Weierstrass filtration on Teichm\"uller curves and Lyapunov exponents: Upper bounds

TL;DR

This paper establishes upper bounds for the slope of the Harder-Narasimhan filtration on the Hodge bundle of Teichmüller curves, leading to a new upper bound for the sum of Lyapunov exponents and unifying various known results.

Contribution

It provides the first upper bounds for the slope of the Harder-Narasimhan filtration on Teichmüller curves and characterizes cases of equality for Lyapunov exponents.

Findings

Sum of Lyapunov exponents does not exceed (g+1)/2.

Equality occurs for curves in the hyperelliptic locus or special Teichmüller curves.

Unified interpretation of partial sums of Lyapunov exponents.

Abstract

We get an upper bound of the slope of each graded quotient for the Harder-Narasimhan filtration of the Hodge bundle of a Teichm\"{u}ller curve. As an application, we show that the sum of Lyapunov exponents of a Teichm\"{u}ller curve does not exceed ${(g+1)}/{2}$, with equality reached if and only if the curve lies in the hyperelliptic locus induced from $\mathcal{Q}(2k_1,...,2k_n,-1^{2g+2})$ or it is a special Teichm\"{u}ller curve in $\Omega\mathcal{M}_g(1^{2g-2})$. It also gives an unified interpretation for many known results about the special partial sums of Lyapunov exponents on Teichm\"uller curves.