### Contractible 3-manifolds and positive scalar curvature, II

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"

### Isometry groups of hyperbolic surfaces

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"

### Spaces of positively curved Riemannian metrics

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"