TITLE: Relative K-stability and 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. We apply this to certain hypersurface singularities with a fairly large Sasaki cone, showing that extremal Sasaki metrics are obstructed in the entire Sasaki cone. This is based on joint work with Craig van Coevering.