The dual Thurston norm and the geometry of closed 3-manifolds

Mitsuhiro Itoh and Takahisa Yamase
Source: Osaka J. Math. Volume 43, Number 1 (2006), 121-129.

Abstract

We investigate closed Riemannian 3-manifolds which satisfy an extremal condition. Using monopole equations and considering the action of the covering transformations, we decide the geometric structure of such 3-manifolds. As a result, we characterize the geometry of 3-manifolds with monopole classes whose dual Thurston norm is equal to one.

First Page:
Primary Subjects: 14J80
Secondary Subjects: 57M50
Full-text: Open access

Permanent link to this document: http://projecteuclid.org/euclid.ojm/1146242997
Mathematical Reviews number (MathSciNet): MR2222404
Zentralblatt MATH identifier: 1107.57021

References

D. Auckly: The Thurston norm and three-dimensional Seiberg-Witten theory, Osaka J. Math. 33 (1996), 737--750.
Mathematical Reviews (MathSciNet): MR1424683
R. Bott and L.W. Tu: Differential Forms in Algebraic Topology, Springer-Verlag, New York-Berlin, 1982.
Mathematical Reviews (MathSciNet): MR658304
Zentralblatt MATH: 0496.55001
A.L. Carey and B.L. Wang: Notes on Seiberg-Witten-Floer Theory, Contemp. Math. 258 (2000), 71--85.
Mathematical Reviews (MathSciNet): MR1778097
Zentralblatt MATH: 0977.57036
R. Friedman and J.W. Morgan: Smooth Four-Manifolds and Complex Surfaces, Springer-Verlag, Berlin, 1994.
Mathematical Reviews (MathSciNet): MR1288304
Zentralblatt MATH: 0817.14017
M. Itoh: Almost Kähler $4$-manifold, $L^2$-scalar curvature functional and Seiberg-Witten equations, Internat. J. Math. 15 (2004), 573--580.
Mathematical Reviews (MathSciNet): MR2078882
Digital Object Identifier: doi:10.1142/S0129167X04002478
M. Itoh and T. Satou: Circle bundle metrics and the conformal flatness, preprint (1999).
Mathematical Reviews (MathSciNet): MR1692909
S. Kobayashi and K. Nomizu: Foundations of Differential Geometry, I, John Wiley & Sons, Inc., New York, 1963.
P.B. Kronheimer: Embedded surfaces and gauge theory in three and four dimensions; in Surveys in Differential Geometry, III, Int. Press, Boston, MA, 1998, 243--298.
Mathematical Reviews (MathSciNet): MR1677890
Zentralblatt MATH: 0965.57030
P.B. Kronheimer and T.S. Mrowka: The genus of embedded surfaces in the projective plane, Math. Res. Lett. 1 (1994), 797--808.
Mathematical Reviews (MathSciNet): MR1306022
P.B. Kronheimer and T.S. Mrowka: Scalar curvature and the Thurston norm, Math. Res. Lett. 4 (1997), 931--937.
Mathematical Reviews (MathSciNet): MR1492131
C. LeBrun: Kodaira Dimension and the Yamabe Problem, Comm. in Anal. Geom. 7 (1999), 133--156.
Mathematical Reviews (MathSciNet): MR1674105
Y. Matsushima: Differentiable Manifolds, Marcel Dekker, Inc., New York, 1972.
Mathematical Reviews (MathSciNet): MR346831
Zentralblatt MATH: 0233.58001
C.T. McMullen: The Alexander polynomial of a $3$-manifold and the Thurston norm on cohomology, Ann. Sci. École Norm. Sup. (4) 35 (2002), 153--171.
Mathematical Reviews (MathSciNet): MR1914929
Digital Object Identifier: doi:10.1016/S0012-9593(02)01086-8
P. Ozsváth and Z. Szabó: Holomorphic disks and genus bounds, Geom. Topol. 8 (2004), 311--334.
Mathematical Reviews (MathSciNet): MR2023281
P. Scott: The geometries of $3$-manifolds, Bull. London Math. Soc. 15 (1983), 401--487.
Mathematical Reviews (MathSciNet): MR705527
W.P. Thurston: Three-Dimensional Geometry and Topology, I, Princeton University Press, Princeton, NJ, 1997.
Mathematical Reviews (MathSciNet): MR1435975
Zentralblatt MATH: 0873.57001
S. Vidussi: Norms on the cohomology of a $3$-manifold and SW theory, Pacific J. Math. 208 (2003), 169--186.
Mathematical Reviews (MathSciNet): MR1979378