Stability of Sasaki-extremal metrics under complex deformations

TL;DR

This paper investigates the stability of Sasaki-extremal metrics under complex structure deformations, providing conditions involving the Futaki invariant, and constructs new examples on 3-Sasaki 7-manifolds.

Contribution

It establishes sufficient conditions for the persistence of Sasaki-extremal metrics under deformations and introduces new families of such metrics on specific manifolds.

Findings

Sasaki-extremal metrics are stable under certain complex deformations.

Nondegeneracy of the relative Futaki invariant ensures existence of deformed Sasaki-extremal structures.

New Sasaki-Einstein and extremal metrics are constructed on 3-Sasaki 7-manifolds.

Abstract

We consider the stability of Sasaki-extremal metrics under deformations of the complex structure on the Reeb foliation. Given such a deformation preserving the action of a compact subgroup of the automorphism group of a Sasaki-extremal structure, a sufficient condition is given involving the nondegeneracy of the relative Futaki invariant for the deformations to contain Sasaki-extremal structures. Deformations of Sasaki-Einstein metrics are also considered, where it suffices that the deformation preserve a maximal torus. As an application, new families of Sasaki-Einstein and Sasaki-extremal metrics are given on deformations of well known 3-Sasaki 7-manifolds.