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 a bi-invariant metric on G is fixed, left-invariant metrics on G are in correspondence with m x 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 S of the space of left-invariant metricsMon G such that there exists a positive real number C depending on G and S such that lambda(1)(G, g)diam(G, g)(2) <= C for all g epsilon S. The existence of the constant C for S = Mis the original conjecture.