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 × 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 \({\mathscr{M}}\) on G such that there exists a positive real number C depending on G and \(\mathcal {S}\) such that λ₁(G,g)diam(G,g)² ≤ C for all \(g\in \mathcal {S}\). The existence of the constant C for \(\mathcal {S}={\mathscr{M}}\) is the original conjecture.

设G是一个维度为m的紧连通李群。一旦双不变度量ģ是固定的，左不变上度量ģ都与对应米×中号正定对称矩阵。我们根据相应的正定对称矩阵的特征值估计与G上的左不变度量相关联的 Laplace-Beltrami 算子的直径和最小正特征值。因此，我们对 Eldredge、Gordina 和 Saloff-Coste 的猜想给出了部分答案；即，我们给出了左不变度量空间的大子集\(\mathcal {S}\) \({\mathscr{M}}\)在G 上，存在依赖于G和\(\mathcal {S}\)的正实数C，使得λ ₁ ( G , g )diam( G , g ) ² ≤ C对于所有\(g\in \)数学 {S}\)。恒定的存在Ç为\（\ mathcal {S} = {\ mathscr {M}} \）是原始猜想。