Title:Maximal Surfaces, Holomorphic Differentials, and Lightlike Polygons in Pseudo-Hyperbolic Geometry

Speaker:Andrea Tamburelli(Associate Professor)

Time:  Aug 11-13,  2026, 09:00-11:00

Venue: Room 3A02, Building No. 37, Wushan Campus

Abstract: 

This minicourse explores the interplay between maximal surfaces in pseudo-hyperbolic spaces, holomorphic differentials, and the geometry of their boundaries at infinity.

  The first lecture will introduce pseudo-hyperbolic spaces (\mathbb H^{2,n}), their conformal boundary—the Einstein universe—and the asymptotic Plateau problem for maximal surfaces. We will discuss the intrinsic and extrinsic geometry of these surfaces, focusing on three naturally associated metric structures: the induced metric, the ambient spacelike distance, and the singular flat metric determined by a holomorphic quartic differential. Barbot surfaces, whose boundaries are lightlike quadrilaterals, will appear as the fundamental flat models. We will also explain how the Tits metric relates the asymptotic geometry of a maximal surface to its total curvature.

  The second lecture will concentrate on maximal surfaces in anti-de Sitter three-space. We will describe how polynomial quadratic differentials on the complex plane give rise to complete maximal surfaces whose boundaries at infinity are lightlike polygons in the two-dimensional Einstein universe. This construction yields a homeomorphism between the moduli space of polynomial quadratic differentials of fixed degree and an appropriate moduli space of lightlike polygons. As an application, we will discuss the relation with minimal Lagrangian maps between ideal hyperbolic polygons.

  In the final lecture, we will return to maximal surfaces in (\mathbb H^{2,n}) and characterize those with polygonal boundary. We will explain the equivalence between three conditions: having a lightlike polygon as boundary at infinity, having finite total curvature, and being asymptotically flat.