Mean Curvature Flow in Higher Codimension: Introduction and Survey

Smoczyk, Knut

doi:10.1007/978-3-642-22842-1_9

Cited by 61 publications

(45 citation statements)

References 84 publications

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“…Suppose V is any time-dependent vector field on M . As in [30], for any given local coordinates y α on R n,m in a neighbourhood of a point in M we define…”

Section: Flow Quantitiesmentioning

confidence: 99%

Spacelike Mean Curvature Flow

Lambert

Lotay

2019

J Geom Anal

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We prove long-time existence and convergence results for spacelike solutions to mean curvature flow in the pseudo-Euclidean space R n,m , which are entire or defined on bounded domains and satisfying Neumann or Dirichlet boundary conditions. As an application, we prove long-time existence and convergence of the G 2 -Laplacian flow in cases related to coassociative fibrations. ContentsSpacelike mean curvature flow has been studied in codimension 1 by Ecker and Huisken [14], Ecker [11][12] and also Gerhardt [17]. The first author has also worked on boundary conditions for this flow [21] [22]. The elliptic counterpart was studied by Bartnik [3] and Bartnik and Simon [4]. For higher codimensions, less is known. The flow of compact manifolds was investigated by G. Li and Salavessa [24]. The higher codimensional maximal surface equation was recently studied by Y. Li [25].Entire graphs. There are several well-known explicit long-time solutions to spacelike MCF. Throughout the article we let ·, · denote the standard quadratic form with signature (n, m) on R n,m and let |x| 2 = x, x for x ∈ R n,m . Recall that x = 0 is spacelike if |x| 2 > 0, lightlike or null if |x| 2 = 0 and timelike if |x| 2 < 0. The light cone is the set of lightlike vectors.Example 1.1 (Grim Reaper). The Grim Reaper is the unique translating solution to (1.3) in R 1,1 (up to translations, dilations and rotations), given bŷis a self-expander for (1.2) (i.e. X ⊥ = tH) coming out of the light cone. For each t, M t is an embedded hyperbolic space in R n,1 .Explicit solutions may be constructed from Examples 1.1 and 1.2 in higher codimension, simply by evolving in R n,1 ⊂ R n,m .All the examples described thus far are entire graphs, and so it is natural to study this setting, where we have the following existence theorem. Theorem 1.3. Let Ω = R n , so the initial spacelike submanifold M 0 is an entire graph. There exists a spacelike solution M t of mean curvature flow starting at M 0 which is smooth and exists for all t > 0. Furthermore, if the mean curvature of M 0 is bounded, then M t attains the initial data M 0 smoothly as t → 0. See Theorem 5.2 for further details. Notice that we make no assumption on the spacelike condition at infinity, so we can start with initial data that asymptotically develops lightlike directions (like the Grim Reaper), and that we obtain long-time existence even without an initial bound on the mean curvature. This theorem is an extension and improvement of the codimension 1 result proven in [11, Theorem 4.2] (see also Remark 4.4).Remark 1.4. As is to be expected for entire flows we make no statement about uniqueness in Theorem 1.3, and solutions to (1.3) are not unique in general. For example, if we take M 0 to be the Grim Reaper in Example 1.1 at t = 0, which is a translating solution, then any solution constructed by our proof of Theorem 1.3 cannot remain a translator (since it would satisfy |û(x, t) −û 0 (x)| ≤ √ 2nt).We are also prove results on the qualitative behaviour of entire flows. In Section 8 we develop further e...

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“…Suppose V is any time-dependent vector field on M . As in [30], for any given local coordinates y α on R n,m in a neighbourhood of a point in M we define…”

Section: Flow Quantitiesmentioning

confidence: 99%

Spacelike Mean Curvature Flow

Lambert

Lotay

2019

J Geom Anal

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“…The latter theorem was later generalized to MCFs in both spherical and hyperbolic space forms, see Baker ([4]) and Liu-Xu-Ye-Zhao ( [32] and [33]). For other progresses on the MCFs, we refer the readers to the references [2], [31], [42] and [44] etc.…”

Section: Introductionmentioning

confidence: 99%

The blow-up of the conformal mean curvature flow

Zhang

2019

Sci. China Math.

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In this paper, we introduce and study the conformal mean curvature flow of submanifolds of higher codimension in the Euclidean space R n . This kind of flow is a special case of a general modified mean curvature flow which is of various origination. As the main result, we prove a blow-up theorem concluding that, under the conformal mean curvature flow in R n , the maximum of the square norm of the second fundamental form of any compact submanifold tends to infinity in finite time. Furthermore, by using the idea of Andrews and Baker for studying the mean curvature flow of submanifolds in the Euclidean space, we also derive some more evolution formulas and inequalities which we believe to be useful in our further study of conformal mean curvature flow. Presently, these computations together with our main theorem are applied to provide a direct proof of a convergence theorem concluding that the external conformal forced mean curvature flow of a compact submanifold in R n with the same pinched condition as Andrews-Baker's will be convergent to a round point in finite time.2000 Mathematics Subject Classification. Primary 53A30; Secondary 53B25. Key words and phrases. conformal mean curvature flow, conformal external force, blow-up of the curvature, round point.

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“…For example, if L 0 is a compact submanifold in Euclidean space, it follows from [16,Proposition 3.10] that the maximal time of existence T of the mean curvature flow is finite. For any generalized isoparametric foliation, we prove in Proposition 3.3 that there is a neighbourhood around the singular leaves in which the MCF has always finite time singularities.…”

Section: Introductionmentioning

confidence: 99%

Mean Curvature Flow of Singular Riemannian Foliations

Alexandrino

Radeschi

2015

J Geom Anal

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Abstract. Given a singular Riemannian foliation on a compact Riemannian manifold, we study the mean curvature flow equation with a regular leaf as initial datum. We prove that if the leaves are compact and the mean curvature vector field is basic, then any finite time singularity is a singular leaf, and the singularity is of type I. These results generalize previous results of Liu and Terng, Pacini and Koike. In particular our results can be applied to partitions of Riemannian manifolds into orbits of actions of compact groups of isometries.

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Mean Curvature Flow in Higher Codimension: Introduction and Survey

Cited by 61 publications

References 84 publications

Spacelike Mean Curvature Flow

Spacelike Mean Curvature Flow

The blow-up of the conformal mean curvature flow

Mean Curvature Flow of Singular Riemannian Foliations

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