First Laplace eigenvalue of strongly isotropy irreducible spaces

TL;DR

This work analyzes the first nonzero eigenvalue of the Laplace–Beltrami operator on compact strongly isotropy irreducible spaces, showing that every such space is Einstein with Einstein constant $E$ and establishing the universal bound $E<\lambda_1\le 16E$. It provides explicit expressions for $\lambda_1$ in all simply connected non-symmetric cases by exploiting Lie-theoretic embeddings, highest-weight representations, and branching rules across ten infinite families and six isolated cases, and compiles detailed multiplicities. The paper also derives sharp lower bounds for non-symmetric spaces and connects these spectral quantities to the Einstein constant via a uniform upper bound on $\lambda_1/E$, yielding a unified view of spectral geometry for these homogeneous spaces. Overall, it advances the Saloff-Coste program by giving concrete, representation-theoretic controls on Laplacian spectra for a wide class of homogeneous spaces and clarifies how topology, symmetry, and isotropy affect eigenvalue bounds with potential implications for heat kernel and geometric analysis on these manifolds.

Abstract

We study the smallest positive eigenvalue $λ_1$ 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 $E<λ_1\leq 16E$, where $E$ denotes the corresponding Einstein constant.

Theorems & Definitions (76)

• Theorem 1.1
• Remark 1.2
• Theorem 2.1
• Lemma 2.2
• proof
• Remark 2.3
• Remark 2.5: Embedding of $H$ into $G$
• Lemma 2.6
• Remark 2.7: Exterior and symmetric products of the standard representation
• Remark 2.8