A note on characterizing pluriharmonic functions via the Ohsawa--Takegoshi extension theorem

TL;DR

This paper characterizes pluriharmonic functions through the equality condition in the Ohsawa--Takegoshi extension theorem, providing a partial resolution to a conjecture in complex analysis.

Contribution

It establishes a new characterization of pluriharmonic functions using the optimal Ohsawa--Takegoshi extension theorem's equality condition.

Findings

Pluriharmonic functions are characterized by the equality in the extension theorem.

The result partially resolves a conjecture in the field.

Provides a new perspective on the relation between function properties and extension theorems.

Abstract

For a continuous function, we prove that the function is pluriharmonic if and only if the equality part of the optimal Ohsawa--Takegoshi $L^2$-extension theorem is satisfied with respect to the metric having the function as a weight. This partially resolves the conjecture proposed by the author.