Ricci Flow on ALF manifolds

TL;DR

This paper demonstrates that Ricci flow preserves the ALF structure on certain manifolds, introduces a new functional related to mass, and analyzes stability and positivity properties of ALF metrics.

Contribution

It develops a weighted Fredholm framework for ALF manifolds, defines a new renormalized functional, and studies stability and mass relations of ALF Ricci-flat metrics.

Findings

Ricci flow preserves ALF structure on manifolds with n≥4.

Introduces a renormalized functional related to relative mass.

Shows certain ALF metrics are dynamically unstable under Ricci flow.

Abstract

We prove that on ALF $n$-manifolds with $n\ge 4$ the Ricci flow preserves the ALF structure, and develop a weighted Fredholm framework adapted to ALF manifolds. Motivated by Perelman's $\lambda$-functional, we define a renormalized functional $\lambda_{\mathrm{ALF}}$ whose gradient flow is the Ricci flow. It is built from a relative mass with respect to a reference Ricci-flat metric at infinity. This yields a natural notion of variational and linear stability for Ricci-flat ALF $4$-metrics and lets us show that the conformally K\"ahler, non-hyperk\"ahler examples are dynamically unstable along Ricci flow. We finally relate the sign of $\lambda_{\mathrm{ALF}}$ to positive relative mass statements for ALF metrics.