On stability and scalar curvature rigidity of quaternion-K\"ahler manifolds

TL;DR

This paper proves that quaternion-K"ahler manifolds with negative scalar curvature are stable and scalar curvature rigid, while those with positive scalar curvature may not be rigid despite semi-stability, highlighting differences based on curvature sign.

Contribution

It establishes stability and rigidity results for quaternion-K"ahler manifolds depending on scalar curvature sign, extending understanding of Einstein manifolds with special holonomy.

Findings

Negative scalar curvature quaternion-K"ahler manifolds are stable and scalar curvature rigid.

Irreducible nonpositive Einstein manifolds with special holonomy are stable.

Existence of positive scalar curvature quaternion-K"ahler manifolds that are semi-stable but not rigid.

Abstract

We show that every quaternion-K\"ahler manifold of negative scalar curvature is stable as an Einstein manifold and therefore scalar curvature rigid. In particular, this implies that every irreducible nonpositive Einstein manifold of special holonomy is stable. In contrast, we demonstrate that there exist quaternion-K\"ahler manifolds of positive scalar curvature which are not scalar curvature rigid even though they are semi-stable.