Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter June 3, 2017

Gravitational instantons with faster than quadratic curvature decay (II)

  • Gao Chen ORCID logo EMAIL logo and Xiuxiong Chen

Abstract

This is our second paper in a series to study gravitational instantons, i.e. complete hyperkähler 4-manifolds with faster than quadratic curvature decay. We prove two main theorems: (i) The asymptotic rate of gravitational instantons to the standard models can be improved automatically. (ii) Any ALF-Dk gravitational instanton must be the Cherkis–Hitchin–Ivanov–Kapustin–Lindström–Roček metric.

Award Identifier / Grant number: 1515795

Funding statement: The second author is partially supported by the National Science Foundation through grant no. 1515795.

Acknowledgements

Both authors are grateful to the insightful and helpful discussions with Sir Simon Donaldson, Blaine Lawson, Claude LeBrun and Martin Roček. We also thank the referees and editors for polishing our article.

References

[1] M. Atiyah, V. G. Drinfeld, N. J. Hitchin and Y. I. Manin, Construction of instantons, Phys. Lett. A 65 (1978), no. 3, 185–187. 10.1142/9789812794345_0018Search in Google Scholar

[2] M. Atiyah and N. J. Hitchin, The geometry and dynamics of magnetic monopoles, Princeton University Press, Princeton 1988. 10.1515/9781400859306Search in Google Scholar

[3] S. Bando, A. Kasue and H. Nakajima, On a construction of coordinate at infinity on manifolds with fast curvature decay and maximal volume growth, Invent. Math. 97 (1989), no. 2, 313–349. 10.1007/BF01389045Search in Google Scholar

[4] P. Boalch, Hyperkahler manifolds and nonabelian Hodge theory of (irregular) curves, preprint (2012), http://arxiv.org/abs/1203.6607. Search in Google Scholar

[5] G. Chalmers, M. Roček and S. Wiles, Degeneration of ALF Dn metrics, J. High Energy Phys. 1999 (1999), no. 1, Paper No. 9. Search in Google Scholar

[6] J. Cheeger and D. Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differential Geom. 6 (1971/72), 119–128. 10.4310/jdg/1214430220Search in Google Scholar

[7] J. Cheeger and G. Tian, Curvature and injectivity radius estimates for Einstein 4-manifolds, J. Amer. Math. Soc. 19 (2006), no. 2, 487–525. 10.1090/S0894-0347-05-00511-4Search in Google Scholar

[8] G. Chen and X. X. Chen, Gravitational instantons with faster than quadratic curvature decay (I), preprint (2015), http://arxiv.org/abs/1505.01790. 10.4310/ACTA.2021.v227.n2.a2Search in Google Scholar

[9] S. A. Cherkis, Instantons on gravitons, Comm. Math. Phys. 306 (2011), no. 2, 449–483. 10.1007/s00220-011-1293-ySearch in Google Scholar

[10] S. A. Cherkis and N. J. Hitchin, Gravitational instantons of type Dk, Comm. Math. Phys. 260 (2005), no. 2, 299–317. 10.1007/s00220-005-1404-8Search in Google Scholar

[11] S. A. Cherkis and A. Kapustin, Singular monopoles and gravitational instantons, Comm. Math. Phys. 203 (1999), no. 3, 713–728. 10.1007/s002200050632Search in Google Scholar

[12] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Reprint of the 1998 edition, Springer, Berlin 2001. 10.1007/978-3-642-61798-0Search in Google Scholar

[13] M. Gromov, Volume and bounded cohomology, Publ. Math. Inst. Hautes Études Sci. 56 (1982), 5–99. Search in Google Scholar

[14] T. Hausel, E. Hunsicker and R. Mazzeo, Hodge cohomology of gravitational instantons, Duke Math. J. 122 (2004), no. 3, 485–548. 10.1215/S0012-7094-04-12233-XSearch in Google Scholar

[15] H.-J. Hein, Gravitational instantons from rational elliptic surfaces, J. Amer. Math. Soc. 25 (2012), no. 2, 355–393. 10.1090/S0894-0347-2011-00723-6Search in Google Scholar

[16] N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. Lond. Math. Soc. (3) 55 (1987), no. 1, 59–126. 10.1112/plms/s3-55.1.59Search in Google Scholar

[17] N. J. Hitchin, A. Karlhede, U. Lindström and M. Roček, Hyper-Kähler metrics and supersymmetry, Comm. Math. Phys. 108 (1987), no. 4, 535–589. 10.1007/BF01214418Search in Google Scholar

[18] R. A. Ionas, Elliptic constructions of hyperkähler metrics, Ph.D. thesis, State University of New York at Stony Brook, 2007. Search in Google Scholar

[19] I. T. Ivanov and M. Roček, Supersymmetric σ-models, twistors, and the Atiyah–Hitchin metric, Comm. Math. Phys. 182 (1996), no. 2, 291–302. 10.1007/BF02517891Search in Google Scholar

[20] A. Kapustin and E. Witten, Electric-magnetic duality and the geometric Langlands program, Commun. Number Theory Phys. 1 (2007), no. 1, 1–236. 10.4310/CNTP.2007.v1.n1.a1Search in Google Scholar

[21] K. Kodaira, On compact analytic surface II, Ann. of Math. (2) 77 (1963), 563–626. 10.2307/1970131Search in Google Scholar

[22] P. B. Kronheimer, A Torelli-type theorem for gravitational instantons, J. Differential Geom. 29 (1989), no. 3, 685–697. 10.4310/jdg/1214443067Search in Google Scholar

[23] P. B. Kronheimer, The construction of ALE spaces as hyper-Kähler quotients, J. Differential Geom. 29 (1989), no. 3, 665–683. 10.1142/9789814539395_0034Search in Google Scholar

[24] P. B. Kronheimer and H. Nakajima, Yang–Mills instantons on ALE gravitational instantons, Math. Ann. 288 (1990), no. 2, 263–307. 10.1007/BF01444534Search in Google Scholar

[25] G. Laumon and B. C. Ngô, Le lemme fondamental pour les groupes unitaires, Ann. of Math. (2) 168 (2008), no. 2, 477–573. 10.4007/annals.2008.168.477Search in Google Scholar

[26] H. B. Lawson, Jr. and M.-L. Michelsohn, Spin geometry, Princeton University Press, Princeton 1989. Search in Google Scholar

[27] C. LeBrun, Complete Ricci-flat Kähler metrics on Cn need not be flat, Several complex variables and complex geometry (Santa Cruz 1989), Proc. Sympos. Pure Math. 52 Part 2, American Mathematical Society, Providence (1991), 297–304. 10.1090/pspum/052.2/1128554Search in Google Scholar

[28] U. Lindström and M. Roček, New hyper-Kähler metrics and new supermultiplets, Comm. Math. Phys. 115 (1988), no. 1, 21–29. 10.1007/BF01238851Search in Google Scholar

[29] V. Minerbe, A mass for ALF manifolds, Comm. Math. Phys. 289 (2009), no. 3, 925–955. 10.1007/s00220-009-0823-3Search in Google Scholar

[30] V. Minerbe, Rigidity for multi-Taub-NUT metrics, J. reine angew. Math. 656 (2011), 47–58.10.1515/crelle.2011.042Search in Google Scholar

[31] H. Nakajima, Instantons on ALE spaces, quiver varieties, and Kac–Moody algebras, Duke Math. J. 76 (1994), no. 2, 365–416. 10.1215/S0012-7094-94-07613-8Search in Google Scholar

[32] A. Newlander and L. Nirenberg, Complex analytic coordinates in almost complex manifolds, Ann. of Math. (2) 65 (1957), 391–404. 10.2307/1970051Search in Google Scholar

[33] B. C. Ngô, Fibration de Hitchin et endoscopie, Invent. Math. 164 (2006), no. 2, 399–453. 10.1007/s00222-005-0483-7Search in Google Scholar

[34] N. Seiberg and E. Witten, Gauge dynamics and compactification to three dimensions, The mathematical beauty of physics (Saclay 1996), Adv. Ser. Math. Phys. 24, World Scientific, Singapore (1997), 333–366. Search in Google Scholar

[35] A. Sen, Dyon-monopole bound states, self-dual harmonic forms on the multi-monopole moduli space, and SL(2,) invariance in string theory, Phys. Lett. B 329 (1994), no. 2–3, 217–221. 10.1016/0370-2693(94)90763-3Search in Google Scholar

[36] A. Sen, Strong-weak coupling duality in four-dimensional string theory, Internat. J. Modern Phys. A 9 (1994), no. 21, 3707–3750. 10.1142/S0217751X94001497Search in Google Scholar

[37] A. Sen, A note on enhanced gauge symmetries in M- and string theory, J. High Energy Phys. 1997 (1997), no. 9, Paper No. 1. 10.1088/1126-6708/1997/09/001Search in Google Scholar

[38] E. Witten, Geometric Langlands from six dimensions, A celebration of the mathematical legacy of Raoul Bott, CRM Proc. Lecture Notes 50, American Mathematical Society, Providence (2010), 281–310. 10.1090/crmp/050/23Search in Google Scholar

[39] S. T. Yau, The role of partial differential equations in differential geometry, Proceedings of the international congress of mathematicians (Helsinki 1978), Annales Academiæ Scientiarum Fennicæ, Helsinki (1980), 237–250. Search in Google Scholar

Received: 2016-12-09
Published Online: 2017-06-03
Published in Print: 2019-11-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 29.3.2024 from https://www.degruyter.com/document/doi/10.1515/crelle-2017-0026/html
Scroll to top button