Degenerating hyperbolic surfaces and spectral gaps for large genus

In this article we study the differences of two consecutive eigenvalues λ i -λ i-1 up to i = 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 1 4 as genus goes to infinity.A min-max principle for eigenvalues on degenerating hyperbolic surfaces is also established.