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Convergence of Einstein Yang-Mills systems

Author: Hongliang Shao
Journal: Proc. Amer. Math. Soc. 142 (2014), 969-979
MSC (2010): Primary 53C21, 53C23
Published electronically: December 10, 2013
MathSciNet review: 3148531
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Abstract: In this paper, we prove a convergence theorem for sequences of Einstein Yang-Mills systems on $ U(1)$-bundles over closed $ n$-manifolds with some bounds for volumes, diameters, $ L^{2}$-norms of bundle curvatures, and
$ L^{\frac {n}{2}}$-norms of curvature tensors. This result is a generalization of earlier compactness theorems for Einstein manifolds.

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  • [1] Michael T. Anderson, Ricci curvature bounds and Einstein metrics on compact manifolds, J. Amer. Math. Soc. 2 (1989), no. 3, 455-490. MR 999661 (90g:53052),
  • [2] Michael T. Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds, Invent. Math. 102 (1990), no. 2, 429-445. MR 1074481 (92c:53024),
  • [3] Michael T. Anderson, Einstein metrics and metrics with bounds on Ricci curvature, Vols. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 443-452. MR 1403944 (97g:53046)
  • [4] Shigetoshi Bando, Atsushi Kasue, and Hiraku Nakajima, On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth, Invent. Math. 97 (1989), no. 2, 313-349. MR 1001844 (90c:53098),
  • [5] Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987. MR 867684 (88f:53087)
  • [6] Jeff Cheeger, Finiteness theorems for Riemannian manifolds, Amer. J. Math. 92 (1970), 61-74. MR 0263092 (41 #7697)
  • [7] Jeff Cheeger and Gang Tian, Curvature and injectivity radius estimates for Einstein 4-manifolds, J. Amer. Math. Soc. 19 (2006), no. 2, 487-525 (electronic). MR 2188134 (2006i:53042),
  • [8] Shiing Shen Chern, Circle bundles, Geometry and topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976), Lecture Notes in Math., Vol. 597, Springer, Berlin, 1977, pp. 114-131. MR 0461353 (57 #1338)
  • [9] Misha Gromov, Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics, vol. 152, Birkhäuser Boston Inc., Boston, MA, 1999. Based on the 1981 French original [MR0682063 (85e:53051)]; With appendices by M. Katz, P. Pansu and S. Semmes; Translated from the French by Sean Michael Bates. MR 1699320 (2000d:53065)
  • [10] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin, 1977. MR 0473443 (57 #13109)
  • [11] R. E. Greene and H. Wu, Lipschitz convergence of Riemannian manifolds, Pacific J. Math. 131 (1988), no. 1, 119-141. MR 917868 (89g:53063)
  • [12] Nigel Hitchin, Compact four-dimensional Einstein manifolds, J. Differential Geometry 9 (1974), 435-441. MR 0350657 (50 #3149)
  • [13] Claude 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. MR 2681684 (2011h:53098),
  • [14] Claude LeBrun, Einstein metrics, four-manifolds, and differential topology, Surveys in differential geometry, Vol. VIII (Boston, MA, 2002) Surv. Differ. Geom., VIII, Int. Press, Somerville, MA, 2003, pp. 235-255. MR 2039991 (2005g:53078)
  • [15] P. Peterson, Riemannian Geometry, Science Press (2007).
  • [16] Jeffrey D. Streets, Ricci Yang-Mills flow, ProQuest LLC, Ann Arbor, MI, 2007. Thesis (Ph.D.)-Duke University. MR 2709943
  • [17] G. Tian, On Calabi's conjecture for complex surfaces with positive first Chern class, Invent. Math. 101 (1990), no. 1, 101-172. MR 1055713 (91d:32042),
  • [18] Andrea Nicole Young, Modified Ricci flow on a principal bundle, ProQuest LLC, Ann Arbor, MI, 2008. Thesis (Ph.D.)-The University of Texas at Austin. MR 2712036

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Additional Information

Hongliang Shao
Affiliation: Department of Mathematics, Capital Normal University, Beijing, People’s Republic of China 100048
Address at time of publication: College of Mathematics and Statistics, Chongqing University, 55, Daxuecheng South Road, Shapingba, Chongqing, People’s Republic of China 401331

Keywords: Einstein Yang-Mills system, Gromov-Hausdorff convergence
Received by editor(s): January 2, 2012
Received by editor(s) in revised form: April 16, 2012
Published electronically: December 10, 2013
Communicated by: Lei Ni
Article copyright: © Copyright 2013 American Mathematical Society

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