Lectures on Mean Curvature Flows

Zhu, Xi-Ping

doi:10.1090/amsip/032

Cited by 85 publications

(86 citation statements)

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“…Hence we can apply the standard regularity theory of uniformly parabolic equations (cf. [10] or [2,17]) to conclude that the solution to (3.3) can not be singular at t = T max , which is a contradiction. Therefore X(·, t) must converge to a point as t → T max .…”

Section: Lemma 6 When T ≥ Tmentioning

confidence: 96%

“…Since the hypersurface is convex and compact, i.e. the Gauss map is everywhere non-degenerate, we use the Gauss map to reparametrize the convex hypersurface (see [2,16,17])…”

Section: Preliminariesmentioning

confidence: 99%

“…We now choose r * is less than n/h + . We follow an idea in [2,17] to prove the following lemma which implies that when t is very near T max , M t is in fact contracting.…”

Section: Case (I) λ = ∞mentioning

confidence: 99%

“…We want to bound the curvature H of Mt, for this purpose, we will use a trick of Chow (Tso) [14] (see also [2,13,17]) to consider the function…”

Section: Similarly By Proposition 1 the Inner And Outer Radii Of Thementioning

confidence: 99%

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Forced convex mean curvature flow in Euclidean spaces

Salavessa

2008

manuscripta math.

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In this paper, we consider the mean curvature flow of convex hypersurfaces in Euclidean spaces with a general forcing term. We show that the flow may shrink to a point in finite time if the forcing term is small, or exist for all times and expand to infinity if the forcing term is large enough. The flow can also converge to a round sphere for some special forcing term and initial hypersurface. Furthermore, the normalization of the flow is carried out so that long time existence and convergence of the rescaled flow are studied. Our work extends Huisken's well-known mean curvature flow and McCoy's mixed volume preserving mean curvature flow.

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Section: Lemma 6 When T ≥ Tmentioning

confidence: 96%

“…Since the hypersurface is convex and compact, i.e. the Gauss map is everywhere non-degenerate, we use the Gauss map to reparametrize the convex hypersurface (see [2,16,17])…”

Section: Preliminariesmentioning

confidence: 99%

“…We now choose r * is less than n/h + . We follow an idea in [2,17] to prove the following lemma which implies that when t is very near T max , M t is in fact contracting.…”

Section: Case (I) λ = ∞mentioning

confidence: 99%

“…We want to bound the curvature H of Mt, for this purpose, we will use a trick of Chow (Tso) [14] (see also [2,13,17]) to consider the function…”

Section: Similarly By Proposition 1 the Inner And Outer Radii Of Thementioning

confidence: 99%

See 2 more Smart Citations

Forced convex mean curvature flow in Euclidean spaces

Salavessa

2008

manuscripta math.

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“…Since the pioneering works of GageHamilton [10] and Grayson [12], there have been many results for this problem and one may consult the books Cao [6], Chou-Zhu [9], Giga [11] and Zhu [23].…”

Section: Introductionmentioning

confidence: 99%