The Curve Shortening Problem

Chou, Kai Seng; Zhu, Xi-Ping

doi:10.1201/9781420035704

Cited by 200 publications

(203 citation statements)

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“…It is expected intuitively that, depending on the convex initial curve, the flow (1) will either converge to a point, converge to a round circle S 1 , or expand to infinity, with each γ t remaining smooth and convex. This is indeed true due to Theorem 3.12 (see also Remark 3.14) of [Chou and Zhu 2001]. Moreover, for given initial data X 0 (ϕ) : S 1 → ‫ޒ‬ 2 , if we consider its homothetic class there exists a unique number > 0 (for convenience we call it the critical number of X 0 ) such that under the flow (1) with initial data λX 0 (ϕ), λ = , γ t will converge to the unit circle S 1 (without rescaling) smoothly as t → ∞.…”

Section: Introductionmentioning

confidence: 53%

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Remarks on some isoperimetric properties of thek− 1 flow

Lin¹,

Tsai

2012

Pacific J. Math.

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We consider the evolution of a convex closed plane curve γ 0 along its inward normal direction with speed k − 1, where k is the curvature. This flow has the feature that it is the gradient flow of the (length − area) functional and has been previously studied by Chou and Zhu, and Yagisita. We revisit the flow and point out some interesting isoperimetric properties not discussed before.We first prove that if the curve γ t converges to the unit circle S 1 (without rescaling), its length L(t) and area A(t) must satisfy certain monotonicity properties and inequalities.On the other hand, if the curve γ t (assume γ 0 is not a circle) expands to infinity as t → ∞ and we interpret Yagisita's result in the right way, the isoperimetric difference L 2 (t) − 4π A(t) of γ t will decrease to a positive constant as t → ∞. Hence, without rescaling, the expanding curve γ t will not become circular. It is asymptotically close to some expanding curve C t , where C 0 is not a circle and each C t is parallel to C 0 . The asymptotic speed of C t is given by the constant 1.

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Section: Introductionmentioning

confidence: 53%

“…According to [Chou and Zhu 2001], the critical number is obtained via a contradiction argument and for a given curve X 0 (ϕ) we do not know what it is. However, we can use the following theorem to give an estimate of (see Corollary 2.7).…”

Section: Estimate Of the Critical Numbermentioning

confidence: 99%

Remarks on some isoperimetric properties of thek− 1 flow

Lin¹,

Tsai

2012

Pacific J. Math.

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“…First we need to establish the following lemma. If M were orientable, this lemma would follow from basic results on curve shortening flow by curvature [17,8]; however, when M is not orientable, the geodesic curvature is not well-defined along the entire curve; hence we use the "disk flow" method devised in [19] which does not require orientability.…”

Section: 3mentioning

confidence: 99%

Vertices of closed curves in Riemannian surfaces

Ghomi¹

2013

Comment. Math. Helv.

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Abstract. We uncover some connections between the topology of a complete Riemannian surface M and the minimum number of vertices, i.e., critical points of geodesic curvature, of closed curves in M . In particular we show that the space forms with finite fundamental group are the only surfaces in which every simple closed curve has more than two vertices. Further we characterize the simply connected space forms as the only surfaces in which every closed curve bounding a compact immersed surface has more than two vertices.

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“…Contracting (t < 0) and expanding (t > 0) self-similar solutions (AbreshLanger [1], Brakke [10] and [14]) correspond to the vector field tuy. All these solutions are discussed in Chou-Zhu [13]. Finally, the contracting\expanding spirals corresponding to ws are discussed in [8] and Li [20].…”

Section: V(t) = [R-(a + L)t]^t Rx)mentioning

confidence: 99%