Strictly quaternionic plurisubharmonic function

Definition

A real-valued $C^2$ function $u$ on $\Omega\subseteq\mathbb H^n$ is strictly quaternionic plurisubharmonic if its quaternionic Hessian is positive definite at every point. Equivalently, the restriction of $u$ to every affine right quaternionic line is strictly subharmonic.

Quantitative local form

On each relatively compact coordinate neighborhood, strictness is equivalent to the existence of $\varepsilon>0$ such that

u(q)-\varepsilon|q|^2

is quaternionic plurisubharmonic. This formulation makes clear that strictness is an open positivity condition.

Geometric roles

Strictly quaternionic PSH defining functions characterize strictly quaternionically pseudoconvex domains. On a hypercomplex manifold, smooth strictly quaternionic PSH functions are precisely the local potentials of HKT metrics.

References

Semyon Alesker, “Quaternionic Monge–Ampère equations,” Journal of Geometric Analysis 13 (2003), 205–238. arXiv record.
Semyon Alesker and Misha Verbitsky, “Plurisubharmonic functions on hypercomplex manifolds and HKT-geometry,” Journal of Geometric Analysis 16 (2006), 375–399. arXiv record.