Critical Kahler toric metrics for the invariant first eigenvalue

TL;DR

This paper investigates the behavior of the invariant first eigenvalue of the Laplacian on toric Kähler manifolds, showing it has no analytic critical points and exploring similar properties on the sphere.

Contribution

It demonstrates that the invariant first eigenvalue function has no analytic critical points on toric Kähler manifolds and the sphere, revealing new insights into its geometric properties.

Findings

Invariant first eigenvalue is unbounded as a function of the metric.

No analytic critical points exist for the invariant first eigenvalue on toric Kähler manifolds.

The first eigenvalue on S^2 with fixed equivariance also admits no critical points.

Abstract

In [LS], it is shown shown that the first eigenvalue of the Laplacian restricted to the space of invariant functions on a toric K\"ahler manifold (i.e. $\lambda_1^\mathbb{T}$, the invariant first eigenvalue) is an unbounded function of the toric K\"ahler metric. In this note we show that, seen as a function on the space of toric K\"ahler metrics on a fixed toric manifold, $\lambda_1^\mathbb{T}$ admits no analytic critical points. We also show that on $S^2$, the first eigenvalue of the Laplacian restricted to the space of $S^1$-equivariant functions of any given integer weight admits no critical points.