Special Lagrangian cones

Haskins, Mark

doi:10.1353/ajm.2004.0029

Cited by 80 publications

(129 citation statements)

References 22 publications

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“…[5], the author [8,9], and others. Special Lagrangian cones in C 3 are a special case, which may be treated using the theory of integrable systems.…”

Section: Examples Of Special Lagrangian Conesmentioning

confidence: 99%

Special Lagrangian Submanifolds with Isolated Conical Singularities. v. Survey and Applications

Joyce¹

2003

J. Differential Geom.

153

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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 fifth in a series of five papers [15,16,17,18] 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. New readers of the series are advised to begin with this paper. We shall survey the major results of [15,16,17,18], giving explanations, but avoiding the long, technical analytic proofs of previous papers. We also integrate the results to give an (incomplete) description of the boundary of a moduli space of compact SL m-folds, and apply them to prove some conjectures in [7,11] on connected sums of SL m-folds, and T 2 -cone singularities of SL 3-folds. 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 [27], 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.We begin in §2 with an introduction to almost Calabi-Yau and special Lagrangian geometry, and the deformation theory of compact SL m-folds. Section 3 defines SL m-folds with conical singularities, our subject, gives examples of special Lagrangian cones, and some basics on homology and cohomology.Section 4 describes the first paper [15] on the regularity of SL m-folds X with conical singularities x 1 , . . . , x n . We study the asymptotic behaviour of X and its derivatives near x i , how quickly it converges to the cone C i .In §5 we discuss the second paper [16] on the deformation theory of compact

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“…[5], the author [8,9], and others. Special Lagrangian cones in C 3 are a special case, which may be treated using the theory of integrable systems.…”

Section: Examples Of Special Lagrangian Conesmentioning

confidence: 99%

Special Lagrangian Submanifolds with Isolated Conical Singularities. v. Survey and Applications

Joyce¹

2003

J. Differential Geom.

153

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“…Lawson constructed the first examples in C n , where we point out the Lagrangian catenoid ([1, Remark 1], [3,Theorem A] and [5,Theorem 6.4]) to show a method of construction of special Lagrangian submanifolds of C n using an (n-1)-dimensional oriented minimal Legendrian submanifold of S 2n−1 and certain plane curves (for a better understanding, see Proposition A in section 2). Making use of this method, we include in section 2 three new examples of special Lagrangian submanifolds which have not been exposed yet.…”

Section: Introductionmentioning

confidence: 99%

On a new construction of special Lagrangian immersions in complex Euclidean space

Castro¹

2004

The Quarterly Journal of Mathematics

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“…Like Delaunay surfaces, SO(p) × SO(q)-invariant special Legendrians are cohomogeneity one objects and are therefore governed by an appropriate system of ODEs. For (p, q) = (1, 2) the SO(p) × SO(q)-invariant special Legendrians are precisely the SO(2)-invariant ones studied previously in [6,8,11] and used as building blocks in our gluing construction of higher genus special Legendrian surfaces [11]. To our knowledge, for general (p, q), SO(p) × SO(q)-invariant special Legendrians were first studied by Castro-Li-Urbano [4].…”

Section: Introductionmentioning

confidence: 65%

“…SL cones in C n with isolated singularities (or equivalently special Legendrians in S 2n−1 ) form the simplest class of singular special Lagrangians, and significant progress on understanding SL cones has been made in the last 10 years. In particular, the situation in dimension three has been clarified considerably [3,6,8,11,15,21]. By comparison the situation in higher dimensions is more complicated and less systematically explored.…”

Section: Introductionmentioning

confidence: 99%

The geometry of SO(<i>p</i>) × SO(<i>q</i>)-invariant special Lagrangian cones

Haskins

Kapouleas

2013

Communications in Analysis and Geometry

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SO(p) × SO(q)-invariant special Lagrangian cones in C p+q(equivalently, SO(p) × SO(q)-invariant special Legendrians in S 2(p+q)−1 ) are an important family of special Lagrangians (SL) whose basic features were studied in our previous paper [13]. In some ways, they play a role analogous to that of Delaunay surfaces in the geometry of CMC surfaces in R 3 ; in particular, they are natural building blocks for our gluing constructions of higherdimensional SL cones [9,10,12]. In this article, we study in detail their geometry paying special attention to features needed in our gluing constructions. In particular, we classify them up to congruence; we determine their full group of symmetries (including various discrete symmetries) in all cases; we prove that many of them are closed and embedded; and finally understand the limiting singular geometry with detailed asymptotics. In understanding the detailed asymptotics a fundamental role is played by a certain conserved quantity (a component of the torque) considered in [13].

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Special Lagrangian cones

Cited by 80 publications

References 22 publications

Special Lagrangian Submanifolds with Isolated Conical Singularities. v. Survey and Applications

Special Lagrangian Submanifolds with Isolated Conical Singularities. v. Survey and Applications

On a new construction of special Lagrangian immersions in complex Euclidean space

The geometry of SO(<i>p</i>) × SO(<i>q</i>)-invariant special Lagrangian cones

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