First Laplace eigenvalue of strongly isotropy irreducible spaces

TL;DR

This paper investigates the smallest positive Laplace eigenvalue on compact strongly isotropy irreducible spaces, providing explicit formulas for simply connected cases and bounds relating it to the Einstein constant.

Contribution

It offers explicit eigenvalue formulas for simply connected spaces and establishes bounds linking the eigenvalue to the Einstein constant for all such spaces.

Findings

Explicit eigenvalue formulas for simply connected cases.

Bound: Einstein constant < eigenvalue ≤ 16 × Einstein constant.

All strongly isotropy irreducible spaces are Einstein manifolds.

Abstract

We study the smallest positive eigenvalue $\lambda_1$ of the Laplace-Beltrami operator associated with any compact strongly isotropy irreducible space. We provide an explicit expression for all simply connected cases. Furthermore, every strongly isotropy irreducible space is automatically an Einstein manifold, and we prove for each of them that $E<\lambda_1\leq 16E$, where $E$ denotes the corresponding Einstein constant.