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

Abstract

We study minimal Lagrangian immersions from an indefinite real space form M s n ( c ) M^n_s(c) into an indefinite complex space form M ~ s n ( 4 c ~ ) {\tilde {\mathbb {M}}}^n_s(4\tilde c) . Provided that c ≠ c ~ c \ne \tilde c , we show that M s n ( c ) 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 = c ~ 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).