×

Tight Lagrangian surfaces in \(S ^{2} \times S ^{2}\). (English) Zbl 1187.53082

Summary: We determine all tight Lagrangian surfaces in \(S ^{2} \times S ^{2}\). In particular, globally tight Lagrangian surfaces in \(S ^{2} \times S ^{2}\) are nothing but real forms of this symmetric space.

MSC:

53D12 Lagrangian submanifolds; Maslov index
53C40 Global submanifolds
53C65 Integral geometry

References:

[1] Castro I., Urbano F.: Minimal Lagrangian surfaces in S 2 {\(\times\)} S 2. Comm. Anal. Geom. 15(2), 217– 248 (2007) · Zbl 1185.53063
[2] Givental, A.B.: Lagrangian imbeddings of surfaces and unfolded Whitney umbrella. Funkt. Anal. Prilozhen 20, 35–41 (1986); English transl., Funct. Anal. Appl. 20, 197–203 (1986) · Zbl 0621.58025
[3] Gotoh T.: The nullity of a compact minimal real hypersurface in a quaternion projective space. Geom. Dedicata 76, 53–64 (1999) · Zbl 0990.53061 · doi:10.1023/A:1005177220062
[4] Howard R.: The kinematic formula in Riemannian homogeneous spaces. Mem. Amer. Math. Soc. 106 (509), vi + 69 pp (1993) · Zbl 0810.53057
[5] Iriyeh, H., Ono, H., Sakai, T.: Integral geometry and Hamiltonian volume minimizing property of a totally geodesic Lagrangian torus in S 2 {\(\times\)} S 2. Proc. Japan Acad. 79, Ser. A 167–170 (2003). arXiv:mathDG/0310432 · Zbl 1055.53063
[6] Kuiper N.H.: Minimal total absolute curvature for immersions. Invent. Math. 10, 209–238 (1970) · Zbl 0195.51102 · doi:10.1007/BF01403250
[7] Kuiper N.H.: On convex maps. Nieuw Archief voor Wisk 10, 147–164 (1962) · Zbl 0112.36502
[8] Little J.A., Pohl W.F.: On tight immersions of maximal codimension. Invent. Math. 13, 179–204 (1971) · Zbl 0217.19101 · doi:10.1007/BF01404629
[9] Ma H., Ohnita Y.: On Lagrangian submanifolds in complex hyperquadrics and isoparametric hypersurfaces in spheres. Math. Z. 261, 749–785 (2009) · Zbl 1165.53037 · doi:10.1007/s00209-008-0350-5
[10] Oh Y.-G.: Second variation and stabilities of minimal lagrangian submanifolds in Kähler manifolds. Invent. Math. 101, 501–519 (1990) · Zbl 0721.53060 · doi:10.1007/BF01231513
[11] Oh Y.-G.: Tight Lagrangian submanifolds in \({\mathbb{C}P^n}\) . Math. Z. 207, 409–416 (1991) · doi:10.1007/BF02571398
[12] Oh Y.-G.: Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks, I. Comm. Pure Appl. Math. 46, 949–993 (1993) · Zbl 0795.58019 · doi:10.1002/cpa.3160460702
[13] Oh Y.-G.: Addendum to ”Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks, I”. Comm. Pure Appl. Math. 48, 1299–1302 (1995) · Zbl 0847.58036 · doi:10.1002/cpa.3160481104
[14] Oh Y.-G.: Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks, III: Arnold-Givental conjecture, The Floer Memorial volume, Progr. Math., vol. 133, pp. 555–573. Birkhäuser, Basel (1995) · Zbl 0842.58033
[15] Takeuchi M., Kobayashi S.: Minimal embeddings of R-symmetric spaces. J. Differ. Geom. 2, 203–215 (1968) · Zbl 0165.24901
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.