On the smallest Laplace eigenvalue for naturally reductive metrics on compact simple Lie groups.

Abstract

Eldredge, Gordina and Saloff-Coste recently conjectured that, for a given compact connected Lie group $G$, there is a positive real number $C$ such that $\\lambda_1(G,g)\\operatorname{diam}(G,g)^2\\leq C$ for all left-invariant metrics $g$ on $G$. In this short note, we establish the conjecture for the small subclass of naturally reductive left-invariant metrics on a compact simple Lie group.