Modified mean curvature flow of entire locally Lipschitz radial graphs in hyperbolic space

Abstract

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 ℍ +1 . The modified mean curvature flow (MMCF)was firstly introduced by Xiao and the second author a few years back in [15], and it provides a tool using geometric flow to find such hypersurfaces with constant mean curvature in ℍ +1 . Similar to the usual mean curvature flow, the MMCF is the natural negative 2 -gradient flow of the area-volume functional (Σ) = (Σ) + (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). K E Y W O R D S constant mean curvature, hyperbolic space, interior gradient eatimates, modified mean curvature flow M S C ( 2 0 1 0 ) 35K20, 53C44, 58J351 INTRODUCTION Mean curvature flow (MCF) was first studied by Brakke [4] in the context of geometric measure theory. Later, smooth compact surfaces evolved by MCF in Euclidean space were investigated by Huisken in [11] and [12], and in arbitrary ambient manifolds in [13]. The evolution of entire graphs by MCF in ℝ +1 was also studied in [6], the result being improved in [7]. Lately, the MCF in Euclidean space has attracted much attention. See, e.g., the survey of various aspects of the MCF of hypersurfaces by Colding, Minicozzi and Pedersen [5] and the references therein. In [19], Unterberger considered the MCF in hyperbolic space ℍ +1 and proved that if the initial surface Σ 0 has bounded hyperbolic height over + , (i.e., Σ 0 = + ), then under the MCF, Σ converges in ∞ to + , which is minimal. The Asymptotic Plateau Problem of finding smooth complete hypersurfaces of constant mean curvature in hyperbolic space ℍ +1 with prescribed asymptotic boundary at infinity has also been studied over the years, see [1], [9], [14], [18] and [16].This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.