On a minimal Lagrangian submanifold of \(\mathbf C^n\) foliated by spheres
type: Item
instance of: scholarly article
title: On a minimal Lagrangian submanifold of \(\mathbf C^n\) foliated by spheres
zbMATH Open document ID: 0974.53059
author: Ildefonso Castro
author: Francisco Urbano
published in: Michigan Mathematical Journal
publication date: 2001-01-01T00:00:00Z
review text: The Lagrangian catenoid in \(\mathbb{C}^n\) is defined by \(M_0=\{(x,y)\in\mathbb{C}^n=\mathbb{R}^n \times \mathbb{R}^n : |x|y=|y|x\), \(\text{Im}(|x|+i|y|)^n=1\), \(|y|<|x|\tan(\pi/n)\}\). The authors prove the following interesting characterizations of the Lagrangian catenoid. Theorem 1. Let \(\phi: M\to \mathbb{C}^n\) be a minimal (nonflat) Lagrangian immersion of an \(n\)-manifold. Then \(M\) is foliated by pieces of round \((n-1)\)-spheres of \(\mathbb{C}^n\) if and only if \(\phi\) is congruent (up to dilations) to an open subset of the Lagrangian catenoid. Theorem 2. Let \(\phi: M\to \mathbb{C}^m\) be a (nonflat) complex immersion of a complex \(n\)-dimensional Kähler manifold \(M\). Then \(M\) is foliated by pieces of round \((2n-1)\)-spheres of \(\mathbb{C}^m\) if and only if \(n=1\) and \(\phi\) is congruent (up to dilations) to an open subset of the Lagrangian catenoid. Theorem 3. Let \(M\) be an \(n\)-dimensional \((n\geq 3)\) complete minimal (nonflat) submanifold with finite total scalar curvature immersed in Euclidean space \({\mathbf E}^{2n}\). Then the compactification by the inversion of \(M\) is Lagrangian for a certain orthogonal complex structure on \({\mathbf E}^{2n}\) if and only if \(M\) is (up to dilations) the Lagrangian catenoid.
Mathematics Subject Classification ID: 53D12
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 1548302
zbMATH Keywords: Lagrangian submanifold
zbMATH Keywords: Lagrangian catenoid
reviewed by: Bang-Yen Chen
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1030132359
OpenAlex ID: W2003835974
DOI: 10.1307/MMJ/1030132359
description: scientific article; zbMATH DE number 1548302