Geometry and applications of deformations of Riemann surfaces

University Of Maryland, College Park, College Park MD

Investigators

Abstract

The investigator will continue his study of geodesics (shortest paths), convexity and curvature (the fundamental descriptor for a geometry) for the Weil-Petersson geometry for Teichmueller space. The investigator has already shown that Teichmueller space is an infinite polyhedron with an infinite number of vertices and geodesic faces meeting at right angles. The description gives a short proof of the Masur-Wolf result determining the full symmetry group of the polyhedron. The investigator has provided detailed information on the geometry near the faces. Lengths of closed geodesics (shortest closed paths) on a hyperbolic surface are parameters for locating a point in Teichmueller space. The investigator will continue his program of using analysis of variations of the lengths to describe the synthetic and differential geometry of Teichmueller space. The description is simpler than the investigator's prior approach, which is presently used by a number of researchers. The investigator is developing a description of curvature in terms of variations of lengths. The investigator will also study harmonic (least total square energy) mappings to and from the infinite polyhedron. He will also continue to study the Weil-Petersson geodesic dynamical system, as well as intersection theory for the moduli space of Riemann surfaces. A Riemann surface is a two-dimensional surface with a notion of angle measure at each point. A Riemann surface with at least two handles is endowed with non Euclidean (hyperbolic) geometry determining distance, shortest paths and a vibrating membrane operator. Riemann surfaces come in various shapes, described by the relative locations of thick and thin subregions. Teichmueller space is the space of all possible shapes for a Riemann surface with a given number of handles. The hyperbolic geometry of individual Riemann surfaces leads to the Weil-Petersson geometry for Teichmueller space. Riemann surfaces provide models for vibrating membranes, propagating particles and fractals.