Diameter and Laplace Eigenvalue Estimates for Left-invariant Metrics on Compact Lie Groups

LAURET, EMILIO A.

POTENTIAL ANALYSIS

SPRINGER

Año: 2023 vol. 58 p. 37 - 70

0926-2601

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 $mimes m$ 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 $lambda_1(G,g)operatorname{diam}(G,g)^2leq C$ for all $ginmathcal S$. The existence of the constant $C$ for $mathcal S=mathcal M$ is the original conjecture.