Degenerating hyperbolic surfaces and spectral gaps for large genus

We study the differences of two consecutive eigenvalues ${\lambda }_i-{\lambda }_{i-1}$, $i$ up to $2g-2$, for the Laplacian on hyperbolic surfaces of genus $g$, and show that the supremum of such spectral gaps over the moduli space has infimum limit at least $\frac{1}{4}$ as the genus goes to infinity. A min-max principle for eigenvalues on degenerating hyperbolic surfaces is also established.