Eigenvalue Comparison and Integral Curvature

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Various problems of mathematical physics can be modeled by the Laplacian or more generally Schrodinger equations. The difference between 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. In the first project, the principal investigator will estimate the fundamental gap for various spaces. The second project relates to volume entropy which is a fundamental geometric invariant for compact smooth manifolds. This concept is closely related to other notions of entropy found in dynamical systems and plays an essential role in differential geometry and geometric group theory among others. The work on entropy rigidity is related to optimal transport, information geometry and discrete geometry. The project will also support educational activities and diversity through mentoring graduate students and postdocs; recruiting women and other underrepresented groups; organizing seminars, workshops and research programs promoting young scholars. The material discussed in the proposal will be the subject of an advanced graduate course at UCSB in Fall 2021.The project has three parts. The first is about eigenvalue and fundamental gap estimates of the Laplacian with Dirichlet boundary conditions on a convex domain in locally symmetric spaces by comparing with some suitable 1-dim model. The second concerns volume entropy comparison and rigidity for metric measure spaces with curvature lower bounds. The last is to study integral curvature pinching for the critical power using Ricci flow.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.