex in $T_{x}M$. The conclusion allows conjugate cut endpoints and assumes no prior convexity. It does not assert strict convexity or smoothness of the boundary. (OpenAI, 2026)

Uniform regularity asserted by the companion

Fix the manifold and numbers $0<\lambda \le \Lambda <\infty$. For every pair of probability densities with $\lambda \le {\rho }_i\le \Lambda$ almost everywhere, the optimal map has a homeomorphic representative. One exponent $\alpha >0$ and one finite constant $C$, depending only on the fixed geometry and these bounds, give Hölder estimates for both that representative and its inverse. The densities need only be measurable. (OpenAI, 2026b)

The exponent is obtained by contradiction and is not explicit. No uniformity over varying metrics is asserted. The first theorem needs no densities at all; the second uses their lower and upper bounds essentially.

🏷️ Potentials, subgradients, and global supports

The papers use a negative-cost convention for dual potentials:

$$\begin{array}{c}u(x)=\underset{y}{\max }\{-c(x,y)-v(y)\},\\ v(y)=\underset{x}{\max }\{-c(x,y)-u(x)\}.\end{array}$$

Their gap $u(x)+c(x,y)+v(y)$ is nonnegative; a pair with zero gap is a contact. At a differentiable contact, $\nabla u(x)$ is a minimizing velocity to the target.

These potentials are Lipschitz and semiconvex: locally, adding a fixed positive quadratic function makes them convex. The ordinary subdifferential $\partial u(x)$ consists of vectors defining first-order lower supports. It can contain several vectors at a nonsmooth point.

An active logarithm is any minimizing velocity to an endpoint attaining the maximum defining $u(x)$. The source proves

$$\partial u(x)=\mathrm{conv}{\mathcal{V}}_u(x),$$

where ${\mathcal{V}}_u(x)$ includes every minimizing logarithm to each active endpoint. Carathéodory’s theorem represents any one subgradient using at most $n+1$ active logarithms. This finite representation does not require the entire potential to have finitely many targets.

The foundation’s stronger conclusion is that every ordinary subgradient is itself a minimizing velocity and has a global cost support. For $p\in \partial u(x)$ and $y={\exp }_{x}p$,

$$\begin{array}{c}p\in G_x,\\ u(x^{\prime })\ge u(x)+c(x,y)-c(x^{\prime },y),\\ x^{\prime }\in M.\end{array}$$

Such a supporting negative-cost function is often called a mountain. A local first-order support has become a global inequality. Establishing this upgrade at cut endpoints is the main geometric task. (OpenAI, 2026a)

🏷️ Shortening a geodesic supplies smooth supports

If $p\in G_x$, put $y={\exp }_{x}p$ and $y_s={\exp }_x(sp)$ for $0<s<1$. Splitting the geodesic’s energy gives

$$\begin{array}{rl}c(z,y)& \le \frac{c(z,y_s)}{s}+\frac{1-s}{2}|p{|}^2,\\ c(x,y_s)& =\frac{s^2}{2}|p{|}^2.\end{array}$$

Equality holds in the first line at $z=x$. Its right side is smooth near $x$, even when the original endpoint is conjugate: the shortened velocity $sp$ lies before cut. Taking $s=1/2$ gives uniform upper supports for the cost and hence uniform semiconvexity of the potentials.

There is also a strict Hessian gain from shortening. Write $H_s(p)=A_x(sp)/s$. The index form of a geodesic is the second variation of its energy, expressed using a vector field along the curve. Its minimum over fields with initial value $\xi$ and final value zero equals $H_s(p)(\xi ,\xi )$ on duration $s$.

Extending such a minimizing field by zero to a longer interval gives an admissible competitor. Equality cannot hold unless the extension is a Jacobi field, which would force it to vanish identically. Thus $H_t(p)>H_s(p)$ for $t<s$. The Jacobi-matrix calculation makes this quantitative:

$$\begin{array}{c}\frac{d}{ds}{H}_s(p)=-S_p(s{)}^{-1}{S}_p(s{)}^{-T},\\ H_t(p)-H_s(p)\ge {\kappa }_g(s-t)\mathrm{Id}.\end{array}$$

Here $S_p$ solves the Jacobi equation with $S_p(0)=0$, $S_p^{\prime }(0)=\mathrm{Id}$; compactness bounds its norm, giving ${\kappa }_g>0$. Weak MTW combines this radial gain with transverse concavity. That combination supplies the strict inequality needed when geodesics might collide. (OpenAI, 2026a)

🏷️ Why a minimizing segment cannot hide conjugacy

One particularly informative lemma says: if an entire velocity segment lies in $G_x$ and its two endpoints are nonconjugate, then every point on it is nonconjugate. The hypothesis that the whole segment is minimizing is crucial; convexity of $G_x$ has not yet been proved.

Suppose instead that an interior velocity $z$ is conjugate, and write the segment as $z+te$ with $|e|=1$. A nonzero kernel vector $k$ of $d{\exp }_x$ produces a Jacobi field vanishing at both ends. Testing the index-form minimum with a multiple of this field shows that

$$A_x(a)(\xi ,\xi )\le C-\frac{c}{P(a)}$$

near $z$, whenever $⟨\xi ,k⟩\ne 0$. The smooth index quantity $P$ is positive before cut and vanishes at $z$.

Apply transverse concavity to the properly shortened segment $r(z+te)$, $r<1$. Its nonconjugate endpoint Hessians give a lower bound in every direction perpendicular to $e$. The displayed blowup therefore forces every kernel vector to be parallel to $e$. Jacobi orthogonality also gives $z\perp e$.

Now use both parameters. Nonnegativity of $P(z+te)$ and its minimum at $t=0$ imply quadratic tangential growth. Smooth radial dependence gives

$$\begin{array}{c}P(r(z+te))\le C(t^2+1-r),\\ A_x(r(z+te))(e,e)\\ \le C_1-\frac{c_1}{t^2+1-r}.\end{array}$$

The radial identity $A_x(a)a=a$ supplies the competing estimate. Put $B_x=\mathrm{Id}-A_x$ and $f_r(t)=B_x(r(z+te))(z,z)$. Transverse MTW makes $f_r$ convex, the radial identity gives $f_r(0)=0$, and the nonsingular endpoint bounds $f_r(t)\le C_{2}t$. Also

$$f_r(t)=t^2{B}_x(r(z+te))(e,e).$$

Consequently $A_x(r(z+te))(e,e)\ge 1-C_2/t$. Choosing $r=1-t^2$ compares a possible negative blowup of order $t^{-1}$ with a forced one of order $t^{-2}$, a contradiction. Induction extends the lemma from a segment to a finite minimizing convex hull. (OpenAI, 2026a)

🏷️ Continuing the full subgradient graph

Define the infimum regularization and the graph projection

$$\begin{array}{c}{Q}_{t}u(z)=\underset{x}{\min }\left\{u(x)+\frac{c(x,z)}{t}\right\},\\ {\Gamma }_u=\{(x,p):p\in \partial u(x)\},\\ F_t(x,p)={\exp }_x(tp).\end{array}$$

For sufficiently small $t$, the cost term dominates the potential’s possible negative Hessian. The minimizer, called the pole, is unique, and $F_t$ is a homeomorphism from the graph to $M$.

The source continues this property up to every $t<1$. At a hypothetical first failing time, closedness puts all scaled subgradients in $G_x$. Their finite active representations form minimizing convex hulls with properly shortened, nonconjugate generators. The previous lemma excludes conjugacy throughout these hulls.

The remaining issue is local collision of graph points. Expanding the two competing values using finite active representations first forces their positive-weight limits into a hyperplane perpendicular to the collision direction. Weak MTW then applies in exactly that transverse direction. The strict radial Hessian gain supplies a positive quadratic gap.

The linear terms must be canceled before taking limits: their weighted sum is exactly zero at each finite stage. Otherwise dividing by the squared collision distance could magnify an uncontrolled error. Uniform Taylor remainders and bounded Hessians let zero-weight limiting slots disappear without any rate assumption on their weights.

Local injectivity, compactness, and the homotopy to the short-time homeomorphism make the projection a one-sheeted covering. A separate minimizing-pole argument captures every minimizing logarithm, excluding a second geodesic to the same endpoint. This rules out the first failure. Letting $t\uparrow 1$ gives the global supporting inequality.

Convexity now follows quickly. For $p_0,p_1\in G_x$, take the maximum of their two mountains touching at $x$. Both velocities are subgradients there, hence so is their segment. Global support puts the entire segment in $G_x$. Taking interiors gives convexity of $I(x)$.

The intermediate conclusion remains strictly intermediate: $F_t$ is invertible and $Q_{t}u=-Q_{1-t}v$ is $C^{1,1}$ for $0<t<1$. At time one an arbitrary potential can collapse many poles onto one target; $u(x)=-c(x,y_0)$ is an example. Density assumptions enter only in the companion’s argument excluding this behavior for transport between positive bounded densities. (OpenAI, 2026a)

🏷️ From small gap sections to Hölder maps

The lifted gap section is

$$\begin{array}{c}{\mathcal{S}}_x(r)=\{p\in G_x:h_x(p)\le r\},\\ h_x(p)=u(x)+\frac{1}{2}|p{|}^2+v({\exp }_{x}p).\end{array}$$

Global support makes these sets compact and convex, including their cut endpoints. The midpoint identity for $|p{|}^2$ yields

$$\begin{array}{c}\mathrm{diam}{\mathcal{S}}_x(r{)}^2\\ \le 8\left(r+{\mathrm{osc}}_{{\exp }_x{\mathcal{S}}_x(r)}v\right).\end{array}$$

Here oscillation means maximum minus minimum. This elementary inequality is the bridge from supporting geometry to quantitative regularity.

Let $F(r)$ be the supremum of that oscillation over all poles and all density pairs between the fixed bounds $\lambda ,\Lambda$. If $F(r)\le C_0{r}^{\beta }$ near zero, the section’s endpoint diameter is at most $C\sqrt{r+r^{\beta }}$, since exponential maps have uniformly bounded derivatives on minimizing velocities.

Contacts $(x,y)$ and $(x^{\prime },y^{\prime })$ have gap at $x$ bounded by $Cd(x,x^{\prime })$, using the common Lipschitz constants of the potentials and cost. Thus both targets lie in one small section and

$$\begin{array}{c}d(y,y^{\prime })\le Cd(x,x^{\prime }{)}^{\alpha },\\ \alpha =\frac{1}{2}\min \{\beta ,1\}.\end{array}$$

At zero height the contact is unique. Applying the same argument to the reversed density pair gives a unique continuous inverse with the same class of estimates. This proves the stated homeomorphism conclusion once the power bound for $F$ is established. (OpenAI, 2026b)

🏷️ How the companion forces a power bound

Failure of every power bound produces logarithmic plateaus: two scales $r_j\ll R_j\to 0$ and actual section oscillations $D_j>0$ satisfying

$$\begin{array}{c}F(R_j)\le (1+o(1))D_j,\\ R_j/D_j^k\to 0\text{for each fixed }k>0.\end{array}$$

The proof chooses intervals where the monotone function $\log (C/F(e^{-s}))$ grows very little despite their increasing lengths. It does not require $D_j\to 0$.

Two scalar modifications of the target potential then create a comparison maximum: a fixed center profile and an outer profile whose curvature has arbitrarily large magnitude $K$ in a controlled range of values. An amplitude variation yields an inequality for every positive active barycentric representation. It selects an outward endpoint with weight bounded below independently of $K$. The order is essential: fix the center and weight constants, choose $K$ and its outer profile, and only then take sufficiently small scales.

There are two major obstacles between that maximum and a determinant contradiction.

Conjugate endpoints. Write ${\sigma }_x(p)=|\det (d{\exp }_x{)}_p|$. The companion proves estimates for individual singular values that remain uniform near conjugacy. Proper radial shortening by $1-\delta$ replaces a singular value $s_j$ by a quantity comparable to $s_j+\delta$. Weak MTW then compares singular values along an extended affine segment. For the selected outward endpoint $p_i$, the positive barycentric weight supplies the needed extension in the opposite direction, giving

$${\sigma }_x(p)\ge c{\sigma }_x(p_i),$$

with $c$ fixed before $K$. A bound merely on nonsingular compact subsets would be insufficient here.

Exceptional sets of the density equation. A Lipschitz pole map can send positive volume into a null set, so one cannot simply apply an almost-everywhere Jacobian equation at a point selected by a maximum. The proof first uses the original contact correspondence’s volume bounds. A smooth lower test $\varphi$ for the original potential, with true log $r=\nabla \varphi$ before cut, obeys

$$\begin{array}{c}W=\mathrm{Hess}\varphi +A_y(r)>0,\\ {\sigma }_y(r)\det W\le \Lambda /\lambda .\end{array}$$

This follows by mapping a small contact sublevel through $z\mapsto {\exp }_z\nabla \varphi (z)$ and comparing its volume with the original contact image. No density bounds are assigned to the modified potentials.

Lower second-order tests transferred to the selected outer endpoint first exclude conjugacy there and bound an associated determinant. A positive baseline then bounds the whole pole matrix and the center Hessian. Only after these bounds are available does the short-time minimizer have nonsingular differential. The area formula now justifies sampling centers where the original density equation holds. This order avoids assuming the desired regularity in the intermediate steps. (OpenAI, 2026b)

🏷️ The missing direction in the determinant estimate

The final linear-algebra step can be isolated from the preceding construction. Its limiting matrices satisfy $V,V_i>0$, $L\ge 0$, $Le=0$, and $p_i=p+de$ with $d>0$, $q=p\cdot e>0$. The center estimates imply

$$\begin{array}{c}L\ge V\text{on }{p}^{\perp },\\ V(e,e)\le \beta q^2,\beta =Cb/D^2,\end{array}$$

where $b$ is the modification amplitude. The outer estimate supplies a rank-one term of size $\gamma =cKb/D^2$ and a weight $m_i\ge m_{\ast }>0$.

Use the oblique projection $P=\mathrm{Id}-e{p}^{\ast }/q$ onto $p^{\perp }$ along $e$, where $p^{\ast }$ denotes inner product with $p$. Since $Le=0$, $P^{\ast }LP=L$, and the inequalities imply

$$V_i\ge m_i{P}^{\ast }VP+\gamma p_i{p}_i^{\ast }.$$

The first term controls the transverse directions; the second restores the missing direction. In a basis consisting of an orthonormal basis of $p^{\perp }$ followed by $e$, the Schur complement $S$ of $V$ satisfies $0<S\le V(e,e)\le \beta q^2$. Taking determinants in that same basis gives

$$\begin{array}{rl}\frac{\det V_i}{\det V}& \ge m_i^{n-1}\frac{\gamma (p_i\cdot e{)}^2}{S}\\ & \ge cK.\end{array}$$

The nonorthogonal change-of-basis factors cancel in the ratio. The source’s contact determinant bounds and radial comparisons convert this gain into $cK{\sigma }_x(p)\le C{\sigma }_x(p_i)$. The earlier affine-segment comparison bounds the right side by $C^{\prime }{\sigma }_x(p)$. The center branch is now known to be nonconjugate, so its positive determinant can be canceled. Choosing $K$ larger than this fixed bound gives the contradiction and hence the power estimate. (OpenAI, 2026b)

🏷️ Reading the family and the review boundary

Read the foundation first: its supporting theorem supplies more than convexity alone. The companion needs supports at every ordinary subgradient, convex lifted sections, and quantitative regularization at intermediate times.

The Brenier-map stability family asks how a transport map changes when its target measure changes. The present family instead derives spatial regularity of each map, with constants shared by a fixed class of density pairs on a curved manifold.

This article checked the geometric continuation argument, the segment nonconjugacy proof, the section-to-Hölder reduction, and selected spectral and determinant mechanisms. The companion’s full scalar-template construction, spectral comparison proof, and long limiting matrix construction were not independently verified. Both research conclusions are attributed to their manuscripts; the displayed elementary deductions explain how the stated interfaces fit together.

🏷️ Key steps and proof outlines

Radial shortening supplies strictness that weak MTW lacks

The transverse MTW condition gives a non-strict concavity inequality. The strict positive term needed in continuation comes from shortening geodesics. With the notation above, put $H_r(p)=A_x(rp)/r$ and let $S_p$ be the Jacobi tensor with $S_p(0)=0$, $S_p^{\prime }(0)=\mathrm{Id}$. A Jacobi field with specified initial value and zero terminal value represents the Hessian of the divided action. Differentiating that representation and using its Wronskian gives

$$\frac{d}{dr}{H}_r(p)=-S_p(r{)}^{-1}{S}_p(r{)}^{-T}.$$

The compact family of minimizing velocities bounds $\Vert S_p(r)\Vert$ from above. Thus the negative derivative has a uniform positive magnitude before conjugacy, and

$$H_t(p)-H_s(p)⪰{\kappa }_g(s-t)\mathrm{Id}(0<t<s<1).$$

Now let the $p_i$ be minimizing velocities, with $m_i>0$ and ${\sum }_i{m}_i=1$, and put $p=\sum m_i{p}_i$ and $E=\mathrm{span}\{p_i-p\}$. Assume the contracted hull $t\mathrm{conv}\{p_i\}$ lies in the injectivity domain $I(x)$. This is a hypothesis at this continuation step, ensuring smooth Hessians along its segments. For $\xi \in E^{\perp }$, MTW applies along those segments because their directions are perpendicular to $\xi$. Jensen’s inequality at time $t$, followed by shortening on each original ray, yields

$$\left(H_t(p)-\sum _i{m}_i{H}_s(p_i)\right)(\xi ,\xi )\ge {\kappa }_g(s-t)|\xi {|}^2.$$

This is the exact strict transverse gap used to rule out two nearby graph points with the same projection. It separates the roles of curvature and radial variation. (OpenAI, 2026a“Supports, divided action, and subgradients”)

A lower test excludes a conjugate modified endpoint

For the companion, consider an active outer contact and let $r$ be its minimizing reverse log. The modification has slope $l>1$ and second derivative ${\phi }_o^{\prime \prime }<0$. Its true contact log is $r/l$, strictly before cut. Shorten the modified ray to time $s$, where $1/l<s<1$, and invert the scalar modification on the resulting smooth support. The scaled relative Hessian of this lower test is

$$W_s=l{A}_y(r/l)-s^{-1}{A}_y(sr)-\frac{{\phi }_o^{\prime \prime }}{l^2}r{r}^T.$$

Divided-action monotonicity gives a positive definite baseline independent of $s$ near one; the rank-one last term is nonnegative. If the full modified endpoint were conjugate, an eigenvalue of $s^{-1}{A}_y(sr)$ would tend to $-\infty$. One eigenvalue of $W_s$ would diverge while the others stay positive, so $\det W_s\to \infty$. The original contact-volume inequality at the fixed true log instead bounds this determinant by

$$\det W_s\le \frac{l^n\Lambda }{\lambda {\sigma }_y(r/l)}<\infty .$$

This proves nonconjugacy without inserting a density equation for the modified potential.

Differentiating the pole map without assuming nearby smoothness

A final regularity tool is that an Alexandrov expansion controls every nearby subgradient. After adding a quadratic to make $f$ convex, write its remainder as $|R(h)|\le \omega (|h|)|h{|}^2$. Compare its subgradient inequality at $h$ with values at $h\pm \alpha |h|e$. For $\zeta \in \partial f(h)$ this gives

$$|(\zeta -p-Ah)\cdot e|\le C\alpha |h|+C\omega (2|h|)|h|/\alpha .$$

Choosing $\alpha \to 0$ with $\omega (2|h|)/\alpha \to 0$ proves $\zeta =p+Ah+o(|h|)$ uniformly. This permits differentiation through the short-time minimizer even though its nearby endpoints need not be differentiability points. Together with the determinant bounds, it justifies the area-formula step in the stated order. (OpenAI, 2026b“The original contact measure and lower tests” and subsequent lower-jet argument)

🏷️ Manuscripts and verification scope

• Global Support and Convex Injectivity Domains under Weak MTW — pinned manuscript and companion source.
• Uniform Bi-Hölder Transport from Weak MTW — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The documented selection covers convex open tangent injectivity domains, global support, convex lifted sections and their diameter bound, and uniform bi-Hölder transport on a fixed manifold. Its additional scale and local-injectivity result is stated only for finite target families; the scope page explicitly says this does not cover the manuscript’s arbitrary-potential local-injectivity assertion. Uniformity is over each fixed density class, not varying metrics. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026a. Global Support and Convex Injectivity Domains under Weak MTW.

🐻  OpenAI 2026b. Uniform Bi-Hölder Transport from Weak MTW.