Differential Geometry and Partial Differential Equations

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Many physical phenomena are modeled by equations that are essentially geometric in nature. A prime example is the theory of General Relativity, which was invented to make gravity compatible with Special Relativity. The Einstein equations are a system of evolution equations that describe the spacetime geometry and curvature, which arise in specific situations. Like many other geometric/physical equations the Einstein equations are nonlinear and thus can produce solutions that develop very high curvature regions and lead to mathematical singularities, which are expected to be black holes. Singularities also arise in surfaces that minimize volume, so called minimal surfaces. Such minimizing surfaces in a curved spacetime are intimately connected with the geometric structure of the spacetime, and their existence and properties reflect properties of the spacetime such as the energy distribution of the gravitational field, and the prediction of gravitational collapse to a black hole. A major goal of this project is to better understand the stability properties of such singularities, especially those that can arise in volume minimizing surfaces in spacetime. When we study curved geometries, there are naturally associated numbers that are the fundamental frequencies with which such a geometry would vibrate if it were thought of as an idealized drumhead. This sequence of numbers is called the spectrum. A second goal of this project is to understand the special geometries that have the largest fundamental frequencies per unit volume. The project also includes significant training of PhD students and post-doctoral fellows. The PI also plans to deliver outreach lectures in geometry.The focus of this research is in the areas of differential geometry, general relativity, and partial differential equations. In work that is motivated both by differential geometry and general relativity the PI will study the singularities, which may form in higher dimensional volume minimizing hypersurfaces. This issue comes up in the study of manifolds of positive scalar curvature and the related positive mass theorem of general relativity. He will study the question of whether singularities can be perturbed away by a small perturbation of the boundary or the ambient metric. He poses some explicit test cases for this question. The second main theme is the study and comparison of three quasi-local masses that have arisen in general relativity. These are the Hawking, Bartnik, and Brown-York masses. Several questions are posed concerning the relative sizes of these quantities. The connections between the Brown-York mass and recent polyhedral comparison theorems will be investigated with a suggestion for solidifying the connections between the two. The third main theme of his research in geometry will be the study of spectral geometry and related questions on minimal submanifolds. The PI has recently analyzed sharp bounds on the high Steklov eigenvalues of the disk and will extend that study to the next simplest cases of the annulus and the Mobius band. Finally he plans to investigate the equivariant eigenvalue maximization problem for surfaces.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.