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Curvature, cones and characteristic numbers

Published online by Cambridge University Press:  25 April 2013

MICHAEL ATIYAH  and
CLAUDE LEBRUN
MICHAEL ATIYAH
Affiliation:
School of Mathematics, University of Edinburgh, Mayfield Road, Edinburgh EH9 3JZ. e-mail: m.atiyah@ed.ac.uk
CLAUDE LEBRUN
Affiliation:
Department of Mathematics, SUNY, Stony Brook, NY 11794-3651, U.S.A. e-mail: claude@math.sunysb.edu
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Abstract

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We study Einstein metrics on smooth compact 4-manifolds with an edge-cone singularity of specified cone angle along an embedded 2-manifold. To do so, we first derive modified versions of the Gauss–Bonnet and signature theorems for arbitrary Riemannian 4-manifolds with edge-cone singularities, and then show that these yield non-trivial obstructions in the Einstein case. We then use these integral formulæ to obtain interesting information regarding gravitational instantons which arise as limits of such edge-cone manifolds.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 155 , Issue 1 , July 2013 , pp. 13 - 37
Copyright
Copyright © Cambridge Philosophical Society 2013 

References

REFERENCES

[1]Abreu, M.Kähler metrics on toric orbifolds. J. Differential Geom. 58 (2001), 151–187.CrossRefGoogle Scholar
[2]Anderson, M. T.Ricci curvature bounds and Einstein metrics on compact manifolds. J. Amer. Math. Soc. 2 (1989), 455–490.CrossRefGoogle Scholar
[3]Atiyah, M. F.Elliptic operators and compact groups. Lecture Notes in Math., vol. 401 (Springer-Verlag, Berlin, 1974).CrossRefGoogle Scholar
[4]Atiyah, M. F. and Hitchin, N. J.The geometry and dynamics of magnetic monopoles. M. B. Porter Lectures (Princeton University Press, 1988).CrossRefGoogle Scholar
[5]Atiyah, M. F. and Singer, I. M.The index of elliptic operators. III. Ann. of Math. (2) 87 (1968), 546–604.CrossRefGoogle Scholar
[6]Atiyah, M. F., Patodi, V. K. and Singer, I. M.Spectral asymmetry and Riemannian geometry. III. Math. Proc. Camb. Phil. Soc. 79 (1976), 71–99.CrossRefGoogle Scholar
[7]Bando, S. and Kobayashi, R.Ricci-flat Kähler metrics on affine algebraic manifolds. II. Math. Ann. 287 (1990), 175–180.CrossRefGoogle Scholar
[8]Besse, A. L.Einstein manifolds. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) vol. 10 (Springer-Verlag, 1987).CrossRefGoogle Scholar
[9]Bielawski, R.Monopoles and the Gibbons–Manton metric. Comm. Math. Phys. 194 (1998), 297–321.CrossRefGoogle Scholar
[10]Boyer, C. P. and Galicki, K.Sasakian geometry. Oxford Mathematical Monographs (Oxford University Press, 2008).Google Scholar
[11]Brendle, S. Ricci flat Kahler metrics with edge singularities. E-print arXiv:1103.5454 [math.DG] (2011).Google Scholar
[12]Cheeger, J. and Tian, G.Curvature and injectivity radius estimates for Einstein 4-manifolds. J. Amer. Math. Soc. 19 (2006), 487–525.CrossRefGoogle Scholar
[13]Chen, X., LeBrun, C. and Weber, B.On conformally Kähler, Einstein manifolds. J. Amer. Math. Soc. 21 (2008), 1137–1168.CrossRefGoogle Scholar
[14]Cherkis, S. A.Instantons on gravitons. Comm. Math. Phys. 306 (2011), 449–483.CrossRefGoogle Scholar
[15]Cherkis, S. A. and Hitchin, N. J.Gravitational instantons of type Dk. Comm. Math. Phys. 260 (2005), 299–317.CrossRefGoogle Scholar
[16]Cherkis, S. A. and Kapustin, A.Dk gravitational instantons and Nahm equations. Adv. Theor. Math. Phys. 2 (1998), 1287–1306 (1999).CrossRefGoogle Scholar
[17]Chern, S.-S.A simple intrinsic proof of the Gauss–Bonnet formula for closed Riemannian manifolds. Ann. of Math. (2) 45 (1944), 747–752.CrossRefGoogle Scholar
[18]Conner, P. E. and Floyd, E. E.Differentiable periodic maps. Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F., Band 33 (Academic Press Inc., New York, 1964).Google Scholar
[19]Dai, X. and Wei, G.Hitchin–Thorpe inequality for noncompact Einstein 4-manifolds. Adv. Math. 214 (2007), 551–570.CrossRefGoogle Scholar
[20]Donaldson, S. K.Kähler metrics with cone singularities along a divisor. In Essays in Mathematics and its Applications (Springer, Heidelberg, 2012), pp. 49–79.CrossRefGoogle Scholar
[21]Eguchi, T. and Hanson, A. J.Self-dual solutions to Euclidean gravity. Ann. Physics 120 (1979), 82–106.CrossRefGoogle Scholar
[22]Gibbons, G. W. and Hawking, S. W.Classification of gravitational instanton symmetries. Comm. Math. Phys. 66 (1979), 291–310.CrossRefGoogle Scholar
[23]Gibbons, G. W. and Manton, N. S.The moduli space metric for well-separated BPS monopoles. Phys. Lett. B 356 (1995), 32–38.CrossRefGoogle Scholar
[24]Hirzebruch, F.The signature of ramified coverings. In Global Analysis (Papers in Honor of K. Kodaira), pp. 253–265 (Tokyo Press, 1969).Google Scholar
[25]Hirzebruch, F.The signature theorem: reminiscences and recreation. In Prospects in Mathematics, (Proc. Sympos., Princeton Univ., Princeton, N.J., 1970), pp. 3–31. Ann. of Math. Studies, no. 70 (Princeton University Press, 1971).Google Scholar
[26]Hitchin, N. J.Compact four-dimensional Einstein manifolds. J. Differential Geometry 9 (1974), 435–441.CrossRefGoogle Scholar
[27]Hitchin, N. J.Polygons and gravitons. Math. Proc. Camb. Phil. Soc. 85 (1979), 465–476.CrossRefGoogle Scholar
[28]Hitchin, N. J.Poncelet polygons and the Painlevé equations. In Geometry and Analysis (Bombay, 1992), pp. 151–185 (Tata Inst. Fund. Res., Bombay, 1995).Google Scholar
[29]Hitchin, N. J.Twistor spaces, Einstein metrics and isomonodromic deformations. J. Differential Geom. 42 (1995), 30–112.CrossRefGoogle Scholar
[30]Hitchin, N. J.A new family of Einstein metrics. In Manifolds and Geometry (Pisa, 1993). Sympos. Math., XXXVI, pp. 190–222 (Cambridge Univ. Press, Cambridge, 1996).Google Scholar
[31]Izawa, T.Note on the Riemann–Hurwitz type formula for multiplicative sequences. Proc. Amer. Math. Soc. 131 (2003), 3583–3588.CrossRefGoogle Scholar
[32]Jeffres, T., Mazzeo, R. and Rubinstein, Y. Kähler–Einstein metrics with edge singularities. E-print arXiv:1105.5216 [math.DG] (2011).Google Scholar
[33]Kasue, A.A convergence theorem for Riemannian manifolds and some applications. Nagoya Math. J. 114 (1989), 21–51.CrossRefGoogle Scholar
[34]Kawasaki, T.The signature theorem for V-manifolds. Topology 17 (1978), 75–83.CrossRefGoogle Scholar
[35]Kawasaki, T.The index of elliptic operators over V-manifolds. Nagoya Math. J. 84 (1981), 135–157.CrossRefGoogle Scholar
[36]Kronheimer, P. B.Instantons gravitationnels et singularités de Klein. C. R. Acad. Sci. Paris Sér. I Math. 303 (1986), 53–55.Google Scholar
[37]Kronheimer, P. B.A Torelli-type theorem for gravitational instantons. J. Differential Geom. 29 (1989), 685–697.CrossRefGoogle Scholar
[38]LeBrun, C.Complete Ricci-flat Kähler metrics on Cn need not be flat. In Several complex variables and complex geometry, part 2 (Santa Cruz, CA, 1989), volume 52 of Proc. Sympos. Pure Math. (Amer. Math. Soc., Providence, RI, 1991), pp. 297–304.CrossRefGoogle Scholar
[39]LeBrun, C.Explicit self-dual metrics on ℂℙ2#. . .ℂℙ2. J. Differential Geom. 34 (1991), 223–253.Google Scholar
[40]LeBrun, C., Nayatani, S. and Nitta, T.Self-dual manifolds with positive Ricci curvature. Math. Z. 224 (1997), 49–63.CrossRefGoogle Scholar
[41]Minerbe, V.Rigidity for multi-Taub-NUT metrics. J. Reine Angew. Math. 656 (2011), 47–58.Google Scholar
[42]Nakajima, H.Self-duality of ALE Ricci-flat 4-manifolds and positive mass theorem. In Recent topics in differential and analytic geometry, volume 18 of Adv. Stud. Pure Math. (Academic Press, Boston, MA, 1990), pp. 385–396.Google Scholar
[43]Pedersen, H.Einstein metrics, spinning top motions and monopoles. Math. Ann. 274 (1986), 35–59.CrossRefGoogle Scholar
[44]Satake, I.The Gauss-Bonnet theorem for V-manifolds. J. Math. Soc. Japan 9 (1957), 464–492.CrossRefGoogle Scholar
[45]Thorpe, J. A.Some remarks on the Gauss–Bonnet formula. J. Math. Mech. 18 (1969), 779–786.Google Scholar
[46]Tian, G. and Yau, S.-T.Complete Kähler manifolds with zero Ricci curvature. I. J. Amer. Math. Soc. 3 (1990), 579–609.Google Scholar
[47]Tod, K. P.Self-dual Einstein metrics from the Painlevé VI equation. Phys. Lett. A 190 (1994), 221–224.CrossRefGoogle Scholar
[48]Tod, K. P.The SU(∞)-Toda field equation and special four-dimensional metrics. In Geometry and physics (Aarhus, 1995), vol. 184 of Lecture Notes in Pure and Appl. Math. (Dekker, New York, 1997), pp. 307–312.Google Scholar