A convergence theorem for Riemannian submanifolds
HTML articles powered by AMS MathViewer
- by Zhong Min Shen PDF
- Trans. Amer. Math. Soc. 347 (1995), 1343-1350 Request permission
Abstract:
In this paper we study the convergence of Riemannian submanifolds. In particular, we prove that any sequence of closed submanifolds with bounded normal curvature and volume in a closed Riemannian manifold subconverge to a closed submanifold in the ${C^{1,\alpha }}$ topology. We also obtain some applications to irreducible homogeneous manifolds and pseudo-holomorphic curves in symplectic manifolds.References
-
L. Andersson, The Pogorelov-Klingenberg theorem for submanifolds with bounded normal curvature, Report UMINF-87-80, Univ. of UMEA, 1980.
- 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
- Jeff Cheeger, Mikhail Gromov, and Michael Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geometry 17 (1982), no. 1, 15–53. MR 658471
- Jeff Cheeger, Finiteness theorems for Riemannian manifolds, Amer. J. Math. 92 (1970), 61–74. MR 263092, DOI 10.2307/2373498
- Shiu Yuen Cheng, Eigenvalue comparison theorems and its geometric applications, Math. Z. 143 (1975), no. 3, 289–297. MR 378001, DOI 10.1007/BF01214381
- Mikhael Gromov, Structures métriques pour les variétés riemanniennes, Textes Mathématiques [Mathematical Texts], vol. 1, CEDIC, Paris, 1981 (French). Edited by J. Lafontaine and P. Pansu. MR 682063 —, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307-347.
- 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 R. Howard, Private communication.
- David Hoffman and Joel Spruck, Sobolev and isoperimetric inequalities for Riemannian submanifolds, Comm. Pure Appl. Math. 27 (1974), 715–727. MR 365424, DOI 10.1002/cpa.3160270601
- Jürgen Jost and Hermann Karcher, Geometrische Methoden zur Gewinnung von a-priori-Schranken für harmonische Abbildungen, Manuscripta Math. 40 (1982), no. 1, 27–77 (German, with English summary). MR 679120, DOI 10.1007/BF01168235
- Atsushi Kasue, A convergence theorem for Riemannian manifolds and some applications, Nagoya Math. J. 114 (1989), 21–51. MR 1001487, DOI 10.1017/S0027763000001380
- Peter Li, Eigenvalue estimates on homogeneous manifolds, Comment. Math. Helv. 55 (1980), no. 3, 347–363. MR 593051, DOI 10.1007/BF02566692
- Peter Li, Minimal immersions of compact irreducible homogeneous Riemannian manifolds, J. Differential Geometry 16 (1981), no. 1, 105–115. MR 633629
- J. H. Michael and L. M. Simon, Sobolev and mean-value inequalities on generalized submanifolds of $R^{n}$, Comm. Pure Appl. Math. 26 (1973), 361–379. MR 344978, DOI 10.1002/cpa.3160260305
- I. G. Nikolaev, Parallel translation and smoothness of the metric of spaces with bounded curvature, Dokl. Akad. Nauk SSSR 250 (1980), no. 5, 1056–1058 (Russian). MR 561573 —, Smoothness of the metric of spaces with bilaterally bounded curvature in the space of A. D. Aleksandrov, Siberian Math. J. 24 (1983), 247-263.
- Stefan Peters, Convergence of Riemannian manifolds, Compositio Math. 62 (1987), no. 1, 3–16. MR 892147 T. Parker and J. Wolfson, A compactness theorem for Gromov’s moduli space, preprint, 1991.
- Tsunero Takahashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan 18 (1966), 380–385. MR 198393, DOI 10.2969/jmsj/01840380 R. Ye, Gromov’s compactness theorem for pseudo-holomorphic curves, preprint, 1991.
Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 347 (1995), 1343-1350
- MSC: Primary 53C20; Secondary 53C15, 53C23, 53C30, 53C40
- DOI: https://doi.org/10.1090/S0002-9947-1995-1254853-0
- MathSciNet review: 1254853