Smooth Approximation of Plurisubharmonic Functions on Almost Complex Manifolds

This note establishes smooth approximation from above for J-plurisubharmonic functions on an almost complex manifold (X,J). The following theorem is proved. Suppose X is J-pseudoconvex, i.e., X admits a smooth strictly J-plurisubharmonic exhaustion function. Let u be an (upper semi-continuous) J-plurisubharmonic function on X. Then there exists a sequence {u_j} of smooth, strictly J-plurisubharmonic functions point-wise decreasing down to u. On any almost complex manifold (X,J) each point has a fundamental neighborhood system of J-pseudoconvex domains, and so the theorem above establishes local smooth approximation on X. This result was proved in complex dimension 2 by the third author, who also showed that the result would hold in general dimensions if a parallel result for continuous approximation were known. This paper establishes the required step by solving the obstacle problem.

Tangents to subsolutions -- existence and uniqueness, Part II

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 $Omegacap W$ 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.

Tangents to subsolutions -- existence and uniqueness, Part I

There is an interesting potential theory associated to each degenerate elliptic, fully nonlinear equation $f(D^2u) = 0$. These include all the potential theories attached to calibrated geometries. This paper begins the study of tangents to the subsolutions in these theories, a topic inspired by the results of Kiselman in the classical plurisubharmonic case. Fundamental to this study is a new invariant of the equation, called the Riesz characteristic, which governs asymptotic structures. The existence of tangents to subsolutions is established in general, as is the existence of an upper semi-continuous density function. Two theorems establishing the strong uniqueness of tangents (which means every tangent is a Riesz kernel) are proved. They cover all O(n)-invariant convex cone equations and their complex and quaternionic analogues, with the exception of the homogeneous Monge-Amp`ere equations, where uniqueness fails. They also cover a large class of geometrically defined subequations which includes those coming from calibrations. A discreteness result for the sets where the density is $geq c > 0$ is also established in any case where strong uniqueness holds. A further result (which is sharp) asserts the Holder continuity of subsolutions when the Riesz characteristic p satisfies $1 leq p < 2$. Many explicit examples are examined. The second part of this paper is devoted to the geometric cases. A Homogeneity Theorem and a Second Strong Uniqueness Theorem are proved, and the tangents in the Monge-Amp`ere case are completely classified.