Spectra, rigidity and stability of sine-cones

TL;DR

This paper analyzes the spectral properties of geometric operators on sine-cones over Einstein manifolds to determine conditions for their stability and rigidity under Ricci flow.

Contribution

It provides explicit spectral computations and stability criteria for sine-cones over Einstein manifolds, linking geometric spectra to stability analysis.

Findings

Spectra of Laplace-Beltrami, connection Laplacian, and Einstein operator computed

Conditions identified for sine-cone stability under Ricci-de Turck flow

Sine-cones shown to be rigid Einstein manifolds under certain conditions

Abstract

We compute the spectra of the Laplace-Beltrami operator, the connection Laplacian on 1-forms and the Einstein operator on symmetric 2-tensors on the sine-cone over a positive Einstein manifold $(M, g)$. We conclude under which conditions on $(M,g)$, the sine-cone is dynamically stable under the singular Ricci-de Turck flow and rigid as a singular Einstein manifold