On the residual Monge-Amp\`{e}re mass of plurisubharmonic functions, III: uniformly directional Lipschitz

TL;DR

This paper investigates the residual Monge-Ampère mass of plurisubharmonic functions with isolated unbounded loci, providing estimates and confirming the zero mass conjecture under uniform directional Lipschitz conditions.

Contribution

It introduces a decomposition formula under Sasakian structure, derives an $L^{1}$-estimate in complex dimension two, and confirms the zero mass conjecture for functions with specific directional separation.

Findings

Decomposition formula for residual Monge-Ampère mass under Sasakian structure

An $L^{1}$-apriori estimate in complex dimension two

Confirmation of the zero mass conjecture under uniform directional Lipschitz continuity

Abstract

The purpose of this article is to study the (residual) Monge-Amp\`{e}re mass of a plurisubharmonic function with an isolated unbounded locus. A general decomposition formula is obtained under the Sasakian structure of the unit sphere. In complex dimension two, we obtain an $L^{1}$-apriori estimate on the complex Monge-Amp\`{e}re operator. This induces an upper-bound estimate on the residual mass, provided with the uniform directional Lipschitz continuity. As an application, the zero mass conjecture is confirmed, if the function further separates the circular direction in its alternating part.