Construction of Lagrangian Self-similar Solutions to the Mean Curvature Flow in $$\mathbb{C}^n$$

Anciaux, Henri

doi:10.1007/s10711-006-9082-z

Cited by 47 publications

(83 citation statements)

References 14 publications

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“…As special Lagrangians are volume minimizing, it is natural to use mean curvature flow to construct special Lagrangians. Equation (1) implies that mean curvature flow is a Lagrangian deformation, that is, a Lagrangian submanifold remains Lagrangian under mean curvature flow, as in Smoczyk [24].…”

Section: Background Materialsmentioning

confidence: 99%

“…Our Lagrangians L in C n are the total space of a 1-parameter family Q s , s ∈ I, where I is an open interval in R, and each Q s is a quadric in a Lagrangian plane R n in C n , which evolve according to an o.d.e. in s. The construction includes and generalizes examples of Lagrangian solitons or special Lagrangians due to Anciaux [1], Joyce [12], Lawlor [15], and Lee and Wang [18,19].…”

Section: Introductionmentioning

confidence: 99%

“…These examples include those of Anciaux [1] and Lee and Wang [18,19], and approach the special Lagrangians of Joyce [12] and Lawlor [15] in a limit.…”

Section: Introductionmentioning

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: Background Materialsmentioning

confidence: 99%

Section: Introductionmentioning

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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“…In [13], Lee-Lue found an equivalent condition to the F-stabiliy. Moreover, they proved that in some cases the closed Lagrangian self-shrinkers given by Anciaux in [1] are Lagrangian F-unstable. See [2,3,4,5,13,14,17], etc.…”

Section: Introductionmentioning

confidence: 99%

“…Lagrangian self-shrinkers are very important examples of self-shrinkers in higher codimension. Anciaux [1] and Joyce-Lee-Tsui [12] constructed some examples of Lagrangian self-shrinkers.…”

Section: Introductionmentioning

confidence: 99%

Hamiltonian F-Stability of Complete Lagrangian Self-Shrinkers

Yang

2015

J Geom Anal

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Abstract. In this paper, we study the Lagrangian F-stability and Hamiltonian F-stability of Lagrangian self-shrinkers. We prove a characterization theorem for the Hamiltonian F-stability of n-dimensional complete Lagrangian self-shrinkers without boundary, with polynomial volume growth and with the second fundamental form satisfying the condition that there exist constants C0 > 0 and ε < 1 16n such that |A| 2 ≤ C0 +ε|x| 2 . We characterize the Hamiltonian F-stablity by the eigenvalues and eigenspaces of the drifted Laplacian.Mathematics Subject Classification (2000): 53C44 (primary), 53C21 (secondary). IntroductionAn n-dimensional submanifold Σ n of R n+p is called a self-shrinker if it is the time t = −1 slice of a self-shrinking mean curvature flow that disappears at (0, 0), i.e. of a mean curvature flow satisfying Σ t = √ −tΣ −1 . We can also consider a self-shrinker as a submanifold that satisfiesSelf-shrinkers are very important singularities of the mean curvature flow. According to the blow up rate of the second fundamental form, Huisken [11] classified the singularities of mean curvature flows into two types: Type I and Type II. In 1984, Huisken [10] showed that, if the initial hypersurface in R n+1 is strictly convex, then along the mean curvature flow, the surface will be strictly convex at each time, and the mean curvature flow will contract to a point at a finite time T . Moreover, the normalized mean curvature flow will converge to a round sphere. In 1990, Huisken [11] proved that any Type I singularity of the mean curvature flow must be a self-shrinker by using the monotonicity formula. He also proved that the only compact self-shrinkers with nonnegative mean curvature are spheres.In [6], Colding-Minicozzi introduced the concept of F-stability and entropy-stability of a self-shrinker, and gave a classification of self-shrinkers in the hypersurface case. The definitions of many concepts in their paper can be naturally generalized to the higher codimension case (cf. [2,3,13]). Given x 0 ∈ R n+p and t 0 > 0, F x 0 ,t 0 is defined by dµ.

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Examples of Equivariant Lagrangian Mean Curvature Flow

Lotay¹

2021

Trends in Mathematics

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Construction of Lagrangian Self-similar Solutions to the Mean Curvature Flow in $$\mathbb{C}^n$$

Cited by 47 publications

References 14 publications

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

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

Hamiltonian F-Stability of Complete Lagrangian Self-Shrinkers

Examples of Equivariant Lagrangian Mean Curvature Flow

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