Length spectrum compactification of the $\mathrm{SO}_{0}(2,3)$-Hitchin component

TL;DR

This paper develops a compactification of the $ ext{SO}_0(2,3)$-Hitchin component by analyzing metric degenerations on maximal surfaces, linking it to geodesic currents and entropy behavior of Hitchin representations.

Contribution

It introduces a new compactification of the Hitchin component via length spectrum degeneration and relates it to geodesic currents and entropy analysis.

Findings

Established closure of flat metrics induced by holomorphic quartic differentials in geodesic current space.

Described the entropy behavior of Hitchin representations along quartic differential rays.

Connected metric degenerations with the boundary structure of the Hitchin component.

Abstract

We find a compactification of the $\mathrm{SO}_{0}(2,3)$-Hitchin component by studying the degeneration of the induced metric on the unique equivariant maximal surface in the 4-dimensional pseudo-hyperbolic space $\mathbb{H}^{2,2}$. In the process, we establish the closure in the space of projectivized geodesic currents of the space of flat metrics induced by holomorphic quartic differentials on a Riemann surface. As an application, we describe the behavior of the entropy of Hitchin representations along rays of quartic differentials.