Bounding $$\lambda _{2}$$ λ 2 for Kähler–Einstein metrics with large symmetry groups

Abstract

We calculate an upper bound for the second non-zero eigenvalue of the scalar Laplacian, \(\lambda _{2}\), for toric-Kahler–Einstein metrics in terms of the polytope data. We also give a similar upper bound for Koiso–Sakane type Kahler–Einstein metrics. We provide some detailed examples in complex dimensions 1, 2 and 3.