Available in electronic format
Available in print format		 Transacrions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

The monopole equations and $J$-holomorphic curves on weakly convex almost Kähler 4-manifolds

Author(s): Yutaka Kanda
Journal: Trans. Amer. Math. Soc. 353 (2001), 2215-2243.
MSC (2000): Primary 57R57
Posted: February 15, 2001
Retrieve article in: PDF DVI PostScript
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract:

We prove that a weakly convex almost Kähler 4-manifold contains a compact, non-constant $J$-holomorphic curve if the corresponding monopole invariant is not zero and if the corresponding line bundle is non-trivial.

References:

[E]
Y.Eliashberg, Contact 3-manifolds, twenty years since Martinet's work, Ann. Inst. Fourier. 42 (1992), 165-192. MR 93k:57029

[F]
K.Fløyshov, The Seiberg-Witten equations and 4-manifolds with boundary, Math. Res. Lett. 3 (1996), 373-390. MR 97i:57037

[Gr]
M.Gromov, Pseudo-holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307-347. MR 87j:53053

[G-T]
D.Gilbarg and N.S.Trudinger, Elliptic partial differential equalities of second order, 2nd ed., Springer-Verlag, 1983. MR 86c:35035

[K-M1]
P.B.Kronheimer and T.S.Mrowka, The genus of embedded surfaces in the projective plane, Math. Res. Lett. 1 (1994), 797-808. MR 96a:57073

[K-M2]
-, Monopoles and contact structures, Inv. Math. 130 (1997), 209-255. MR 98h:57058

[Ko]
D.Kotschick, The Seiberg-Witten invariants of symplectic four-manifolds [after C.H.Taubes], $\text{S}\acute {\text{e}}\text{minaire Bourbaki 48}\grave {\text{e}}\text{ne ann}\acute {\text{e}}\text{e}$ $\text{n}^{\circ }812$ (1995-1996). MR 98h:57057

[Ma]
R.Mandelbaum, Irrational connected sums, Trans. Amer. Math. Soc. 247 (1979), 137-156. MR 80e:57023

[M-S-T]
J.W.Morgan, Z.Szab $\acute {\text{o}}$ and C.H.Taubes, A product formula for the Seiberg-Witten invariants and the generalized Thom conjecture, Jour. Diff. Geom. 44 (1996), 706-788. MR 97m:57052

[O-O]
H.Ohta and K.Ono, Simple singularities and topology of symplectically filling 4-manifold, Comment. Math. Helv. 74 (1999), 575-590. CMP 2000:06

[T1]
C.H.Taubes, SW $\Rightarrow $ Gr: From the Seiberg-Witten equations to pseudo-holomorphic curves, J. Amer. Math. Soc. 9 (1996), 845-918. MR 97a:57053

[T2]
-, Gr $\Rightarrow $ SW: From pseudo-holomorphic curves to Seiberg-Witten solutions, J. Differential Geom. 51 (1999), 203-334. MR 20001:53123

[T3]
-, Counting pseudo-holomorphic curves in dimension 4, preprint.

[W]
E.Wittem, Monopoles and four-manifolds, Math. Res. Lett. 1 (1994), 769-796. MR 96d:57035

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 57R57

Retrieve articles in all Journals with MSC (2000): 57R57

Additional Information:

Yutaka Kanda
Affiliation: Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan
Email: kanda@math.sci.hokudai.ac.jp

DOI: 10.1090/S0002-9947-01-02697-6
PII: S 0002-9947(01)02697-6
Keywords: Symplectic structure, monopole equation, $J$-holomorphic curve
Received by editor(s): March 8, 1999
Received by editor(s) in revised form: February 28, 2000
Posted: February 15, 2001
Copyright of article: Copyright 2001, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google