# 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} }

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…

## 4 Citations

A Canonical Neighborhood Theorem for Mean Curvature Flow in Higher Codimension

- MathematicsInternational Mathematics Research Notices
- 2022

In dimensions $n \geq 5$, we prove a canonical neighborhood theorem for the mean curvature flow of compact $n$-dimensional submanifolds in $\mathbb {R}^N$ satisfying a pinching condition $|A|^2 <…

Uniqueness of convex ancient solutions to hypersurface flows

- MathematicsJournal für die reine und angewandte Mathematik (Crelles Journal)
- 2022

Abstract We show that every convex ancient solution of mean curvature flow with Type I curvature growth is either spherical, cylindrical, or planar. We then prove the corresponding statement for…

Collapsing and noncollapsing in convex ancient mean curvature flow

- Mathematics
- 2021

We prove a slab theorem for convex ancient solutions to mean curvature flow without any additional hypotheses (such as concavity of the arrival time, bounded curvature on compact time intervals, or…

Convexity Estimates for High Codimension Mean Curvature Flow

- Mathematics
- 2020

We consider the evolution by mean curvature of smooth $n$-dimensional submanifolds in $\mathbb{R}^{n+k}$ which are compact and quadratically pinched. We will be primarily interested in flows of high…

## References

SHOWING 1-10 OF 32 REFERENCES

Mean curvature flow with surgeries of two–convex hypersurfaces

- Mathematics
- 2009

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…

Convexity estimates for mean curvature flow and singularities of mean convex surfaces

- Mathematics
- 1999

Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions

- MathematicsGeometry & Topology
- 2021

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

- Mathematics
- 1984

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

- Mathematics
- 2018

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

- Mathematics
- 2010

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

- Mathematics
- 2019

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 =…