ADS 3-manifolds and Higgs bundles

TL;DR

This paper explores the connection between closed AdS 3-manifolds and Higgs bundles, introducing a new construction method that elucidates properties like volume formulas and applications to minimal immersions into quadrics.

Contribution

It presents a novel approach to constructing AdS structures, enabling explicit analysis and applications to minimal immersions and volume computations.

Findings

New construction method for AdS structures

Recovery of Tholozan's volume formula

Characterization of minimal immersions into quadrics

Abstract

In this paper we investigate the relationships between closed AdS 3-manifolds and Higgs bundles. We have a new way to construct AdS structures that allows us to see many of their properties explicitly, for example we can recover the very recent formula by Tholozan for the volumes. We also find applications to the theory of minimal immersions into quadrics with their natural pseudo-Riemannian structure: using the geometry of the AdS manifolds we can characterize the representations admitting equivariant minimal immersions of the Poincare disc into the Klein quadric, the Grassmannian Gr(2,4), and understand the geometry of these minimal immersions.