The Journal of Geometric Analysis

, Volume 7, Issue 1, pp 149-159

First online:

On genericity for holomorphic curves in four-dimensional almost-complex manifolds

  • Helmut HoferAffiliated withCourant Institute, New-York University Email author 
  • , Véronique LizanAffiliated withU.F.R. Sciences, Mathématiques, I.U.F.M. de Toulouse
  • , Jean-Claude SikoravAffiliated withLaboratoire Emile Picard, UMR CNRS 5578, Université Paul Sabatier

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We consider spaces of immersed (pseudo-)holomorphic curves in an almost complex manifold of dimension four. We assume that they are either closed or compact with boundary in a fixed totally real surface, so that the equation for these curves is elliptic and has a Fredholm index. We prove that this equation is regular if the Chern class is ≥ 1 (in the case with boundary, if the ambient Maslov number is ≥ 1). Then the spaces of holomorphic curves considered will be manifolds of dimension equal to the index.

Math Subject Classification

30G20 35J60 47A53 53C15 58G03

Key Words and Phrases

Holomorphic curve almost complex manifold nonlinear elliptic PDE Fredholm index