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Math-2657 · mathematics · Geometry

Uniform bi-Hölder transport under weak MTW

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Fix a compact connected smooth boundaryless Riemannian manifold of dimension at least two satisfying weak MTW for half-squared distance, and positive numbers $\lambda \leq \Lambda $. Do constants $\alpha >0$ and $C<\infty $ depending only on that metric and $\lambda ,\Lambda $ guarantee that for every pair of probability densities between $\lambda $ and $\Lambda $ almost everywhere, the optimal map has a homeomorphic representative and it and its inverse are globally $\alpha$-Hölder with constant C?

Short English statement

Fix a compact connected smooth boundaryless Riemannian manifold of dimension at least two satisfying weak MTW for half-squared distance, and positive numbers $\lambda \leq \Lambda $. Do constants $\alpha >0$ and $C<\infty $ depending only on that metric and $\lambda ,\Lambda $ guarantee that for every pair of probability densities between $\lambda $ and $\Lambda $ almost everywhere, the optimal map has a homeomorphic representative and it and its inverse are globally $\alpha$-Hölder with constant C?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2657-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2657

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. The parameters, regularity assumptions and normalization in this question are retained in applying that assertion.

With the hypotheses and conventions above, the claimed result addresses the whole mathematical assertion on this page.

Uniform bi-Hölder transport under weak MTW

OpenAI's claim

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.
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

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.

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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