Spectrally distinguishing symmetric spaces I

Abstract

We prove that the irreducible symmetric space of complex structures on ${\mathbb{R}}^{2n}$ (resp.\ quaternionic structures on ${\mathbb{C}}^{2n}$) is spectrally unique within a $2$-parameter (resp.\ $3$-parameter) family of homogeneous metrics on the underlying differentiable manifold. Such families are strong candidates to contain all homogeneous metrics admitted on the corresponding manifolds. The main tool in the proof is an explicit expression for the smallest positive eigenvalue of the Laplace-Beltrami operator associated to each homogeneous metric involved. As a second consequence of this expression, we prove that any non-symmetric Einstein metric in the homogeneous families mentioned above is $\nu$-unstable.

References in corpus (6)

• Stability and instability of Ricci solitons
• Bifurcation and local rigidity of homogeneous solutions to the Yamabe problem on spheres
• The smallest Laplace eigenvalue of homogeneous 3-spheres
• Full Laplace spectrum of distance spheres in symmetric spaces of rank one
• Spectral uniqueness of bi-invariant metrics on symplectic groups
• Spectrally distinguishing symmetric spaces I

Cited by in corpus (2)

• Spectrally distinguishing symmetric spaces I
• Linear stability of Perelman's $\nu$-entropy of standard Einstein manifolds