Higgs bundles and geometric structures on manifolds

TL;DR

This paper surveys how Higgs bundles can be used to construct and understand geometric structures on manifolds, highlighting recent advances and their connections to flat bundles and higher Teichmüller theory.

Contribution

It explains the main ideas and recent developments in using Higgs bundles to study geometric structures on manifolds, building on Baraglia's work.

Findings

Higgs bundles provide explicit descriptions of flat bundles.

Recent results connect Higgs bundles with geometric structures.

Survey of joint work with Qiongling Li on this topic.

Abstract

Geometric structures on manifolds became popular when Thurston used them in his work on the geometrization conjecture. They were studied by many people and they play an important role in higher Teichm\"uller theory. Geometric structures on a manifold are closely related with representations of the fundamental group and with flat bundles. Higgs bundles can be very useful in describing flat bundles explicitly, via solutions of Hitchin's equations. Baraglia has shown in his Ph.D. Thesis that Higgs bundles can also be used to construct geometric structures in some interesting cases. In this paper, we will explain the main ideas behind this theory and we will survey some recent results in this direction, which are joint work with Qiongling Li.