Tangents to subsolutions -- existence and uniqueness, Part II

TL;DR

This paper investigates the existence, uniqueness, and classification of tangent functions to G-plurisubharmonic functions, establishing conditions for strong uniqueness and analyzing cases where tangents are non-unique, especially in geometric contexts.

Contribution

It provides new results on the existence and strong uniqueness of tangents for G-plurisubharmonic functions, and classifies tangents in cases of non-uniqueness related to calibrated geometries.

Findings

Tangents always exist and are G-harmonic at points of continuity.

Strong uniqueness of tangents holds in many classical symmetry cases.

Classifications of non-unique tangents are achieved for Monge-Ampère related functions.

Abstract

This part II of the paper is concerned with questions of existence and uniqueness of tangents in the special case of G-plurisubharmonic functions, where G is a compact subset of the Grassmannian of p-planes in ${\mathbb R}^n$. An upper semi-continuous function u on an open set $\Omega$ in ${\mathbb R}^n$ is G-plurisubharmonic if its restriction to $\Omega\cap W$ is subharmonic for every affine G-plane $W$. Here G is assumed to be invariant under a subgroup K of O(n) which acts transitively on the sphere $S^{n-1}$. Tangents to u at a point x are the cluster points of u under a natural flow (or blow-up) at x. They always exist and are G-harmonic at all points of continuity. A homogeneity property is established for all tangents in these geometric cases. This leads to principal results concerning the Strong Uniqueness of Tangents, which means that all tangents are unique and of the form $\Theta K_p$ where $K_p$ is the Riesz kernel and $\Theta$ is the density of u at the point. Strong uniqueness is a form of regularity which implies that the sets $\{\Theta(u,x)\geq c\}$ for $c>0$ are discrete. When the invariance group K= O(n), U(n) or Sp(n), strong uniqueness holds for all but a small handful of cases. It also holds for essentially all interesting G which arise in calibrated geometry. When strong uniqueness fails, homogeneity implies that tangents are characterized by a subequation on the sphere, which is worked out in detail. In the cases corresponding to the real, complex and quaternionic Monge-Amp\`ere equations (convex functions, and complex and quaternionic plurisubharmonic functions) tangents, which are far from unique, are then systematically studied and classified.