Circle actions and scalar curvature

TL;DR

This paper develops methods to construct positive scalar curvature metrics on manifolds with circle symmetries, showing existence results for invariant and non-invariant metrics under various topological conditions.

Contribution

It introduces new techniques for constructing positive scalar curvature metrics on S^1-manifolds, including invariant and non-invariant cases, using equivariant bordism and generalized genus invariants.

Findings

Existence of S^1-invariant positive scalar curvature metrics on manifolds with fixed point components of codimension 2.

Non-invariant metrics of positive scalar curvature exist on many S^1-manifolds.

Connected sums of certain S^1-manifolds admit invariant positive scalar curvature metrics under specific conditions.

Abstract

We construct metrics of positive scalar curvature on manifolds with circle actions. One of our main results is that there exist $S^1$-invariant metrics of positive scalar curvature on every $S^1$-manifold which has a fixed point component of codimension 2. As a consequence we can prove that there are non-invariant metrics of positive scalar curvature on many manifolds with circle actions. Results from equivariant bordism allow us to show that there is an invariant metric of positive scalar curvature on the connected sum of two copies of a simply connected semi-free $S^1$-manifold $M$ of dimension at least six provided that $M$ is not $\text{spin}$ or that $M$ is $\text{spin}$ and the $S^1$-action is of odd type. If $M$ is spin and the $S^1$-action of even type then there is a $k>0$ such that the equivariant connected sum of $2^k$ copies of $M$ admits an invariant metric of positive scalar curvature if and only if a generalized $\hat{A}$-genus of $M/S^1$ vanishes.