Improved Hebey-Vaugon conjecture on equivariant Yamabe invariants in dimension 3

TL;DR

This paper establishes an improved upper bound for the G-equivariant Yamabe invariant on certain 3-manifolds with group actions, advancing the Hebey-Vaugon conjecture in geometric analysis.

Contribution

The authors provide a refined upper bound for the G-equivariant Yamabe invariant in dimension 3, under specific topological conditions, improving upon previous conjectures.

Findings

Established an upper bound for the G-equivariant Yamabe invariant.

Extended the Hebey-Vaugon conjecture in the context of 3-manifolds.

Applied topological assumptions to refine geometric invariant estimates.

Abstract

Consider a closed connected $3$-manifold $M$ acted diffeomorphically on by a compact Lie group $G$ with at least one orbit of finite cardinality. We show an upper bound for the $G$-equivariant Yamabe invariant $\sigma_G(M)$ under certain topological assumptions, which improved a conjecture of Hebey-Vaugon.