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

LAURET, EMILIO A.

Evento online con chalas en YouTube

Conferencia; 3rd Geometric Analysis Festival; 2021

Jeonbuk National University

In 1980, Peter Li proved that $\lambda_1(M,g)\geq \tfrac{\pi^2}{4}\operatorname{diam}(M,g)^{-2}$for every compact connected homogeneous Riemannian manifold $(M,g)$ (of arbitrary dimension).Here, $\lambda_1(M,g)$ stands for the smallest positive eigenvalue of the Laplace-Beltrami operator of $(M,g)$ and $\operatorname{diam}(M,g)$ denotes the diameter of $(M,g)$. Although it is easy to see that $\lambda_1(M,g) \operatorname{diam}(M,g)^2$ is not uniformly bounded by above among the same class of spaces, it is not know what occurs when the dimension of $M$ is fixed. Eldredge, Gordina and Saloff-Coste [\href{https://doi.org/10.1007/s00039-018-0457-8}{GAFA {\bf 28}, 1321--1367 (2018)}] conjectured the following particular case of the mentioned open problem: \begin{quote}\it for any compact connected Lie group $G$, there is $C>0$ depending only on $G$ such that $\lambda_1(G,g) \operatorname{diam}(G,g)^2\leq C$ for every left-invariant metric $g$ on $G$.\end{quote}This was (affirmative) solved in the mentioned article for $G$ a tori, $\operatorname{SU}(2)$ and $\operatorname{SO}(3)$, and these are the only known cases so far. In this work [\href{https://doi.org/10.1007/s11118-021-09932-1}{Potential Anal.\ (2021), in press}], given $G$ a compact connected Lie group and $g$ a left-invariant metric on $G$, we independently estimate $\lambda_1(G,g)$ and ${\operatorname{diam}(M,g)}$ in terms of the {\it metric eigenvalues} of $g$, that is, the eigenvalues of the corresponding positive definite symmetric matrix (w.r.t.\ a fixed bi-invariant metric) associated $g$. As a consequence, we construct a large subset $\mathcal S$ of the space of left-invariant metrics on $G$ such that there is $C=C(\mathcal S)>0$ satisfying that $\lambda_1(G,g) \operatorname{diam}(G,g)^2\leq C$ for all $g\in\mathcal S$.