Weil-Petersson Geometry, Renormalized Volume and Higher Teichmuller Theory

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A topological surface is a space which is allowed to change its shape by stretching or bending but without tearing or performing any discontinuous actions. One can study the properties of the surface by considering the space of all shapes it can have. This space of shapes is called the moduli space of the surface. One example would be if a circle is allowed to change its shape but remain an ellipse, then the moduli space would be described by two numbers, the length of the short axis and the length of the long axis and therefore be two dimensional. Topological surfaces (and higher dimensional objects) can be studied by considering the shape or geometry of its moduli space. One such geometry is the Weil-Petersson geometry which plays an important role in mathematics and physics. This NSF award supports a project with a focus on the Weil-Petersson geometry of a moduli space. In prior work, the PI and collaborators introduced a flow on the moduli space of a surface, called the Weil-Petersson renormalized volume gradient flow. This flow reveals much of the structure of the moduli space of three-dimensional spaces. For a large class of three-dimensional spaces this is a uniformizing flow, flowing any shape to make it as symmetric as possible: in the circle analogy, making the ellipse become a round circle. One of the major directions is to show that this flow is uniformizing for all spaces of a certain type. This work is at the intersection of mathematics and physics and is expected to lead to new connections between the two fields. The project will support a graduate student and allow the PI to disseminate the work through conferences and seminars. The project focuses on two main areas of research, 1) renormalized volume and its Weil-Petersson gradient flow and 2) the Weil-Petersson geometry of higher Teichmuller spaces. These two areas are relatively new, having developed over the last fifteen years. In 1) the PI plans to use renormalized volume to study the structure of hyperbolic three-manifolds. The renormalized volume of a hyperbolic manifold is closely related to its convex core volume but has nicer analytic properties such as being a smooth function on moduli space. In prior work, the PI and collaborators introduced the Weil-Petersson gradient flow of renormalized volume to study the geometry of the deformation space of convex cocompact hyperbolic structures on a three dimensional manifold. In particular this work showed that when the space is acylindrical then the flowlines are Weil-Petersson quasigeodesics and that the renormalized volume is minimized at the unique structure which has convex core boundary totally geodesic. Furthermore, a surgered version of the flow is a uniformizing flow, flowing every point to the unique structure which has convex core boundary totally geodesic. A major project is to show that in the boundary incompressible case, the flow limits to the conjectured decomposition along its windows and acylindrical pieces. In higher Teichmuller theory the PI and collaborators consider extending the analytic and metric structure of classical Teichmuller theory to geometric representations into higher rank Lie groups. In earlier work, the PI and collaborators introduced a natural extension of the Weil-Petersson metric to higher Teichmuller theory. More recently they have generalized the construction to define extensions based on the simple roots of the associated Lie algebra. The PI will investigate these Weil-Petersson extensions and study their geometric structure. This has already led to a number of rigidity results, related to simple spectral length and the Liouville volume for Hitchin representations.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.