Bounding the first invariant eigenvalue of toric K\"ahler manifolds

TL;DR

This paper extends bounds on the first invariant eigenvalue from specific metrics on spheres to general toric K"ahler manifolds with non-negative scalar curvature, providing explicit bounds and analyzing their accuracy for extremal metrics.

Contribution

It generalizes existing eigenvalue bounds to all toric K"ahler manifolds with non-negative scalar curvature, including extremal metrics, and studies their sharpness.

Findings

Derived upper bounds for the first invariant eigenvalue on complex projective spaces.

Extended bounds to extremal toric K"ahler metrics.

Analyzed the accuracy of bounds for Calabi's extremal metrics on blow-ups of complex projective space.

Abstract

We generalise a theorem of Engman and Abreu--Freitas on the first invariant eigenvalue of non-negatively curved $S^{1}$-invariant metrics on $\mathbb{CP}^{1}$ to general toric K\"ahler metrics with non-negative scalar curvature. In particular, a simple upper bound of the first non-zero invariant eigenvalue for such metrics on complex projective space $\mathbb{C}P^{n}$ is exhibited. We 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 $\mathbb{C}P^{2}\sharp -\mathbb{C}P^{2}$.