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Transactions of the American Mathematical Society

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



Geometric curvature bounds in Riemannian manifolds with boundary

Authors: Stephanie B. Alexander, I. David Berg and Richard L. Bishop
Journal: Trans. Amer. Math. Soc. 339 (1993), 703-716
MSC: Primary 53C21; Secondary 53C20
MathSciNet review: 1113693
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Abstract: An Alexandrov upper bound on curvature for a Riemannian manifold with boundary is proved to be the same as an upper bound on sectional curvature of interior sections and of sections of the boundary which bend away from the interior. As corollaries those same sectional curvatures are related to estimates for convexity and conjugate radii; the Hadamard-Cartan theorem and Yau's isoperimetric inequality for spaces with negative curvature are generalized.

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  • [Av1] A. D. Alexandrov, A theorem on triangles in a metric space and some of its applications, Trudy Mat. Inst. Steklov. 38 (1951), 5-23. (Russian) (Much of [Av1] is translated in [Av2].)
  • [Av2] -, Über eine Verallgemeinerung der Riemannschen Geometrie, Schr. Forschungsinst. Math. 1 (1957), 33-84. MR 0087119 (19:304h)
  • [ABN] A. D. Alexandrov, V. N. Berestovskii, and I. G. Nikolaev, Generalized Riemannian spaces, Russian Math. Surveys 41 (1986), 1-54. MR 854238 (88e:53103)
  • [ABB1] S. B. Alexander, I. D. Berg, and R. L. Bishop, The Riemannian obstacle problem, Illinois J. Math. 31 (1987), 167-184. MR 869484 (88a:53038)
  • [ABB2] -, Cut loci, minimizers and wave fronts in Riemannian manifolds with boundary, Michigan Math. J. 40 (1993) (to appear).
  • [ArBp] S. B. Alexander and R. L. Bishop, The Hadamard-Cartan theorem in locally convex spaces, Enseign. Math. 36 (1990), 309-320. MR 1096422 (92c:53044)
  • [Bn] W. Ballmann, Singular spaces of non-positive curvature. (E. Ghys and P. de la Harpe, eds.), Sur les Groupes Hyperboliques d'apres Mikhael Gromov, Birkhäuser, Boston, Basel, and Stuttgart, 1990. MR 1086658
  • [BpCn] R. L. Bishop and R. J. Crittenden, Geometry of manifolds, Academic Press, New York and London, 1964. MR 0169148 (29:6401)
  • [BoZr] Yu. D. Burago and V. A. Zalgaller, Geometric inequalities, Springer-Verlag, Berlin, Heidelberg, and New York, 1988. MR 936419 (89b:52020)
  • [C-V] S. Cohn-Vossen, Existenz Kurzester Wege, Doklady SSSR 8 (1935), 339-342.
  • [Fr] H. Federer, Geometric measure theory, Springer-Verlag, Berlin, Heidelberg, and New York, 1969. MR 0257325 (41:1976)
  • [GlMr] D. Gromoll and W. T. Meyer, Examples of complete manifolds with positive Ricci curvature, J. Differential Geom. 21 (1985), 195-211. MR 816669 (87b:53068)
  • [Gv1] M. Gromov, Hyperbolic manifolds, groups and actions. (I. Kra and B. Maskit, eds.), Riemann Surfaces and Related Topics (Proa., Stony Brook, 1978), Ann. of Math. Studies, no. 97, Princeton Univ. Press, 1981, pp. 183-213. MR 624814 (82m:53035)
  • [Gv2] -, Hyperbolic groups. (S. M. Gersten, ed.), Essays in Group Theory, Math. Sci. Res. Inst. Publ., no. 8, Springer-Verlag, New York, Berlin, Heidelberg, 1987, pp. 75-264. MR 919829 (89e:20070)
  • [Rk] Yu. G. Reshetnyak, Nonexpanding mappings in a space of curvature no greater than $ K$, Sibirsk. Mat. Z. 9 (1968), 918-927; English transl, in Siberian Math. J. 9 (1968), 683-689. MR 0244922 (39:6235)
  • [Si] D. Scollozi, Un risultato di locale unicita per le geodetiche su varieta con bordo, Boll. Un. Mat. Ital. B(6) 5 (1986), 309-327.
  • [Tv] M. Troyanov, Espaces a courbure negative et groupes hyperboliques. (E. Ghys and P. de la Harpe, eds.), Sur les Groupes Hyperboliques d'apres Mikhael Gromov, Birkhäuser, Boston, Basel, and Stuttgart, 1990. MR 1086651
  • [Wr] F.-E. Wolter, Cut loci in bordered and unbordered Riemannian manifolds, Technische Universität Berlin, FB Mathematik, Dissertation 249S, 1985.
  • [Yu] S.-T. Yau, Isoperimetric constants and the first eigenvalue of a compact Riemannian manifold, Ann. Sci. École. Norm. Sup. 8 (1975), 487-507. MR 0397619 (53:1478)

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