Generic regularity of homologically area minimizing hypersurfaces in eight dimensional manifolds
type: Item
instance of: scholarly article
title: Generic regularity of homologically area minimizing hypersurfaces in eight dimensional manifolds
zbMATH Open document ID: 0843.58027
author: Nathan Smale
published in: Communications in Analysis and Geometry
publication date: 1996-08-21T00:00:00Z
review text: Consider a smooth, compact manifold \(N\) of dimension \(n+1\) with nontrivial homology class \(H_n (N, \mathbb{Z})\). For \(k\geq 3\) let \(M^k\) denote the class of \(C^k\) metrics on \(N\), endowed with the \(C^k\) topology, and let \(|\cdot |_k\) be a norm defining the topology. Given \(\alpha\in H_n (N, \mathbb{Z})\), \(\alpha\neq 0\), the subclass \({\mathcal F}^k_\alpha\) of \(M^k\) consists of all metrics \(g\) such that there exists a smooth area minimizing (relative to \(g\)) \(n\)-dimensional, integer multiplicity current \(T\) homologous to \(\alpha\). The main result of this paper states that for \(n=7\), \({\mathcal F}^k_\alpha\) contains an open subset being dense in \(M^k\).
Mathematics Subject Classification ID: 58E12
Mathematics Subject Classification ID: 49Q05
Mathematics Subject Classification ID: 53A10
zbMATH DE Number: 831710
zbMATH Keywords: minimal currents
zbMATH Keywords: regularity
zbMATH Keywords: homology classes
reviewed by: M. Fuchs
MaRDI profile type: Publication
DOI: 10.4310/CAG.1993.V1.N2.A2
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description: scientific article; zbMATH DE number 831710