Weak solutions for the Ricci flow I

Abstract

This is the first of a series of papers, where we introduce a new class of estimates for the Ricci flow, and use them both to characterize solutions of the Ricci flow and to provide a notion of weak solutions to the Ricci flow in the nonsmooth setting. In this first paper, we prove various new estimates for the Ricci flow, and show that they in fact characterize solutions of the Ricci flow. Namely, given a family $(M,g_t)_{t\in I}$ of Riemannian manifolds, we consider the path space $P\mathcal{M}$ of its space time $\mathcal{M}=M\times I$. Our first characterization says that $(M,g_t)_{t\in I}$ evolves by Ricci flow if and only if an infinite dimensional gradient estimate holds for all functions on $P\mathcal{M}$. We prove additional characterizations in terms of the $C^{1/2}$-regularity of martingales on path space, as well as characterizations in terms of log-Sobolev and spectral gap inequalities for a family of Ornstein-Uhlenbeck type operators. Our estimates are infinite dimensional generalizations of much more elementary estimates for the linear heat equation on $(M,g_t)_{t\in I}$, which themselves generalize the Bakry-Emery-Ledoux estimates for spaces with lower Ricci curvature bounds. Based on our characterizations we can define a notion of weak solutions for the Ricci flow. We will develop the structure theory of these weak solutions in subsequent papers.