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Math-2674 · mathematics · Geometry

Weak Hessian bounds along every RCD geodesic

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
In a complete separable full-support RCD(K,N) metric measure space with finite $N>1$ and measure finite on bounded sets, let F be bounded and globally Lipschitz and G bounded and continuous. If Hess $F\leq G\cdot g$ weakly against compact test gradients and nonnegative compactly supported Lipschitz weights, does each constant-speed minimizing geodesic $\gamma :[0,1]\to X$ satisfy $(F\circ \gamma )\prime\prime \leq d(\gamma (0),\gamma (1))^{2}G\circ \gamma $ in distributions?

Short English statement

In a complete separable full-support RCD(K,N) metric measure space with finite $N>1$ and measure finite on bounded sets, let F be bounded and globally Lipschitz and G bounded and continuous. If Hess $F\leq G\cdot g$ weakly against compact test gradients and nonnegative compactly supported Lipschitz weights, does each constant-speed minimizing geodesic $\gamma :[0,1]\to X$ satisfy $(F\circ \gamma )\prime\prime \leq d(\gamma (0),\gamma (1))^{2}G\circ \gamma $ in distributions?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2674-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2674

Weak Hessians on every geodesic: The manuscript claims that an upper distributional Hessian bound for a bounded Lipschitz function on a finite-dimensional RCD space gives the corresponding second-derivative bound along every minimizing geodesic. The upper-bound function is required to be bounded and continuous. 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.

Weak Hessian bounds along every RCD geodesic

OpenAI's claim

Weak Hessians on every geodesic: The manuscript claims that an upper distributional Hessian bound for a bounded Lipschitz function on a finite-dimensional RCD space gives the corresponding second-derivative bound along every minimizing geodesic. The upper-bound function is required to be bounded and continuous.

Mathematical statement of the claimed result

Mathematical claim 1.
On a full-support unreduced $\operatorname{RCD}(K,N)$ space, $K\in\mathbb R$, $1<N<\infty$, let $F$ be bounded and globally Lipschitz, and $G$ bounded and continuous. For compactly supported test $g\in D(\Delta)$, bounded and globally Lipschitz with $\Delta g\in W^{1,2}$, put $Y=\nabla g$ and \[H_F(Y,Y)(h)=-\int\operatorname{div}(hY)\langle\nabla F,Y\rangle\,dm-\int h\langle\nabla F,\nabla_YY\rangle\,dm.\] Suppose $H_F(\nabla g,\nabla g)(h)\le\int hG|\nabla g|^2\,dm$ for every such $g$ and nonnegative compactly supported Lipschitz $h$. Then, for any minimizing geodesic $\sigma:[0,1]\to M$ parameterized at constant speed $\ell$, the claim says it is claimed to satisfy \[(F\circ\sigma)^{\prime\prime}\le\ell^2G\circ\sigma\] in distributions on $(0,1)$. The conclusion is for every geodesic, not only almost every geodesic of a test plan.

Outline of the argument

Weak Hessians on every geodesic: The measure is multiplied by an exponential of the function, giving a weighted Bochner inequality with shifted curvature. Resolvent regularity and a common graph core place this inequality on the actual weighted generator domain. Curvature-dimension comparison for the weighted measures and a large-parameter limit extract the desired inequality on any prescribed geodesic.

Formal verification

Family 356 describes Lean coverage for the weak-Hessian inequality on every geodesic; the full sectional-curvature characterization is separate.

Weak Hessians on every geodesic: The manuscript links WeakHessian.lean (declaration OAI.WeakHessian.every_geodesic).

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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