https://neckpinch.com/authors/patrick-allmann/New arXiv papers by Patrick Allmann, from Neckpinch Tue, 10 Nov 2026 04:10:28 GMT https://neckpinch.com/papers/1707.07087/arxiv:1707.07087Sat, 22 Jul 2017 00:00:00 GMTPatrick Allmann, Longzhi Lin, Jingyong Zhu — The asymptotic Plateau problem asks for the existence of smooth complete hypersurfaces of constant mean curvature with prescribed asymptotic boundary at infinity in the hyperbolic space ℍⁿ⁺¹. The modified mean curvature flow (MMCF) was firstly introduced by Xiao and the second author a few years back, and it provides a tool using geometric flow to find such hypersurfaces with constant mean curvature in ℍⁿ⁺¹. Similar to the usual mean curvature flow, the MMCF is the natural negative L²-gradient flow of the area-volume functional 𝓘(Σ)=A(Σ)+σV(Σ) associated to a hypersurface Σ. In this paper, we prove that the MMCF starting from an entire locally Lipschitz continuous radial graph exists and stays radially graphic for all time. In general one cannot expect the convergence of the flow as it can be seen from the flow starting from a horosphere (whose asymptotic boundary is degenerate to a point).Mean curvature flowMinimal & CMC surfaces