A sharp lower bound on the small eigenvalues of surfaces

Renan Gross, Guy Lachman, Asaf Nachmias

Let $S$ be a compact hyperbolic surface of genus $g\ge 2$ and let $I(S)=\frac{1}{\mathrm{Vol}(S)}{\int }_S\frac{1}{\mathrm{Inj}(x{)}^2\wedge 1}dx$, where $\mathrm{Inj}(x)$ is th…