Minimal Lagrangian submanifolds with constant sectional curvature in indefinite complex space forms

Vrancken, Luc

doi:10.1090/S0002-9939-01-06213-X

Minimal Lagrangian submanifolds with constant sectional curvature in indefinite complex space forms
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Proc. Amer. Math. Soc. 130 (2002), 1459-1466 Request permission

Abstract:

We study minimal Lagrangian immersions from an indefinite real space form $M^n_s(c)$ into an indefinite complex space form ${\tilde {\mathbb {M}}}^n_s(4\tilde c)$. Provided that $c \ne \tilde c$, we show that $M^n_s(c)$ has to be flat and we obtain an explicit description of the immersion. In the case when the metric is positive definite or Lorentzian, this result was respectively obtained by Ejiri (1982) and by Kriele and the author (1999). In the case that $c = \tilde c$, this theorem is no longer true; see for instance the examples discovered by Chen and the author (accepted for publication in the Tôhoku Mathematical Journal).

References

Kinetsu Abe and Martin A. Magid, Indefinite rigidity of complex submanifolds and maximal surfaces, Math. Proc. Cambridge Philos. Soc. 106 (1989), no. 3, 481–494. MR 1010373, DOI 10.1017/S0305004100068225
Manuel Barros and Alfonso Romero, Indefinite Kähler manifolds, Math. Ann. 261 (1982), no. 1, 55–62. MR 675207, DOI 10.1007/BF01456410
B. Y. Chen and L. Vrancken. Lagrangian minimal immersions of a lorentzian real space form $M^n_1(c)$ into a lorentzian complex space form $\tilde M^n_1(4 c)$, accepted for publication in Tôhoku Mathematical Journal.
Norio Ejiri, Totally real minimal immersions of $n$-dimensional real space forms into $n$-dimensional complex space forms, Proc. Amer. Math. Soc. 84 (1982), no. 2, 243–246. MR 637177, DOI 10.1090/S0002-9939-1982-0637177-8
Marcus Kriele and Luc Vrancken, Minimal Lagrangian submanifolds of Lorentzian complex space forms with constant sectional curvature, Arch. Math. (Basel) 72 (1999), no. 3, 223–232. MR 1671282, DOI 10.1007/s000130050326
Jean-Marie Morvan, Minimal Lagrangian submanifolds. A survey, Geometry and topology of submanifolds, III (Leeds, 1990) World Sci. Publ., River Edge, NJ, 1991, pp. 206–226. MR 1344469
Helmut Reckziegel, Horizontal lifts of isometric immersions into the bundle space of a pseudo-Riemannian submersion, Global differential geometry and global analysis 1984 (Berlin, 1984) Lecture Notes in Math., vol. 1156, Springer, Berlin, 1985, pp. 264–279. MR 824074, DOI 10.1007/BFb0075098
Alfonso Romero, On a certain class of complex Einstein hypersurfaces in indefinite complex space forms, Math. Z. 192 (1986), no. 4, 627–635. MR 847011, DOI 10.1007/BF01162709
Alfonso Romero, Some examples of indefinite complete complex Einstein hypersurfaces not locally symmetric, Proc. Amer. Math. Soc. 98 (1986), no. 2, 283–286. MR 854034, DOI 10.1090/S0002-9939-1986-0854034-6
Alfonso Romero, An extension of Calabi’s rigidity theorem to complex submanifolds of indefinite complex space forms, Manuscripta Math. 59 (1987), no. 3, 261–276. MR 909845, DOI 10.1007/BF01174794
Masaaki Umehara, Diastases and real analytic functions on complex manifolds, J. Math. Soc. Japan 40 (1988), no. 3, 519–539. MR 945350, DOI 10.2969/jmsj/04030519
L. Vrancken. The Magid-Ryan conjecture for equiaffine hyperspheres with constant sectional curvature. Journal of Differential Geometry, 54:99–138, 2000.