• Corpus ID: 203952196

Singularity models of pinched solutions of mean curvature flow in higher codimension

@article{Naff2019SingularityMO,
  title={Singularity models of pinched solutions of mean curvature flow in higher codimension},
  author={Keaton Naff},
  journal={arXiv: Differential Geometry},
  year={2019}
}
  • Keaton Naff
  • Published 9 October 2019
  • Mathematics
  • arXiv: Differential Geometry
We consider noncompact ancient solutions to the mean curvature flow in $\mathbb{R}^{n+1}$ ($n \geq 3$) that are strictly convex, uniformly two-convex, and satisfy derivative estimates $|\nabla A| \leq \gamma_1 |H|^2, |\nabla^2 A| \leq \gamma_2 |H|^3$. We show that such an ancient solution must the translating bowl soliton. As an application, in arbitrary codimension, we consider compact $n$-dimensional ($n \geq 5$) solutions to the mean curvature flow in $\mathbb{R}^N$ that satisfy the pinching… 
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References

SHOWING 1-10 OF 32 REFERENCES
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 curvature
Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions
In this paper, we consider noncompact ancient solutions to the mean curvature flow in R (n ≥ 3) which are strictly convex, uniformly two-convex, and noncollapsed. We prove that such an ancient
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],
Cylindrical Estimates for High Codimension Mean Curvature Flow
We study high codimension mean curvature flow of a submanifold M of dimension n in Euclidean space R subject to the quadratic curvature condition |A| ≤ cn|H | , cn = min{ 4 3n , 1 n−2 }. This
Mean curvature flow of Pinched submanifolds to spheres
The evolution of hypersurfaces by their mean curvature has been studied by many authors since the appearance of Gerhard Huisken’s seminal paper [Hu1]. More recently, mean curvature flow of higher
Codimension estimates in mean curvature flow
We show that the blow-ups of compact solutions to the mean curvature flow in $\mathbb{R}^N$ initially satisfying the pinching condition $|H| > 0$ and $|A|^2 < c |H|^2$ for some suitable constant $c =
Interior estimates for hypersurfaces moving by mean curvature
On Hypersurfaces with no Negative Sectional Curvatures
...
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