Stability of sin-cones and cosh-cylinders

TL;DR

This paper investigates the stability and instability of Einstein warped products, specifically sin-cones and cosh-cylinders, linking stability to spectral properties and analyzing Ricci flow convergence.

Contribution

It provides a comprehensive spectral characterization of stability for sin-cones and cosh-cylinders, including new results on Ricci flow convergence and stability over symmetric spaces.

Findings

Stability of sin-cones is determined by Laplacian and Einstein operator spectra.

Cosh-cylinders exhibit convergence under Ricci flow from small perturbations.

Complete stability classification for sin-cones over symmetric spaces of compact type.

Abstract

This work concerns stability and instability of Einstein warped products with an Einsteinian fiber of codimension 1. We study the cases where the scalar curvature of the warped product and of the fiber are either both positive or both negative to complement the results in [Kr\"o16]. Up to a small gap in the case of sin-cones, the stability properties of such warped products are now completely determined by spectral properties of the Laplacian and the Einstein operator of the fiber. For cosh-cylinders, we are furthermore able to prove a convergence result for the Ricci flow starting in a small neighbourhood. As an interesting class of examples, we determine the stability properties of sin-cones over symmetric spaces of compact type.