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 ${\lambda }_1(G,g)\mathrm{diam}(G,g{)}^2\le C$ 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.