Tangents to Subsolutions Existence and Uniqueness, II

Abstract

This second part of the paper (see Ann Math de Toulouse, arXiv:1408.5797 for part I) is concerned with questions of existence and uniqueness of tangents in the special case of $$\varvec{\mathbb {G}}$$ -plurisubharmonic functions, where $$\varvec{\mathbb {G}}\subset G(p,\mathbf{R}^n)$$ is a compact subset of the Grassmannian of p-planes in $$\mathbf{R}^n$$ . An u.s.c. function u on an open set $$\Omega \subset \mathbf{R}^n$$ is $$\varvec{\mathbb {G}}$$ -plurisubharmonic if its restriction to $$\Omega \cap W$$ is subharmonic for every affine $$\varvec{\mathbb {G}}$$ -plane W. Here $$\varvec{\mathbb {G}}$$ is assumed to be invariant under a subgroup $$K\subset \mathrm{O}(n)$$ which acts transitively on $$S^{n-1}$$ . 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 $$\varvec{\mathbb {G}}$$ -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 $$\{\Theta (u,x)\ge c\}$$ for $$c>0$$ are discrete. When the invariance group $$K= \mathrm{O}(n), \mathrm{U}(n)$$ or Sp(n) strong uniqueness holds for all but a small handful of cases. It also holds for essentially all interesting $$\varvec{\mathbb {G}}$$ 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–Ampère equations (convex functions, and complex and quaternionic plurisubharmonic functions), tangents, which are far from unique, are then systematically studied and classified.