Minimal surfaces in foliated manifolds
type: Item
instance of: scholarly article
title: Minimal surfaces in foliated manifolds
zbMATH Open document ID: 0601.53024
DOI: 10.1007/BF02621899
author: Joel R. Hass
published in: Commentarii Mathematici Helvetici
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/140037
review text: The main object of these studies is a 2-dimensional foliation of a 3- manifold by leaves minimal with respect to some Riemannian metric. Among other results the following considerations are made: Using some local and global properties of minimal surfaces the author collects several topological conditions preventing a foliated compact 3-manifold from the existence of a Riemannian metric making all the leaves minimal. Some of these conditions are well known, some follow immediately from the Novikov theorem. However, the proofs given here are new, interesting and independent of the Novikov theorem which is proved in a new way in the Appendix. Also, the author proves that given a region \(Q\subset R^ 3\) equipped with a minimal foliation \({\mathcal F}\) and a closed curve \(\Gamma\) contained with its convex hull in Q we have \[ \chi (S)\leq k/2+\sum_{x\in A}[1- \deg (x)], \] where S is a branched minimal surface spanned by \(\Gamma\), k is the number of tangencies of \(\Gamma\) to \({\mathcal F}\) and A is the set of branch points of S. The analogous result for higher dimensions is obtained provided that \({\mathcal F}\) is totally geodesic.
Mathematics Subject Classification ID: 53C12
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 3969563
zbMATH Keywords: branched immersion
zbMATH Keywords: Novikov theorem
zbMATH Keywords: minimal foliation
zbMATH Keywords: minimal surface
zbMATH Keywords: totally geodesic
reviewed by: Paweł Walczak
MaRDI profile type: Publication
OpenAlex ID: W2024259583
description: scientific article; zbMATH DE number 3969563