https://neckpinch.com/authors/markus-wolff/New arXiv papers by Markus Wolff, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT 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/2312.07317arxiv:2312.07317Tue, 12 Dec 2023 00:00:00 GMTMarkus Wolff — We consider null mean curvature flow along the standard lightcone in the de Sitter spacetime. This flow was first studied by Roesch–Scheuer along null hypersurfaces for the detection of MOTS, and independently by the author in the specific case of the standard Minkowski lightcone. Similar to the Minkowski case, null mean curvature flow along the de Sitter lightcone can be related to 2d-Ricci flow for surfaces of genus 0 by an appropriate rescaling. Building on this rescaling procedure, we analyse singularity formation, asymptotic behavior and ancient solutions to the flow.Ricci flowMean curvature flowAncient solutionsSingularitiesPhysics https://arxiv.org/abs/2306.10892arxiv:2306.10892Mon, 19 Jun 2023 00:00:00 GMTMarkus Wolff — We prove an analogue statement to an estimate by De Lellis-Müller in ℝ³ on the standard Minkowski lightcone. More precisely, we show that under some additional assumptions, any spacelike cross section of the standard lightcone is W^(2,2)-close to a round surface provided the trace-free part of a scalar second fundamental form A is sufficiently small in L². To determine the correct intrinsically round cross section of reference, we define an associated 4-vector, which transforms equivariantly under Lorentz transformations in the restricted Lorentz group. A key step in the proof consists of a geometric, scaling invariant estimate, and we give two different proofs. One utilizes a recent characterization of singularity models of null mean curvature flow along the standard lightcone by the author, while the other is heavily inspired by an almost-Schur lemma by De Lellis-Topping.Mean curvature flowSingularitiesPhysics https://arxiv.org/abs/2208.05709arxiv:2208.05709Thu, 11 Aug 2022 00:00:00 GMTGerhard Huisken, Markus Wolff — We construct weak solutions for the evolution of hypersurfaces along their inverse space-time mean curvature in asymptotically flat maximal initial data sets. As the speed of the new flow is given by a space-time invariant, it can detect both future- and past-trapped apparent horizons. The weak solution extends concepts developed by Huisken-Ilmanen for inverse mean curvature flow and by Moore for inverse null mean curvature flow.Mean curvature flowOther curvature flowsInverse mean curvature flowWeak & singular flowsBoundaries & non-compactPhysics https://arxiv.org/abs/2206.02621arxiv:2206.02621Mon, 06 Jun 2022 00:00:00 GMTMarkus Wolff — Identifying any conformally round metric on the 2-sphere with a unique cross section on the standard lightcone in the 3+1-Minkowski spacetime, we gain a new perspective on 2d-Ricci flow on topological spheres. It turns out that in this setting, Ricci flow is equivalent to a null mean curvature flow first studied by Roesch–Scheuer along null hypersurfaces. Exploiting this equivalence, we can translate well-known results from 2d-Ricci flow first proven by Hamilton into a full classification of the singularity models for null mean curvature flow in the Minkowski lightcone. Conversely, we obtain a new proof of Hamilton's classical result using only the maximum principle.Ricci flowMean curvature flowSingularitiesSurfaces