Bounded pluriharmonic functions and holomorphic functions on Teichm\"uller space
Abstract
In this paper, we discuss the boundary behavior of bounded pluriharmonic functions on the Teichm\"uller space. We will show a version of the Fatou theorem that every bounded pluriharmonic function admits the radial limits along the Teichm\"uller 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 on the space of projective measured foliations.
Cite
@article{arxiv.2312.13535,
title = {Bounded pluriharmonic functions and holomorphic functions on Teichm\"uller space},
author = {Hideki Miyachi},
journal= {arXiv preprint arXiv:2312.13535},
year = {2024}
}
Comments
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)