Convex functions on Alexandrov surfaces

Mashiko, Yukihiro

doi:10.1090/S0002-9947-99-02193-5

Convex functions on Alexandrov surfaces
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Trans. Amer. Math. Soc. 351 (1999), 3549-3567 Request permission

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

A.D.Alexandrov, Über eine Verallgemeinerung der Riemannshen Geometrie, Schriftenreiche der Institut für Mathematik (1957), 33–84.
Victor Bangert, Totally convex sets in complete Riemannian manifolds, J. Differential Geometry 16 (1981), no. 2, 333–345. MR 638797
Yu. Burago, M. Gromov, and G. Perel′man, A. D. Aleksandrov spaces with curvatures bounded below, Uspekhi Mat. Nauk 47 (1992), no. 2(284), 3–51, 222 (Russian, with Russian summary); English transl., Russian Math. Surveys 47 (1992), no. 2, 1–58. MR 1185284, DOI 10.1070/RM1992v047n02ABEH000877
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
Jeff Cheeger and Detlef Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. of Math. (2) 96 (1972), 413–443. MR 309010, DOI 10.2307/1970819
R. E. Greene and K. Shiohama, Convex functions on complete noncompact manifolds: topological structure, Invent. Math. 63 (1981), no. 1, 129–157. MR 608531, DOI 10.1007/BF01389196
R. E. Greene and K. Shiohama, Convex functions on complete noncompact manifolds: differentiable structure, Ann. Sci. École Norm. Sup. (4) 14 (1981), no. 4, 357–367 (1982). MR 654202, DOI 10.24033/asens.1410
Robert E. Greene and Katsuhiro Shiohama, The isometry groups of manifolds admitting nonconstant convex functions, J. Math. Soc. Japan 39 (1987), no. 1, 1–16. MR 867983, DOI 10.2969/jmsj/03910001
Nobuhiro Innami, A classification of Busemann $G$-surfaces which possess convex functions, Acta Math. 148 (1982), 15–29. MR 666106, DOI 10.1007/BF02392724
Y. Machigashira, The Gaussian curvature of Alexandrov surfaces, to appear.
Yukio Otsu and Takashi Shioya, The Riemannian structure of Alexandrov spaces, J. Differential Geom. 39 (1994), no. 3, 629–658. MR 1274133
G. Perelman, Alexandrov’s spaces with curvatures bounded from below II, preprint.
Katsuhiro Shiohama, An introduction to the geometry of Alexandrov spaces, Lecture Notes Series, vol. 8, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1993. MR 1320267
Katsuhiro Shiohama and Minoru Tanaka, Cut loci and distance spheres on Alexandrov surfaces, Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992) Sémin. Congr., vol. 1, Soc. Math. France, Paris, 1996, pp. 531–559 (English, with English and French summaries). MR 1427770