  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.
