Geometric Variational Problems and Scalar Curvature

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One aspect of the proposed research has to do with the geometry and topology of manifolds with scalar curvature lower bounds. Scalar curvature is the simplest curvature invariant of a Riemannian manifold. It represents the amount by which the volume of a small geodesic ball in a Riemannian manifold deviates from that of the standard ball in Euclidean space. Scalar curvature also arises in natural sciences. For instance, in general relativity, it is the Lagrangian density of the Einstein-Hilbert action. A natural and deep question in geometry, topology and mathematical physics is to understand the affect of scalar curvature conditions on a manifold. The other main area of investigation concerns minimal surfaces. Minimal surfaces arise as the mathematical model of a number of interfaces in nature. In mathematical model of general relativity, minimal surfaces occur as “apparent horizons” of black holes; soap films and capillary interfaces also provide examples of minimal surfaces. The PI will investigate the existence, regularity and topology of minimal surfaces. The two aspects proposed here are deeply connected via geometric variational theory.The project concerns a range topics on differential geometry, geometric measure theory and partial differential equations. A main theme of the research in geometry will be a geometric comparison theorem for scalar curvature using Riemannian polyhedra, with the aim to define weak notions of positive scalar curvature on spaces with low regularity. The PI plans to continue his investigations into such a theorem for more general polytopes, especially simplexes of higher dimensions, and its connection to quasi-local mass in general relativity. The PI also plans to continue his investigation on the structure of moduli spaces of manifolds with positive scalar curvature and mean convex boundary, including studying its high homotopy groups, and the structure of moduli spaces defined by other related curvature conditions. In addition, the PI will study singular spaces with scalar curvature lower bounds, and understand when such a singular manifold arises as a certain limit of smooth manifolds with same assumptions. A central tool in the PI’s research is the theory of minimal varieties. The PI plans to understand the existence and regularity of minimal surfaces with free boundary and capillary boundary conditions in general Lipschitz domains, especially in locally convex polyhedral domains. He also plans to establish a general existence theory of capillary surfaces via a min-max construction.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.