Higgs Bundles and Geometric Structures on Surfaces

TL;DR

This paper explores the connection between geometric structures on surfaces and Higgs bundle moduli, highlighting uniformization and recent advances in the field.

Contribution

It provides a survey of the relationship between geometric structures and Higgs bundles, emphasizing uniformization and recent developments.

Findings

Link between geometric structures and Higgs bundle moduli clarified

Survey of uniformization via hyperbolic geometry

Recent theoretical advancements discussed

Abstract

This paper concerns the relationship between locally homogeneous geometric structures on topological surfaces and the moduli of polystable Higgs bundles on Riemann surfaces, due to Hitchin and Simpson. In particular we discuss the uniformization of Riemann surfaces by hyperbolic geometry from this viewpoint, and survey more recent developments in this theory.