On an expansion of the special Lagrangian form
type: Item
instance of: scholarly article
title: On an expansion of the special Lagrangian form
zbMATH Open document ID: 0911.49024
author: Dao Trong Thi
author: Nguyen Duy Binh
published in: Acta Mathematica Vietnamica
publication date: 1999-04-22T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 49Q15
Mathematics Subject Classification ID: 49Q20
zbMATH DE Number: 1145965
zbMATH Keywords: area-minimizing
zbMATH Keywords: calibrated geometries
zbMATH Keywords: Lagrangian calibration
reviewed by: Harold Parks
MaRDI profile type: Publication
Recommended article: Complexes à \(n\) dimensions et intégrales abéliennes.
Recommended article: Symplectic direct decomposition of complex symplectic linear spaces
Recommended article: The Sturm-Liouville group
Recommended article: Integral geometry under the action of the first symplectic group
Recommended article: Separable calibrations and minimal surfaces
Recommended article: Symplectic classification of quadratic forms, and general Mehler formulas
Recommended article: Opérateurs de Dirac et équations de Maxwell.
Recommended article: On functions of several split-quaternionic variables
Recommended article: Q3473775
Recommended article: Über Algebren, die mit der Komplexgruppe in Zusammenhang stehen, und ihre Darstellungen.
description: scientific article; zbMATH DE number 1145965