Oliver Lindblad Petersen

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. Our decay rates are strong enough to prove positive scalar curvature rigidity in $L^p$, for each $p\in \left[1,\frac{n}{n-2}\right)$, generalizing a result by Appleton.

We consider the heat equation associated to Schrödinger operators acting on vector bundles on asymptotically locally Euclidean (ALE) manifolds. Novel $L^p-L^q$ decay estimates are established, allowing the Schrödinger operator to have a non-trivial $L^2$-kernel. We also prove new decay estimates for spatial derivatives of arbitrary order, in a general geometric setting. Our main motivation is the application to stability of non-linear geometric equations, primarily Ricci flow, which will be presented in a companion paper. The arguments in this paper use that many geometric Schrödinger operators can be written as the square of Dirac type operators. By a remarkable result of Wang, this is even true for the Lichnerowicz Laplacian, under the assumption of a parallel spinor. Our analysis is based on a novel combination of the Fredholm theory for Dirac type operators on ALE manifolds and recent advances in the study of the heat kernel on non-compact manifolds.

We consider the long-time behaviour of the mean curvature flow of spacelike hypersurfaces in the Lorentzian product manifold $M\times \mathbb{R}$, where $M$ is asymptotically flat. If the initial hypersurface $F_0\subset M\times \mathbb{R}$ is uniformly spacelike and asymptotic to $M\times \left\{s\right\}$ for some $s\in \mathbb{R}$ at infinity, we show that a mean curvature flow starting at $F_0$ exists for all times and converges uniformly to $M\times \left\{s\right\}$ as $t\to \infty$.