Variation of extremal length functions on Teichmuller space

TL;DR

This paper investigates how extremal length functions behave on Teichmuller space, showing they are plurisubharmonic and analyzing their second variation along Weil-Petersson geodesics, linking conformal invariants and harmonic maps.

Contribution

It establishes the plurisubharmonicity of extremal length functions and relates their second variation to harmonic map energy, providing new insights into Teichmuller geometry.

Findings

Extremal length functions are plurisubharmonic on Teichmuller space.

Second variation of extremal length relates to harmonic map energy.

Provides a new perspective on the geometry of Teichmuller space.

Abstract

Extremal length is an important conformal invariant on Riemann surface. It is closely related to the geometry of Teichmuller metric on Teichmuller space. By identifying extremal length functions with energy of harmonic maps from Riemann surfaces to $\mathbb{R}$-trees, we study the second variation of extremal length functions along Weil-Petersson geodesics. We show that the extremal length of any measured foliation is a pluri-subharmonic function on Teichmuller space.