https://neckpinch.com/authors/klaus-kroncke/New arXiv papers by Klaus Kröncke, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2510.24529arxiv:2510.24529Tue, 28 Oct 2025 00:00:00 GMTKlaus Kroencke — 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.Ricci flowEinstein metricsScalar curvatureStability & uniquenessBoundaries & non-compactPhysics https://arxiv.org/abs/2412.17563arxiv:2412.17563Mon, 23 Dec 2024 00:00:00 GMTKlaus Kroencke, Markus Wolff — We construct asymptotic foliations of asymtotically Schwarzschildean lightcones by surfaces of constant spacetime mean curvature (STCMC). Our construction is motivated by the approach of Huisken-Yau for the Riemannian setting in employing a geometric flow. We prove that initial data within a sufficient a-priori class converges exponentially to an STCMC surface under area preserving null mean curvature flow. Further, we show that the resulting STCMC surfaces form an asymptotic foliation that is unique within the a-priori class.Mean curvature flowLong-time existence & convergence https://arxiv.org/abs/2312.13011arxiv:2312.13011Wed, 20 Dec 2023 00:00:00 GMTKlaus Kroencke, Louis Yudowitz — We prove dynamical stability and instability theorems for Poincaré-Einstein metrics under the Ricci flow. Our key tool is a variant of the expander entropy for asymptotically hyperbolic manifolds, which Dahl, McCormick and the first author established in a recent article. It allows us to characterize stability and instability in terms of a local positive mass theorem and in terms of volume comparison for nearby metrics.Ricci flowRicci solitonsEinstein metricsStability & uniquenessBoundaries & non-compactPhysics https://arxiv.org/abs/2307.06196arxiv:2307.06196Wed, 12 Jul 2023 00:00:00 GMTMattias Dahl, Klaus Kroencke, Stephen McCormick — We define a geometric quantity for asymptotically hyperbolic manifolds, which we call the volume-renormalized mass. It is essentially a linear combination of the ADM mass surface integral and a renormalization of the volume. We show that the volume-renormalized mass is well-defined and diffeomorphism invariant under weaker fall-off conditions than required to ensure that the renormalized volume and the ADM mass surface integral are well-defined separately. We prove several positivity results for the volume-renormalized mass. We also use it to define a renormalized Einstein–Hilbert action and a renormalized expander entropy which is nondecreasing under the Ricci flow. Further, we show that local maximizers of the entropy are local minimizers of the volume-renormalized mass.Ricci flowRicci solitonsBoundaries & non-compactPhysics