Constant Gaussian curvature metrics as maximizers of the bottom of the spectrum

A characterization of noncompact ball quotients by Munteanu (2009) implies that only complete Kähler-Einstein metrics on certain noncompact ball quotients maximize the bottom of the spectrum among complete Kähler metrics with Ricci curvature bounded below by -1. In this talk, I will report my recent observation, which was made in an attempt to generalize the result of Munteanu for general noncompact Kähler-Einstein manifolds, that a similar result also holds for any noncompact hyperbolic Riemann surface on which the normalized Kähler-Ricci flow deforms the initial metric to the constant Gaussian curvature metric.