Journal of the Mathematical Society of Japan

The perturbation of the Seiberg–Witten equations revisited

Mikio FURUTA and Shinichiroh MATSUO

Full-text: Open access


We introduce a new class of perturbations of the Seiberg–Witten equations. Our perturbations offer flexibility in the way the Seiberg–Witten invariants are constructed and also shed a new light to LeBrun's curvature inequalities.

Article information

J. Math. Soc. Japan, Volume 68, Number 4 (2016), 1655-1668.

First available in Project Euclid: 24 October 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R57: Applications of global analysis to structures on manifolds, Donaldson and Seiberg-Witten invariants [See also 58-XX]
Secondary: 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60]

Seiberg–Witten equations scalar curvature self-dual Weyl curvature


FURUTA, Mikio; MATSUO, Shinichiroh. The perturbation of the Seiberg–Witten equations revisited. J. Math. Soc. Japan 68 (2016), no. 4, 1655--1668. doi:10.2969/jmsj/06841655.

Export citation


  • C. Bär, On nodal sets for Dirac and Laplace operators, Comm. Math. Phys., 188 (1997), 709–721.
  • S. Bauer, Intersection Forms of Spin Four-Manifolds, available at arXiv:1211.7092v1.
  • S. Bauer and M. Furuta, A stable cohomotopy refinement of Seiberg–Witten invariants, I, Invent. Math., 155 (2004), 1–19.
  • R. Fry and S. McManus, Smooth bump functions and the geometry of Banach spaces: a brief survey, Expo. Math., 20 (2002), 143–183.
  • P. Kronheimer and T. Mrowka, Monopoles and three-manifolds, New Mathematical Monographs, 10, Cambridge University Press, Cambridge, 2007.
  • C. LeBrun, Curvature and smooth topology in dimension four, Global analysis and harmonic analysis (Marseille–Luminy, 1999), Sémin. Congr., 4, Soc. Math. France, Paris, 2000, pp.,179–200 (English, with English and French summaries).
  • C. LeBrun, The Einstein–Maxwell equations, extremal Kähler metrics, and Seiberg–Witten theory, The many facets of geometry, Oxford Univ. Press, Oxford, 2010, pp.,17–33.
  • C. LeBrun, Einstein metrics, four-manifolds, and differential topology, Surveys in differential geometry, VIII. (Boston, MA, 2002), Surv. Differ. Geom., VIII, Int. Press, Somerville, MA, 2003, pp.,235–255.
  • C. LeBrun, Hyperbolic manifolds, harmonic forms, and Seiberg–Witten invariants, Proceedings of the Euroconference on Partial Differential Equations and their Applications to Geometry and Physics, (Castelvecchio Pascoli, 2000), 2002, pp.,137–154.
  • C. LeBrun, Polarized 4-manifolds, extremal Kähler metrics, and Seiberg–Witten theory, Math. Res. Lett., 2 (1995), 653–662.
  • C. LeBrun, Ricci curvature, minimal volumes, and Seiberg–Witten theory, Invent. Math., 145 (2001), 279–316.
  • C. LeBrun, Yamabe constants and the perturbed Seiberg–Witten equations, Comm. Anal. Geom., 5 (1997), 535–553.
  • T. Mrowka, P. Ozsváth and B. Yu, Seiberg–Witten monopoles on Seifert fibered spaces, Comm. Anal. Geom., 5 (1997), 685–791.
  • P. Ozsváth and Z. Szabó, The symplectic Thom conjecture, Ann. of Math. (2), 151 (2000), 93–124.
  • Y. Ruan, Virtual neighborhoods and the monopole equations, Topics in symplectic 4-manifolds, (Irvine, CA, 1996), First Int. Press Lect. Ser., I, Int. Press, Cambridge, MA, 1998, pp.,101–116.
  • C. Sung, Extremal almost-Kähler metrics and Seiberg–Witten theory, Ann. Global Anal. Geom., 22 (2002), 155–166.
  • C. H. Taubes, Seiberg–Witten and Gromov invariants for symplectic 4-manifolds, First International Press Lecture Series, 2, (Ed. R. Wentworth), International Press, Somerville, MA, 2000.
  • C. H. Taubes, The Seiberg–Witten invariants and symplectic forms, Math. Res. Lett., 1 (1994), 809–822.
  • E. Witten, Monopoles and four-manifolds, Math. Res. Lett., 1 (1994), 769–796.