On indefinite special Lagrangian submanifolds in indefinite complex Euclidean spaces (Q1021780)

Following very close the study of special Lagrangian submanifolds of \(\mathbb{C}^m\) in the calibration context of \textit{R.\ Harvey} and \textit{H. B.\ Blaine} in [``Calibrated geometries'', Acta Math. 148, 47--157 (1982; Zbl 0584.53021)], the author gives a step by step analogy for the case of special indefinite Lagrangian submanifolds of the pseudo-Riemannian case \(\mathbb{C}^m_k= \mathbb{R}^m_k \times \mathbb{R}^m_k\). Analogous examples are also given. It is observed that these ``calibrated'' submanifolds of index \(k\) are neither volume minimizing nor volume maximizing, as already observed by \textit{V. P. Gorokh} in [``The stability of a minimal surface in a pseudo-Euclidean space'', J.\ Sov.\ Math.\ 53, No. 5, 491--493 (1991); translation from Ukr.\ Geom.\ Sb.\ 33, 41--45 (1990; Zbl 0783.53011)]. This contrasts with the case of the space-like calibration defined by \textit{M.\ Warren} [arXiv:math/070229, 2007, to appear in Trans.\ Am.\ Math.\ Soc.], calibrating special Lagrangian submanifolds, with volume maximizing property.