Differential Geometry, PDE and Mathematical Physics
Department of Mathematics and Statistics
Texas Tech University
Renormalized area of minimal submanifolds of Poincaré-Einstein manifolds has been the focus of much study in conformal geometry over the past 20 years. As well as in the AdS/CFT correspondence in physics. In 2008, Alexakis and Mazzeo showed that the renormalized area of 2D minimal surfaces in hyperbolic space has many interesting functional properties. Not least of which being it’s connection to the asymptotic Plateau problem and the Willmore functional. In 2001, Anderson proved a rigidity result in terms of the renormalized volume for 4D Poincaré-Einstein spaces. More recently, Bernstein proved a rigidity result for 2D minimal surfaces in hyperbolic space. We will report on a partial rigidity result for minimal hypersurfaces of 5D hyperbolic space in terms of the renormalized area.
We review a form of the Gauss-Bonnet theorem on a four-manifold with corners for which the curvatures have nice conformal transformation properties. A natural question of Escobar-Yamabe type is: can one make a conformal change so that all of the Gauss-Bonnet integrands vanish except on the corner, and are there constant? We answer this in the affirmative for the half-ball in four-space. This is joint work with several authors.
Abstract pdf
The theory of p-elastic curves is a classical topic that dates back to the beginnings of the Calculus of Variations. Since then, these variational problems have been ubiquitous in Differential Geometry, Geometric Analysis and Mathematical Physics. In this talk, after reviewing the latest results on the theory, we will suggest open problems and discuss potential approaches to solve them.