Lagrangian submanifolds with zero scalar curvature in complex Euclidean space (Q5948681)

An isometric immersion \(\varphi:M^n \to\mathbb{C}^n\) into complex Euclidean space \(\mathbb{C}^n\) with standard complex structure \(J\) is called Lagrangian if \(J\) maps each tangent space of \(M\) into its corresponding normal space. In the family of the Lagrangian submanifolds invariant under the standard action of \(SO(n)\) on \(\mathbb{C}^n\) the Ricci flat ones are shown to be also flat; the latter have been studied in [\textit{B.-Y. Chen}, Tohoku Math. J. (2) 49, 277-297 (1997; Zbl 0877.53041)]. In the present paper a geometric description is given for such submanifolds of this family which have zero scalar curvature. They are totally geodesic. When \(n=3\), they correspond to a time slice of the Schwarzschild's spacetime that models the outer space around a massive star. It is remarked that they belong to a wide class of Lagrangian submanifolds of \(\mathbb{C}^n\) introduced in [\textit{A. Ros} and \textit{F. Urbano}, J. Math. Soc. Japan 50, 203-226 (1998; Zbl 0906.53037), Def. 1], and after an inversion they correspond with the Lagrangian \(n\)-sphere, studied in [\textit{I. Castro}, Geom. Dedicata 70, 197-208 (1998; Zbl 0904.53036)].