Bochner formulas, functional inequalities and generalized Ricci flow

TL;DR

This paper establishes sharp functional inequalities along generalized Ricci flow using Bochner formulas and introduces a twisted connection to analyze stochastic properties, providing new characterizations of the flow.

Contribution

It introduces a novel Bochner formula for the Bismut connection and defines a twisted connection to characterize generalized Ricci flow via functional inequalities.

Findings

Sharp Poincaré and log-Sobolev inequalities along Ricci flow

A new Bochner formula for the Bismut connection

Characterization of Ricci flow through Malliavin calculus

Abstract

As a consequence of the Bochner formula for the Bismut connection acting on gradients, we show sharp universal Poincar\'e and log-Sobolev inequalities along solutions to generalized Ricci flow. Using the two-form potential we define a twisted connection on spacetime which determines an adapted Brownian motion on the frame bundle, yielding an adapted Malliavin gradient on path space. We show a Bochner formula for this operator, leading to characterizations of generalized Ricci flow in terms of universal Poincar\'e and log-Sobolev type inequalities for the associated Malliavin gradient and Ornstein-Uhlenbeck operator.