Pinched ancient solutions to the high codimension mean curvature flow

Lynch, Stephen

doi:10.1007/s00526-020-01888-1

Cited by 7 publications

(3 citation statements)

References 24 publications

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“…We also find that the only quadratically pinched ancient solutions are the totally umbilic ones (cf. [2,11,17,19]).…”

Section: The Key Estimates For Smooth Flowsmentioning

confidence: 99%

Quadratically pinched submanifolds of the sphere via mean curvature flow with surgery

Langford,

Lynch,

Nguyen

2021

Preprint

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We study mean curvature flow of n-dimensional submanifolds of S n+ℓ K , the round (n + ℓ)-sphere of sectional curvature K > 0, under the quadratic curvature pinching condition3(n+1) K when n = 5 or 6. This condition is related to a theorem of Li and Li [Arch. Math., 58:582-594, 1992] which states that the only n-dimensional minimal submanifolds of S n+ℓ K satisfying |A| 2 < 2n 3 K are the totally geodesic n-spheres. We prove the existence of a suitable mean curvature flow with surgeries starting from initial data satisfying the pinching condition. As a result, we conclude that any smoothly, properly immersed submanifold of S n+1 K satisfying the pinching condition is diffeomorphic either to the sphere S n or to the connected sum of a finite number of handles S 1 × S n−1 . The results are sharp when n ≥ 8 due to hypersurface counterexamples.

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“…We also find that the only quadratically pinched ancient solutions are the totally umbilic ones (cf. [2,11,17,19]).…”

Section: The Key Estimates For Smooth Flowsmentioning

confidence: 99%

Quadratically pinched submanifolds of the sphere via mean curvature flow with surgery

Langford,

Lynch,

Nguyen

2021

Preprint

Self Cite

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“…Finally, we want to mention that after the first version of this paper appeared on arXiv.org in 2016, many other results on ancient solutions to curvature flows appeared, see [5,24] for a classification of ancient solutions using Aleksandrov reflection and [6,7,19,20] for other related results.…”

Section: Introductionmentioning

confidence: 99%

On the classification of ancient solutions to curvature flows on the sphere

Bryan¹,

Ivaki²,

Scheuer³

2024

Annali Scuola Normale Superiore - Classe Di Scienze

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We consider the evolution of hypersurfaces on the unit sphere S n+1 by smooth functions of the Weingarten map. We introduce the notion of 'quasi-ancient' solutions for flows that do not admit non-trivial, convex, ancient solutions. Such solutions are somewhat analogous to ancient solutions for flows such as the mean curvature flow, or 1-homogeneous flows. The techniques presented here allow us to prove that any convex, quasi-ancient solution of a curvature flow which satisfies a backwards in time uniform bound on mean curvature must be stationary or a family of shrinking geodesic spheres. The main tools are geometric, employing the maximum principle, a rigidity result in the sphere and an Aleksandrov reflection argument. We emphasize that no homogeneity or convexity/concavity restrictions are placed on the speed, though we do also offer a short classification proof for several such restricted cases.

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“…These pinching estimates were greatly refined by Naff in [30]. Ancient solutions were recently studied by Lynch and Nguyen [28]. In the survey article [33] Smoczyk presents results on: short-time existence and uniqueness, long-time existence and convergence, and singularities.…”

Section: Introductionmentioning

confidence: 99%

A convergent finite element algorithm for mean curvature flow in arbitrary codimension

Binz¹,

Kovács²

2021

Preprint

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Optimal-order uniform-in-time H 1 -norm error estimates are given for semi-and full discretizations of mean curvature flow of surfaces in arbitrarily high codimension. The proposed and studied numerical method is based on a parabolic system coupling the surface flow to evolution equations for the mean curvature vector and for the orthogonal projection onto the tangent space. The algorithm uses evolving surface finite elements and linearly implicit backward difference formulae. This numerical method admits a convergence analysis in the case of finite elements of polynomial degree at least two and backward difference formulae of orders two to five. Numerical experiments in codimension 2 illustrate and complement our theoretical results.

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Pinched ancient solutions to the high codimension mean curvature flow

Cited by 7 publications

References 24 publications

Quadratically pinched submanifolds of the sphere via mean curvature flow with surgery

Quadratically pinched submanifolds of the sphere via mean curvature flow with surgery

On the classification of ancient solutions to curvature flows on the sphere

A convergent finite element algorithm for mean curvature flow in arbitrary codimension

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