Toric aspects of the first eigenvalue

TL;DR

This paper investigates the first non-zero eigenvalue of the Laplacian on toric Kähler manifolds, providing bounds, characterizations, and exploring the spectral properties of torus-invariant functions.

Contribution

It establishes an explicit upper bound for the eigenvalue based on polytope data, characterizes when this bound is attained, and analyzes the unboundedness of the equivariant eigenvalue.

Findings

The upper bound for $mbda_1$ is attained only by $P^n$ with Fubini-Study metric.

$mbda_1^T$ is unbounded among toric Kähler metrics.

$mbda_1^T$ and $mbda_1$ generally differ.

Abstract

In this paper we study the smallest non-zero eigenvalue $\lambda_1$ of the Laplacian on toric K\"ahler 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 K\"ahler 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 K\"ahler metrics thus generalizing a result of Abreu-Freitas on $S^2$. In particular, $\lambda_1^T$ and $\lambda_1$ do not coincide in general.