Dynamical functionals on ancient ARF Ricci flows

TL;DR

This paper introduces a new dynamical energy functional for ancient Ricci flows, establishing rigidity results and bounds, and derives local eigenvalue estimates for coupled Ricci and heat flows.

Contribution

It presents a novel energy functional for ancient Ricci flows and extends eigenvalue estimates to coupled Ricci and heat flows, advancing understanding of Ricci flow behavior.

Findings

The energy functional satisfies a steady Ricci breather-type rigidity.

Provides an upper bound for the $\lambda$-functional.

Derives local eigenvalue estimates for coupled Ricci and heat flows.

Abstract

We introduce a dynamical energy functional on compact ancient asymptotically Ricci-flat Ricci flows with modest decay using limits of conjugate heat flows. This functional satisfies a steady Ricci breather-type rigidity and provides an upper bound for the ordinary $\lambda$-functional while retaining many of its properties. In addition, motivated by work of Colding and Minicozzi, we derive local eigenvalue estimates for normalized Ricci flows coupled with conjugate heat flows.