Diameter and Laplace eigenvalue estimates for left-invariant metrics on compact Lie groups

Abstract

Let $G$ be a compact connected Lie group of dimension $m$. Once a bi-invariant metric on $G$ is fixed, left-invariant metrics on $G$ are in correspondence with $m\times m$ positive definite symmetric matrices. We estimate the diameter and the smallest positive eigenvalue of the Laplace-Beltrami operator associated to a left-invariant metric on $G$ in terms of the eigenvalues of the corresponding positive definite symmetric matrix. As a consequence, we give partial answers to a conjecture by Eldredge, Gordina and Saloff-Coste; namely, we give large subsets $\mathcal{S}$ of the space of left-invariant metrics $\mathcal{M}$ on $G$ such that there exists a positive real number $C$ depending on $G$ and $\mathcal{S}$ such that ${\lambda }_1(G,g)\mathrm{diam}(G,g{)}^2\le C$ for all $g\in \mathcal{S}$. The existence of the constant $C$ for $\mathcal{S}=\mathcal{M}$ is the original conjecture.

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References in corpus (1)

• The first eigenvalue of a homogeneous CROSS

Cited by in corpus (1)

• Diameter and Laplace eigenvalue estimates for compact homogeneous Riemannian manifolds