Bounding the first invariant eigenvalue of toric Kähler manifolds (Q1707223)

The authors investigate the properties of the first invariant non-zero eigenvalue of the ordinary Laplacian for toric Kähler metrics on a closed manifold \(M\). Given a particular toric Kähler metric, we will denote this quantity \(\lambda_{1}^{\mathbb{T}}\) to distinguish it from the first non-zero eigenvalue \(\lambda_{1}\). It is trivial to see that \({\lambda_{1}\leq \lambda_{1}^{\mathbb{T}}}\). \textit{M. Abreu} and \textit{P. Freitas} proved that [Proc. Lond. Math. Soc. (3) 84, No. 1, 213--230 (2002; Zbl 1015.58012)], amongst \(\mathbb{S}^{1}\)-invariant metrics with a fixed volume, \(\lambda_{1}^{\mathbb{T}}\) can take any value in \((0,\infty)\). Under the assumption that \(g\) on \(\mathbb{C}P^{1}\) has non-negative Gauss curvature, both \textit{M. Engman} [Can. Math. Bull. 49, No. 2, 226--236 (2006; Zbl 1112.58032)] and Abreu and Freitas [loc. cit.] proved that \(\lambda_{1}^{\mathbb{T}}\) is bounded. The main purpose of this article is a generalisation of this result to higher-dimensional compact toric Kähler metrics. In this paper, the authors obtain the following main result: Let \((M,\omega)\) be a compact toric Kähler metric with non-negative scalar curvature. Then \(\lambda_{1}^{\mathbb{T}}\) is bounded above by a quantity that only depends upon the cohomology class \({[\omega]}\). In particular, on the complex projective space \(\mathbb{C}P^{n}\), \(\lambda_{1}^{\mathbb{T}}\leq n+2\). The authors 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 \(\mathbb{C}P^{2}\sharp -\mathbb{C}P^{2}\).