Minimal Lagrangian submanifolds of Lorentzian complex space forms with constant sectional curvature
For \(n\in\{3,4,\dots\}\), \(s\in \{1,\dots, n\}\) and \(\widetilde c\in\mathbb{R}\setminus\{0\}\) there is defined a semi-Euclidean inner product on \(\mathbb{C}^{n-1}\) by \(\beta_s(z,w):= \text{Re}(-\sum^s_{k=1} \overline{z_k} w_k+ \sum^{n+1}_{k= s+1} \overline{z_k} w_k)\), and a hypersurface \(N_s(\widetilde c):= \{z\in \mathbb{C}^{n+1}|\widetilde c\cdot\beta_s(z, z)= 1\}\). The image \(\widetilde M^n_1(4\widetilde c):= \pi(N_s(\widetilde c))\subset \mathbb{C} P^n\) with \(s=1\) for \(\widetilde c>0\) and \(s=2\) for \(\widetilde c<0\) under the usual projection \(\pi: \mathbb{C}^{n+1}\setminus \{0\}\to\mathbb{C} P^n\) can canonically be endowed with the structure of a complex Lorentzian manifold of constant holomorphical sectional curvature \(4\widetilde c\). The authors classify the minimal submanifolds \(M^n(c)\subset\widetilde M^n_1(4\widetilde c)\) of constant curvature \(c\neq\widetilde c\) which are Lagrangian, i.e.: \(JT_pM^n= \perp_p M^n\) for \(p\in M^n\) (where \(J\) is the complex structure of \(\mathbb{C} P^n\)). In particular, they derive \(c=0\); moreover, for \(\widetilde c<0\) only one such submanifold exists, namely for \(n=3\). An essential step in the proof is related to a formal similarity of the structure of the authors' work on Lorentzian affine hyperspheres with constant affine sectional curvature [Trans. Am. Math. Soc. 352, 1581-1599 (2000)].
scientific article; zbMATH DE number 1295495