Bounded Pluriharmonic Functions and Holomorphic Functions on Teichm\"uller Space II -- Poisson integral formula --

TL;DR

This paper develops a Poisson integral formula for bounded pluriharmonic functions on Teichmüller space, extending classical results and providing new insights into the value distribution and mean value properties of these functions.

Contribution

It establishes the Poisson integral formula for pluriharmonic functions on Teichmüller space and explores related value distribution and mean value theorems.

Findings

Poisson integral formula for pluriharmonic functions on Teichmüller space

A version of the F. and M. Riesz theorem for plurisubharmonic functions

Teichmüller-theoretic interpretation of the mean value theorem

Abstract

In this paper, we establish the Poisson integral formula for bounded pluriharmonic functions on the Teichm\"uller space of analytically finite Riemann surfaces of type $(g,m)$ with $2g-2+m>0$. We also discuss a version of the F. and M. Riesz theorem concerning the value distribution of plurisubharmonic functions on the Teichm\"uller space, as well as a Teichm\"uller-theoretic interpretation of the mean value theorem for pluriharmonic functions.