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The fundamental gap of a domain is the difference between the first twoeigenvalues of the Laplace operator. In a series of recent and celebratedworks, 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 theseresults is to establish a strong concavity estimate for the logarithm of thefirst eigenfunction. In this article, we prove corresponding log-concavity andfundamental gap estimates for surfaces with non-constant positive curvature viaa two-point maximum principle. However, the curvature not being constantgreatly increases the difficulty of the computation.

Modulus of Concavity and Fundamental Gap Estimates on Surfaces