Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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
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] Werner Ballmann, Mikhael Gromov, and Viktor Schroeder, Manifolds of nonpositive curvature, Progress in Mathematics, vol. 61, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 823981
  • [2] Jeff Cheeger and Detlef Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. of Math. (2) 96 (1972), 413–443. MR 0309010
  • [3] Stefan Cohn-Vossen, Kürzeste Wege und Totalkrümmung auf Flächen, Compositio Math. 2 (1935), 69–133 (German). MR 1556908
  • [4] -, Totalkümmung und geodätischeLinien auf einfach zusammenhängenden offenenvolständigen Flächenstücken, Recueil Math. Moscow 43 (1936), 139-163.
  • [5] Jost-Hinrich Eschenburg, Horospheres and the stable part of the geodesic flow, Math. Z. 153 (1977), no. 3, 237–251. MR 0440605
  • [6] M. Gromov, Hyperbolic manifolds, groups and actions, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 183–213. MR 624814
  • [7] Atsushi Kasue, A compactification of a manifold with asymptotically nonnegative curvature, Ann. Sci. École Norm. Sup. (4) 21 (1988), no. 4, 593–622. MR 982335
  • [8] Masao Maeda, A geometric significance of total curvature on complete open surfaces, Geometry of geodesics and related topics (Tokyo, 1982) Adv. Stud. Pure Math., vol. 3, North-Holland, Amsterdam, 1984, pp. 451–458. MR 758663
  • [9] Katsuhiro Shiohama, Topology of complete noncompact manifolds, Geometry of geodesics and related topics (Tokyo, 1982) Adv. Stud. Pure Math., vol. 3, North-Holland, Amsterdam, 1984, pp. 423–450. MR 758662
  • [10] Katsuhiro Shiohama, An integral formula for the measure of rays on complete open surfaces, J. Differential Geom. 23 (1986), no. 2, 197–205. MR 845705
  • [11] Takashi Shioya, The ideal boundaries of complete open surfaces, Tohoku Math. J. (2) 43 (1991), no. 1, 37–59. MR 1088713, 10.2748/tmj/1178227534

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