Selfsimilar solutions to the mean curvature flow

Stathopoulos, Nikolaos

doi:10.1515/crll.1998.057

Cited by 28 publications

(23 citation statements)

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“…where α > 0 and k 1 > k 0 , with k 0 the constant appearing in (19). We claim that ϕ is a supersolution of (18) if α large enough.…”

Section: Maximum Principlementioning

confidence: 80%

“…As in the case of the mean curvature flow, see [9,16,19], it turns out that M t moves out to infinity as time increases, and that it converges to a smooth limit after an appropriate rescaling. As in [9], the convergence of the rescaled flow will be proved under an additional assumption on the initial value which controls the possible oscillations at infinity: there exist constants 0 < δ < 1 and K 0 > 0 such that…”

Section: Asymptotic Behaviourmentioning

confidence: 80%

“…The same authors [10] showed that global time existence holds for any initial data which is an entire graph over R n , regardless of its growth. After this, Stavrou [19] proved the convergence to a selfsimilar profile of Lipschitz graphs having a unique cone at infinity, while Rasul [16] obtained a convergence result under a weaker oscillation condition than in [9]. The long time behaviour without rescaling has been investigated for special classes of solutions: Clutterbuck, Schnürer and Schulze [8] have proved stability of translating solutions of the graphical mean curvature flow, while Clutterbuck and Schnürer [7] have considered the stability of mean convex cones.…”

Section: ∂F ∂T (P T) = −S (P T) ν (P T)mentioning

confidence: 99%

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Evolution of convex entire graphs by curvature flows

Alessandroni¹,

Sinestrari²

2015

Geometric Flows

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Abstract:We consider the evolution of an entire convex graph in euclidean space with speed given by a symmetric function of the principal curvatures. Under suitable assumptions on the speed and on the initial data, we prove that the solution exists for all times and it remains a graph. In addition, after appropriate rescaling, it converges to a homothetically expanding solution of the flow. In this way, we extend to a class of nonlinear speeds the well known results of Ecker and Huisken for the mean curvature flow. MSC: 53C44

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“…where α > 0 and k 1 > k 0 , with k 0 the constant appearing in (19). We claim that ϕ is a supersolution of (18) if α large enough.…”

Section: Maximum Principlementioning

confidence: 80%

Section: Asymptotic Behaviourmentioning

confidence: 80%

Section: ∂F ∂T (P T) = −S (P T) ν (P T)mentioning

confidence: 99%

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Evolution of convex entire graphs by curvature flows

Alessandroni¹,

Sinestrari²

2015

Geometric Flows

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“…Further they show using this condition that the convergence is exponentially fast in time to an expanding self-similar solution. Stavrou in his paper [16] proves asymptotic convergence using a much weaker condition to that in [6]. In their later paper Ecker and Huisken [7] used the local properties of mean curvature ow to obtain interior estimates, to deduce the fact that if the initial graph was only Lipschitz continuous, then it would have a smooth solution for all time under mean curvature ow, without the need for any assumption on the growth and curvature of the graph at in nity.…”

Section: A(t) := Maxmentioning

confidence: 99%

“…Lemma 6.1 ensures that f 2ρ vanishes at in nity which enables us to apply the parabolic maximum principle to conclude that f 2ρ is uniformly bounded by its initial data. Finally we use the result of Stavrou [16] to conclude uniform convergence to self-similar solutions, since our assumption is stronger than his. …”

Section: Chaptermentioning

confidence: 99%

Slow convergence of graphs under mean curvature flow

Rasul

2010

Communications in Analysis and Geometry

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In loving memory of my mother Zahida Perveen ErklärungIch bestätige hiermit, dass ich diese Arbeit selbständig verfasst und keine anderen als die angegebenen Hilfsmittel und Quellen verwendet habe. Kashif RasulTag der mündlichen Qualifikation: 27. Januar 2010 1. Gutachter: Prof. Dr. Klaus Ecker 2. Gutachterin: Dr. Maria Athanassenas AbstractIn this thesis we study the mean curvature flow of entire graphs in Euclidean space. From the work of Ecker and Huisken, we know that given some initial growth condition at infinity, such graphs become self-similar under the evolution and the convergence is exponentially fast in time.In this work, we propose an alternative condition at infinity, motivated by looking at the heat equation, and show that under mean curvature flow such a growth condition is preserved for the height and gradient of the graph. For the curvature we propose an analogous result to that of Ecker and Huisken, by proving a spatial decay estimate with slightly stronger condition.Our main result then says that under mean curvature flow and our condition, the graph also becomes self similar, but slower than in the exponential case. ZusammenfassungIn dieser Doktorarbeit wird der mittlere Krümmungsfluss von ganzen Graphen im euklidischen Raum betrachtet. Ecker und Huisken zeigen, dass unter gewissen Wachstumsbedingungen im Unendlichen, solche Graphen unter dem Fluss selbst ähnlich werden, wobei die Konvergenz in der Zeit exponentiell schnell ist.Durch die Wärmeleitungsgleichung motiviert, schlagen wir in dieser Arbeit eine alternative Bedingung im Unendlichen vor und zeigen, dass solch eine Wachstumsbedingung für die Höhe und den Gradienten unter dem mittleren Krümmungsfluss erhalten bleibt.Unser Hauptresultat besagt, dass unter der alternativen Bedingung, der Graph einer Lösung des mittleren Krümmungsflusses ebenfalls selbstähnlich wird, allerdings mit einer in der Zeit langsameren Konvergenzrate.

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Geometric Flow Equations

Schnürer

2018

Tutorials, Schools, and Workshops in the Mathematical Sciences

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Selfsimilar solutions to the mean curvature flow

Cited by 28 publications

References 0 publications

Evolution of convex entire graphs by curvature flows

Evolution of convex entire graphs by curvature flows

Slow convergence of graphs under mean curvature flow

Geometric Flow Equations

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