Modified mean curvature flow of star-shaped hypersurfaces in hyperbolic space

Lin, Longzhi; Xiao, Ling

doi:10.4310/cag.2012.v20.n5.a6

Cited by 8 publications

(20 citation statements)

References 37 publications

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“…For non-convex sets, most available results appears to concern (1.4), and for dimensions larger than two the discussion focuses on classification of singularities that could occur before extinction ( [2], [31], [32], [39], [35]). In two dimensions, for (1.4), Angenent proves in [4]- [5] that the number of intersections of a pair of curves does not increase over the evolution, and in particular simply connected domains stay so until it shrinks to a point.…”

Section: Literaturementioning

confidence: 99%

On mean curvature flow with forcing

Kim¹,

Kwon²

2019

Communications in Partial Differential Equations

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This paper investigates geometric properties and well-posedness of a mean curvature flow with volume-dependent forcing. We prove that the flow preserves the ρ-reflection property, which corresponds to a quantitative Lipschitz property of the set with respect to the nearest ball. Based on this property we show that the problem is well-posed and its solutions starting with ρ-reflection property become smooth in finite time. Lastly, for a model problem, we will discuss the flow's exponential convergence to the unique equilibrium in Hausdorff topology. For the analysis, we adopt the approach developed in [24] to combine viscosity solutions approach and variational method. The main challenge lies in the lack of comparison principle, which accompanies forcing terms that penalize small volume.

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Section: Literaturementioning

confidence: 99%

On mean curvature flow with forcing

Kim¹,

Kwon²

2019

Communications in Partial Differential Equations

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“…Hence for any x ∈ Ω there exists a sequence t k → ∞ such that (F − σ)u(x, t k ) → 0. On the other hand, u(x, ·) is monotone increasing and bounded (see Lemma 3.3 of [LX10]). Therefore exists, and is of class C ∞ (Ω) ∩ C 1 (Ω).…”

Section: Convergence To a Stationary Solutionmentioning

confidence: 99%

“…In this paper, we continue our study of modified curvature flow problems in hyperbolic space. Consider a complete (locally strictly) convex hypersurface in H n+1 with a prescribed asymptotic boundary Γ at infinity, whose principal curvature is greater than σ (e.g in our earlier work [LX10] section 8 we gave an example of such "good" initial surfaces.) and is given by an embedding X(0) : Ω → H n+1 , where Ω ⊂ ∂ ∞ H n+1 .…”

Section: Introductionmentioning

confidence: 99%

Curvature Flow of Complete Convex Hypersurfaces in Hyperbolic Space

Xiao

2012

J Geom Anal

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“…In [17], among others, De Silva and Spruck recovered this result using the method of calculus of variations. In the previous joint work [15] of Xiao and the second author, the following modified mean curvature flow (MMCF) was first introduced, which is the natural negative L 2 ‐gradient flow of the area‐volume functional

I false(normalΣ false) = {scriptI}_{normalΩ} false(v false) = A_{normalΩ} false(v false) + σ V_{normalΩ} false(v false)

associated to Σ as in [17]. It can be used to continuously deform hypersurfaces in

H^{n + 1}

into constant mean curvature hypersurfaces with prescribed asymptotic boundary at infinity.…”

Section: Introductionmentioning

confidence: 99%

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

Allmann

Lin

Zhu

2020

Mathematische Nachrichten

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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.

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Modified mean curvature flow of star-shaped hypersurfaces in hyperbolic space

Cited by 8 publications

References 37 publications

On mean curvature flow with forcing

On mean curvature flow with forcing

Curvature Flow of Complete Convex Hypersurfaces in Hyperbolic Space

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

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