Ricci Flow on ALF manifolds

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.