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Math-2339 · mathematics · Geometry

Convexity of Injectivity Loci Under MTW

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
For a complete Riemannian manifold satisfying the Ma--Trudinger--Wang curvature condition for squared distance, must every tangent injectivity locus be convex? At $x$, this locus consists of tangent vectors for which the radial geodesic remains minimizing up to its endpoint before the cut locus; convexity is in the vector space $T_xM$.

Formulation references: [S1], [S2].

Short English statement

For a complete Riemannian manifold satisfying the Ma--Trudinger--Wang curvature condition for squared distance, must every tangent injectivity locus be convex? At $x$, this locus consists of tangent vectors for which the radial geodesic remains minimizing up to its endpoint before the cut locus; convexity is in the vector space $T_xM$.

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2339-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2339

Weak MTW support and injectivity-domain convexity: Every compact connected boundaryless manifold in dimension at least two satisfying weak MTW has convex tangent injectivity domains.

Uniform bi-Hölder transport under weak MTW: The original source resolves the catalogue statement shorthand: OWR31/2008 p.1762 uses the weak cost-orthogonal MTW condition, and Figalli–Gallouët–Rifford arXiv:1404.0141v1 pp.1–3 states Villani’s conjecture for smooth compact manifolds of dimension n>=2, with exactly the open pre-cut I(x), no nonfocality, and weak MTW as Definition1.1. The manuscript geo:convexity supplies exactly this source-level convexity conclusion. Its compactness is the original conjecture scope rather than an unsupported restriction of the intended record.

This page records partial support. Complete resolution of the catalogue question is not established by this record.

Convexity of Injectivity Loci Under MTW

OpenAI's claim

Weak MTW support and injectivity-domain convexity: On a compact smooth manifold, weak MTW curvature is claimed to imply convexity of every tangent injectivity domain; the paper also gives global supporting mountains for ordinary subgradients of squared-distance potentials, without a density assumption.

Uniform bi-Hölder transport under weak MTW: For a fixed compact weak-MTW manifold, the manuscript claims one Hölder modulus for optimal maps and their inverses over all density pairs with common positive lower and finite upper bounds; no density smoothness or absence of conjugate cut points is assumed.

Mathematical statement of the claimed result

Mathematical claim 1.
Let $(M,g)$ be smooth, connected, compact and boundaryless with $\dim M\ge2$. Set $c(x,y)=d(x,y)^2/2$ and $I(x)=\{v\in T_xM:d(x,\exp_x(av))=a|v|\text{ for some }a>1\}$. Assume the weak MTW inequality \[-\frac32\left.\frac{\partial^4}{\partial s^2\partial t^2}c(\exp_x(t\xi),\exp_x(v+s\eta))\right|_{s=t=0}\ge0\quad(v\in I(x),\ \langle\xi,\eta\rangle=0).\] Then the claimed conclusion is $v_0,v_1\in I(x)\Longrightarrow(1-\theta)v_0+\theta v_1\in I(x)$ for every $0\le\theta\le1$. No strict MTW bound, nonfocality or prior convexity is assumed.
Mathematical claim 2.
On a smooth connected compact boundaryless $n$-manifold, $n\ge2$, use $c=d^2/2$ and the weak MTW assumption $-\tfrac32\partial_s^2\partial_t^2c(\exp_x(t\xi),\exp_x(v+s\eta))|_0\ge0$ for every $v$ in the open pre-cut domain and $\xi\perp\eta$. Fix $0<\lambda\le\Lambda<\infty$ such that normalized densities $\lambda\le\rho_i\le\Lambda$ exist. There are $0<\alpha\le1,C<\infty$, depending only on $(M,g),\lambda,\Lambda$, such that the almost-everywhere optimal map for $c$ from $\rho_0dV$ to $\rho_1dV$ has a homeomorphic representative $T$ with \[d(Tx,Tx^\prime)\le Cd(x,x^\prime)^\alpha,\qquad d(T^{-1}y,T^{-1}y^\prime)\le Cd(y,y^\prime)^\alpha.\] The two densities are merely measurable, have total mass one and obey the bounds almost everywhere. Its geometric convexity result also asserts convexity of every open tangent injectivity domain.

Outline of the argument

Weak MTW support and injectivity-domain convexity: Proper radial shortening gives smooth distance supports and strict time monotonicity of the branch Hessian. The full ordinary-subgradient graph is represented locally by convex combinations of all active minimizing logarithms. At the first scale where support could fail, limiting convex hulls remain minimizing; a conjugacy-kernel analysis and the radial Hessian identity rule out the alleged failure.

Uniform bi-Hölder transport under weak MTW: Lifted gap sections of transport potentials are studied in the tangent injectivity domain. Convexity turns a uniform power estimate on dual-potential oscillation into a diameter estimate on their endpoint images. Taking the supremum over all admissible densities gives one class-wide bound, and comparison at nearby poles supplies Hölder continuity in both transport directions.

Formal verification

Family 360 describes Lean coverage for injectivity-domain convexity, global support and the listed finite-family estimates, as well as uniform bi-Hölder transport; arbitrary-potential local injectivity is broader.

Weak MTW support and injectivity-domain convexity: The manuscript links BiholderTransport.lean (declaration OAI.WeakMTWTransport.uniform_biHolder_transport), WeakMTWGlobalSupport.lean (declaration OAI.WeakMTWGlobalSupport.current_main_convexity).

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

Uniform bi-Hölder transport under weak MTW: The manuscript links BiholderTransport.lean (declaration OAI.WeakMTWTransport.uniform_biHolder_transport), WeakMTWGlobalSupport.lean (declaration OAI.WeakMTWGlobalSupport.current_main_convexity).

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

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