Convex ancient solutions of the mean curvature flow

@article{Huisken2015ConvexAS,
  title={Convex ancient solutions of the mean curvature flow},
  author={Gerhard Huisken and Carlo Sinestrari},
  journal={Journal of Differential Geometry},
  year={2015},
  volume={101},
  pages={267-287}
}
We study solutions of the mean curvature flow which are defined for all negative times, usually called ancient solutions. We give various conditions ensuring that a closed convex ancient solution is a shrinking sphere. Examples of such conditions are: a uniform pinching condition on the curvatures, a suitable growth bound on the diameter, or a reverse isoperimetric inequality. We also study the behaviour of uniformly k-convex solutions, and consider generalizations to ancient solutions immersed… Expand
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References

SHOWING 1-10 OF 22 REFERENCES
Convex solutions to the mean curvature flow
In this paper we study the classication of ancient convex solutions to the mean curvature ow in R n+1 . An open problem related to the classication of type II singularities is whether a convexExpand
Ancient solutions of the mean curvature flow
In this short article, we prove the existence of ancient solutions of the mean curvature flow that for t -> 0 collapse to a round point, but for t -> -infinity become more and more oval: near theExpand
Noncollapsing in mean-convex mean curvature flow
We provide a direct proof of a noncollapsing estimate for compact hypersurfaces with positive mean curvature moving under the mean curvature flow: Precisely, if every point on the initialExpand
Mean curvature flow singularities for mean convex surfaces
Abstract. We study the evolution by mean curvature of a smooth n–dimensional surface ${\cal M}\subset{\Bbb R}^{n+1}$, compact and with positive mean curvature. We first prove an estimate on theExpand
Mean curvature flow with surgeries of two–convex hypersurfaces
We consider a closed smooth hypersurface immersed in euclidean space evolving by mean curvature flow. It is well known that the solution exists up to a finite singular time at which the curvatureExpand
Flow by mean curvature of convex surfaces into spheres
The motion of surfaces by their mean curvature has been studied by Brakke [1] from the viewpoint of geometric measure theory. Other authors investigated the corresponding nonparametric problem [2],Expand
Mean curvature flow of mean convex hypersurfaces
In the last 15 years, White and Huisken-Sinestrari developed a far-reaching structure theory for the mean curvature flow of mean convex hypersurfaces. Their papers provide a package of estimates andExpand
Contraction of convex hypersurfaces in Euclidean space
We consider a class of fully nonlinear parabolic evolution equations for hypersurfaces in Euclidean space. A new geometrical lemma is used to prove that any strictly convex compact initialExpand
Classification of compact ancient solutions to the curve shortening flow
We consider an embedded convex ancient solution $\Gamma_t$ to the curve shortening flow in $\mathbb{R}^2$. We prove that there are only two possibilities: the family $\Gamma_t$ is either the familyExpand
Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature
which shrink towards the center of the initial sphere in finite time. It was shown in [3], that this behaviour is very typical: If the initial hypersurface M o o R , + 1 is uniformly convex, then theExpand
...
1
2
3
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