The normalized curve shortening flow and homothetic solutions

Abresch, Uwe; Langer, Joel

doi:10.4310/jdg/1214440025

Cited by 300 publications

(425 citation statements)

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“…For special Lagrangian submanifolds, with θ ≡ 0 in (ii) and α = 0 in (iii), the ansatz includes the examples of Lawlor [15] with λ 1 = · · · = λ n = C = 1, and Joyce [12, §5- §6]. For more general Lagrangian self-similar solutions, it includes the examples of Abresch and Langer [2] when n = 1, the examples of Anciaux [1], which have λ 1 = · · · = λ n = C = 1 and are symmetric under the action of SO(n) on C n , and the examples of Lee and Wang [18, §6], [19], which have w j (s) ≡ e iλ j s . M.-T. Wang and the second author also tried to study an ansatz of a similar form to (2) before.…”

Section: Statements Of Main Resultsmentioning

confidence: 99%

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Self-similar solutions and translating solitons for Lagrangian mean curvature flow

Joyce

Lee

Tsui

2010

J. Differential Geom.

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Abstract. We construct many self-similar and translating solitons for Lagrangian mean curvature flow, including self-expanders and translating solitons with arbitrarily small oscillation on the Lagrangian angle. Our translating solitons play the same role as cigar solitons in Ricci flow, and are important in studying the regularity of Lagrangian mean curvature flow.Given two transverse Lagrangian planes R n in C n with sum of characteristic angles less than π, we show there exists a Lagrangian selfexpander asymptotic to this pair of planes. The Maslov class of these self-expanders is zero. Thus they can serve as local models for surgeries on Lagrangian mean curvature flow. Families of self-shrinkers and self-expanders with different topologies are also constructed. This paper generalizes the work of Anciaux [1], Joyce [12], Lawlor [15] and Lee and Wang [18,19].

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Section: Statements Of Main Resultsmentioning

confidence: 99%

“…, α n , A) in the domain of Ψ m,n , the following derivative is an isomorphism: We shall use Proposition 6.4 and induction on n to eliminate possibility (B), so that (A) holds. The first step n = 1 is studied by Abresch and Langer [2], and translated into our notation, [2, Th. A & Prop.…”

Section: Conversely Ifmentioning

confidence: 99%

Self-similar solutions and translating solitons for Lagrangian mean curvature flow

Joyce

Lee

Tsui

2010

J. Differential Geom.

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“…All of these examples are mean convex (i.e., have H ≥ 0) and, in fact, these are the only mean convex examples under mild assumptions; see [Hui90], [Hui93], [AL86], and Theorem 0.17. Without the assumption on mean convexity, then there are expected to be many more examples of self-shrinkers in R 3 .…”

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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“…where Γ is one of the homothetically (convex immersed) shrinking curves in I R 2 found by Abresch and Langer [1].…”

Section: Blow-up Rate Of the Mean Curvaturementioning

confidence: 91%

Blow-up rate of the mean curvature during the mean curvature flow and a gap theorem for self-shrinkers

Šešum

2011

Communications in Analysis and Geometry

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In this paper, we prove that the mean curvature blows up at the same rate as the second fundamental form at the first singular time T of any compact, type-I mean curvature flow. For the mean curvature flow of surfaces, we obtain similar result provided that the Gaussian density is less than two. Our proofs are based on continuous rescaling and the classification of self-shrinkers. We show that all notions of singular sets defined in [19] coincide for any type-I mean curvature flow, thus generalizing the result of Stone who established that for any mean convex type-I mean curvature flow. We also establish a gap theorem for self-shrinkers.

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The normalized curve shortening flow and homothetic solutions

Cited by 300 publications

References 5 publications

Self-similar solutions and translating solitons for Lagrangian mean curvature flow

Self-similar solutions and translating solitons for Lagrangian mean curvature flow

Generic mean curvature flow I; generic singularities

Blow-up rate of the mean curvature during the mean curvature flow and a gap theorem for self-shrinkers

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