2025/03/27 by Ilka Agricola, Agricola, Ilka, Henkel, Jonas · 1 citation
Mathematics · #43A85 #43A90 (secondary) #53C21 #53C25 #53C30 #58J50 (primary) #Advanced Algebra and Geometry #Differential Geometry (math.DG) #FOS: Mathematics #Mathematical Analysis and Transform Methods #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2503.21416
openalex publication_date 2025/03/27 · openalex created_date 2025/10/11 · openalex updated_date 2026/07/28
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 W1,1=SU(3)/S1; 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 W1,1 as a limiting case and cover all the positively curved SU(3)× 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.