Deformations of Special Lagrangian Submanifolds

Salur, Sema

doi:10.1142/s0219199700000177

Cited by 27 publications

(35 citation statements)

References 14 publications

(31 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A. Butscher [3] has recently generalized this to the case of the intersection of two special Lagrangian submanifolds with boundary in a general Calabi-Yau manifold. Gluing techniques applied to special Lagrangian submanifolds of Calabi-Yau threefolds have also been used by S. Salur in [20,21]. Finally, we would like to mention the recent work of J. Isenberg, R. Mazzeo, and D. Pollack [10], where a technical linear analysis similar to ours is developed.…”

Section: Introductionmentioning

confidence: 94%

See 1 more Smart Citation

Complete, embedded, minimal n‐dimensional submanifolds in ℂⁿ

Arezzo

Pacard

2003

Comm Pure Appl Math

View full text Add to dashboard Cite

Section: Introductionmentioning

confidence: 94%

“…Not unrelated is the problem of determining whether one can desingularize a configuration of special Lagrangian planes through special Lagrangian submanifolds, possibly leaving free the phase of the calibration to change as in [19]. In the case of two planes, Lawlor's examples obviously answer the question.…”

Section: Introductionmentioning

confidence: 99%

Complete, embedded, minimal n‐dimensional submanifolds in ℂⁿ

Arezzo

Pacard

2003

Comm Pure Appl Math

View full text Add to dashboard Cite

“…Moreover, we assume that the action is equivariant with respect to isomorphism (19). Suppose that we have sections η i of L i , which are invariant under the T -action.…”

Section: Proofmentioning

confidence: 99%

“…McLean has shown in [16,Cor. 3.9] that if L is a compact SLag submanifold of M, then the moduli space of SLag submanifolds passing through L is smooth of dimension equal to the first Betti number of L. (This can be also generalized to almost Calabi-Yau manifolds and to symplectic manifolds with trivialized canonical bundle; see [6,Lem. 3.1.1] and [19].) If b 1 (L) = n, one can ask if the moduli space of SLag submanifolds through L gives a fibration of some neighbourhood of L in M. Now SLag submanifolds are in particular Lagrangian submanifolds of (M, ω CY ).…”

Section: Introductionmentioning

confidence: 99%

Calibrated fibrations on noncompact manifolds via group actions

Goldstein¹

2001

Duke Math. J.

View full text Add to dashboard Cite

In this paper we use Lie group actions on noncompact Riemannian manifolds with calibrations to construct calibrated submanifolds. In particular, if we have an (n −1)torus acting on a noncompact Calabi-Yau n-fold with a trivial first cohomology, then we have a special Lagrangian fibration on that n-fold. We produce several families of examples for this construction and give some applications to special Lagrangian geometry on compact almost Calabi-Yau manifolds. We also use group actions on noncompact G 2 -manifolds to construct coassociative submanifolds, and we exhibit some new examples of coassociative submanifolds via this setup.

show abstract

“…Salur [18,19] considers a nonsingular, connected, immersed SL 3-fold N in a Calabi-Yau 3-fold with a codimension two self-intersection along an S 1 , and constructs new SL 3-folds by smoothing along the S 1 . Butscher [3] studies SL m-folds N in C m with boundary in a symplectic submanifold W 2m−2 ⊂ C m .…”

mentioning

confidence: 99%

Special Lagrangian Submanifolds with Isolated Conical Singularities. III. Desingularization, The Unobstructed Case

Joyce¹

2004

Annals of Global Analysis and Geometry

151

View full text Add to dashboard Cite

Special Lagrangian m-folds (SL m-folds) are a distinguished class of real mdimensional minimal submanifolds which may be defined in C m , or in CalabiYau m-folds, or more generally in almost Calabi-Yau m-folds (compact Kähler m-folds with trivial canonical bundle). We write an almost Calabi-Yau m-fold as M or (M, J, ω, Ω), where the manifold M has complex structure J, Kähler form ω and holomorphic volume form Ω. This is the third in a series of five papers [10,11,12,13] studying SL m-folds with isolated conical singularities. That is, we consider an SL m-fold X in an almost Calabi-Yau m-fold M for m > 2 with singularities at x 1 , . . . , x n in M , such that for some special Lagrangian cones C i in T xi M ∼ = C m with C i \ {0} nonsingular, X approaches C i near x i in an asymptotic C 1 sense. Readers are advised to begin with the final paper [13], which surveys the series, and applies the results to prove some conjectures.The first paper [10] laid the foundations for the series, and studied the regularity of SL m-folds with conical singularities near their singular points. The second paper [11] discussed the deformation theory of compact SL m-folds X with conical singularities in an almost Calabi-Yau m-fold M . This paper and the sequel [12] study desingularizations of compact SL mfolds X with conical singularities. That is, we construct a family of compact, nonsingular SL m-foldsÑ t in M for t ∈ (0, ǫ] such thatÑ t → X as t → 0, in the sense of currents.Having a good understanding of the singularities of special Lagrangian submanifolds will be essential in clarifying the Strominger-Yau-Zaslow conjecture on the Mirror Symmetry of Calabi-Yau 3-folds [20], and also in resolving conjectures made by the author [7] on defining new invariants of Calabi-Yau 3-folds by counting special Lagrangian homology 3-spheres with weights. The series aims to develop such an understanding for simple singularities of SL m-folds.Here is the basic idea of the paper. Let X be a compact SL m-fold with conical singularities x 1 , . . . , x n in an almost Calabi-Yau m-fold (M, J, ω, Ω). 1Choose an isomorphism υ i : C m → T xi M for i = 1, . . . , n. Then there is a unique SL cone C i in C m with X asymptotic to υ i (C i ) at x i . Let L i be an Asymptotically Conical SL m-fold (AC SL m-fold ) in C m , asymptotic to C i at infinity. As C i is a cone it is invariant under dilations, so tC i = C i for all t > 0. Thus tL i = {t x : x ∈ L i } is also an AC SL m-fold asymptotic to C i for t > 0. We explicitly construct a 1-parameter family of compact, nonsingular Lagrangian m-folds N t in (M, ω) for t ∈ (0, δ) by gluing tL i into X at x i , using a partition of unity.When t is small, N t is close to being special Lagrangian (its phase is nearly constant), but also close to being singular (it has large curvature and small injectivity radius). We prove using analysis that for small ǫ ∈ (0, δ) we can deform N t to a special Lagrangian m-foldÑ t in M for all t ∈ (0, ǫ], using a small Hamiltonian deformation. The proof involves a delicate balancin...

show abstract

Deformations of Special Lagrangian Submanifolds

Cited by 27 publications

References 14 publications

Complete, embedded, minimal n‐dimensional submanifolds in ℂⁿ

Complete, embedded, minimal n‐dimensional submanifolds in ℂⁿ

Calibrated fibrations on noncompact manifolds via group actions

Special Lagrangian Submanifolds with Isolated Conical Singularities. III. Desingularization, The Unobstructed Case

Contact Info

Product

Resources

About

Deformations of Special Lagrangian Submanifolds

Cited by 27 publications

References 14 publications

Complete, embedded, minimal n‐dimensional submanifolds in ℂn

Complete, embedded, minimal n‐dimensional submanifolds in ℂn

Calibrated fibrations on noncompact manifolds via group actions

Special Lagrangian Submanifolds with Isolated Conical Singularities. III. Desingularization, The Unobstructed Case

Contact Info

Product

Resources

About

Complete, embedded, minimal n‐dimensional submanifolds in ℂⁿ

Complete, embedded, minimal n‐dimensional submanifolds in ℂⁿ