Relative K-stability and Extremal Sasaki metrics

TL;DR

This paper introduces a notion of K-stability for polarized Sasakian manifolds relative to automorphisms, linking it to the existence of extremal Sasaki metrics and providing examples where no extremal metrics exist.

Contribution

It extends the concept of K-stability to Sasakian manifolds, relates it to extremal metrics, and identifies new obstructions and examples of Sasaki cones lacking extremal metrics.

Findings

K-stability implies the existence of Sasaki-extremal metrics.

Computed the invariant for deformation to the normal cone, extending Lichnerowicz obstruction.

Provided examples of Sasaki cones with no extremal Sasaki metrics.

Abstract

We define K-stability of a polarized Sasakian manifold relative to a maximal torus of automorphisms. The existence of a Sasaki-extremal metric in the polarization is shown to imply that the polarization is K-semistable. Computing this invariant for the deformation to the normal cone gives an extention of the Lichnerowicz obstruction, due to Gauntlett, Martelli, Sparks, and Yau, to an obstruction of Sasaki-extremal metrics. We use this to give a list of examples of Sasakian manifolds whose Sasaki cone contains no extremal representatives. These give the first examples of Sasaki cones of dimension greater than one that contain no extremal Sasaki metrics whatsoever. In the process we compute the unreduced Sasaki cone for an arbitrary smooth link of a weighted homogeneous polynomial.