K-Semistability for irregular Sasakian manifolds

TL;DR

This paper introduces a new notion of K-semistability for irregular Sasakian manifolds, linking geometric stability to constant scalar curvature and recovering key volume minimization and obstruction results.

Contribution

It extends orbifold K-semistability to irregular Sasakian manifolds and establishes a connection between constant scalar curvature and K-semistability.

Findings

Sasakian manifolds with constant scalar curvature are K-semistable.

Reveals how volume minimization results follow from K-semistability.

Shows the Lichnerowicz obstruction can be derived from this framework.

Abstract

We introduce a notion of K-semistability for Sasakian manifolds. This extends to the irregular case the orbifold K-semistability of Ross-Thomas. Our main result is that a Sasakian manifold with constant scalar curvature is necessarily K-semistable. As an application, we show how one can recover the volume minimization results of Martelli-Sparks-Yau, and the Lichnerowicz obstruction of Gauntlett-Martelli-Sparks-Yau from this point of view.