Extremal K-contact metrics

TL;DR

This paper extends the understanding of K-contact manifolds by interpreting scalar curvature as a moment map, generalizing invariants, and exploring deformation theory in dimension five under weaker conditions.

Contribution

It generalizes the Sasaki-Futaki invariant to K-contact geometry and relates curvature to moment maps, expanding the theoretical framework.

Findings

Transverse Hermitian scalar curvature as a moment map

Generalization of Sasaki-Futaki invariant

Deformation results in 5D under weaker conditions

Abstract

Extending a result of He to the non-integrable case of K-contact manifolds, it is shown that transverse Hermitian scalar curvature may be interpreted as a moment map for the strict contactomorphism group. As a consequence, we may generalize the Sasaki-Futaki invariant to K-contact geometry and establish a number of elementary properties. Moreover, we prove that in dimension 5 certain deformation-theoretic results can be established also under weaker integrability conditions by exploiting the relationship between J-anti-invariant and self-dual 2-forms.