Convergence of Einstein Yang-Mills systems

Shao, Hongliang

doi:10.1090/S0002-9939-2013-11817-4

Convergence of Einstein Yang-Mills systems
HTML articles powered by AMS MathViewer

by Hongliang Shao PDF

Proc. Amer. Math. Soc. 142 (2014), 969-979 Request permission

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.

References

Michael T. Anderson, Ricci curvature bounds and Einstein metrics on compact manifolds, J. Amer. Math. Soc. 2 (1989), no. 3, 455–490. MR 999661, DOI 10.1090/S0894-0347-1989-0999661-1
Michael T. Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds, Invent. Math. 102 (1990), no. 2, 429–445. MR 1074481, DOI 10.1007/BF01233434
Michael T. Anderson, Einstein metrics and metrics with bounds on Ricci curvature, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 443–452. MR 1403944
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, DOI 10.1007/BF01389045
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, DOI 10.1007/978-3-540-74311-8
Jeff Cheeger, Finiteness theorems for Riemannian manifolds, Amer. J. Math. 92 (1970), 61–74. MR 263092, DOI 10.2307/2373498
Jeff Cheeger and Gang Tian, Curvature and injectivity radius estimates for Einstein 4-manifolds, J. Amer. Math. Soc. 19 (2006), no. 2, 487–525. MR 2188134, DOI 10.1090/S0894-0347-05-00511-4
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
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
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin-New York, 1977. MR 0473443, DOI 10.1007/978-3-642-96379-7
R. E. Greene and H. Wu, Lipschitz convergence of Riemannian manifolds, Pacific J. Math. 131 (1988), no. 1, 119–141. MR 917868, DOI 10.2140/pjm.1988.131.119
Nigel Hitchin, Compact four-dimensional Einstein manifolds, J. Differential Geometry 9 (1974), 435–441. MR 350657
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, DOI 10.1093/acprof:oso/9780199534920.003.0003
Claude LeBrun, Einstein metrics, four-manifolds, and differential topology, Surveys in differential geometry, Vol. VIII (Boston, MA, 2002) Surv. Differ. Geom., vol. 8, Int. Press, Somerville, MA, 2003, pp. 235–255. MR 2039991, DOI 10.4310/SDG.2003.v8.n1.a8
P. Peterson, Riemannian Geometry, Science Press (2007).
Jeffrey D. Streets, Ricci Yang-Mills flow, ProQuest LLC, Ann Arbor, MI, 2007. Thesis (Ph.D.)–Duke University. MR 2709943
G. Tian, On Calabi’s conjecture for complex surfaces with positive first Chern class, Invent. Math. 101 (1990), no. 1, 101–172. MR 1055713, DOI 10.1007/BF01231499
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