Quaternionic plurisubharmonic functions and their applications to convexity

Abstract

Abstract. The paper is a survey of the recent theory of plurisubharmonic functions of quaternionic variables, together with its applications to the theory of valuations on convex sets and HKT-geometry (Hyper-Kähler with Torsion

New Structures on Valuations and Applications

“…There is yet another construction of valuations based on complex and quaternionic pluripotential theory. It is somewhat more specialized and will not be discussed here; we refer to [7], [13], and the survey [8].…”

Plurisubharmonic Functions on the Octonionic Plane and Spin(9)-Invariant Valuations on Convex Sets

“…Section 3.4 of [18]. Note also that a nice determinant does exist for quaternionic hermitian matrices of any size: see the survey [17], the article [29], and for applications to quaternionic plurisubharmonic functions see [6], [7], [11], [13].…”

“…Then part of this theory has been generalized to more general context of (not necessarily flat) hypercomplex manifolds by M. Verbitsky and the author [16]. We refer also to [13] for a survey of these results. Very recently, other classes of plurisubharmonic functions have been introduced in the context of calibrated geometries [32].…”