Ricci flow and the scalar curvature rigidity of Einstein manifolds

2025/10/28 by Kroencke, Klaus
#53C21 #53C25 #53E20 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2510.24529

Abstract

We review recent results relating linear stability to dynamical stability and the scalar curvature rigidity of Einstein manifolds. We discuss closed and open Einstein manifolds as well as complete noncompact Einstein manifolds which are asymptotically locally Euclidean and asymptotically hyperbolic. For these classes, the relation to the positive mass theorem will also be explained.

Citations

• Einstein metrics, their moduli spaces and stability
• The Volume-Renormalized Mass from a Hamiltonian Perspective
• Green functions and a positive mass theorem for asymptotically hyperbolic 3-manifolds
• Dynamical Stability and Instability of Poincaré--Einstein Manifolds
• A volume-renormalized mass for asymptotically hyperbolic manifolds
• Dynamical (in)stability of Ricci-flat ALE metrics along Ricci flow
• Convergence of the Ricci flow to Ricci-flat ALE manifolds and positive scalar curvature rigidity
• A Łojasiewicz inequality for ALE metrics
• Eguchi-Hanson singularities in U(2)-invariant Ricci flow
• Holonomy rigidity for Ricci-flat metrics
• The stability inequality for Ricci-flat cones
• Stability of symmetric spaces of noncompact type under Ricci flow
• Stability of hyperbolic space under Ricci flow
• Geometric flows with rough initial data
• Stability of Euclidean space under Ricci flow
• On the Stability of Riemannian Manifold with Parallel Spinors
• The mass of asymptotically hyperbolic Riemannian manifolds
• The mass of spacelike hypersurfaces in asymptotically anti-de Sitter space-times
• Einstein metrics and complex structures
• A new proof of the positive energy theorem
• On the proof of the positive mass conjecture in general relativity

Related

• Dynamical Stability and Instability of Poincaré--Einstein Manifolds
• Quasi-Einstein Metrics and a curvature identity assoicated to the Ricci flow
• Convergence of the Ricci flow to Ricci-flat ALE manifolds and positive scalar curvature rigidity
• Scalar curvature rigidity and Ricci Deturck flow on perturbations of Euclidean Space
• Volume comparison of conformally compact manifolds with scalar curvature R≥ -n(n-1)