Skip to main content
×
×
Home
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 1
  • Cited by
    This chapter has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Оревков, Степан Юрьевич and Orevkov, Stepan Yur'evich 1996. Диаграммы Рудольфа и аналитическая реализация накрытия Витушкина. Математические заметки, Vol. 60, Issue. 2, p. 206.

    ×
  • Print publication year: 1991
  • Online publication date: June 2011

Filling by holomorphic discs and its applications

Summary

The survey is devoted to application of the technique of filling by holomorphic discs to different symplectic and complex analytic problems.

COMPLEX AND SYMPLECTIC RECOLLECTIONS

J-Convexity

Let X, J be an almost complex manifold of the real dimension 4 and Σ be an oriented hypersurface in X of the real codimension 1. Each tangent plane Tx(Σ), x ∈ Σ, contains a unique complex line ξxTx(Σ) which we will call a complex tangency to Σ at x. The complex tangency is canonically oriented and, therefore, cooriented. Hence the tangent plane distribution ξ on Σ can be defined by an equation α = 0 where the 1-form α is unique up to multiplication by a positive function. The 2-form dα ∣ξ is defined up to the multiplication by the same positive factor. We say that Σ is J-convex (or pseudo-convex) if dα(T, JT) > 0 for any non-zero vector T ∈ ξx, x ∈ Σ. We use the word “pseudo-convex” when the almost complex structure J is not specified.

An important property of a J-convex hypersurface Σ is that it cannot be touched inside (according to the canonical coorientation of Σ) by a J-holomorphic curve. In particular, if Ω is a domain in X bounded by a smooth J-convex boundary ∂Ω then all interior points of a J-holomorphic curve CX with ∂C ⊂ ∂Ω belong to IntΩ. Moreover, C is transversal to ∂Ω in all regular points of its boundary ∂C.

Recommend this book

Email your librarian or administrator to recommend adding this book to your organisation's collection.

Geometry of Low-Dimensional Manifolds
  • Online ISBN: 9780511629341
  • Book DOI: https://doi.org/10.1017/CBO9780511629341
Please enter your name
Please enter a valid email address
Who would you like to send this to *
×