Gap theorems for minimal submanifolds in \(\mathbb{R}^{n+1}\)
type: Item
instance of: scholarly article
title: Gap theorems for minimal submanifolds in \(\mathbb{R}^{n+1}\)
zbMATH Open document ID: 1020.53041
author: Lei Ni
published in: Communications in Analysis and Geometry
publication date: 2003-10-05T00:00:00Z
review text: Let \(M^n\subset \mathbb{R}^N\) a complete minimal immersed submanifold of \(\mathbb{R}^N\), \(n<N\), and let \(|A|\) be the norm of the second fundamental form of \(M^n\). The following two gap theorems are proved in this paper. Theorem A. Let \(N=n+1\), \(n\geq 4\). There exists a constant \(C_1(n)\), depending only on \(n\), such that if \[ \left( \int_M|A|^n dM^n \right)^{1/n} < C_1, \] then \(M\) is a hyperplane. Theorem B. Let \(N\) be arbitrary and \(n\geq 3\). There exists a constant \(C_2(n)\), depending only on \(n\), such that if \[ \left( \int_M|A|^n dM^n \right)^{1/n} < C_2, \] then \(M\) has only one end. Theorem B is used to prove Theorem A. The constants \(C_1\) and \(C_2\) depend on appropriate Sobolev inequalities. It is not known whether Theorem A holds for arbitrary \(N\).
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 58E12
Mathematics Subject Classification ID: 58E35
zbMATH DE Number: 1833135
zbMATH Keywords: minimal immersed submanifold
zbMATH Keywords: norm
zbMATH Keywords: second fundamental form
zbMATH Keywords: Sobolev inequalities
reviewed by: Manfredo P. do Carmo
MaRDI profile type: Publication
DOI: 10.4310/CAG.2001.V9.N3.A2
description: scientific article; zbMATH DE number 1833135