On genericity for holomorphic curves in four-dimensional almost-complex manifolds. (English) Zbl 0911.53014

Hofer, Helmut; Lizan, Véronique; Sikorav, Jean-Claude

J. Geom. Anal. 7, No. 1, 149-159 (1997).

Let \((V,J)\) be an almost complex manifold of dimension four and \(C_0\subset (V,J)\) a pseudo-holomorphic curve without boundary. 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.

Reviewer: Nicolai Konstantinovish Smolentsev (Kemerovo)

MSC:

53C15	General geometric structures on manifolds (almost complex, almost product structures, etc.)
35J60	Nonlinear elliptic equations
47A53	(Semi-) Fredholm operators; index theories
58J05	Elliptic equations on manifolds, general theory
30G20	Generalizations of Bers and Vekua type (pseudoanalytic, \(p\)-analytic, etc.)

Keywords:

pseudo-holomorphic curve; almost complex manifold; nonlinear elliptic PDE; Fredholm index

Citations:

Zbl 0592.53025

References:

[1]	Gromov, M., Pseudo-holomorphic curves in symplectic manifolds, Invent. Math., 82, 307-347 (1985) · Zbl 0592.53025
[2]	Griffiths, P.; Harris, J., Principles of Algebraic Geometry (1978), New York: Wiley, New York · Zbl 0408.14001
[3]	Hofer, H., Pseudo-holomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three, Invent. Math., 114, 515-563 (1993) · Zbl 0797.58023
[4]	Ivashkovich, S., and Shevchishin, V. Pseudoholomorphic curves and envelopes of meromorphy of two-spheres in CP^2, preprint, Essen/Bochum/Dusseldorf SFB237, January 1995.
[5]	McDuff, D., Rational and ruled symplectic manifolds, J. Am. Math. Soc., 3, 679-711 (1990) · Zbl 0723.53019
[6]	McDuff, D., The local behaviour of holomorphic curves in almost-complex 4-manifolds, J. Diff. Geom., 34, 143-164 (1991) · Zbl 0736.53038
[7]	McDuff, D., and Salamon, D. Pseudo-holomorphic curves. Manuscript. · Zbl 1272.53002
[8]	McDuff, D., and Salamon, D.J-holomorphic curves and quantum cohomology.Amer. Math. Soc. Lect. Notes6 (1994). · Zbl 0809.53002
[9]	Sikorav, J. C.; Audin, M.; Lafontaine, J., Some properties of holomorphic curves in almost complex manifolds, Chapter V ofHolomorphic Curves in Symplectic Geometry (1994), Basel: Birkhäuser, Basel
[10]	Vekua, N. I., Generalized Analytic Functions (1962), London: Pergamon, London · Zbl 0100.07603