Pluripotential theory on Teichm\"uller space II -- Poisson integral formula

TL;DR

This paper develops a Poisson integral formula for pluriharmonic functions on Teichmüller space, linking boundary behavior and measures, advancing pluripotential theory in complex geometric analysis.

Contribution

It establishes the Poisson integral formula for pluriharmonic functions on Teichmüller space and explores its boundary behavior and measure relationships.

Findings

Poisson integral formula for pluriharmonic functions on Teichmüller space

Boundary Schwarz type theorem for the Poisson integral

Connection between pluriharmonic measures and Patterson-Sullivan measures

Abstract

This is the second paper in a series of investigations of the pluripotential theory on Teichm\"uller space. The main purpose of this paper is to establish the Poisson integral formula for pluriharmonic functions on Teichm\"uller space which are continuous on the Bers compactification. We also observe that the Schwarz type theorem on the boundary behavior of the Poisson integral. We will see a relationship between the pluriharmonic measures and the Patterson-Sullivan measures discussed by Athreya, Bufetov, Eskin and Mirzakhani.