Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hessian estimates for the conjugate heat equation coupled with the Ricci flow

Hong Huang

Abstract

In this short note we obtain some local and global upper bounds for the Hessian of a positive solution to the conjugate heat equation coupled with the Ricci flow.

Hessian estimates for the conjugate heat equation coupled with the Ricci flow

Abstract

In this short note we obtain some local and global upper bounds for the Hessian of a positive solution to the conjugate heat equation coupled with the Ricci flow.

Paper Structure

This paper contains 3 sections, 9 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Gradient estimates
  3. Hessian estimates

Key Result

Theorem 1.1

Let $(M^n,g(t))$, $t\in [0,T]$, be a Ricci flow on a compact manifold, $u$ be a positive solution to the conjugate heat equation coupled with the Ricci flow on $M\times [0,T]$ with $0 < u \leq A$. Then we have and, in particular, where $C$ depends on $n$, the upper bounds of $|Rm|$, $|\nabla Ric|$, and $|\nabla^2 R|$ on $M\times [0,T]$.

Theorems & Definitions (9)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Proposition 2.2
  • Corollary 2.3
  • Proposition 2.4
  • Lemma 3.1
  • Lemma 3.2
  • Corollary 3.3