The Conjugate Linearized Ricci Flow on Closed 3-Manifolds

TL;DR

This paper characterizes the conjugate linearized Ricci flow on closed 3-manifolds, introduces Ricci flow conjugated constraint sets, and provides integral representations and properties that extend to higher dimensions.

Contribution

It introduces the notion of Ricci flow conjugated constraint sets and derives integral representations for the Ricci flow metric and tensor, extending classical results to higher dimensions.

Findings

Characterization of conjugate linearized Ricci flow on closed 3-manifolds

Introduction of Ricci flow conjugated constraint sets

Integral representation formulas for Ricci flow metric and Ricci tensor

Abstract

We characterize the conjugate linearized Ricci flow and the associated backward heat kernel on closed three--manifolds of bounded geometry. We discuss their properties, and introduce the notion of Ricci flow conjugated constraint sets which characterizes a way of Ricci flow averaging metric dependent geometrical data. We also provide an integral representation of the Ricci flow metric itself and of its Ricci tensor in terms of the heat kernel of the conjugate linearized Ricci flow. These results, which readily extend to closed n-dimensional manifolds, yield for various conservation laws, monotonicity and asymptotic formulas for the Ricci flow and its linearization.