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The perturbation of the Seiberg-Witten equations revisited. (English) Zbl 1377.57031

The Seiberg-Witten equations led to a number of new results in topology and geometry in a short amount of time. One of the early applications of the equations was C. LeBrun’s work related to extremal Kähler metrics on \(4\)-manifolds [Math. Res. Lett. 2, No. 5, 653–662 (1995; Zbl 0874.53051)]. LaBrun followed this with several other related results [C. LeBrun, in: Global analysis and harmonic analysis. Papers from the conference, Marseille-Luminy, France, May 1999. Paris: Société Mathématique de France. 179–200 (2000; Zbl 1002.53024); in: The many facets of geometry. A tribute to Nigel Hitchin. Oxford: Oxford University Press. 17–33 (2010; Zbl 1229.53073); in: Surveys in differential geometry. Vol. VIII: Lectures on geometry and topology held in honor of Calabi, Lawson, Siu, and Uhlenbeck at Harvard University, Cambridge, MA, USA, May 3–5, 2002. Somerville, MA: International Press. 235–255 (2003; Zbl 1051.53038); Geom. Dedicata 91, 137–154 (2002; Zbl 1032.53031); Invent. Math. 145, No. 2, 279–316 (2001; Zbl 0999.53027); Commun. Anal. Geom. 5, No. 3, 535–553 (1997; Zbl 0901.53028)]. Motivated to understand and generalize some of LeBrun’s curvature inequalities, Furuta and Matsuo developed a new perturbation of the Seiberg-Witten equations. In this paper, they give a brief list of prior perturbations, use a special case of their new perturbations to give a quick proof of LeBrun’s inequality as motivation. They follow with the description of a full family or perturbations and a proof of generalized curvature inequalities.

MSC:

57R57 Applications of global analysis to structures on manifolds
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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