Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmuller geodesic flow

TL;DR

This paper calculates the sum of positive Lyapunov exponents for the Hodge bundle under Teichmuller flow, using advanced mathematical tools like the Riemann-Roch theorem and Laplacian comparisons during surface degeneration.

Contribution

It provides a novel computation method for Lyapunov exponents in the context of Teichmuller dynamics and Hodge theory, linking geometric analysis with algebraic geometry.

Findings

Explicit formula for the sum of positive Lyapunov exponents

Connection between Laplacian determinants and surface degeneration

Advancement in understanding Teichmuller flow dynamics

Abstract

We compute the sum of the positive Lyapunov exponents of the Hodge bundle with respect to the Teichmuller geodesic flow. The computation is based on the analytic Riemann-Roch Theorem and uses a comparison of determinants of flat and hyperbolic Laplacians when the underlying Riemann surface degenerates.