On balanced HKT manifolds

TL;DR

This paper investigates the properties of balanced HKT structures on compact hypercomplex manifolds, proving the openness of the HKT cone and exploring the hyperholomorphic vector fields' algebraic and harmonic properties.

Contribution

It establishes the openness of the balanced HKT cone and analyzes the structure and harmonicity of hyperholomorphic vector fields in this context.

Findings

Openness of the balanced HKT cone within the HKT structures.

Harmonicity of (1,0)-forms dual to hyperholomorphic vector fields.

Non-existence of hyperholomorphic (1,0)-vector fields on certain manifolds with HKT–Einstein metrics.

Abstract

We prove the openness of the balanced HKT cone within the cone of HKT structures on a compact hypercomplex manifold $(M,I,J,K)$. We also study the Lie algebra of hyperholomorphic vector fields of type (1,0) with respect to $I$, with particular emphasis on the case when there exists a compatible balanced HKT metric. These fields exhibit a strict interplay with the balanced HKT structure, for instance, we prove a harmonicity property for (1,0)-forms dual to hyperholomorphic vector fields. We also show non-existence of hyperholomorphic (1,0)-vector fields on some hypercomplex manifolds admitting a HKT--Einstein metric.