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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. document.getElementById('cloak5bdf2a23e0646fa356dded611af27d18').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy5bdf2a23e0646fa356dded611af27d18 = '&#101;m&#105;l&#105;&#111;.l&#97;&#117;r&#101;t' + '&#64;'; addy5bdf2a23e0646fa356dded611af27d18 = addy5bdf2a23e0646fa356dded611af27d18 + '&#117;ns' + '&#46;' + '&#101;d&#117;' + '&#46;' + '&#97;r'; var addy_text5bdf2a23e0646fa356dded611af27d18 = '&#101;m&#105;l&#105;&#111;.l&#97;&#117;r&#101;t' + '&#64;' + '&#117;ns' + '&#46;' + '&#101;d&#117;' + '&#46;' + '&#97;r';document.getElementById('cloak5bdf2a23e0646fa356dded611af27d18').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy5bdf2a23e0646fa356dded611af27d18 + '\'>'+addy_text5bdf2a23e0646fa356dded611af27d18+'<\/a>';

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}.