Intersections of Quadrics, Moment-Angle Manifolds, and Hamiltonian-Minimal Lagrangian Embeddings

A. E. Mironov ; T. E. Panov

Funkcionalʹnyj analiz i ego priloženiâ, Tome 47 (2013) no. 1, pp. 47-61.

Résumé

We study the topology of Hamiltonian-minimal Lagrangian submanifolds $N$ in $\mathbb{C}^m$ constructed from intersections of real quadrics in a work of the first author. This construction is linked via an embedding criterion to the well-known Delzant construction of Hamiltonian toric manifolds. We establish the following topological properties of $N$: every $N$ embeds as a submanifold in the corresponding moment-angle manifold $\mathcal Z$, and every $N$ is the total space of two different fibrations, one over the torus $T^{m-n}$ with fiber a real moment-angle manifold $\mathcal{R}$ and the other over a quotient of $\mathcal{R}$ by a finite group with fiber a torus. These properties are used to produce new examples of Hamiltonian-minimal Lagrangian submanifolds with quite complicated topology.

Détail

Keywords: moment-angle manifold, simplicial fan, simple polytope.

Bibliographie

[1] H. Anciaux, I. Castro, “Construction of Hamiltonian-minimal Lagrangian submanifolds in complex Euclidean space”, Results Math., 60:1–4 (2011), 325–349 ; http://arxiv.org/abs/0906.4305 | DOI | MR | Zbl

[2] A. Amarzaya, Y. Ohnita, “On Hamiltonian stability of certain $H$-minimal Lagrangian submanifolds and related problems”, General Study on Riemannian Submanifolds, RIMS Kokyuroku, 1292, Kyoto University, Kyoto, 2002, 72–93 | MR

[3] A. Amarzaya, Y. Ohnita, Hamiltonian stability of certan symmetric $R$-spaces embedded in complex Euclidian spaces, Tokyo Metropolitan University preprint series, No. 13, 2002 | MR

[4] P. Biran, “Geometry of symplectic intersections”, Proceedings of ICM, Vol. 2 (Beijing, 2002), Higher Ed. Press, Beijing, 2002, 241–256 | MR

[5] F. Bosio, L. Meersseman, “Real quadrics in $\mathbb C^n$, complex manifolds and convex polytopes”, Acta Math., 197:1 (2006), 53–127 | DOI | MR | Zbl

[6] V. M. Buchstaber, “Lectures on Toric Topology”, Trends in Math., 10, no. 1, Information Center for Math. Sci., KAIST, 2008, 1–64

[7] V. M. Buchstaber, T. E. Panov, Torus Actions and Their Applications in Topology and Combinatorics, Univ. Lecture Series, 24, Amer. Math. Soc., Providence, R.I., 2002 | DOI | MR | Zbl

[8] I. Castro, H. Li, F. Urbano, “Hamiltonian-minimal Lagrangian submanifolds in complex space form”, Pacific J. Math., 227:1 (2006), 43–63 | DOI | MR | Zbl

[9] I. Castro, F. Urbano, “Examples of unstable Hamiltonian-minimal Lagrangian tori in $\mathbb C^2$”, Compositio Math., 111:1 (1998), 1–14 | DOI | MR | Zbl

[10] Bang-Yen Chen, O. J. Garay, “Classification of Hamiltonian-stationary Lagrangian sumbanifolds of constant in $\mathbb C P^3$ with positive nullity”, Nonlinear Anal., 69:2 (2008), 747–762 | DOI | MR | Zbl

[11] M. W. Davis, T. Januszkiewicz, “Convex polytopes, Coxeter orbifolds and torus actions”, Duke Math. J., 62:2 (1991), 417–451 | DOI | MR | Zbl

[12] T. Delzant, “Hamiltoniens périodiques et images convexes de l'application moment”, Bull. Soc. Math. France, 116:3 (1988), 315–339 | DOI | MR | Zbl

[13] Y. Dong, “Hamiltonian-minimal Lagrangian submanifolds in Kaehler manifolds with symmetries”, Nonlinear Anal., 67:3 (2007), 865–882 | DOI | MR | Zbl

[14] S. Gitler, S. Lopez de Medrano, Intersections of quadrics, moment-angle manifolds and connected sums, http://arxiv.org/abs/0901.2580 | MR

[15] M. Gromov, “Pseudoholomorphic curves in symplectic manifolds”, Invent. Math., 82:2 (1985), 307–347 | DOI | MR | Zbl

[16] M. Haskins, N. Kapouleas, Twisted products and $SO(p)\times SO(q)$-invariant special Lagrangian cones, http://arxiv.org/abs/1005.1419 | MR

[17] F. Hélein, P. Romon, “Hamiltonian stationary Lagrangian surfaces in $\mathbb C^2$”, Comm. Anal. Geom., 10:1 (2002), 79–126 | DOI | MR | Zbl

[18] F. Hélein, P. Romon, “Weierstrass representation of Lagrangian surfaces in four-dimensional space using spinors and quaternions”, Comm. Math. Helv., 75:4 (2000), 688–680 | DOI | MR

[19] D. Joyce, “Special Lagrangian $m$-folds in $\mathbb C^m$ with symmetries”, Duke Math. J., 115:1 (2002), 1–51 | DOI | MR | Zbl

[20] S. Lopez de Medrano, “Topology of the intersection of quadrics in $\mathbb R^n$”, Algebraic Topology, Lecture Notes in Math., 1370, Springer-Verlag, Berlin, 1989, 280–292 | DOI | MR

[21] Hui Ma, “Hamiltonian stationary Lagrangian surfaces in $\mathbb CP^2$”, Ann. Global. Anal. Geom., 27:1 (2005), 1–16 | DOI | MR | Zbl

[22] A. E. Mironov, “O gamiltonovo-minimalnykh lagranzhevykh torakh v $\mathbb CP^2$”, Sib. matem. zhurn., 44:6 (2003), 1324–1328 | MR | Zbl

[23] A. E. Mironov, “O novykh primerakh gamiltonovo-minimalnykh i minimalnykh lagranzhevykh podmnogoobrazii v $\mathbb C^n$ i $\mathbb C P^n$”, Matem. sb., 195:1 (2004), 89–102 | DOI | MR | Zbl

[24] A. E. Mironov, Dafeng Zuo, “On a family of conformally flat Hamiltonian-minimal Lagrangian tori in $\mathbb C P^3$”, Internat. Math. Res. Notices, 2008 | DOI | MR

[25] S. Nemirovski, “Lagrangian Klein bottles in $\mathbb R^{2n}$”, Geom. Funct. Anal., 19:3 (2009), 902–909 ; http://arxiv.org/abs/0712.1760 | DOI | MR | Zbl

[26] Yong-Geun Oh, “Volume minimization of Lagrangian submanifolds under Hamiltonian deformations”, Math. Z., 212:2 (1993), 175–192 | MR | Zbl

[27] T. Panov, “Moment-angle manifolds and complexes”, Trends in Math., 12, Information Center for Math. Sci., KAIST, 2010, 43–69; http://arxiv.org/abs/1008.5047

[28] T. Panov, Y. Ustinovsky, “Complex-analytic structures on moment-angle manifolds”, Moscow Math. J. (to appear) | MR

[29] A. Postnikov, “Permutohedra, associahedra, and beyond”, Internat. Math. Res. Notices, 6 (2009), 1026–1106 ; http://arxiv.org/abs/math/0507163 | DOI | MR | Zbl

[30] P. Seidel, “Graded Lagrangian submanifolds”, Bull. Soc. Math. Fr., 128:1 (2000), 103–149 | DOI | MR | Zbl

[31] V. V. Shevchishin, “Lagranzhevy vlozheniya butylki Kleina i kombinatornye svoistva gruppy klassov otobrazhenii”, Izv. RAN. Ser. matem., 73:4 (2009), 153–224 | DOI | MR | Zbl