Strictly quaternionically pseudoconvex domain

Definition

A bounded domain $\Omega\subseteq\mathbb H^n$ with smooth boundary is strictly quaternionically pseudoconvex if, near every $p\in\partial\Omega$ , there is a smooth strictly quaternionic plurisubharmonic function $\rho$ such that

\Omega=\{\rho<0\},\qquad \rho(p)=0, \qquad d\rho(p)\ne0

within that neighborhood.

Role of the defining function

The nonvanishing differential makes $\rho=0$ a smooth local boundary, while positive definiteness of its quaternionic Hessian supplies the strict pseudoconvexity. The condition is independent of the particular admissible defining function.

Examples and comparison

The Euclidean ball in $\mathbb H^n$ is strictly quaternionically pseudoconvex, with defining function $\rho(q)=|q|^2-1$ . The definition is parallel to strict pseudoconvexity in several complex variables, but its positivity is tested on quaternionic rather than complex lines.

References

Semyon Alesker, “Quaternionic Monge–Ampère equations,” Journal of Geometric Analysis 13 (2003), 205–238. arXiv record. Relevant: Definition 0.1.1.
Semyon Alesker, “Quaternionic plurisubharmonic functions and their applications to convexity,” 2016 revision. arXiv record. Relevant: §5.