Geometry & Topology Seminar

On convex domains in R^n and S^n, the first Dirichlet eigenfunction is known to be log-concave, a fact is crucial to estimate the spectral gap, which is the difference between the second and first Dirichlet eigenvalue. I will present power-concavity results for Dirichlet eigenfunctions: on convex domains in and the first eigenfunction is p-concave (i.e., u^p  is concave; p = 0 recovers log-concavity) with an exponent determined by the geometry of the domain. I will also discuss concavity estimates on horoconvex domains in hyperbolic space (which are domains whose boundaries second fundamental form is greater or equal than 1), which yield new spectral-gap bounds in H^n. This is based on joint work with Zhiqin Lu and on work with Gabriel Khan.