Explicit Laplace Spectra of Homogeneous Principal Bundles

TL;DR

This paper develops a representation-theoretic method to explicitly compute the Laplace spectra of homogeneous principal bundles, applying it to classify stability and bifurcation phenomena in specific geometric families.

Contribution

It introduces a unified approach for spectral computation on homogeneous bundles and derives explicit spectra for classical 3-$( heta,eta)$-Sasaki and Stiefel manifolds, advancing geometric analysis.

Findings

Computed full spectra for classical 3-$( heta,eta)$-Sasaki manifolds.

Derived explicit eigenvalues and multiplicities for Stiefel manifolds.

Classified scalar stability and identified Yamabe bifurcation thresholds.

Abstract

We present a unified representation-theoretic method to compute the Laplace-Beltrami spectrum on homogeneous principal bundles. For this setting, we introduce a multi-parameter family of metric deformations called generalized canonical variations. Building upon the geometric realization of such fibrations as naturally reductive spaces, we establish a simplified spectral branching criterion. We apply this method to derive the full spectra (yielding all eigenvalues and multiplicities) for several prominent geometric families. Specifically, we compute the full spectra for the entire classical series of homogeneous 3-$(\alpha,\delta)$-Sasaki manifolds (Types A, B, C, and D) and for real and complex Stiefel manifolds over Grassmannians. These explicit formulas provide the analytical data to investigate related problems in geometric analysis. As an application, we classify the scalar stability of these spaces under Perelman's $\nu$-entropy and, for the 3-$(\alpha,\delta)$-Sasaki manifolds, determine the exact thresholds for Yamabe bifurcations.