Special Lagrangian submanifolds in the complex sphere
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 16 (2007) no. 2, p. 215-227

We construct a family of Lagrangian submanifolds in the complex sphere which are foliated by (n-1)-dimensional spheres. Among them we find those which are special Lagrangian with respect to the Calabi-Yau structure induced by the Stenzel metric.

Nous construisons une famille de sous-variétés lagrangiennes dans la sphère complexe qui sont feuilletées par des sphères de dimension n-1. Nous décrivons celles qui sont de plus lagrangiennes spéciales pour la structure de Calabi-Yau induite par la métrique de Stenzel.

@article{AFST_2007_6_16_2_215_0,
     author = {Anciaux, Henri},
     title = {Special Lagrangian submanifolds in the complex sphere},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 16},
     number = {2},
     year = {2007},
     pages = {215-227},
     doi = {10.5802/afst.1145},
     mrnumber = {2331538},
     zbl = {pre05236223},
     language = {en},
     url = {http://www.numdam.org/item/AFST_2007_6_16_2_215_0}
}
Anciaux, Henri. Special Lagrangian submanifolds in the complex sphere. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 16 (2007) no. 2, pp. 215-227. doi : 10.5802/afst.1145. http://www.numdam.org/item/AFST_2007_6_16_2_215_0/

[Au] Audin (M.).— Lagrangian submanifolds, in Symplectic geometry of integrable Hamiltonian systems, M. Audin, A. Cannas da Silva, E. Lerman, Advanced courses in Mathematics CRM Barcelona, Birkhäuser (2003). | MR 2000744

[Br] Bryant (R.).— Some examples of special Lagrangian tori, Adv. Theor. Math. Phys. 3, no. 1, p. 83-90 (1999). | MR 1704218 | Zbl 0980.32006

[CBLP] Cvetič (M.), Gibbons (G. W.), Lü (H.) & Pope (C. N.).— Ricci-flat metrics, harmonic forms and brane resolutions, Comm. Math. Phys. 232, no. 3, p. 457-500 (2003). | MR 1952474 | Zbl 1027.53044

[CMU] Castro (I.), Montealegre (C. R.) & Urbano (F.).— Minimal Lagrangian submanifolds in the complex hyperbolic space, Illinois J. Math. 46, no. 3, p. 695-721 (2002). | MR 1951236 | Zbl 1032.53052

[CU1] Castro (I.), Urbano (F.).— On a Minimal Lagrangian Submanifold of n Foliated by Spheres, Mich. Math. J., 46, p. 71-82 (1999). | MR 1682888 | Zbl 0974.53059

[CU2] Castro (I.), Urbano (F.).— On a new construction of special Lagrangian immersions in complex Euclidean space, Q. J. Math. 55, no. 3, p. 253-265 (2004). | MR 2082092 | Zbl 1086.53074

[Ha] Haskins (M.).— Special Lagrangian cones, Amer. J. Math. 126, no. 4, p. 845-871 (2004). | MR 2075484 | Zbl 1074.53067

[HL] Harvey (R.), Lawson (H. B.).— Calibrated geometries, Acta Mathematica, 148, p. 47-157 (1982). | MR 666108 | Zbl 0584.53021

[Jo1] Joyce (D.).— U(1)-invariant special Lagrangian 3-folds in 3 and special Lagrangian fibrations. Turkish J. Math. 27, no. 1, p. 99-114 (2003). | MR 1975333 | Zbl 1040.53091

[Jo2] Joyce (D.).— Riemannian holonomy groups and calibrated geometry, in Calabi-Yau manifolds and related geometries. Lectures from the Summer School held in Nordfjordeid, June 2001. Universitext. Springer-Verlag, Berlin (2003). | MR 1963560 | Zbl 1016.53041

[Oh] Oh (Y.-G.).— Second variation and stabilities of minimal Lagrangian submanifolds in Kähler manifolds, Invent. Math. 101, p. 501-519 (1990). | MR 1062973 | Zbl 0721.53060

[St] Stenzel (M.).— Ricci-flat metrics on the complexification of a compact rank one symmetric space, Manuscripta Math. 80, no. 2, p. 151-163 (1993). | MR 1233478 | Zbl 0811.53049

[SYZ] Strominger (A.), Yau (S.-T.) & Zaslow (E.).— Mirror symmetry is T-duality, Nuclear Physics, B479, hep-th/9606040 (1996). | MR 1429831 | Zbl 0896.14024

[Y] Yau (S.-T.).— On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equations I, Comm. Pure Appl. Math. 31, p. 339-411 (1978). | MR 480350 | Zbl 0369.53059