An entropy formula for the heat equation on manifolds with time-dependent metric, application to ancient solutions

TL;DR

This paper introduces a new entropy functional for the heat equation on manifolds with evolving metrics, demonstrating its non-decreasing nature and applying it to classify ancient solutions, revealing conditions for constancy.

Contribution

The paper presents a novel entropy functional for heat equations on time-dependent manifolds and applies it to classify ancient solutions based on entropy growth.

Findings

Entropy is non-decreasing under certain conditions.

Constant solutions are characterized by sublinear entropy growth.

Existence of nonconstant solutions with linear entropy growth.

Abstract

We introduce a new entropy functional for nonnegative solutions of the heat equation on a manifold with time-dependent Riemannian metric. Under certain integral assumptions, we show that this entropy is non-decreasing, and moreover convex if the metric evolves under super Ricci flow (which includes Ricci flow and fixed metrics with nonnegative Ricci curvature). As applications, we classify nonnegative ancient solutions to the heat equation according to their entropies. In particular, we show that a nonnegative ancient solution whose entropy grows sublinearly on a manifold evolving under super Ricci flow must be constant. The assumption is sharp in the sense that there do exist nonconstant positive eternal solutions whose entropies grow exactly linearly in time. Some other results are also obtained.