Uniqueness of unbounded solutions of the Lagrangian mean curvature flow equation for graphs

Chen, Jingyi; Pang, Chao

doi:10.1016/j.crma.2009.06.020

Cited by 7 publications

(3 citation statements)

References 6 publications

(18 reference statements)

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“…By using existence [1] and uniqueness [3] results for the Lagrangian mean curvature flow, the rigidity of self-expanding, self-shrinking and translating solutions for the Lagrangian mean curvature flow was studied in [2] when the Hessian of the potential function is strictly bounded between −1 and 1. The same rigidity for self-shrinking and translating solutions with arbitrarily bounded Hessian was derived from a Liouville type property for ancient solutions to parabolic equations [8] (for self-shrinking solutions, a special case of [8] was treated recently in [6]).…”

Section: Introductionmentioning

confidence: 99%

Rigidity of entire self-shrinking solutions to curvature flows

Chau

Chen

Yuan

2012

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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We show that (a) any entire graphic self-shrinking solution to the Lagrangian mean curvature flow in C m with the Euclidean metric is flat; (b) any space-like entire graphic self-shrinking solution to the Lagrangian mean curvature flow in C m with the pseudo-Euclidean metric is flat if the Hessian of the potential is bounded below quadratically; and (c) the Hermitian counterpart of (b) for the Kähler Ricci flow.

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Section: Introductionmentioning

confidence: 99%

Rigidity of entire self-shrinking solutions to curvature flows

Chau

Chen

Yuan

2012

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…Recently Chen and Yin [CY07] proved that uniqueness for complete manifolds M still holds within the class of smooth solutions with bounded second fundamental tensor, if the ambient Riemannian manifold (N, g) has bounded geometry in a certain sense. Chen and Pang [CP09] considered uniqueness of unbounded solutions of the Lagrangian mean curvature flow equation for graphs.…”

Section: Short-time Existence and Uniquenessmentioning

confidence: 99%

Mean Curvature Flow in Higher Codimension: Introduction and Survey

Smoczyk

2011

Global Differential Geometry

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In this text we outline the major techniques, concepts and results in mean curvature flow with a focus on higher codimension. In addition we include a few novel results and some material that cannot be found elsewhere. Mean curvature flowMean curvature flow is perhaps the most important geometric evolution equation of submanifolds in Riemannian manifolds. Intuitively, a family of smooth submanifolds evolves under mean curvature flow, if the velocity at each point of the submanifold is given by the mean curvature vector at that point. For example, round spheres in euclidean space evolve under mean curvature flow while concentrically shrinking inward until they collapse in finite time to a single point, the common center of the spheres. Mullins [Mul56] proposed mean curvature flow to model the formation of grain boundaries in annealing metals. Later the evolution of submanifolds by their mean curvature has been studied by Brakke [Bra78] from the viewpoint of geometric measure theory. Among the first authors who studied the corresponding nonparametric problem were Temam [Tem76] in the late 1970's and Gerhardt [Ger80] and Ecker [Eck82] in the early 1980's. Pioneering work was done by Gage [Gag84], Gage & Hamilton [GH86] and Grayson [Gra87] who proved that the curve shortening flow (more precisely, the "mean" curvature flow of curves in R 2 ) shrinks embedded closed curves to "round" points. In his seminal paper Huisken [Hui84] proved that closed convex hypersurfaces in euclidean space R m+1 , m > 1 contract to single round points in finite time (later he extended his result to hypersurfaces in Riemannian manifolds that satisfy a suitable stronger convexity, see [Hui86]). Then, until the mid 1990's, most authors who studied mean curvature flow mainly considered hypersurfaces, both in euclidean and Riemannian manifolds, whereas mean curvature flow in higher codimension did not play a great role. There are various reasons for this, one of them is certainly the much different geometric situation of submanifolds in higher codimension since the normal bundle and the second fundamental tensor

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“…Koch and Lamm show uniqueness of MCF [11] for entire graph with small Lipschitz bound in any codimension. Chen and Peng prove in [7] that any viscosity solution of the graphical Lagrangian MCF with a continuous initial data is unique. For general immersions, Chen and Yin show in [5] the uniqueness of MCF among flows with uniformly bounded second fundamental forms.…”

Section: Introductionmentioning

confidence: 99%

Uniqueness Theorem for non-compact mean curvature flow with possibly unbounded curvatures

Lee¹,

Ma²

2017

Preprint

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In this paper, we discuss uniqueness and backward uniqueness for mean curvature flow of non-compact manifolds. We use an energy argument to prove two uniqueness theorems for mean curvature flow with possibly unbounded curvatures. These generalize the results in [5]. Using similar method, we also obtain a uniqueness result on Ricci flows. A backward uniqueness theorem is also proved for mean curvature flow with bounded curvatures.

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Uniqueness of unbounded solutions of the Lagrangian mean curvature flow equation for graphs

Cited by 7 publications

References 6 publications

Rigidity of entire self-shrinking solutions to curvature flows

Rigidity of entire self-shrinking solutions to curvature flows

Mean Curvature Flow in Higher Codimension: Introduction and Survey

Uniqueness Theorem for non-compact mean curvature flow with possibly unbounded curvatures

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