Extremal length functions are log-plurisubharmonic

TL;DR

This paper proves that extremal length functions on Teichmüller space are log-plurisubharmonic, providing new proofs for existing results and explicit formulas for their Levi forms, enhancing understanding of Teichmüller geometry.

Contribution

It establishes the log-plurisubharmonicity of extremal length functions and offers explicit Levi form formulas, with alternative proofs for related properties in Teichmüller theory.

Findings

Extremal length functions are log-plurisubharmonic on Teichmüller space.

Provides explicit Levi form formulas for extremal length functions.

Confirms hyperconvexity of Teichmüller space.

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.