Differential tests for plurisubharmonic functions and Koch curves

TL;DR

This paper investigates the properties of singular plurisubharmonic functions, develops an algorithm for identifying certain curves as minimum sets, and proves measure-zero results for upper non-contact points, advancing understanding in complex analysis.

Contribution

It introduces an algorithm to determine when a parametrized curve is a minimum set for plurisubharmonic functions and proves measure-zero properties of upper non-contact sets.

Findings

The set of upper non-contact points has Lebesgue measure zero.

Developed an algorithm for identifying minimum sets related to curves.

Proved existence results for upper tests with controlled opening similar to convex functions.

Abstract

We study minimum sets of singular plurisubharmonic functions and their relation to upper contact sets. In particular we develop an algorithm checking when a naturally parametrized curve is such a minimum set. The case of Koch curves is studied in detail. We also study the size of the set of upper non-contact points. We show that this set is always of Lebesgue measure zero thus answering an open problem in the viscosity approach to the complex Monge-Amp\`ere equation. Finally, we prove that similarly to the case of convex functions, strictly plurisubharmonic lower tests yield existence of upper tests with a control on the opening.