Mean curvature flow with surgeries of two–convex hypersurfaces

Huisken, Gerhard; Sinestrari, Carlo

doi:10.1007/s00222-008-0148-4

Cited by 152 publications

(295 citation statements)

References 26 publications

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“…Our construction should also be compared with that of G Huisken and C Sinestrari [14] for Mean Curvature Flow, where in particular there is a similar argument for nonaccumulation of surgeries. Needless to say, we rely heavily on Perelman's work, in particular the notions of Ä -noncollapsing and canonical neighbourhoods.…”

Section: Introductionmentioning

confidence: 94%

Ricci flow on open 3–manifolds and positive scalar curvature

2011

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Section: Introductionmentioning

confidence: 94%

Ricci flow on open 3–manifolds and positive scalar curvature

2011

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“…In [Hui84], Huisken showed that MCF starting at any smooth compact convex initial hypersurface in R n+1 remains smooth and convex until it becomes extinct at a point and if we rescale the flow about the point in space-time where it becomes extinct, then the rescalings converge to round spheres. Huisken-Sinestrari, [HS99a], [HS99b], and White, [Whi00], [Whi03], have proven a number of striking and important results about MCF of mean convex hypersurfaces and their singularities and Huisken-Sinestrari, [HS09], have developed a theory for MCF with surgery for two-convex hypersurfaces in R n+1 (n ≥ 3) using their analysis of singularities (and their blow ups).…”

Section: Introductionmentioning

confidence: 99%

Generic mean curvature flow I; generic singularities

2012

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It has long been conjectured that starting at a generic smooth closed embedded surface in R 3 , the mean curvature flow remains smooth until it arrives at a singularity in a neighborhood of which the flow looks like concentric spheres or cylinders. That is, the only singularities of a generic flow are spherical or cylindrical. We will address this conjecture here and in a sequel. The higher dimensional case will be addressed elsewhere. The key to showing this conjecture is to show that shrinking spheres, cylinders, and planes are the only stable self-shrinkers under the mean curvature flow. We prove this here in all dimensions. An easy consequence of this is that every singularity other than spheres and cylinders can be perturbed away.

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“…In any case, our examples evidence that a singularity analysis following the traditional approaches ( [14], [23] and references therein) is not possible for the constrained versions of mcf. The only remaining hope is to produce completely different techniques to face the study of singularities or to find another curvature condition (stronger than mean convexity but weaker than convexity) which is preserved under (1.1).…”

Section: Introduction and Main Resultsmentioning

confidence: 87%

Non-preserved curvature conditions under constrained mean curvature flows

Cabezas-Rivas

2016

Differential Geometry and its Applications

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Abstract. We provide explicit examples which show that mean convexity (i.e. positivity of the mean curvature) and positivity of the scalar curvature are non-preserved curvature conditions for hypersurfaces of the Euclidean space evolving under either the volume-or the area preserving mean curvature flow. The relevance of our examples is that they disprove some statements of the previous literature, overshadow a widespread folklore conjecture about the behaviour of these flows and bring out the discouraging news that a traditional singularity analysis is not possible for constrained versions of the mean curvature flow.Mathematics Subject Classification (2010) 53C44

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Mean curvature flow with surgeries of two–convex hypersurfaces

Cited by 152 publications

References 26 publications

Ricci flow on open 3–manifolds and positive scalar curvature

Ricci flow on open 3–manifolds and positive scalar curvature

Generic mean curvature flow I; generic singularities

Non-preserved curvature conditions under constrained mean curvature flows

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