Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The limit spaces of two-dimensional manifolds with uniformly bounded integral curvature


Author: Takashi Shioya
Journal: Trans. Amer. Math. Soc. 351 (1999), 1765-1801
MSC (1991): Primary 53C20
Published electronically: January 27, 1999
MathSciNet review: 1458311
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the class of closed $2$-dimensional Riemannian manifolds with uniformly bounded diameter and total absolute curvature. Our first theorem states that this class of manifolds is precompact with respect to the Gromov-Hausdorff distance. Our goal in this paper is to completely characterize the topological structure of all the limit spaces of the class of manifolds, which are, in general, not topological manifolds and even may not be locally $2$-connected. We also study the limit of $2$-manifolds with $L^p$-curvature bound for $p \ge 1$.


References [Enhancements On Off] (What's this?)

  • 1. A. D. Aleksandrov, A theorem on triangles in a metric space and some of its applications, Trudy Mat. Inst. Steklov., v 38, Trudy Mat. Inst. Steklov., v 38, Izdat. Akad. Nauk SSSR, Moscow, 1951, pp. 5–23 (Russian). MR 0049584 (14,198a)
  • 2. -, Über eine Verallgemeinerung der Riemannschen Geometrie, Schriften Forschungsinst. Math. 1 (1957), 33-84.
  • 3. A. D. Aleksandrov, V. N. Berestovskiĭ, and I. G. Nikolaev, Generalized Riemannian spaces, Uspekhi Mat. Nauk 41 (1986), no. 3(249), 3–44, 240 (Russian). MR 854238 (88e:53103)
  • 4. A. D. Aleksandrov and V. A. Zalgaller, Intrinsic geometry of surfaces, Translated from the Russian by J. M. Danskin. Translations of Mathematical Monographs, Vol. 15, American Mathematical Society, Providence, R.I., 1967. MR 0216434 (35 #7267)
  • 5. Yu. D. Burago, Closure of the class of manifolds of bounded curvature, Proc. Steklov Inst. Math. 76 (1967), 175-183.
  • 6. Yu. Burago, M. Gromov, and G. Perel′man, A. D. Aleksandrov spaces with curvatures bounded below, Uspekhi Mat. Nauk 47 (1992), no. 2(284), 3–51, 222 (Russian, with Russian summary); English transl., Russian Math. Surveys 47 (1992), no. 2, 1–58. MR 1185284 (93m:53035), http://dx.doi.org/10.1070/RM1992v047n02ABEH000877
  • 7. Herbert Busemann, The geometry of geodesics, Academic Press Inc., New York, N. Y., 1955. MR 0075623 (17,779a)
  • 8. Mark Cassorla, Approximating compact inner metric spaces by surfaces, Indiana Univ. Math. J. 41 (1992), no. 2, 505–513. MR 1183356 (93i:53042), http://dx.doi.org/10.1512/iumj.1992.41.41029
  • 9. X. Chen, Limit of metrics in Riemannian surfaces and the uniformization theorem, preprint, 1996.
  • 10. Kenji Fukaya, Collapsing of Riemannian manifolds and eigenvalues of Laplace operator, Invent. Math. 87 (1987), no. 3, 517–547. MR 874035 (88d:58125), http://dx.doi.org/10.1007/BF01389241
  • 11. 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 (85e:53051)
  • 12. Philip Hartman, Geodesic parallel coordinates in the large, Amer. J. Math. 86 (1964), 705–727. MR 0173222 (30 #3435)
  • 13. Y. Machigashira and F. Ohtsuka, Total excess on length surfaces, preprint, 1995.
  • 14. G. Ya. Perel′man, Elements of Morse theory on Aleksandrov spaces, Algebra i Analiz 5 (1993), no. 1, 232–241 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 5 (1994), no. 1, 205–213. MR 1220498 (94h:53054)
  • 15. P. Petersen, S. Shteingold, and G. Wei, Comparison geometry with integral curvature bounds, Geom. Funct. Anal. 7 (1997), no. 6, 1011-1030. CMP 98:06
  • 16. P. Petersen and G. Wei, Almost maximal volume and integral Ricci curvature bounds, preprint, 1996.
  • 17. -, Relative volume comparison with integral curvature bounds, Geom. Funct. Anal. 7 (1997), no. 6, 1031-1045. CMP 98:06
  • 18. Geometry. IV, Encyclopaedia of Mathematical Sciences, vol. 70, Springer-Verlag, Berlin, 1993. Nonregular Riemannian geometry; A translation of Geometry, 4 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989 [ MR1099201 (91k:53003)]; Translation by E. Primrose. MR 1263963 (94i:53038)
  • 19. -, Two-Dimensional Manifolds of Bounded Curvature, 3-163, Vol. 70 of Encyclopaedia of Math. Sci. [18], 1993, pp. 3-163.
  • 20. K. Shiohama and M. Tanaka, An isoperimetric problem for infinitely connected complete open surfaces, Geometry of manifolds (Matsumoto, 1988) Perspect. Math., vol. 8, Academic Press, Boston, MA, 1989, pp. 317–343. MR 1040533 (91b:53049)
  • 21. T. Shioya, The Gromov-Hausdorff limits of two-dimensional manifolds under integral curvature bound, Geometry and Topology: Proceedings of Workshops in Pure Mathematics, Vol. 16, Part III (1996) (Young Wook Kim, Sung-Eun Ko, Yongjin Song, Younggi Choi, eds.).
  • 22. Takashi Shioya, Diameter and area estimates for 𝑆² and 𝑃² with non-negatively curved metrics, Progress in differential geometry, Adv. Stud. Pure Math., vol. 22, Math. Soc. Japan, Tokyo, 1993, pp. 309–319. MR 1274956 (95h:53055)
  • 23. M. Troyanov, Un principe de concentration-compacité pour les suites de surfaces riemanniennes, Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991), no. 5, 419–441 (French, with English summary). MR 1136350 (92m:53077)
  • 24. Deane Yang, Convergence of Riemannian manifolds with integral bounds on curvature. I, Ann. Sci. École Norm. Sup. (4) 25 (1992), no. 1, 77–105. MR 1152614 (93a:53037)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 53C20

Retrieve articles in all journals with MSC (1991): 53C20


Additional Information

Takashi Shioya
Affiliation: Graduate School of Mathematics, Kyushu University, Fukuoka 812-8581, Japan
Email: shioya@math.kyushu-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02103-0
PII: S 0002-9947(99)02103-0
Keywords: Total curvature, $L^p$-curvature bound, triangle comparison, the Gromov-Hausdorff convergence
Received by editor(s): October 30, 1996
Received by editor(s) in revised form: March 26, 1997
Published electronically: January 27, 1999
Additional Notes: This work was partially supported by a Grant-in-Aid for Scientific Research from the Japan Ministry of Education, Science and Culture.
Article copyright: © Copyright 1999 American Mathematical Society