Variational Problems in Geometry

University Of Pennsylvania, Philadelphia PA

Investigators

Abstract

The proposed research focuses on minimal surfaces, which are surfaces that locally minimize area. Classical examples of minimal surfaces can be obtained physically by immersing a wire frame into a soap solution and forming a soap film surface whose boundary is the wire frame. The theme of minimization permeates all of the natural sciences and minimal surfaces are thus an important model for several phenomena in nature. They have been linked to molecular engineering, materials science, and to the theory of black holes in general relativity. The proposed research aims to understand how such surfaces can be locally minimizing but not globally so and, moreover, how this can reveal the shape of the space they live in. Minimal submanifolds are critical points to the most fundamental variational problem in the geometry, that of minimizing area. They have been an essential object in mathematical research since the work of Euler and Lagrange, and many of the ideas developed in their study turned out to be key in the development of calculus of variations, nonlinear PDEs, and mathematical physics. In addition, minimal submanifolds have also been a crucial method in understanding curvature, yielding many striking applications to geometry, low-dimensional topology, and general relativity. Recent advances on the existence theory of minimal hypersurfaces suggest that this is a very exciting time for the field. A major component of the proposed research is to investigate new Morse index estimates in light of the new existence results. One goal is to study what are the geometrical and topological properties of the new minimal hypersurfaces produced by the theory, which in turn can lead to several applications. 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.