Hamilton's Gradient Estimates and A Monotonicity Formula for Heat Flows on Metric Measure Spaces

TL;DR

This paper extends Hamilton's gradient estimates and a monotonicity formula for heat flows from smooth manifolds to non-smooth metric measure spaces satisfying curvature-dimension conditions.

Contribution

It generalizes key heat flow estimates and entropy formulas to non-smooth spaces with Riemannian curvature-dimension conditions, broadening their applicability.

Findings

Hamilton's gradient estimates are valid on metric measure spaces.

A monotonicity formula for entropy is established in this setting.

The results connect heat flow analysis with geometric measure theory.

Abstract

In this paper, we extend the Hamilton's gradient estimates \cite{har93} and a monotonicity formula of entropy \cite{ni04} for heat flows from smooth Riemannian manifolds to (non-smooth) metric measure spaces with appropriate Riemannian curvature-dimension condition.