Usually when I blog here about positive scalar curvature, the manifolds I consider are assumed to have no boundary. In this post I want to explain the basics of what happens when the manifolds actually do have a boundary. So first of all, one has to mention that any compact manifold with boundary admits a … Continue reading "Positive scalar curvature metrics on manifolds with boundary"

Mark Kac asked in a paper from 1966 the following question: Can one hear the shape of a drum? The mathematically precise question is the following: Assume that \((M,g)\) and \((N,h)\) are two compact Riemannian surfaces (thought of as the heads of two drums). If the Laplace operators of \((M,g)\) and of \((N,h)\) have the … Continue reading "Can one hear orientability?"

Let \((M,g)\) be a complete, contractible Riemannian \(3\)-manifold (without boundary). Chang-Weinberger-Yu (link) proved that if \((M,g)\) has uniformly positive scalar curvature, then \(M\) must be homeomorphic to \(\mathbb{R}^3\). Recently (arXiv:1906.04128), Wang proved that if \((M,g)\) has positive scalar curvature and \(M\) has trivial fundamental group at infinity, then \(M\) must be homeomorphic to \(\mathbb{R}^3\). Jiang … Continue reading "Contractible 3-manifolds and positive scalar curvature, II"

A month ago Aougab, Patel and Vlamis posted a preprint on the arXiv (arXiv:2007.01982) about the question which groups, for a fixed orientable surface of infinite genus, can be realized as the full isometry group of a Riemannian metric of constant negative curvature on that surface. To my surprise, they stated in the introduction that … Continue reading "Isometry groups of hyperbolic surfaces"