The minimum sets and free boundaries of strictly plurisubharmonic functions

Abstract

We study the minimum sets of plurisubharmonic functions with strictly positive Monge-Ampère densities. We investigate the relationship between their Hausdorff dimension and the regularity of the function. Under suitable assumptions we prove that the minimum set cannot contain analytic subvarieties of large dimension. In the planar case we analyze the influence on the regularity of the right hand side and consider the corresponding free boundary problem with irregular data. We provide sharp examples for the Hausdorff dimension of the minimum set and the related free boundary. We also draw several analogues with the corresponding real results.

Differential Tests for Plurisubharmonic Functions and Koch Curves

“…In the current note we continue our study initiated in [11]. Our aim is twofold: first we provide more explicit examples of fractal minimum sets and develop an algorithm for producing these.…”

“…In [11] we showed an example of a fractal Julia set which is of Hausdorff dimension strictly larger than one but is nevertheless a minimum set of a strictly subharmonic function. This raises a question how generic such examples are.…”

“…Minimum sets of (strictly) plurisubharmonic functions appear naturally in several branches of complex analysis, such as El Mir's theory of extension of closed positive currents or in the ∂-Neumann theory (see [11] and the references therein). We would like to point out that they appear also in polynomial convexity (see [31]), where they are called totally real sets, and as the common boundary of two disjoint strongly pseudoconvex domains (see [29]).…”

“…Their result states that such minimum sets are contained in C 1 totally real subvarieties. In [11] we investigated these sets under the weaker condition that the possibly singular plurisubharmonic function has strictly positive Monge-Ampère measure. In particular we showed that in this setting the Hausdorff dimension can be larger than in the case considered by Harvey and Wells.…”

“…pass from quasiconformality to conformality). The same can be done for the corresponding exteriors; • Express the Green function by means of the conformal mapping, with control of the decay at the boundary; • Having this, use the criterion from [11] to decide whether the set is a minimum set of a strictly subharmonic function and construct one.…”