The covariance metric in the Blaschke locus

TL;DR

This paper investigates the geometric structure of the Blaschke locus, demonstrating it as a smooth manifold away from Teichmüller space and analyzing its Riemannian properties using the covariance metric, with applications to Hitchin representations.

Contribution

It establishes the smooth manifold structure of the Blaschke locus outside Teichmüller space and explores its Riemannian geometry with respect to the covariance metric, including geodesic analysis.

Findings

The Blaschke locus is a finite-dimensional smooth manifold away from Teichmüller space.

Certain geodesics from Hitchin representations have infinite length in the covariance metric.

The covariance metric provides a new perspective on the geometric structure of the Blaschke locus.

Abstract

We prove that the Blaschke locus has the structure of a finite dimensional smooth manifold away from the Teichm{\"u}ller space and study its Riemannian manifold structure with respect to the covariance metric introduced by Guillarmou, Knieper and Lefeuvre in \cite{GeodesicStretch}. We also identify some families of geodesics in the Blaschke locus arising from Hitchin representations for orbifolds and show that they have infinite length with respect to the covariance metric.