Journal of the Mathematical Society of Japan

Total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below, III

Kei KONDO and Minoru TANAKA

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Abstract

This article is the third in a series of our investigation on a complete non-compact connected Riemannian manifold M. In the first series [KT1], we showed that all Busemann functions on an M which is not less curved than a von Mangoldt surface of revolution M ˜ are exhaustions, if the total curvature of M ˜ is greater than π. A von Mangoldt surface of revolution is, by definition, a complete surface of revolution homeomorphic to R2 whose Gaussian curvature is non-increasing along each meridian. Our purpose of this series is to generalize the main theorem in [KT1] to an M which is not less curved than a more general surface of revolution.

Article information

Source
J. Math. Soc. Japan, Volume 64, Number 1 (2012), 185-200.

Dates
First available in Project Euclid: 26 January 2012

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1327586979

Digital Object Identifier
doi:10.2969/jmsj/06410185

Mathematical Reviews number (MathSciNet)
MR2836657

Zentralblatt MATH identifier
1246.53055

Subjects
Primary: 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20] 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60]

Keywords
Busemann function radial curvature total curvature

Citation

KONDO, Kei; TANAKA, Minoru. Total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below, III. J. Math. Soc. Japan 64 (2012), no. 1, 185--200. doi:10.2969/jmsj/06410185. https://projecteuclid.org/euclid.jmsj/1327586979


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References

  • J. Cheeger and D. Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. of Math. (2), 96 (1972), 415–443.
  • S. Cohn-Vossen, Kürzeste Wege und Totalkrümmung auf Flächen, Compositio Math., 2 (1935), 63–113.
  • S.Cohn-Vossen, Totalkrümmung und geodätische Linien auf einfach zusammenhängenden offenen volständigen Flächenstücken, Recueil Math. Moscow, 43 (1936), 139–163.
  • U. Dini, Fondamenti per la teorica delle funzioni di variabili reali, Pisa, 1878.
  • D. Gromoll and W. Meyer, On complete manifolds of positive curvature, Ann. of Math. (2), 75 (1969), 75–90.
  • T. Hawkins, Lebesgue's Theory of Integration: Its origins and development, University of Wisconsin Press, Madison 1970.
  • A. Kasue, A compactification of a manifold with asymptotically nonnegative curvature, Ann. Sci. École Norm. Sup. (4), 21 (1988), 593–622.
  • N. N. Katz and K. Kondo, Generalized space forms, Trans. Amer. Math. Soc., 354 (2002), 2279–2284.
  • K. Kondo and M. Tanaka, Total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below.,I, Math. Ann., 351 (2011), 251–266.
  • K. Kondo and M. Tanaka, Total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below.,II, Trans. Amer. Math. Soc., 362 (2010), 6293–6324.
  • K. Kondo and M. Tanaka, Toponogov comparison theorem for open triangles, Tohoku Math. J., 63 (2011), 363–396.
  • F. Morgan, Geometric Measure Theory, A Beginer's Guide, Academic Press, 1988.
  • K. Shiohama, The role of total curvature on complete noncompact Riemannian 2-manifolds, Illinois J. Math., 28 (1984), 597–620.
  • K. Shiohama, T. Shioya and M. Tanaka, The Geometry of Total Curvature on Complete Open Surfaces, Cambridge Tracts in Math., 159, Cambridge University Press, Cambridge, 2003.
  • M. Tanaka, On the cut loci of a von Mangoldt's surface of revolution, J. Math. Soc. Japan, 44 (1992), 631–641.
  • M. Tanaka and K. Kondo, The Gaussian curvature of a model surface with finite total curvature is not always bounded., arXiv:1102.0852v2
  • V. A. Toponogov, Riemannian spaces containing straight lines (in Russian), Dokl. Akad. Nauk SSSR, 127 (1959), 977–979.
  • R. L. Wheeden and A. Zygmund, Measure and Integral, Marcel Decker, New York, 1977.