Local HKT potential theorem

Statement

Let $(M,I,J,K)$ be a hypercomplex manifold. If $u$ is a smooth strictly quaternionic plurisubharmonic function, then the hyperhermitian form

g_u=t(\partial\partial_Ju)

is an HKT metric. Conversely, every HKT metric is locally of this form around each point.

Meaning of the formula

The map $t$ identifies real positive $(2,0)$ -forms with hyperhermitian forms. Strict positivity of $\partial\partial_Ju$ makes $g_u$ a Riemannian metric, while $\partial^2=0$ gives the HKT closure condition.

Flat normalization

On $\mathbb H^n$ , the metric matrix is a fixed scalar multiple— $1/4$ in the Alesker–Verbitsky convention—of the quaternionic Hessian of $u$ . Other normalizations of the Cauchy–Fueter operators move this scalar but do not change positivity or the local-potential statement.

Kähler analogy

The theorem parallels the fact that a smooth strictly complex plurisubharmonic function is a local Kähler potential. The HKT formula uses $\partial\partial_J$ rather than $i\partial\bar\partial$ , and the resulting metric may have nonzero torsion.

References

Semyon Alesker and Misha Verbitsky, “Plurisubharmonic functions on hypercomplex manifolds and HKT-geometry,” Journal of Geometric Analysis 16 (2006), 375–399. arXiv record. Relevant: Proposition 1.14.
Semyon Alesker, “Quaternionic plurisubharmonic functions and their applications to convexity,” 2016 revision. arXiv record. Relevant: Theorem 6.13.