Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

Convex functions on Alexandrov surfaces


Author: Yukihiro Mashiko
Journal: Trans. Amer. Math. Soc. 351 (1999), 3549-3567
MSC (1991): Primary 53C20
DOI: https://doi.org/10.1090/S0002-9947-99-02193-5
Published electronically: February 5, 1999
MathSciNet review: 1473452
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We investigate the topological structure of Alexandrov surfaces of curvature bounded below which possess convex functions. We do not assume the continuities of these functions. Nevertheless, if the convex functions satisfy a condition of local nonconstancy, then the topological structures of Alexandrov surfaces and the level sets configurations of these functions in question are determined.


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

  • 1. A.D.Alexandrov, Über eine Verallgemeinerung der Riemannshen Geometrie, Schriftenreiche der Institut für Mathematik (1957), 33-84.
  • 2. Bangert, V., Totally convex sets in complete Riemannian manifolds, J. Diff. Geom. 16 (1981), 333-345. MR 83e:53041
  • 3. Y. Burago, M. Gromov, and G. Perelman, A. D. Alexandrov's spaces with curvatures bounded below, Russian math. surveys 47:2 (1992), 1-58. MR 93m:53035
  • 4. H.Busemann, The Geometry of Geodesics, Academic Press, New York-London (1955). MR 17:779a
  • 5. J. Cheeger and D. Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. of Math. 96 (1972), 413-443. MR 46:8121
  • 6. R. E. Greene and K. Shiohama, Convex functions on complete noncompact manifolds: Topological structure, Invent. Math. 63 (1981), 129-157. MR 82e:53065
  • 7. R. E. Greene and K. Shiohama, Convex functions on complete noncompact manifolds: Differential structure, Ann. Sci. École Norm. Sup. (4),14 (1981), 357-367. MR 83m:53057
  • 8. R. E. Greene and K. Shiohama, The isometry groups of manifolds admitting nonconstant convex functions, J. Math. Soc. Japan 39 (No.1) (1987). MR 88a:53032
  • 9. N. Innami, A classification of Busemann G-surfaces which possess convex functions, Acta Math. 148 (1982), 15-29. MR 84f:53063
  • 10. Y. Machigashira, The Gaussian curvature of Alexandrov surfaces, to appear.
  • 11. Y. Otsu and T. Shioya, The Riemannian structure of Alexandrov spaces, J. Diff. Geom. 39 (1994), 629-658. MR 95e:53062
  • 12. G. Perelman, Alexandrov's spaces with curvatures bounded from below II, preprint.
  • 13. K. Shiohama, An Introduction to the Geometry of Alexandrov Spaces, Seoul National univ. Lecture notes series 8. MR 96c:53064
  • 14. K. Shiohama and M. Tanaka, Cut loci and Distance spheres on Alexandrov surfaces, Proc. Round Table in Diff. Goem.-in honor of Marcel Berger-Collection SMF Séminaires & Congrès no.1 (1996), 531-560. MR 98a:53062

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

Yukihiro Mashiko
Affiliation: Graduate School of Mathematics, Kyushu University, 6-10-1 Hakozaki, Higashi-ku, Fukuoka, 812-81 Japan
Email: mashiko@math.kyushu-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-99-02193-5
Keywords: Alexandrov spaces of curvature bounded below, convex functions
Received by editor(s): April 10, 1997
Published electronically: February 5, 1999
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society