Minimizing volume among Lagrangian submanifolds

Schoen, Richard; Wolfson, Jon

doi:10.1090/pspum/065/1662755

Cited by 28 publications

(43 citation statements)

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“…It is known (see [SW99], p. 186) that any Lagrangian submanifold of R 2n through the origin which is exact (this condition is always satisfied locally) can be lifted to a Legendrian submanifold of R 2n+1 through the origin. The lift of this Lagrangian subspace will be a Legendrian bisection nearby g, because it will be transversal to both T g s The induced bracket still satisfies the Jacobi identity and (2).…”

Section: Moreover the Jacobi Foliation Is Given Exactly By (The Connmentioning

confidence: 99%

Contact reduction and groupoid actions

Zambon

Zhu

2005

Trans. Amer. Math. Soc.

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Abstract. We introduce a new method to perform reduction of contact manifolds that extends Willett's and Albert's results. To carry out our reduction procedure all we need is a complete Jacobi map J : M → Γ 0 from a contact manifold to a Jacobi manifold. This naturally generates the action of the contact groupoid of Γ 0 on M , and we show that the quotients of fibers J −1 (x) by suitable Lie subgroups Γ x are either contact or locally conformal symplectic manifolds with structures induced by the one on M .We show that Willett's reduced spaces are prequantizations of our reduced spaces; hence the former are completely determined by the latter. Since a symplectic manifold is prequantizable iff the symplectic form is integral, this explains why Willett's reduction can be performed only at distinguished points. As an application we obtain Kostant's prequantizations of coadjoint orbits. Finally we present several examples where we obtain classical contact manifolds as reduced spaces.

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Section: Moreover the Jacobi Foliation Is Given Exactly By (The Connmentioning

confidence: 99%

Contact reduction and groupoid actions

Zambon

Zhu

2005

Trans. Amer. Math. Soc.

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“…H-minimal Lagrangian cones in ‫ރ‬ 2 were studied in [Schoen and Wolfson 1999]. Hélein and Romon [2000;2002a] derived a Weierstrass-type representation formula to describe all H-minimal Lagrangian tori and Klein bottles in ‫ރ‬ 2 .…”

Section: Introductionmentioning

confidence: 99%

Hamiltonian-minimal Lagrangian submanifolds in complex space forms

Castro

Urbano

2006

Pacific J. Math.

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and, in particular, Legendre curves in odddimensional spheres and anti-de Sitter spaces, we construct new examples of Hamiltonian-minimal Lagrangian submanifolds in complex projective and hyperbolic spaces, including explicit one-parameter families of embeddings of quotients of certain product manifolds. We also give new examples of minimal Lagrangian submanifolds in complex projective and hyperbolic spaces. Making use of all these constructions, we get Hamiltonian-minimal and special Lagrangian cones in complex Euclidean space as well.

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“…Harvey and Lawson themselves produced several examples of minimal Lagrangian submanifolds and gave certain general constructions of such objects. More recently, Schoen and Wolfson [18] have been working on questions of existence of special Lagrangian submanifolds using variational techniques. Current developments in the mathematical foundations of string theory, in the form of mirror symmetry and the Strominger-Yau-Zaslow conjecture [17], have greatly stimulated further investigation into special Lagrangian submanifolds in the hopes of understanding the moduli spaces of these objects.…”

Section: Introductionmentioning

confidence: 99%

Regularizing a singular special Lagrangian variety

Butscher¹

2004

Communications in Analysis and Geometry

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Suppose M 1 and M 2 are two special Lagrangian submanifolds with boundary of R 2n , n ≥ 3, that intersect transversally at one point p. The set M 1 ∪ M 2 is a singular special Lagrangian variety with an isolated singularity at the point of intersection. Suppose further that the tangent planes at the intersection satisfy an angle criterion (which always holds in dimension n = 3). Then, M 1 ∪ M 2 is regularizable; in other words, there exists a family of smooth, minimal Lagrangian submanifolds M α with boundary that converges to M 1 ∪ M 2 in a suitable topology. This result is obtained by first gluing a smooth neck into a neighbourhood of M 1 ∩ M 2 and then by perturbing this approximate solution until it becomes minimal and Lagrangian.

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Minimizing volume among Lagrangian submanifolds

Cited by 28 publications

References 0 publications

Contact reduction and groupoid actions

Contact reduction and groupoid actions

Hamiltonian-minimal Lagrangian submanifolds in complex space forms

Regularizing a singular special Lagrangian variety

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