Modulus of Concavity and Fundamental Gap Estimates on Surfaces

Abstract

The fundamental gap of a domain is the difference between the first two eigenvalues of the Laplace operator. In a series of recent and celebrated works, it was shown that for convex domains in ${\mathbb{R}}^n$ and ${\mathbb{S}}^n$ with Dirichlet boundary condition the fundamental gap is at least $\frac{3{\pi }^2}{D^2}$ where $D$ denotes the diameter of the domain. The key to these results is to establish a strong concavity estimate for the logarithm of the first eigenfunction. In this article, we prove corresponding log-concavity and fundamental gap estimates for surfaces with non-constant positive curvature via a two-point maximum principle. However, the curvature not being constant greatly increases the difficulty of the computation.

42 pages