Hitchin Components, Riemannian Metrics and Asymptotics
Doctor of Philosophy
Higher Teichm\"uller spaces are deformation spaces arising from subsets of the space of representations of a surface group into a general Lie group, e.g., $$PSL(n,\RR)$$, which share some of the properties of classical Teichmueller space. By the non-abelian Hodge theory, such representation spaces correspond to the space of Higgs bundles. We focus on two aspects on the Higher Teichm\"uller space: Riemannian geometry and dynamics. First, we construct a new Riemannian metric on the deformation space for $$PSL(3,\RR)$$, and then prove Teichmueller space endowed with Weil-Petersson metric is totally geodesic in deformation space for $$PSL(3,\RR)$$ with the new metric. Secondly, in a joint work with Brian Collier, we are able to obtain asymptotic behaviors and related properties of representations for certain families of Higgs bundles of rank n.
Hitchin Components; Higgs Bundles