Algebra & Geometry Seminar 04/24,

Speaker: Xavier Ramos Olivé (Smith College),   https://www.smith.edu/people/xavier-ramos-olive

Eigenvalue Estimates Under Integral Curvature Conditions

The relationship between curvature bounds and the eigenvalues of the Laplacian is a widely studied subject in Geometric Analysis. The first eigenvalue of a manifold with a positive lower bound on the Ricci curvature is controlled from below by the first eigenvalue of a sphere. Similarly, given a manifold with non-negative Ricci curvature and with bounded diameter, the first eigenvalue of the Laplacian is always larger than the first eigenvalue of a circle. These estimates, however, are very sensitive to perturbations of the geometry. In this talk we will discuss eigenvalue estimates under integral curvature conditions, a weaker kind of curvature assumption that is less sensitive to perturbations. We will discuss Zhong-Yang type eigenvalue estimates for closed manifolds and for closed smooth metric measure spaces, and if time permits, we will also explore estimates on the first Neumann eigenvalue and on the Dirichlet eigenvalue gap on non-convex domains. Some of these results are joint work with Olaf Post, Christian Rose, Shoo Seto, Lili Wang, Guofang Wei and Qi S. Zhang.