On the existence of higher dimensional Enneper's surface
type: Item
instance of: scholarly article
title: On the existence of higher dimensional Enneper's surface
zbMATH Open document ID: 0869.53005
DOI: 10.1007/BF02566436
author: Jaigyoung Choe
published in: Commentarii Mathematici Helvetici
publication date: 1997-05-12T00:00:00Z
full work available at URL: https://eudml.org/doc/140413
review text: The author shows the existence of complete minimal immersions of \(\mathbb{R}^n\) into \(\mathbb{R}^{n+1}\) for \(n=3,4,5,6\), which are generalizations of the classical Enneper's minimal surface in \(\mathbb{R}^3\). His proof consists of three steps. Firstly, he solves a Dirichlet problem for the minimal surface equation and gets a certain \(n\)-dimensional minimal hypersurface in \(\mathbb{R}^{n+1}\). Secondly, he obtains a compact Enneper type surface by reflecting the minimal hypersurface across the totally geodesic part of its boundary. Thirdly, by blowing up the compact Enneper type surface he constructs the desired complete minimal hypersurface in \(\mathbb{R}^{n+1}\). The process of blowing up requires some curvature estimates. For this reason, the construction is restricted to the case \(n\leq 6\).
Mathematics Subject Classification ID: 53A10
zbMATH DE Number: 1008569
zbMATH Keywords: Enneper hypersurfaces
zbMATH Keywords: blow-up
zbMATH Keywords: minimal immersions
zbMATH Keywords: curvature estimates
reviewed by: Erich Hoy
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1008569