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Existence of mean curvature flow singularities with bounded mean curvature [PDF]
Duke Mathematical Journal, 2020 In [Vel94], Velazquez constructed a countable collection of mean curvature flow solutions in $\mathbb{R}^N$ in every dimension $N \ge 8$. Each of these solutions becomes singular in finite time at which time the second fundamental form blows up.
M. Stolarski
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A remark on soliton equation of mean curvature flow [PDF]
Anais da Academia Brasileira de Ciências, 2004 In this note, we consider self-similar immersions of the mean curvature flow and show that a graph solution of the soliton equation, provided it has bounded derivative, converges smoothly to a function which has some special properties (see Theorem 1.1 ...
Li Ma, Yang Yang
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Self-Expanders of the Mean Curvature Flow [PDF]
Vietnam Journal of Mathematics, 2020 We study self-expanding solutions M m ⊂ ℝ n $M^{m}\subset \mathbb {R}^{n}$ of the mean curvature flow. One of our main results is, that complete mean convex self-expanding hypersurfaces are products of self-expanding curves and flat subspaces, if and ...
Knut Smoczyk
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Width and mean curvature flow [PDF]
Geometry & Topology, 2007 Given a Riemannian metric on a homotopy $n$-sphere, sweep it out by a continuous one-parameter family of closed curves starting and ending at point curves. Pull the sweepout tight by, in a continuous way, pulling each curve as tight as possible yet preserving the sweepout.
Tobias Colding, William P. Minicozzi
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The hyperbolic mean curvature flow
Journal de Mathématiques Pures et Appliquées, 2007 26 ...
Philippe G. LeFloch, Knut Smoczyk
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On the extension of the mean curvature flow [PDF]
Mathematische Zeitschrift, 2009 Consider a family of smooth immersions $F(\cdot,t): M^n\to \mathbb{R}^{n+1}$ of closed hypersurfaces in $\mathbb{R}^{n+1}$ moving by the mean curvature flow $\frac{\partial F(p,t)}{\partial t} = -H(p,t)\cdot (p,t)$, for $t\in [0,T)$. In \cite{Cooper} Cooper has recently proved that the mean curvature blows up at the singular time $T$. We show that if
Nam Q. Le, Nataša Šešum
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Singularities of mean curvature flow and flow with surgeries [PDF]
Surveys in Differential Geometry, 2007 We collect in this paper several results on the formation of singularities in the mean curvature flow of hypersurfaces in euclidean space, under various kinds of convexity assumptions. We include some recent estimates for the flow of 2-convex surfaces, i.e. the surfaces where the sum of the two smallest principal curvatures is positive everywhere. Such
Carlo Sinestrari
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Inverse mean curvature flow and Ricci-pinched three-manifolds [PDF]
Journal für die Reine und Angewandte Mathematik, 2023 Let ( M , g ) (M,g) be a noncompact, connected, complete Riemannian three-manifold with nonnegative Ricci curvature satisfying Ric ≥ ε tr ( Ric ) g \mathrm{Ric}\geq\varepsilon\operatorname{tr}(\mathrm{Ric})g for some ε > 0 \varepsilon>0 .
G. Huisken, T. Koerber
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Weak-strong uniqueness for volume-preserving mean curvature flow [PDF]
Revista matemática iberoamericana, 2022 In this note, we derive a stability and weak-strong uniqueness principle for volume-preserving mean curvature flow. The proof is based on a new notion of volume-preserving gradient flow calibrations, which is a natural extension of the concept in the ...
Tim Laux
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Numerical analysis for the interaction of mean curvature flow and diffusion on closed surfaces [PDF]
Numerische Mathematik, 2022 An evolving surface finite element discretisation is analysed for the evolution of a closed two-dimensional surface governed by a system coupling a generalised forced mean curvature flow and a reaction–diffusion process on the surface, inspired by a ...
C. M. Elliott +2 more
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