Surface group seminar

This is a continuation of last year's informal seminar. This year, we plan to venture into "slightly-elevated" Teichmüller theory and discuss convex projective structures on surfaces.

A convex real projective structure on a surface S (of genus at least two) is a geometric structure modelled on the real projective plane (RP^2) with transition maps in PSL(3,R) , which develops to a convex domain in the projective plane. From the work of Goldman and Choi, we know that the corresponding monodromy representations are the so called "Hitchin representations" of the fundamental group of S to PSL(3,R). A special case of this are Fuchsian representations coming from hyperbolic structures, and the other convex RP^2-structures can be thought of as deformations of these. There is a connection with analysis since Labourie and independently Loftin showed that these structures can be parametrised by holomorphic cubic differentials on S. Their approach for this is to use Cheng-Yau's solution of a conjecture of Calabi on the existence of hyperbolic affine spheres.

These weekly lectures will aim to give an exposition of this theory, emphasizing the relation between geometry and analysis. We shall start with recalling some basic Teichmüller theory and surface-group representations into PSL(2,R). Some relevant papers are listed below.

Pre-requisites: Topology (including fundamental groups and homology) , complex analysis, and a working knowledge of the geometry of the hyperbolic plane.

Relevant papers:

• Convex real projective structures on compact surfaces (W. Goldman, JDG 1990): https://mathscinet.ams.org/mathscinet-getitem?mr=1053346

• Convex real projective structures on closed surfaces are closed (W. Goldman and S.Choi, PAMS 1993): https://mathscinet.ams.org/mathscinet-getitem?mr=1145415

• Collar Lemma for Hitchin Representations (G-S. Lee and T. Zhang, G&T 2017): https://mathscinet.ams.org/mathscinet-getitem?mr=3654108

Some references for the next semester:

• Affine spheres and convex RPⁿ-manifolds (J. Loftin, AMJ 2001): https://mathscinet.ams.org/mathscinet-getitem?mr=1828223

• Flat projective structures on surfaces and cubic holomorphic differentials (F. Labourie, PAMQ 2007): https://mathscinet.ams.org/mathscinet-getitem?mr=2402597

• Espace des modules marqués des surfaces projectives convexes de volume fini (L. Marquis, G&T 2010): https://mathscinet.ams.org/mathscinet-getitem?mr=2740643

• Cubic differentials and finite volume convex projective surfaces (Y. Benoist and D. Hulin, G&T 2013): https://mathscinet.ams.org/mathscinet-getitem?mr=3039771

• Polynomial cubic differentials and convex polygons in the projective plane (D.Dumas, M. Wolf, GAFA 2015): https://mathscinet.ams.org/mathscinet-getitem?mr=3432157

• Poles of cubic differentials and ends of convex RP²-surfaces (X. Nie, preprint): https://arxiv.org/abs/1806.06319