A Bochner Formula on Path Space for the Ricci Flow

TL;DR

This paper extends the Bochner formula to infinite-dimensional path space on evolving manifolds, providing new characterizations and estimates for Ricci flow solutions through inequalities on parabolic path space.

Contribution

It introduces a generalized Bochner formula for martingales on parabolic path space, characterizing Ricci flow solutions and deriving gradient and Hessian estimates.

Findings

Characterization of Ricci flow via Bochner inequalities on path space

Gradient and Hessian estimates for martingales on parabolic path space

Simplified proofs of existing Ricci flow characterizations

Abstract

We generalize the classical Bochner formula for the heat flow on evolving manifolds $(M,g_{t})_{t \in [0,T]}$ to an infinite-dimensional Bochner formula for martingales on parabolic path space $P\mathcal{M}$ of space-time $\mathcal{M} = M \times [0,T]$. Our new Bochner formula and the inequalities that follow from it are strong enough to characterize solutions of the Ricci flow. Specifically, we obtain characterizations of the Ricci flow in terms of Bochner inequalities on parabolic path space. We also obtain gradient and Hessian estimates for martingales on parabolic path space, as well as condensed proofs of the prior characterizations of the Ricci flow from Haslhofer-Naber \cite{HN18a}. Our results are parabolic counterparts of the recent results in the elliptic setting from \cite{HN18b}.