Slow convergence of graphs under mean curvature flow

Abstract

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.

Evolution of convex entire graphs by curvature flows

“…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…”

“…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.…”

“…It is not restrictive to assume thatT 1 /2 satisfies the same bound imposed on t 0 in (16). Then, by the previous step, there existsx ∈ B 1 (0) such that…”

Self-Expanders of the Mean Curvature Flow

“…Later Stavrou [32] proved such a result under the weaker assumption that the initial hypersurface attains a unique tangent cone at infinity. Rasul [25] showed that under an alternative condition at infinity and bounded gradient, the rescaled graphs converge to self-similar solutions but at a slower speed. Clutterbuck and Schnürer [5] considered graphical solutions to mean curvature flow and obtained a stability result for homothetically expanding solutions coming out of cones of positive mean curvature.…”