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

17.12.2020, 16:15  –  Online-Seminar
Forschungsseminar Differentialgeometrie

Klaus Kröncke (Tübingen)

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.

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