Diameter and Laplace eigenvalue estimates for compact homogeneous Riemannian manifolds

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 → λ 1 (G/K, g) diam(G/K, g) 2 is bounded by above 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, SU(2), and 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.