On an expansion of the special Lagrangian form
The main result of this paper states the equality of the comass of a simply separable \(p\)-form in \({\mathbb R}^{n}\) and the comass of the real part of the induced complex form on \({\mathbb C}^{n}\). By a simply separable \(p\)-form is meant a form that can be written \(\omega = \sum_{I} a_{I} dx_{I},\) where \({\mathbb R}_{n} = V_{1}\oplus V_{2}\oplus\dots V_{k}\) is an orthogonal decomposition, the multi-indices \(I=(i_{1},i_{2},\dots,i_{q})\) satisfy \(p=\sum_{j\in{I}}\dim{V_{j}},\) and \(dx_{I}=dx_{V_{i_{1}}} \wedge dx_{V_{i_{2}}} \wedge\dots\wedge dx_{V_{i_{q}}}\) with \(dx_{V_{i_{j}}}\) the volume form on \(V_{j}.\) The complex form induced by \(\omega\) is obtained by replacing each \(dx_{j}\) by \(dz_{j} = dx_{j} + idy_{j}.\) The authors also prove results about the maximal directions associated with induced forms. In particular, they completely describe the maximal directions of the real part of powers of the complex symplectic form on the quaternionic vector space \(H^{n}.\) The latter result generalizes work of \textit{R. Bryant} and \textit{R. Harvey} [J. Am. Math. Soc. 2, No. 1, 1-31 (1989; Zbl 0666.53032)].
scientific article; zbMATH DE number 1145965