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

Author: Luc Vrancken
Journal: Proc. Amer. Math. Soc. 130 (2002), 1459-1466
MSC (2000): Primary 53B25, 53B30
Published electronically: October 17, 2001
MathSciNet review: 1879970
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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).

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Additional Information

Luc Vrancken
Affiliation: Mathematisch Instituut, Universiteit Utrecht, Budapestlaan 6, 3584CD Utrecht, The Netherlands

Keywords: Lagrangian, constant sectional curvature, indefinite complex space forms
Received by editor(s): October 7, 1999
Received by editor(s) in revised form: November 10, 2000
Published electronically: October 17, 2001
Additional Notes: This work was partially supported by a research fellowship of the Alexander von Humboldt Stiftung (Germany)
Communicated by: Christopher Croke
Article copyright: © Copyright 2001 American Mathematical Society