Limits of Blaschke metrics

TL;DR

This paper investigates the degeneration limits of Blaschke metrics within the SL(3,R)-Hitchin component, revealing a compactification related to projectivized geodesic currents and flat metrics from holomorphic cubic differentials.

Contribution

It introduces a new compactification of the Hitchin component by analyzing Blaschke metric degenerations and their relation to geodesic currents and flat metrics.

Findings

Established the closure of flat metrics induced by cubic differentials in the space of geodesic currents.

Connected the degeneration of affine spheres to the boundary of the Hitchin component.

Provided a geometric framework for understanding limits of Blaschke metrics.

Abstract

We find a compactification of the $\mathrm{SL}(3,\mathbb{R})$-Hitchin component by studying the degeneration of the Blaschke metrics on the associated equivariant affine spheres. In the process, we establish the closure in the space of projectivized geodesic currents of the space of flat metrics induced by holomorphic cubic differentials on a Riemann surface.