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# Mathematical Research Letters

## Volume 25 (2018)

### Number 1

### Relative K-stability and extremal Sasaki metrics

Pages: 1 – 19

DOI: https://dx.doi.org/10.4310/MRL.2018.v25.n1.a1

#### Authors

#### 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.

#### Keywords

Sasakian, K-stability, Sasaki-extremal

#### 2010 Mathematics Subject Classification

Primary 53C25. Secondary 32W20.

The first author was partially supported by grant #245002 from the Simons Foundation.

Received 20 August 2016

Accepted 15 March 2017

Published 4 June 2018