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Dynamical functionals on ancient ARF Ricci flows

Isaac M. Lopez, Rio Schillmoeller

TL;DR

The paper develops a canonical dynamical energy framework for compact ancient ARF Ricci flows converging to a Ricci-flat metric and shows how a monotone dynamical functional, denoted $\\lambda_{\\mathrm{dyn}}^{\\infty}$, upper-bounds Perelman’s $\\lambda$-functional and enforces breather-type rigidity. It then extends Colding–Minicozzi style eigenvalue estimates to a drift Laplacian associated with a Ricci flow coupled to a conjugate heat flow, proving sharp local bounds for the first eigenvalue. Central to the construction are uniform a priori bounds for backward conjugate heat flows, leading to a limiting function $f^{\\infty}$ that defines the dynamical functional and underpins monotonicity and rigidity conclusions. Together, these results advance dynamical stability analysis of Ricci-flat metrics by bridging dynamical entropy-type functionals with local spectral control in ARF settings.

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 $λ$-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.

Dynamical functionals on ancient ARF Ricci flows

TL;DR

The paper develops a canonical dynamical energy framework for compact ancient ARF Ricci flows converging to a Ricci-flat metric and shows how a monotone dynamical functional, denoted , upper-bounds Perelman’s -functional and enforces breather-type rigidity. It then extends Colding–Minicozzi style eigenvalue estimates to a drift Laplacian associated with a Ricci flow coupled to a conjugate heat flow, proving sharp local bounds for the first eigenvalue. Central to the construction are uniform a priori bounds for backward conjugate heat flows, leading to a limiting function that defines the dynamical functional and underpins monotonicity and rigidity conclusions. Together, these results advance dynamical stability analysis of Ricci-flat metrics by bridging dynamical entropy-type functionals with local spectral control in ARF settings.

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 -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.

Paper Structure

This paper contains 17 sections, 18 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Organization of the article
  4. Notation and conventions
  5. Acknowledgements
  6. Curvature estimates for ARF Ricci flows
  7. Definitions
  8. Ricci curvature estimates
  9. A dynamical energy functional on ancient ARF Ricci flows
  10. Estimates for conjugate heat flows
  11. Properties of the dynamical energy functional for ARF flows
  12. Eigenvalue estimates for conjugate heat flows
  13. Local upper bounds for the eigenvalues of conjugate heat flows
  14. Sharpness
  15. Evolution equations and inequalities
  16. ...and 2 more sections

Key Result

Theorem 1.1.1

Suppose that $(M^n,g(t))$ is an ancient solution to the Ricci flow equation on a compact manifold for which the Lojasiewicz-Simon inequality for the $\lambda$-functional, holds for some $\theta \in [\frac{2}{5},\frac{1}{2}]$ for all $t\in (-\infty,T_0]$ for some $T_0<0$, where $f_g$ is the minimizer of $\lambda$. Suppose further that there exist constants $C_k,T_k>0$ independent of $t$ such that

Theorems & Definitions (34)

  • Theorem 1.1.1
  • Theorem 1.1.2
  • Definition 2.1: Asymptotically Ricci-flat flow
  • Lemma 2.2.1
  • proof
  • Lemma 2.2.2
  • proof
  • Lemma 2.2.3
  • proof
  • Lemma 3.1.1
  • ...and 24 more