Diameter and Laplace eigenvalue estimates for compact homogeneous Riemannian manifolds

Abstract

Let GG be a compact connected Lie group and let KK be a closed subgroup of GG. In this paper we study whether the functional gλ1(G/K,g)diam(G/K,g)2g\mapsto \lambda_1(G/K,g)\operatorname{diam}(G/K,g)^2 is bounded among GG-invariant metrics gg on G/KG/K. Eldredge, Gordina, and Saloff-Coste conjectured in 2018 that this assertion holds when KK is trivial; the only particular cases known so far are when GG is abelian, SU(2)\operatorname{SU}(2), and SO(3)\operatorname{SO}(3). In this article we prove the existence of the mentioned upper bound for every compact homogeneous space G/KG/K having multiplicity-free isotropy representation.Comment: Accepted for publication in Transformation Groups. arXiv admin note: text overlap with arXiv:2004.0035

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