https://neckpinch.com/authors/wolfgang-maurer/New arXiv papers by Wolfgang Maurer, from Neckpinch Tue, 10 Nov 2026 04:10:28 GMT https://neckpinch.com/papers/2104.00057/arxiv:2104.00057Wed, 31 Mar 2021 00:00:00 GMTWolfgang Maurer — For a mean curvature flow of complete graphical hypersurfaces Mₜ=graph u(·,t) defined over domains Ωₜ, the enveloping cylinder is ∂Ωₜ×ℝ. We prove the smooth convergence of Mₜ−h eₙ₊₁ to the enveloping cylinder under certain circumstances. Moreover, we give examples demonstrating that there is no uniform curvature bound in terms of the inital curvature and the geometry of Ωₜ. Furthermore, we provide an example where the hypersurface increasingly oscillates towards infinity in both space and time. It has unbounded curvature at all times and is not smoothly asymptotic to the enveloping cylinder. We also prove a relation between the initial spatial asymptotics at the boundary and the temporal asymptotics of how the surface vanishes to infinity for certain rates in the case Ωₜ are balls.Mean curvature flow https://neckpinch.com/papers/2103.06072/arxiv:2103.06072Wed, 10 Mar 2021 00:00:00 GMTWolfgang Maurer — By a symmetric double graph we mean a hypersurface which is mirror-symmetric and the two symmetric parts are graphs over the hyperplane of symmetry. We prove that there is a weak solution of mean curvature flow that preserves these properties and singularities only occur on the hyperplane of symmetry. The result can be used to construct smooth solutions to the free Neumann boundary problem on a supporting hyperplane with singular boundary. For the construction we introduce and investigate a notion named "vanity" and which is similar to convexity. Moreover, we rely on Sáez' and Schnürer's "mean curvature flow without singularities" to approximate weak solutions with smooth graphical solutions in one dimension higher.Mean curvature flowSingularitiesWeak & singular flowsFree boundary & capillary https://neckpinch.com/papers/1903.03502/arxiv:1903.03502Fri, 08 Mar 2019 00:00:00 GMTKlaus Kroencke, Oliver Lindblad Petersen, Felix Lubbe, Tobias Marxen, Wolfgang Maurer, Wolfgang Meiser, Oliver C. Schnürer, Áron Szabó, Boris Vertman — We consider the long-time behaviour of the mean curvature flow of spacelike hypersurfaces in the Lorentzian product manifold M×ℝ, where M is asymptotically flat. If the initial hypersurface F₀⊂ M×ℝ is uniformly spacelike and asymptotic to M×s\ for some s∈ℝ at infinity, we show that a mean curvature flow starting at F₀ exists for all times and converges uniformly to M×s\ as t→ ∞.Mean curvature flowBoundaries & non-compactLong-time existence & convergencePhysics https://neckpinch.com/papers/1604.05238/arxiv:1604.05238Mon, 18 Apr 2016 00:00:00 GMTWolfgang Maurer — We consider mean curvature flow of an initial surface that is the graph of a function over some domain of definition in Rⁿ. If the graph is not complete then we impose a constant Dirichlet boundary condition at the boundary of the surface. We establish longtime-existence of the flow and investigate the projection of the flowing surface onto Rⁿ, the shadow of the flow. This moving shadow can be seen as a weak solution for mean curvature flow of hypersurfaces in Rⁿ with a Dirichlet boundary condition. Furthermore, we provide a lemma of independent interest to locally mollify the boundary of an intersection of two smooth open sets in a way that respects curvature conditions.Mean curvature flowWeak & singular flowsBoundaries & non-compactLong-time existence & convergence