The classical singularity theorems of General Relativity show that a Lorentzian manifold with a smooth metric satisfying certain physically reasonable curvature and causality conditions cannot be causal geodesically complete. One drawback of these classical theorems is that they require smoothness of the metric while in many physical models the metric is less regular. In my talk I will present recent work concerning singularity theorems for metrics that are merely continuously differentiable - a regularity where one still has existence but not uniqueness for solutions of the geodesic equation. I will first give a general overview of the new challenges arising in the statements and proofs of singularity theorems for metrics of lower regularity and then discuss the proof of Hawking's theorem for C^1-metrics in a little more detail.

We will establish long-time and derivative estimates for the heat semigroup of various natural Schrödinger operators on asymptotically locally Euclidean (ALE) manifolds. These include the Lichnerowicz Laplacian of a Ricci-flat ALE manifold, provided that it is spin and admits a parallel spinor. The estimates will be used to prove its L^p-stability under the Ricci flow for p < n. A positive scalar curvature rigidity theorem will also be deduced. This is joint work with Oliver Lindblad Petersen.

I consider the heat flow for Yang-Mills connections on R^d in supercritical dimensions 5 \leq d \leq 9, where the model admits self-similar blowup solutions, with an explicit example given by Weinkove. I will discuss the stability of the Weinkove solution and show that it is nonlinearly asymptotically stable under small equivariant perturbations. The talk is based on joint works with Irfan Glogić and Roland Donninger (Vienna)