Guofang Wei
University of California, Santa Barbara
Title
Fundamental Gap and Log-concavity of First Eigenfunction
Abstract
The fundamental (or mass) gap refers to the difference between the first two eigenvalues of the Laplacian or more generally for Schr\"{o}dinger operators. It is a very interesting quantity both in mathematics and physics as the eigenvalues are possible allowed energy values in quantum physics. The log-concavity of the first eigenfunction plays an important role in the fundamental gap estimate. For convex domains in the Euclidean space, Brascamp-Lieb proved the log-concavity of the first Dirichelt eigenfunction back in the 70s. Since then there have been many works in this direction. We will survey many recent fantastic results for convex domains in Euclidean spaces, spheres, hyperbolic spaces and surfaces with positive curvature with Dirichlet boundary conditions, starting with the breakthrough of Andrews-Clutterbuck. Then we will present a very recent work on horoconvex domains in the hyperbolic space. This last result is joint with Ling Xiao. All necessary background information will be introduced in the talk.
