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

LAURET, EMILIO A.

Evento online organizado desde Buenos Aires

Congreso; Mathematical Congress of the Americas 2021; 2021

Universidad de Buenos Aires

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}.