Critical Kahler toric metrics for the invariant first eigenvalue

Abstract

In [LS], it is shown shown that the first eigenvalue of the Laplacian restricted to the space of invariant functions on a toric Kähler manifold (i.e. ${\lambda }_1^{\mathbb{T}}$, the invariant first eigenvalue) is an unbounded function of the toric Kähler metric. In this note we show that, seen as a function on the space of toric Kähler 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.

14 pages