Fundamental Gap Estimates and Geometry /Topology of Ricci Limit Spaces

Various problems of mathematical physics can be modeled by the Laplacian or more general Schrodinger equations. The difference of the first two eigenvalues of the Laplacian is referred to as the fundamental gap, which represents the energy needed to excite a particle from ground level to the next level in quantum mechanics. The principal investigator will estimate the fundamental gap for various spaces. The proposed activities are related to optimal transport, information geometry and discrete geometry. The project will also support educational activities through mentoring undergraduate and graduate students as well as postdocs; organizing seminars, workshops and research programs promoting young scholars. The project is centered around Riemannian geometry and geometric analysis with three parts. The first is about the fundamental gap estimates of the Laplacian with Dirichlet boundary conditions on a horoconvex domain in the hyperbolic space and convex domain in locally symmetric spaces by comparison with some suitable 1-dim model. The second concerns geometry and topology of spaces with Ricci curvature lower bound, especially the fundamental group of noncompact manifolds with nonnegative Ricci curvature; minimal volume entropy rigidity for metric measure spaces with curvature lower bounds. The last is to study integral curvature for the critical power. 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.