Toric aspects of the first eigenvalue

In this paper we study the smallest non-zero eigenvalue $λ_1$ of the Laplacian on toric Kähler manifolds. We find an explicit upper bound for $λ_1$ 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 $λ_1$ which we denote by $λ_1^T$. It is the the smallest non-zero eigenvalue of the Laplacian restricted to torus-invariant functions. We prove that $λ_1^T$ is not bounded among toric Kähler metrics thus generalizing a result of Abreu-Freitas on $S^2$. In particular, $λ_1^T$ and $λ_1$ do not coincide in general.

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