Spectrally distinguishing symmetric spaces I

TL;DR

This paper proves spectral uniqueness of certain symmetric spaces within specific metric families and shows non-symmetric Einstein metrics in these families are unstable, using explicit eigenvalue expressions.

Contribution

It establishes spectral uniqueness of irreducible symmetric spaces of complex and quaternionic structures within parameterized metric families and analyzes their stability.

Findings

Spectral uniqueness of symmetric spaces proven.

Explicit eigenvalue formulas derived for homogeneous metrics.

Non-symmetric Einstein metrics are shown to be $ u$-unstable.

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.