Convex functions on Alexandrov surfaces

Abstract

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. IntroductionTypical examples of convex functions on Riemannian manifolds are Busemann functions on complete Riemannian manifolds of nonnegative sectional curvature. Using filtration by compact totally convex sets, J. Cheeger and D. Gromoll investigated the structure theorem by which the topological structure of complete noncompact Riemannian manifolds of nonnegative sectional curvature is determined ([5]).It is well known that the existence of a locally nonconstant convex function on a complete Riemannian manifold M imposes strong restrictions on the topology of M ([6], Theorems C, D and F). Here, the local nonconstancy of a convex function means that it is not constant on any open set. As is explained on p.130 of [6], every noncompact smooth manifold M admits a complete Riemannian metric g and a nontrivial smooth function ϕ such that ϕ is convex with respect to g. For such a ϕ, the minimum set of ϕ has a nonempty interior in which all the topological information of M is included. Thus it is natural to assume that a convex function is locally nonconstant. Geodesic completeness on M plays an essential role in the above results. In fact, the property of local Lipschitz continuity of a convex function on M is derived from geodesic completeness, and also the noncompactness of M .It is natural to ask if certain topological restrictions can be obtained for geodesic spaces admitting locally nonconstant convex functions. In fact, N. Innami proved in Theorem 3.13 of [9] that if a Busemann G-surface admits a locally nonconstant convex function, then it is homeomorphic to either a plane R 2 , a cylinder S 1 × R 1 , or an open Möbius strip. Here the geodesic completeness holds on Busemann Gsurfaces.

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“…A convex function is said to be affine if and only if the equality in (1-1) holds for all γ and for all λ ∈ (0, 1). A splitting theorem for Riemannian manifolds admitting affine functions has been investigated in [Innami 1982b], while Alexandrov spaces admitting affine functions have been studied in [Innami 1982b;Mashiko 1999a;Mashiko 2002]. An overview on the convexity of Riemannian manifolds can be found in [Burago and Zalgaller 1977].…”

“…Locally nonconstant convex functions, affine functions and peakless functions have been investigated on complete Riemannian manifolds and complete noncompact Busemann G-spaces and Alexandrov spaces in various ways. The topology of Riemannian manifolds admitting convex functions was investigated in [Bangert 1978;Greene and Shiohama 1981b;1981a;1987], and that of Busemann G-surfaces in [Innami 1982a;Mashiko 1999b]. It should be noted that convex functions on complete Alexandrov surfaces are not continuous.…”

Affine functions on Alexandrov spaces

“…The easiest example of an affine function is the projection onto a Euclidean factor. Under some assumptions it is known that a space X with one-sided curvature bound splits as a product X = Y × R if it admits a non-constant affine function [AB05] (see also [I82,Ma99,Ma02] for earlier results). The decisive assumption in [AB05] is that the space X is geodesically complete in the case of an upper curvature bound or does not have boundary in the case of a lower curvature bound.…”

“…In dimension two the answers are yes (see also [Ma99] for a proof of Lipschitz continuity of affine functions on Alexandrov surfaces without boundary).…”

Topology of complete Finsler manifolds admitting convex functions

“…Locally nonconstant convex functions, affine functions and peakless functions have been investigated on complete Riemannian manifolds and complete noncompact Busemann G-spaces and Alexandrov spaces in various ways. The topology of Riemannian manifolds admitting convex functions was investigated in [Bangert 1978;Greene and Shiohama 1981b;1981a;1987], and that of Busemann G-surfaces in [Innami 1982a;Mashiko 1999b]. It should be noted that convex functions on complete Alexandrov surfaces are not continuous.…”