Scalar curvature and geometric variational problems

Curvature describes local bending of a space, and is used to distinguish how two shapes are different. This project concerns a particular notion of curvature, called the scalar curvature. Scalar curvature determines 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. Curvature arises throughout the natural sciences, and particularly in general relativity, where scalar curvature is the Lagrangian density of the Einstein-Hilbert action. A natural yet deep question is to understand the effects of scalar curvature on the global properties of a manifold. In particular, the project will focus on geometric variational problems, including minimal surfaces and soap bubbles. Minimal surfaces arise as the mathematical model of a number of interfaces in nature. In general relativity, minimal surfaces occur as “apparent horizons” of black holes; soap films and capillary interfaces also provide examples of minimal surfaces. Some key questions include the existence, regularity and topology of minimal surfaces. These two aspects of the research are deeply related, and will advance our understanding of the shape of nature. The PI will integrate this research with a variety of knowledge-disseminating activities that include organizing seminars, conferences, mini-courses, and reading groups, and giving public lectures. The proposed research concerns a range of topics in differential geometry, geometric measure theory and partial differential equations. Particularly, the PI would like to focus on the following four main topics. The first topic is the investigation of the obstruction problem for manifolds with positive scalar curvature, including the well-known `K(pi, 1) conjecture’. The second topic is to further understand the geometric comparison theorem for scalar curvature lower bound using Riemannian polyhedra. Minimal surface and soap bubbles are key technical tools for such problems. The third topic is to further investigate the stable Bernstein problem for minimal surfaces in R^n. The fourth topic is to further study the moduli space of positive scalar curvature metrics on 3-manifolds (with or without boundary). Understanding such questions has potential applications in 4-dimensional general relativity and in topology. 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.