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

Let G be a compact connected Lie group of dimension m. Once bi-invariant metric on is fixed, left-invariant metrics are in correspondence with m × positive definite symmetric matrices. We estimate the diameter and smallest eigenvalue Laplace-Beltrami operator associated to terms eigenvalues corresponding matrix. As consequence, we give partial answers conjecture by Eldredge, Gordina Saloff-Coste; namely, large subsets $\mathcal {S}$ space ${\mathscr{M}}$ such that there exists real number C depending λ1(G,g)diam(G,g)2 ≤ for all $g\in \mathcal . The existence constant {S}={\mathscr{M}}$ original conjecture.