The Conjugate Heat Equation and Ancient Solutions of the Ricci Flow

TL;DR

This paper establishes Gaussian bounds for the conjugate heat equation under Ricci flow and characterizes backward limits of certain solutions as gradient shrinking Ricci solitons in dimensions four and higher.

Contribution

It extends Perelman's results to higher dimensions, providing new Gaussian bounds and classifying backward limits of Ricci flow solutions.

Findings

Gaussian bounds for the conjugate heat equation under Ricci flow

Backward limits of type I κ-solutions are gradient shrinking Ricci solitons in dimensions ≥4

Extension of Perelman's 3D results to higher dimensions

Abstract

We prove Gaussian type bounds for the fundamental solution of the conjugate heat equation evolving under the Ricci flow. As a consequence, for dimension 4 and higher, we show that the backward limit of type I $\kappa$-solutions of the Ricci flow must be a non-flat gradient shrinking Ricci soliton. This extends Perelman's previous result on backward limits of $\kappa$-solutions in dimension 3, in which case that the curvature operator is nonnegative (follows from Hamilton-Ivey curvature pinching estimate). The Gaussian bounds that we obtain on the fundamental solution of the conjugate heat equation under evolving metric might be of independent interest.