Diameter and Laplace eigenvalue estimates for compact homogeneous Riemannian manifolds

TL;DR

This paper proves an upper bound on the product of the first Laplace eigenvalue and the square of the diameter for a class of compact homogeneous spaces, extending previous conjectures and known cases.

Contribution

It establishes the boundedness of the eigenvalue-diameter functional for all compact homogeneous spaces with multiplicity-free isotropy representation.

Findings

Boundedness of the functional for spaces with multiplicity-free isotropy.

Extension of conjecture to a broader class of homogeneous spaces.

Provides new geometric estimates for Laplace eigenvalues.

Abstract

Let $G$ be a compact connected Lie group and let $K$ be a closed subgroup of $G$. In this paper we study whether the functional $g\mapsto \lambda_1(G/K,g)\operatorname{diam}(G/K,g)^2$ is bounded among $G$-invariant metrics $g$ on $G/K$. Eldredge, Gordina, and Saloff-Coste conjectured in 2018 that this assertion holds when $K$ is trivial; the only particular cases known so far are when $G$ is abelian, $\operatorname{SU}(2)$, and $\operatorname{SO}(3)$. In this article we prove the existence of the mentioned upper bound for every compact homogeneous space $G/K$ having multiplicity-free isotropy representation.