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The Laplace-Beltrami spectrum on Naturally Reductive Homogeneous Spaces

Ilka Agricola, Jonas Henkel

TL;DR

This work derives a comprehensive framework for the Laplace-Beltrami spectrum on compact naturally reductive homogeneous spaces by introducing a generalized Casimir operator $C_g$ and a Freudenthal-type eigenvalue formula $c_g( ho)=g( ext{λ},2 ho+ ext{λ})$, enabling explicit spectrum calculations via $K$-spherical representations. It then realizes large classes of canonical variations of normal homogeneous metrics as naturally reductive spaces on enlarged groups, producing explicit spectrum formulas and clarifying how deformation parameters govern eigenvalues. The second part specializes to the Aloff-Wallach manifold $W^{1,1}$, computing the full spectrum for a family of $3$-$( ext{α}, ext{δ})$-Sasaki metrics, recovering Urakawa’s normal-homogeneous case as a limit and completing Urakawa’s list of first eigenvalues for positively curved compact spaces. A detailed representation-theoretic treatment for $SU(3) imes SO(3)$ is provided, including branching rules, a Python-based computational supplement, and geometric interpretations via canonical variations over CP^2, with broader implications for the spectra of homogeneous positively curved manifolds and their deformations.

Abstract

We prove a formula for the spectrum of the Laplace-Beltrami operator on functions for compact naturally reductive homogeneous spaces in terms of eigenvalues of a generalized Casimir operator and spherical representations. We apply this result to a large family of canonical variations of normal homogeneous metrics, thus allowing for the first time to study how the spectrum depends on the deformation parameters of the metric. As an application, we provide a formula for the full spectrum of compact positive homogeneous $3$-$(α,δ)$-Sasaki manifolds (a family of metrics which includes, in particular, all homogeneous $3$-Sasaki manifolds). The second part of the paper is devoted to the detailed computation and investigation of the spectrum of this family of metrics on the Aloff-Wallach manifold $W^{1,1}=SU(3)/S^{1}$; in particular, we provide a documented Python script that allows the explicit computation in any desired range. We recover Urakawa's eigenvalue computation for the $SU(3)$-normal homogeneous metric on $W^{1,1}$ as a limiting case and cover all the positively curved $SU(3)\times SO(3)$-normal homogeneous realizations discovered by Wilking. By doing so, we complete Urakawa's list of the first eigenvalue on compact, simply conntected, normal homogeneous spaces with positive sectional curvature.

The Laplace-Beltrami spectrum on Naturally Reductive Homogeneous Spaces

TL;DR

This work derives a comprehensive framework for the Laplace-Beltrami spectrum on compact naturally reductive homogeneous spaces by introducing a generalized Casimir operator and a Freudenthal-type eigenvalue formula , enabling explicit spectrum calculations via -spherical representations. It then realizes large classes of canonical variations of normal homogeneous metrics as naturally reductive spaces on enlarged groups, producing explicit spectrum formulas and clarifying how deformation parameters govern eigenvalues. The second part specializes to the Aloff-Wallach manifold , computing the full spectrum for a family of --Sasaki metrics, recovering Urakawa’s normal-homogeneous case as a limit and completing Urakawa’s list of first eigenvalues for positively curved compact spaces. A detailed representation-theoretic treatment for is provided, including branching rules, a Python-based computational supplement, and geometric interpretations via canonical variations over CP^2, with broader implications for the spectra of homogeneous positively curved manifolds and their deformations.

Abstract

We prove a formula for the spectrum of the Laplace-Beltrami operator on functions for compact naturally reductive homogeneous spaces in terms of eigenvalues of a generalized Casimir operator and spherical representations. We apply this result to a large family of canonical variations of normal homogeneous metrics, thus allowing for the first time to study how the spectrum depends on the deformation parameters of the metric. As an application, we provide a formula for the full spectrum of compact positive homogeneous --Sasaki manifolds (a family of metrics which includes, in particular, all homogeneous -Sasaki manifolds). The second part of the paper is devoted to the detailed computation and investigation of the spectrum of this family of metrics on the Aloff-Wallach manifold ; in particular, we provide a documented Python script that allows the explicit computation in any desired range. We recover Urakawa's eigenvalue computation for the -normal homogeneous metric on as a limiting case and cover all the positively curved -normal homogeneous realizations discovered by Wilking. By doing so, we complete Urakawa's list of the first eigenvalue on compact, simply conntected, normal homogeneous spaces with positive sectional curvature.

Paper Structure

This paper contains 18 sections, 28 theorems, 127 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The spectrum of a naturally reductive homogeneous space
  3. $G$-invariant differential operators and the generalized Casimir operator
  4. The Freudenthal formula for the generalized Casimir operator
  5. Digression: The Laplacian for Riemannian submersions
  6. Naturally reductive canonical variations of normal homogeneous metrics and their spectrum
  7. The spectrum of homogeneous $3$-$(\alpha,\delta)$-Sasaki manifolds
  8. The spectrum of the Aloff-Wallach manifold W1,1 = SU(3)/S1
  9. Representation theory of SU(3)x SO(3)
  10. Formulas for the spectrum of $(W^{1,1},g_{t_{0},t_{1}})$
  11. Proof of \ref{['item: spectrum formula']} in Theorem $\ref{['thm: spectrum Aloff Wallach']}$
  12. Computation of spherical representations
  13. Comments on the other homogenous $3$-$(\alpha,\delta)$-Sasaki manifolds
  14. Geometric interpretation of the spectrum and electronic supplement
  15. Interpretation as a canonical variation of metrics over $\mathbb{CP}^2$
  16. ...and 3 more sections

Key Result

Lemma 2.3

The homomorphism $\mathop{\mathrm{\mathfrak{g}}}\nolimits\rightarrow \mathcal{D}(G)$ extends uniquely to an isomorphism $U(\mathop{\mathrm{\mathfrak{g}}}\nolimits)\rightarrow \mathcal{D}(G)$.

Figures (5)

  • Figure 1: $H$ weights of a $K'$-spherical representation $\varrho_{0}\otimes \varrho_{1}(z)$.
  • Figure 2: Spectrum with $t_{0}=0.5$, $t_{1}$ varying Agricola Henkel 24
  • Figure 3: $\mathop{\mathrm{SU}}\nolimits(3)\times \mathop{\mathrm{SO}}\nolimits(3)$-naturally reductive metrics of $W^{1,1}$.
  • Figure 4: First eigenvalue $\eta_{1}(g_{{0.5},t_{1}})$ depending on $t_{1}$ and its estimates. Nagy, Semmelmann: $f_{1}(t_{1},1)$, Lichnerowicz, Obata: $f_{2}(t_{1},1)$
  • Figure 5: Spectrum with constant volume Agricola Henkel 24

Theorems & Definitions (58)

  • Remark 2.1
  • Definition 2.2
  • Lemma 2.3: Helgason
  • Definition 2.4: Generalized Casimir operator
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Theorem 2.7: Takeuchi
  • Remark 2.8
  • Lemma 2.9
  • ...and 48 more