Bounding the first invariant eigenvalue of toric Kähler manifolds

Abstract

We generalise a theorem of Engman and Abreu--Freitas on the first invariant eigenvalue of non-negatively curved $S^1$-invariant metrics on ${\mathbb{CP}}^1$ to general toric Kähler metrics with non-negative scalar curvature. In particular, a simple upper bound of the first non-zero invariant eigenvalue for such metrics on complex projective space $\mathbb{C}{P}^n$ is exhibited. We derive an analogous bound in the case when the metric is extremal and a detailed study is made of the accuracy of the bound in the case of Calabi's extremal metrics on $\mathbb{C}{P}^2♯-\mathbb{C}{P}^2$.

12 pages, 3 figures