Distribution-Valued Ricci Bounds for Metric Measure Spaces, Singular Time Changes, and Gradient Estimates for Neumann Heat Flows

TL;DR

This paper introduces distribution-valued Ricci bounds for metric measure spaces, establishing their equivalence with gradient estimates, their stability under time changes, and their applicability to Neumann Laplacians on semi-convex subsets, along with new localization and boundary curvature notions.

Contribution

It develops a novel framework of distribution-valued Ricci bounds, proves their key properties, and extends RCD space analysis to subsets with boundary curvature concepts.

Findings

Distribution-valued Ricci bounds are equivalent to sharp gradient estimates.

These bounds are preserved under time changes with Lipschitz functions.

The framework applies to Neumann Laplacians on semi-convex subsets, with bounds expressed via measures.

Abstract

We will study metric measure spaces $(X,d,m)$ beyond the scope of spaces with synthetic lower Ricci bounds. In particular, we introduce distribution-valued lower Ricci bounds BE$_1(\kappa,\infty)$ $\bullet$ for which we prove the equivalence with sharp gradient estimates, $\bullet$ the class of which will be preserved under time changes with arbitrary $\psi\in{\mathrm Lip}_b(X)$, and $\bullet$ which are satisfied for the Neumann Laplacian on arbitrary semi-convex subsets $Y\subset X$. In the latter case, the distribution-valued Ricci bound will be given by the signed measure $\kappa= k\, m_Y + \ell\,\sigma_{\partial Y}$ where $k$ denotes a variable synthetic lower bound for the Ricci curvature of $X$ and $\ell$ denotes a lower bound for the "curvature of the boundary" of $Y$, defined in purely metric terms. We also present a new localization argument which allows us to pass on the RCD property to arbitrary open subsets of RCD spaces. And we introduce new synthetic notions for boundary curvature, second fundamental form, and boundary measure for subsets of RCD spaces.