On a parabolic curvature lower bound generalizing Ricci flows

TL;DR

This paper introduces a new curvature lower bound condition for Ricci flows that generalizes previous concepts, characterized through optimal transport, heat flow estimates, and variational inequalities, extending the understanding of Ricci curvature in evolving manifolds.

Contribution

It provides a novel characterization of a Ricci nonnegativity condition for super Ricci flows using optimal transport and heat flow inequalities, without relying on tensor calculus.

Findings

Equivalent to a Bochner inequality and heat flow gradient estimates

Wasserstein contraction along the adjoint heat flow

Convexity of a modified entropy along Wasserstein geodesics

Abstract

Optimal transport plays a major role in the study of manifolds with Ricci curvature bounded below. Some results in this setting have been extended to super Ricci flows, revealing a unified approach to analysis on Ricci nonnegative manifolds and Ricci flows. However we observe that the monotonicity of Perelman's functionals ($\mathcal{F}$, $\mathcal{W}$, reduced volume), which hold true for Ricci flows and Ricci nonnegative manifolds, cannot be strictly generalized to super Ricci flows. In 2010 Buzano introduced a condition which still generalizes Ricci flows and Ricci nonnegative manifolds, and on which Perelman's monotonicities do hold. We provide characterizations of this condition using optimal transport and understand it heuristically as Ricci nonnegativity of the space-time. This interpretation is consistent with its equivalence to Ricci nonnegativity on Perelman's infinite dimensional manifold. More precisely, we prove that for smooth evolutions of Riemannian manifolds, this condition is equivalent to a Bochner inequality (resembling Perelman's Harnack inequality but for the forward heat flow), a gradient estimate for the heat flow, a Wasserstein contraction along the adjoint heat flow, the convexity of a modified entropy along Wasserstein geodesics, and an Evolutionary Variational Inequality (EVI). The optimal transport statements use Perelman's $L$ distance as cost, as first studied on Ricci flows by Topping and by Lott. We also consider dimensionally improved and weighted versions of these conditions. The dimensional Bochner inequality and all gradient estimates for the forward heat equation, along with the EVIs, appear to be new even for general Ricci flows, and are related to the Hamiltonian perspective on the $L$ distance. Most of our proofs do not use tensor calculus or Jacobi fields, suggesting the possibility of future extensions to more singular settings.