Geometry, analysis and variational methods

Princeton University, Princeton NJ

Investigators

Abstract

The PI investigates questions concerning the variational theory of minimal submanifolds and its applications. Minimal surfaces are among the most natural objects in differential geometry. They have encountered striking applications in many other fields, like three-dimensional topology, mathematical physics, complex and conformal geometry, among others. 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 in nature. Significant progress in this area has always had a great impact in mathematical analysis and the physical sciences. The research of 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 space. This research project contains a number of problems at the interface between geometry, analysis and the calculus of variations. One of the project's goals is to develop a good understanding of the Morse-theoretic properties of the space of minimal varieties in a given Riemannian manifold. This is to be accomplished by a combination of min-max techniques and topological methods, where the relevant spaces of cycles are defined by means of geometric measure theory. The PI will study the existence and basic properties, like the Morse index, of min-max minimal varieties.