# Local Stability of Einstein Metrics Under the Ricci Iteration

## TL;DR

This paper establishes a spectral condition for the local stability of positive Ricci curvature Einstein manifolds under Ricci iteration, and applies it to symmetric spaces of compact type.

## Contribution

It introduces a new spectral criterion for stability and analyzes the stability of various Einstein manifolds using this criterion.

## Key findings

- Positive Ricci Einstein manifolds are stable under Ricci iteration if their Lichnerowicz Laplacian spectrum meets certain conditions.
- Symmetric spaces of compact type satisfy the stability criterion.
- The paper provides a framework to assess stability of Einstein manifolds under Ricci flow.

## Abstract

We provide a sufficient condition for the local stability of closed Einstein manifolds of positive Ricci curvature under the Ricci iteration in terms of the spectrum of the Lichnerowicz Laplacian acting on divergence-free tensor fields. We use this result to consider the stability of several Einstein manifolds under the Ricci iteration, including symmetric spaces of compact type.