Ricci Flow on ALF manifolds

TL;DR

This work develops a robust framework for Ricci flow on ALF manifolds by establishing that the ALF end structure is preserved under the flow and by embedding the dynamics into a variational setting via a renormalized functional λ_ALF built from a relative mass m(g,g_RF). It provides a weighted Fredholm theory for the Laplacian on ALF spaces, enabling a gradient-flow interpretation of Ricci flow and precise stability analyses. The authors prove dynamical instability for non-hyperkähler conformally Kähler Ricci-flat ALF metrics, while hyperkähler ALF metrics are shown to be dynamically and linearly stable, with instability detected through the second variation of λ_ALF. A positive mass theorem in the ALF context is established, relating nonnegative scalar curvature to nonnegative relative mass and rigidity to hyperkähler Hitchin–Page ends, tying together variational, analytic, and geometric aspects of ALF manifolds.

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 $λ$-functional, we define a renormalized functional $λ_{\mathrm{ALF}}$ 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 Kähler, non-hyperkähler examples are dynamically unstable along Ricci flow. We finally relate the sign of $λ_{\mathrm{ALF}}$ to positive relative mass statements for ALF metrics.

Theorems & Definitions (76)

• Theorem 1
• Theorem 3: Dynamical Instability
• Theorem 4
• Theorem 5
• Remark 6
• Proposition 2.2
• Lemma 2.3
• proof
• proof : Proof of \ref{['prop:gr']}
• Proposition 2.5