Equivariant $3$-manifolds with positive scalar curvature

TL;DR

This paper proves that the space of G-invariant positive scalar curvature metrics on closed 3-manifolds is either empty or contractible, and classifies such manifolds for connected G, confirming a conjecture for spherical orbifolds.

Contribution

It establishes the contractibility of the space of G-invariant PSC metrics and classifies these manifolds, extending the understanding of scalar curvature in symmetric 3-manifolds.

Findings

The space of G-invariant PSC metrics is either empty or contractible.

Confirmed the generalized Smale conjecture for spherical three-orbifolds.

Provided a classification of PSC G-invariant three-manifolds for connected G.

Abstract

In this paper, for any compact Lie group $G$, we show that the space of $G$-invariant Riemannian metrics with positive scalar curvature (PSC) on any closed three-manifold is either empty or contractible. In particular, we prove the generalized Smale conjecture for spherical three-orbifolds. Moreover, for connected $G$, we make a classification of all PSC $G$-invariant three-manifolds.