Variation of geodesic length functions over Teichm\"uller space

TL;DR

This paper computes the second variation of geodesic length functions over Teichmüller space, providing new proofs of their plurisubharmonicity and extending results to weighted punctured Riemann surfaces.

Contribution

It introduces a second variation formula depending on harmonic Beltrami differentials and applies it to prove plurisubharmonicity in new settings.

Findings

Second variation formula depending on harmonic Beltrami differentials

New proof of plurisubharmonicity of geodesic length functions

Extension of results to weighted punctured Riemann surfaces

Abstract

In a family of compact, canonically polarized, complex manifolds equipped with K\"ahler-Einstein metrics the first variation of the lengths of closed geodesics was previously shown in by the authors in [arXiv:0808.3741v2] to be the geodesic integral of the harmonic Kodaira-Spencer form. We compute the second variation. For one dimensional fibers we arrive at a formula that only depends upon the harmonic Beltrami differentials. As an application a new proof for the plurisubharmonicity of the geodesic length function and its logarithm (with new upper and lower estimates) follows, which also applies to the previously not known cases of Teichm\"uller spaces of weighted punctured Riemann surfaces, where the methods of Kleinian groups are not available.