L^p-stability and positive scalar curvature rigidity of Ricci-flat ALE manifolds

Abstract:

We prove stability of integrable ALE manifolds with a parallel spinor under Ricci flow, given an initial metric which is close in $L^p\cap L^{\infty}$, for any $p\in (1,n)$, where n is the dimension of the manifold. In particular, our result applies to all known examples of 4-dimensional gravitational instantons. The result is obtained by a fixed point argument, based on novel estimates for the heat kernel of the Lichnerowicz Laplacian. It allows us to give a precise description of the convergence behaviour of the Ricci flow. Our decay rates are strong enough to prove positive scalar curvature rigidity in $L^p$, for each $p\in [1,\frac{n}{n-2})$, generalizing a result by Appleton. This is joint work with Oliver Lindblad Petersen.