An Upper Bound for Hessian Matrices of Positive Solutions of Heat Equations

TL;DR

This paper establishes upper bounds for the Hessian of the logarithm of positive solutions to the heat equation on Riemannian manifolds, considering both fixed metrics and Ricci flow evolution, complementing existing lower bounds.

Contribution

It provides new global and local upper bounds for the Hessian of log solutions to the heat equation under various geometric conditions.

Findings

Upper bounds for Hessian of log solutions established

Bounds apply to fixed and evolving (Ricci flow) metrics

Complements known lower bounds for the same quantities

Abstract

We prove global and local upper bounds for the Hessian of log positive solutions of the heat equation on a Riemannian manifold. The metric is either fixed or evolves under the Ricci flow. These upper bounds supplement the well-known global lower bound.