Toric Aspects of the First Eigenvalue

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 \(\mathbb C\mathbb P^n\) endowed with the Fubini–Study metric and therefore \(\mathbb C\mathbb P^n\) 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 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.