Diameter and Laplace eigenvalue estimates for left-invariant metrics on compact Lie groups. (English) Zbl 1507.58005
Summary: 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 \(m \times m\) 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 \({\mathcal{M}}\) on \(G\) such that there exists a positive real number \(C\) depending on \(G\) and \(\mathcal{S}\) such that \(\lambda_1(G,g)\mathrm{diam}(G,g)^2 \leq C\) for all \(g\in \mathcal{S} \). The existence of the constant \(C\) for \(\mathcal{S}={\mathcal{M}}\) is the original conjecture.
MathOverflow Questions:
On maximal closed connected subgroups of a compact connected semisimple Lie group?MSC:
| 58C40 | Spectral theory; eigenvalue problems on manifolds |
| 58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |
| 22C05 | Compact groups |
| 53C30 | Differential geometry of homogeneous manifolds |
| 53C17 | Sub-Riemannian geometry |
Software:
MathOverflowReferences:
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