The Conjugate Heat Equation and Ancient Solutions of the Ricci Flow

2010/06/03 by Xiaodong Cao, Qi S. Zhang, Cao, Xiaodong +1 · 1 citation
Mathematics · #53C44 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1006.0540

openalex publication_date 2010/06/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 κ-solutions of the Ricci flow must be a non-flat gradient shrinking Ricci soliton. This extends Perelman's previous result on backward limits of κ-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.

Cited by

• Backward Harnack inequality and Hamilton estimates for heat type equations

Related

• Heat kernel bounds, ancient κ solutions and the Poincaré conjecture
• A gradient estimate for all positive solutions of the conjugate heat equation under Ricci flow
• The Entropy of Ricci Flows with Type-I Scalar Curvature Bounds
• Maximum principle and convergence of fundamental solutions for the Ricci flow
• Heat kernel and curvature bounds in Ricci flows with bounded scalar curvature