Toric aspects of the first eigenvalue

Abstract

In this paper we study the smallest non-zero eigenvalue ${\lambda }_1$ of the Laplacian on toric Kähler manifolds. We find an explicit upper bound for ${\lambda }_1$ in terms of moment polytope data. We show that this bound can only be attained for ${\mathbb{CP}}^n$ endowed with the Fubini-Study metric and therefore ${\mathbb{CP}}^n$ endowed with the Fubini-Study metric is spectrally determined among all toric Kähler metrics. We also study the equivariant counterpart of ${\lambda }_1$ which we denote by ${\lambda }_1^T$. It is the the smallest non-zero eigenvalue of the Laplacian restricted to torus-invariant functions. We prove that ${\lambda }_1^T$ is not bounded among toric Kähler metrics thus generalizing a result of Abreu-Freitas on $S^2$. In particular, ${\lambda }_1^T$ and ${\lambda }_1$ do not coincide in general.

24 pages. Added some more details and corrected an estimate. Fixed some typos