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Transactions of the American Mathematical Society

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Convexity of the ideal boundary for complete open surfaces


Author: Jin-Whan Yim
Journal: Trans. Amer. Math. Soc. 347 (1995), 687-700
MSC: Primary 53C20; Secondary 53C45
DOI: https://doi.org/10.1090/S0002-9947-1995-1243176-1
MathSciNet review: 1243176
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Abstract: For complete open surfaces admitting total curvature, we define several kinds of convexity for the ideal boundary, and provide examples of each of them. We also prove that a surface with most strongly convex ideal boundary is in fact a generalization of a Hadamard manifold in the sense that the ideal boundary consists entirely of Busemann functions.


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  • [1] W. Ballmann, M. Gromov, and V. Schroeder, Manifolds of nonpositive curvature, Progr. Math., Vol. 61, Birkhäuser, Boston, Basel, and Stuttgart, 1985. MR 823981 (87h:53050)
  • [2] J. Cheeger and D. Gromoll, Structure of complete manifolds of nonnegative curvature, Ann. of Math. (2) 96 (1972), 413-443. MR 0309010 (46:8121)
  • [3] S. Cohn-Vossen, Kürzeste Wege und Totalkrümmung auf Flächen, Compositio Math. 2 (1935), 63-133. MR 1556908
  • [4] -, Totalkümmung und geodätischeLinien auf einfach zusammenhängenden offenenvolständigen Flächenstücken, Recueil Math. Moscow 43 (1936), 139-163.
  • [5] J.-H. Eschenburg, Horospheres and the stable part of the geodesic flow, Math. Z. 153 (1977), 237-251. MR 0440605 (55:13479)
  • [6] M. Gromov, Hyperbolic manifolds, groups and actions, Riemann Surfaces and Related Topics, Stony Brook Conference, Ann. of Math. Stud., Vol. 97, Princeton Univ. Press, Princeton, NJ. MR 624814 (82m:53035)
  • [7] A. Kasue, A compactification of a manifold with asymptotically nonnegative curvature, Ann. Sci. École Norm. Sup. (4) 21 (1988), 593-622. MR 982335 (90d:53049)
  • [8] M. Maeda, A geometric significance of total curvature on complete open surfaces, Geometry of Geodesies and Related Topics, Adv. Stud. Pure Math., 3, Kinokuniya, Tokyo, 1984, pp. 451-458. MR 758663 (85j:53045)
  • [9] K. Shiohama, Topology of complete noncompact manifolds, Geometry of Geodesics and Related Topics, Adv. Stud. Pure Math., 3, Kinokuniya, Tokyo, 1984, pp. 423-450. MR 758662 (85m:53043)
  • [10] -, An integral formula for the measure of rays on complete open surfaces, J. Differential Geom. 94 (1986), 197-205. MR 845705 (87i:53104)
  • [11] T. Shioya, The ideal boundary of complete open surfaces, Tôhoku Math. J. 43 (1991), 37-59. MR 1088713 (92b:53050)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1995-1243176-1
Article copyright: © Copyright 1995 American Mathematical Society

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