Session S23 - Group actions in Differential Geometry

Friday, July 23, 16:00 ~ 16:30 UTC-3

Diameter and Laplace eigenvalue estimates for left-invariant metrics on compact Lie groups

Emilio Lauret

Universidad Nacional del Sur, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Given $G$ a compact Lie group, we estimate the first Laplace eigenvalue and the diameter of a left-invariant metric on $G$ in terms of its {\it metric eigenvalues}, that is, the eigenvalues of the corresponding positive definite symmetric matrix (w.r.t.\ a fixed bi-invariant metric) associated to a left-invariant metric.

As a consequence, we give a partial answer to the following conjecture by Eldredge, Gordina, and Saloff-Coste [GAFA {\bf 28}, 1321--1367 (2018)]: there exists a positive real number $C$ depending only on $G$ such that the product between the first Laplace eigenvalue and the square of the diameter is bounded by above by $C$ for every left-invariant metric.

The talk is based on the article \url{https://arxiv.org/abs/2004.00350}.

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