Ricci flow and the scalar curvature rigidity of Einstein manifolds
Klaus Kroencke
TL;DR
This survey analyzes how dynamical stability under Ricci flow and scalar curvature rigidity for Einstein manifolds are intertwined across closed, open, ALE, and AH geometries. It highlights the roles of the Einstein operator $\Delta_E$, the $\lambda$-functional, and the Yamabe invariant as unifying variational tools, and introduces mass-type invariants ($m_{ADM}$, $m_{VR,\hat{g}}$, $\lambda_{\text{ALE}}$, $\mu_{AH,\hat{g}}$) that connect rigidity phenomena to positive mass theorems. The results establish equivalences between linear stability, variational extremality, and dynamical stability in multiple settings, with spectral, decay, and holonomy conditions shaping the picture; they extend to noncompact geometries via precise asymptotics and boundary data. Collectively, the work provides a comprehensive framework linking rigidity of scalar curvature to the long-time behavior of Ricci flow and to PMT-type obstructions in ALE and AH contexts, including an AH analogue of Ilmanen’s conjecture.
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.