Geometry, Analysis, and Variational Methods

Princeton University, Princeton NJ

Investigators

Abstract

This project will investigate questions related to the variational theory of minimal surfaces and its applications. Minimal surfaces, of which soap bubbles are an illustrative example, are among the most natural objects in differential geometry. They have applications in many areas, such as three-dimensional topology, mathematical physics, complex and conformal geometry, and materials science. In general relativity, minimal surfaces appear as models for the apparent horizons of black holes. The minimal surface equation plays a very important role as a model for several kinds of nonlinear phenomena. Minimal surfaces have also been recently used in the design of materials with applications in biology and in chemistry. The project also includes training of PhD students and junior researchers. The PI will also disseminate his work through lectures, conferences, and workshops. This project will advance our basic understanding of minimal surfaces and their general existence theory. It concerns foundational questions about when these objects exist and how their properties relate to features of the ambient. The aim is to investigate the Morse-theoretic properties of the space of minimal varieties in a given Riemannian manifold. The idea is to use a combination of min-max methods, with the Almgren-Pitts min-max theory, and topological methods with the existence of homotopically nontrivial families of varieties. PI will study several questions related to this theme, including constructions of minimal hypersurfaces in higher dimensions and in the noncompact case. The project also includes plans for continued training of PhD students and post-doctoral researchers. 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.