# Pseudo holomorphic curves in symplectic manifolds

@article{Gromov1985PseudoHC, title={Pseudo holomorphic curves in symplectic manifolds}, author={Mikhael Gromov}, journal={Inventiones mathematicae}, year={1985}, volume={82}, pages={307-347} }

Definitions. A parametrized (pseudo holomorphic) J-curve in an almost complex manifold (IS, J) is a holomorphic map of a Riemann surface into Is, say f : (S, J3 ~(V, J). The image C=f(S)C V is called a (non-parametrized) J-curve in V. A curve C C V is called closed if it can be (holomorphically !) parametrized by a closed surface S. We call C regular if there is a parametrization f : S ~ V which is a smooth proper embedding. A curve is called rational if one can choose S diffeomorphic to the… Expand

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#### References

SHOWING 1-10 OF 39 REFERENCES

SOME INTEGRATION PROBLEMS IN ALMOST-COMPLEX AND COMPLEX MANIFOLDS

- Mathematics
- 1963

The theory of almost-complex manifolds leads to a number of problems which properly belong to the field of differential equations. This paper deals with three such problems. The first concerns… Expand

A symplectic fixed point theorem for complex projective spaces

- Mathematics
- 1985

1. Arnold's conjecture. An automorphism ^ of a symplectic manifold (P,u;) is homologous to the identity if there is a smooth family tyt (* € [0,1]) of automorphisms such that the time-dependent… Expand

The existence of minimal immersions of two-spheres

- Mathematics
- 1977

In this paper we develop an existence theory for minimal 2-spheres in compact Riemannian manifolds. The spheres we obtain are conformally immersed minimal surfaces except at a finite number of… Expand

Function Theory on Differentiable Submanifolds

- Mathematics
- 1974

Publisher Summary This chapter presents an analytic continuation of holomorphic functions defined on or near a subset of ℂ n to some larger set. This is the Hartogs phenomenon, restricted to function… Expand

The Birkhoff-Lewis fixed point theorem and a conjecture of V.I. Arnold

- Mathematics
- 1983

Abstract : The following conjecture of V. I. Arnold is proved: every measure preserving diffeomorphism of the torus T2, which is homologeous to the identity, and which leaves the center of mass… Expand

Lectures on Symplectic Manifolds

- Mathematics
- 1977

Introduction Symplectic manifolds and lagrangian submanifolds, examples Lagrangian splittings, real and complex polarizations, Kahler manifolds Reduction, the calculus of canonical relations,… Expand

Twistorial construction of harmonic maps of surfaces into four-manifolds

- Mathematics
- 1985

L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions… Expand

Riemann boundary value problems for nonlinear elliptic systems

- Mathematics
- 1983

A boundary value problem is solved for the nonlinear first order complex differential equation in the plane. The complex function H is measurable on , Lipschitz continuous with respect to the last ...

Surfaces minimales et la construction de Calabi-Penrose

- Mathematics
- 1984

© Société mathématique de France, 1985, tous droits réservés. L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisque/) implique l’accord avec les… Expand

Du côté de chez Pu

- Mathematics
- 1972

© Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1972, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www.… Expand