On genericity for holomorphic curves in four-dimensional almost-complex manifolds
type: Item
instance of: scholarly article
title: On genericity for holomorphic curves in four-dimensional almost-complex manifolds
zbMATH Open document ID: 0911.53014
DOI: 10.1007/BF02921708
author: Véronique Lizan
author: Jean-Claude Sikorav
author: Helmut Hofer
published in: The Journal of Geometric Analysis
publication date: 1999-04-28T00:00:00Z
review text: Let \((V,J)\) be an almost complex manifold of dimension four and \(C_0\subset (V,J)\) a pseudo-holomorphic curve without boundary. \textit{M. Gromov} [Invent. Math. 82, 307-347 (1985; Zbl 0592.53025)] showed that the equation of these curves is elliptic and has a Fredholm index. He also proved that, if the Chern class \(c_1(C_0)\geq 1\), then the space of pseudo-holomorphic curves near \(C_0\) is a manifold of dimension equal the index. The purpose of this note is to give a proof of this theorem as well as corresponding one for a curve \(C_0\) with boundary. Let \(C_0\subset V\) be a regular holomorphic curve with boundary \(\partial C_0\) in a totally real surface \(W\subset V\). In this paper, it is shown that, if the ambient Maslov number \(\mu (C_0,\partial C_0)\geq 1\), then the space of pseudo-holomorphic curves \((C,\partial C)\subset (V,W)\) near \((C_0,\partial C_0)\) is a manifold of dimension equal to the index.
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 47A53
Mathematics Subject Classification ID: 58J05
Mathematics Subject Classification ID: 30G20
zbMATH DE Number: 1192507
zbMATH Keywords: pseudo-holomorphic curve
zbMATH Keywords: almost complex manifold
zbMATH Keywords: nonlinear elliptic PDE
zbMATH Keywords: Fredholm index
Wikidata QID: Q56985514
reviewed by: Nicolai K. Smolentsev
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02921708
OpenAlex ID: W2051790307
description: scientific article; zbMATH DE number 1192507