Mathematical Research Letters

Volume 26 (2019)

Number 3

Critical Kähler toric metrics for the invariant first eigenvalue

Pages: 851 – 873

DOI: https://dx.doi.org/10.4310/MRL.2019.v26.n3.a8

Author

Rosa Sena-Dias (Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Departamento de Matemática, Instituto Superior Técnico, Lisboa, Portugal)

Abstract

In [LS] it is shown that the first eigenvalue of the Laplacian restricted to the space of invariant functions on a toric Kähler manifold (i.e. $\lambda^{\mathbb{T}}_1$, 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, $\lambda^{\mathbb{T}}_1$ 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.

Received 3 November 2017

Accepted 30 September 2018

Published 25 October 2019