Friday

A natural way to generalize the Lichnerowicz obstruction to positive scalar curvature on spin manifolds is to find curvature conditions which imply that some twisted Dirac operators have vanishing index. Such vanishing has topological consequences for the manifold, in terms of its rational cobordism type. To make these generalizations most interesting, the curvature conditions should be as weak as possible, easily computable, and, ideally, invariant under appropriate surgeries. Following this scheme and inspired by recent works of Petersen and Wink, we determine linear inequalities on the eigenvalues of curvature operators that imply vanishing of the twisted hat A genus on a closed Riemannian spin manifold, where the twisting bundle is any prescribed parallel bundle of tensors. For instance, this yields a new obstruction to existence of Einstein metrics with 5-positive curvature operator on certain 8-manifolds, e.g., on HP^2. (This is based on joint work with Jackson Goodman.)

This talk will focus on recent developments in the studies of the degenerations of metrics with Ricci curvature bounds. Mainly we will discuss the collapsing geometry of Einstein manifolds with special holonomy and non-smooth metric spaces with Ricci curvature bounds.

I will discuss recent work (joint with Otis Chodosh, Chao Li, and Doug Stryker) showing that every complete two-sided stable minimal hypersurface in R⁵ is flat. The ideas in the proof are motivated by those in the study of manifolds with certain uniformly positive curvature conditions, in our case bi-Ricci curvature.

We present several new stability and rigidity results for scalar curvature. In particular, we solve Gromov's spherical stability problem and show that initial data sets with vanishing mass embed into pp-wave spacetimes. This is based upon joint work with Yiyue Zhang.