Geometry, Analysis, and Variational Methods

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This research project concerns problems in the interface between Geometry, Analysis, and the Calculus of Variations. Questions concerning the variational theory of minimal surfaces and its applications will be investigated. 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. One of the 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. We will study the existence and basic properties, like the Morse index, of min-max minimal varieties.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.