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"

It is by now a classical topic in index theory to study on a (closed) Riemannian (spin) manifold the space of all Riemannian metrics of positive scalar curvature. We have several results showing that this space is usually highly complicated from a homotopy theoretic point of view (provided it is non-empty). Instead of studying positivity … Continue reading "Spaces of positively curved Riemannian metrics"

Recently I saw some papers on the arXiv on conjugation curvature of finitely generated groups (also called medium-scale curvature, transportation curvature, metric Ricci curvature or comparison curvature for Cayley graphs). I got a bit interested in it and so decided to write up a short post about it. Let \(G\) be a finitely generated group … Continue reading "Conjugation Curvature"

Given a natural number n, the Collatz sequence it generates is the following: if n is even, then divide it by 2, if n is odd, then multiply it by 3 and add 1; and now iterate this procedure. The Collatz conjecture states that you will always end up with the number 1 after finitely … Continue reading "Collatz conjecture"