https://neckpinch.com/authors/knut-smoczyk/New arXiv papers by Knut Smoczyk, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2403.10739arxiv:2403.10739Fri, 15 Mar 2024 00:00:00 GMTAndreas Savas-Halilaj, Knut Smoczyk — We consider the graphical mean curvature flow of maps bf f:ℝ^m→ℝⁿ, m≥ 2, and derive estimates on the growth rates of the evolved graphs, based on a new version of the maximum principle for properly immersed submanifolds that extends the well-known maximum principle of Ecker and Huisken derived in their seminal paper [10]. In the case of uniformly area decreasing maps bf f:ℝ^m→ℝ², m≥ 2, we use this maximum principle to show that the graphicality and the area decreasing property are preserved. Moreover, if the initial graph is asymptotically conical at infinity, we prove that the normalized mean curvature flow smoothly converges to a self-expander.Mean curvature flowSelf-shrinkers & translatorsBoundaries & non-compact https://arxiv.org/abs/2201.05523arxiv:2201.05523Fri, 14 Jan 2022 00:00:00 GMTRenan Assimos, Andreas Savas-Halilaj, Knut Smoczyk — We consider the graphical mean curvature flow of strictly area decreasing maps f:M→ N, where M is a compact Riemannian manifold of dimension m>1 and N a complete Riemannian surface of bounded geometry. We prove long-time existence of the flow and that the strictly area decreasing property is preserved, when the bi-Ricci curvature BRic_M of M is bounded from below by the sectional curvature σ_N of N. In addition, we obtain smooth convergence to a minimal map if Ric_M≥sup\0,sup_Nσ_N\. These results significantly improve known results on the graphical mean curvature flow in codimension 2.Mean curvature flowLong-time existence & convergence