no. 3
Hessian surfaces and local Lagrangian embeddings
Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 3, pp. 675-685.

In this paper we prove that any smooth surfaces can be locally isometrically embedded into C2 as Lagrangian surfaces. As a byproduct we obtain that any smooth surfaces are Hessian surfaces.

DOI : 10.1016/j.anihpc.2017.07.003
Classification : 53C42, 35F50, 53A15
Mots clés : Hessian metric, Local embedding, Lagrangian surfaces, Gauss–Codazzi system 
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     title = {Hessian surfaces and local {Lagrangian} embeddings},
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Han, Qing; Wang, Guofang. Hessian surfaces and local Lagrangian embeddings. Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 3, pp. 675-685. doi : 10.1016/j.anihpc.2017.07.003. http://archive.numdam.org/articles/10.1016/j.anihpc.2017.07.003/

[1] Amari, S.; Amstrong, J. Curvature of Hessian manifolds, Differ. Geom. Appl., Volume 33 (2014), pp. 1–12 | DOI | MR | Zbl

[2] Amari, S.; Nagaoka, H. Methods of Information Geometry, Transl. Math. Monogr., vol. 191, American Mathematical Society, Providence, 2000 | MR | Zbl

[3] Ay, N.; Tuschmann, W. Dually flat manifolds and global information geometry, Open Syst. Inf. Dyn., Volume 9 (2002), pp. 195–200 | MR | Zbl

[4] Bryant, R., 2013 http://mathoverflow.net/questions/122308/

[5] Cannas da Silva, A. Lectures on Symplectic Geometry, Lect. Notes Math., vol. 1764, Springer-Verlag, Berlin, 2001 | MR | Zbl

[6] Capogna, L.; Lin, F.-H. Legendrian energy minimizers. I. Heisenberg group target, Calc. Var. Partial Differ. Equ., Volume 12 (2001), pp. 145–171 | DOI | MR | Zbl

[7] Castro, I.; Urbano, C. Lagrangian surfaces in the complex Euclidean plane with conformal Maslov form, Tohoku Math. J., Volume 45 (1993), pp. 565–582 | DOI | MR | Zbl

[8] Chen, B.-Y.; Houh, C.-S. On totally real flat surfaces, Boll. Unione Mat. Ital., A (5), Volume 15 (1978), pp. 370–378 | MR | Zbl

[9] Chen, B.-Y. Riemannian geometry of Lagrangian submanifolds, Taiwan. J. Math., Volume 5 (2001), pp. 681–723 | MR | Zbl

[10] Chen, B.-Y.; Dillen, F.; Verstraelen, L.; Vrancken, L. Lagrangian isometric immersions of a real-space-form Mn(c) into a complex space form M˜n(4c) , Math. Proc. Camb. Philos. Soc., Volume 124 (1998), pp. 107–125 | MR | Zbl

[11] Cheng, S.-Y.; Yau, S.-T. On the regularity of the Monge–Ampère equation det⁡(∂2u/∂xi∂xj)=F(x,u) , Commun. Pure Appl. Math., Volume 30 (1977), pp. 41–68 | MR | Zbl

[12] Furuhata, H.; Matsuzoe, H.; Urakawa, H. Open problems in affine differential geometry and related topics, Interdiscip. Inf. Sci., Volume 4 (1999), pp. 125–127 | MR | Zbl

[13] Freed, D.S. Special Kähler manifolds, Commun. Math. Phys., Volume 203 (1999), pp. 31–52 | DOI | MR | Zbl

[14] Han, Q.; Hong, J.-X. Isometric Embedding of Riemannian Manifolds in Euclidean Spaces, Math. Surv. Monogr., vol. 130, American Mathematical Society, Providence, 2006 | DOI | MR | Zbl

[15] Hitchin, N.J. The moduli space of complex Lagrangian submanifolds, Asian J. Math., Volume 3 (1999), pp. 77–91 | DOI | MR | Zbl

[16] Janet, M. Sur la possibilité de plonger un espace riemannian donné dans un espace euclidien, Ann. Soc. Pol. Math., Volume 5 (1926), pp. 38–43 | JFM

[17] Loftin, J.; Yau, S.-T.; Zaslow, E. Affine manifolds, SYZ geometry and the “Y” vertex, J. Differ. Geom., Volume 71 (2005), pp. 129–158 | DOI | MR | Zbl

[18] Moore, J.D.; Morvan, J.-M. On isometric Lagrangian immersions, Ill. J. Math., Volume 45 (2001), pp. 833–849 | MR | Zbl

[19] Nesterov, Y.; Nemirovskii, A.S.; Ye, Y. Interior-Point Polynomial Algorithms in Convex Programming, vol. 13, SIAM, 1994 | MR | Zbl

[20] Poznjak, E.G. Isometric embeddings of two-dimensional Riemannian metrics in Euclidean space, Russ. Math. Surv., Volume 28 (1973) no. 4, pp. 47–76 | MR | Zbl

[21] Shima, H. The Geometry of Hessian Structures, World Scientific, 2007 | DOI | MR | Zbl

[22] Weinstein, A. Lectures on Symplectic Manifolds, CBMS Reg. Conf. Ser. Math., vol. 29, American Mathematical Society, Providence, 1977 | DOI | MR | Zbl

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