ADS 3-manifolds and Higgs bundles

In this paper we investigate the relationships between closed \mathrmAdS 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 \mathrmAdS manifolds we can characterize the representations admitting equivariant minimal immersions of the Poincaré disc into the Klein quadric, the Grassmannian \mathrmGr(2,4), and understand the geometry of these minimal immersions.