# On the smallest Laplace eigenvalue for naturally reductive metrics on compact simple Lie groups

## TL;DR

This paper proves a conjecture relating the first Laplace eigenvalue and diameter for a specific class of metrics on compact simple Lie groups, confirming a boundedness property.

## Contribution

It establishes the conjecture for naturally reductive left-invariant metrics on compact simple Lie groups, a subclass previously unconfirmed.

## Key findings

- Confirmed the conjecture for naturally reductive metrics
- Bounded the product of eigenvalue and diameter squared
- Extended understanding of spectral geometry on Lie groups

## Abstract

Eldredge, Gordina and Saloff-Coste recently conjectured that, for a given compact connected Lie group $G$, there is a positive real number $C$ such that $\lambda_1(G,g)\operatorname{diam}(G,g)^2\leq C$ for all left-invariant metrics $g$ on $G$. In this short note, we establish the conjecture for the small subclass of naturally reductive left-invariant metrics on a compact simple Lie group.