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.

The present work was initiated when the first author was visiting Mehdi Lejmi at the CUNY Graduate Center of New York, he would like to thank him deeply for the delightful hospitality and financial support. He is also grateful to INdAM as a recipient of a scholarship aimed at carrying out a research project abroad. The first author is indebted to Beatrice Brienza and Luigi Vezzoni for useful discussions and for showing interest in the paper. Finally, he acknowledges Elia Fusi for a careful reading of an early draft of the paper. The authors thank Ethan Addison for his suggestions that improved the exposition. The second author acknowledges support provided by the Simons Foundation Grant #636075.