Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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


Author: Yutaka Kanda
Journal: Trans. Amer. Math. Soc. 353 (2001), 2215-2243
MSC (2000): Primary 57R57
DOI: https://doi.org/10.1090/S0002-9947-01-02697-6
Published electronically: February 15, 2001
MathSciNet review: 1814068
Full-text PDF Free Access

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 [Enhancements On Off] (What's this?)

  • [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: https://doi.org/10.1090/S0002-9947-01-02697-6
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
Published electronically: February 15, 2001
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society