Bounds on the Heat Kernel under the Ricci Flow

Abstract

We establish an estimate for the fundamental solution of the heat equation on a closed Riemannian manifold $M$ of dimension at least 3, evolving under the Ricci flow. The estimate depends on some constants arising from a Sobolev imbedding theorem. Considering the case when the scalar curvature is positive throughout the manifold, at any time, we will obtain, as a corollary, a bound similar to the one known for the fixed metric case.

8 pages

References in corpus (3)

• Gradient estimates for the heat equation under the Ricci flow
• The Conjugate Heat Equation and Ancient Solutions of the Ricci Flow
• Some results for the Perelman LYH-type inequality

Cited by in corpus (2)

• Gaussian upper bounds for the heat kernel on evolving manifolds
• Dimension-free Harnack inequalities for conjugate heat equations and their applications to geometric flows