Courses, Lectures & Working groups

HITCHIN-NGO LABORATORY: HIGGS BUNDLES AND HIGHER TEICHMÜLLER THEORY DAY -- Ray structures in Lower Teichmuller Spaces // Harmonic metrics of Higgs bundles over non-compact surfaces

Speaker:  Michael Wolf (Georgia Tech) // Qiongling Li (Chern Institute of Mathematics, Nankai University) ()
Date:  Thursday, 18 May 2023 - 10:00 - 12:30
Place:  Aula Naranja, ICMAT


10:00 - 11:00. "Ray structures in Lower Teichmuller Spaces", Michael Wolf (Georgia Tech)

Abstract:  We study 'rays' in the Hitchin/Teichmüller component of representations of surface groups in PSL(2,R) and SL(3,R).  These families reflect rays of Hitchin holomorphic differentials associated to the harmonic maps which are equivariant with respect to the representations.  In Teichmuller space, these rays interpolate between Teichmuller geodesics and Thurston geodesics in a way we will explain. By restricting to Thurston geodesics which have an energy minimization feature, we find something like an exponential map in the Thurston metric for Teichmuller space with the Thurston boundary as visibility boundary. In the SL(3,R) Hitchin component, holonomies along the rays have asymptotics predicted by periods (i.e integrals using the local geometry) of the defining holomorphic differentials in a formula with a 'tropical' style, and an endpoint given by an associated real building.

11:30 - 12:30. "Harmonic metrics of Higgs bundles over non-compact surfaces", Qiongling Li (Chern Institute of Mathematics, Nankai University)

Abstract:For a Higgs bundle of degree 0 over a compact hyperbolic Riemann surface, the Hitchin-Kobayashi correspondence says the existence of a harmonic metric is equivalent to the polystability of the Higgs bundle. In this talk, we discuss some recent progress on the existence and uniqueness of harmonic metrics on Higgs bundles over general non-compact Riemann surfaces. This is joint work with Takuro Mochizuki.