Bounded pluriharmonic functions and holomorphic functions on Teichmüller space

Abstract

In this paper, we discuss the boundary behavior of bounded pluriharmonic functions on the Teichmüller space. We will show a version of the Fatou theorem that every bounded pluriharmonic function admits the radial limits along the Teichmüller geodesic rays, and a version of the F. and M. Riesz theorem that the radial limit of a non-constant bounded holomorphic function is not constant on any non-null measurable set on the Bers boundary in terms of the pluriharmonic measure. As a corollary, we obtain the non-ergodicity of the action of the Torelli group for a closed surface of genus $g\ge 2$ on the space of projective measured foliations.

arXiv admin note: text overlap with arXiv:1810.04343 Typos are corrected. I revised mainly the sections "Introduction" and "Conclusion". I added a brief proof of Proposition 2.1 from a comment and revised the proof of Corollary 1.2 for readability. The paper is accepted to International Mathematics Research Notices (IMRN)