The faces of the Grassmannian of 3-planes in \({\mathbb{R}}^ 7\) (calibrated geometries on \({\mathbb{R}}^ 7)\)
type: Item
instance of: scholarly article
title: The faces of the Grassmannian of 3-planes in \({\mathbb{R}}^ 7\) (calibrated geometries on \({\mathbb{R}}^ 7)\)
zbMATH Open document ID: 0556.53036
DOI: 10.1007/BF01388960
author: F. Reese Harvey
author: Frank Morgan
published in: Inventiones Mathematicae
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143309
review text: A classification is given of the faces of the Grassmannian of oriented 3- planes in \({\mathbb{R}}^ 7\), and hence of the calibrated geometries of 3-dimensional area-minimizing surfaces in \({\mathbb{R}}^ 7\). There are five discrete types of faces and five infinite families of types. New phenomena include faces which are not totally geodesic in the Grassmannian: nonround \(S^ 1\)'s, \(S^ 2\)'s, and \(S^ 3\)'s. Classification of calibrated geometries in \({\mathbb{R}}^ 6\), the first nonclassical case, was completed in 1982 by \textit{J. Dadok} and \textit{R. Harvey} [Duke Math. J. 50, 1231-1243 (1983; Zbl 0535.49030)] and \textit{F. Morgan} [Lin. Alg. Appl. 66, 1-28 (1985)]. Results are forthcoming on \({\mathbb{R}}^ 8\) by Dadok, Harvey, and Morgan and on Grassmannians by Gluck, Morgan, and Ziller.
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53A07
Mathematics Subject Classification ID: 53A10
zbMATH DE Number: 3886696
zbMATH Keywords: calibrated geometries
zbMATH Keywords: area-minimizing surfaces
zbMATH Keywords: Grassmannian
MaRDI profile type: Publication
cites work: Calibrations on \(R^ 6\)
cites work: Q5565773
cites work: Normal and integral currents
cites work: Calibrated geometries
cites work: Q3725140
cites work: The exterior algebra \(\Lambda ^ kR^ n\) and area minimization
cites work: On the Singular Structure of Three-Dimensional, Area-Minimizing Surfaces
cites work: On the singular structure of two-dimensional area minimizing surfaces in \({\mathbb{R}}^ n\)
cites work: Q5547252
cites work: Q3679859
description: scientific article; zbMATH DE number 3886696