Systole and ${\lambda }_{2g–2}$ of closed hyperbolic surfaces of genus $g$

Abstract

We apply topological methods to study eigenvalues of the Laplacian on closed hyperbolic surfaces. For any closed hyperbolic surface $S$ of genus $g$, we get a geometric lower bound on ${\lambda }_{2g-2}(S):$ ${\lambda }_{2g-2}(S)>1/4+{ϵ}_0(S)>0,$ where ${ϵ}_0(S)$ is an explicit constant which depends only on the systole of $S$.