Extremal length functions are log-plurisubharmonic
Abstract
In this paper, we show that the extremal length functions on Teichm\"uller space are log-plurisubharmonic. As a corollary, we obtain an alternative proof of L.Liu and W.Su's results on the plurisubharmonicity of extremal length functions. We also obtain alternative proofs of S.Krushkal's results that a function defined by the Teichm\"uller distance from a reference point is plurisubharmonic, and the Teichm\"uller space is hyperconvex. To show the log-plurisubharmonicity, we give an explicit formula of the Levi form of the extremal length functions in generic case.
Cite
@article{arxiv.1505.06785,
title = {Extremal length functions are log-plurisubharmonic},
author = {Hideki Miyachi},
journal= {arXiv preprint arXiv:1505.06785},
year = {2015}
}
Comments
In this revision, we correct typos and some errors. OLD COMMENT: In the second version, I correct typos and add a disk-convexity result and an alternative approach to Gardiner's formulaIn In this version, I add explanations