Geometry, Analysis and Variational Methods

The PI will investigate questions related to the variational theory of minimal surfaces and its applications. Minimal surfaces are among the most natural objects in Differential Geometry. They have encountered applications in many different areas, like 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 been recently used in the design of materials with applications in biology and in chemistry. Significant progress in this area has always had a great impact in mathematical analysis and in the physical sciences. This project will also contribute to training students and postdocs in these studies and to the dissemination of knowledge in the mathematical sciences community. The research of this project will advance the 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. The PI proposes several questions related to this research theme, including rigidity of Riemannian metrics in terms of the variationally defined area widths and new connections with the theory of geodesics. 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.