Convex functions on complete noncompact manifolds: Topological structure

Convex functions on complete noncompact manifolds : differentiable structure

“…To some extent, the results on differentiable structure could be deduced from the topological versions combined with general results on differential topology. For instance, under the hypotheses in Theorem A of this paper, it was shown in [5] that the manifold is homeomorphic to a product of a topological manifold with [R; it follows then from general results ( [13], cf. also [3], [12]) that the manifold is diffeomorphic to a product of a differentiable manifold with (R if the manifold has dimension at least five (cf.…”

“…The purpose of the present paper is to show how, in spite of the possible absence of smooth convex approximations, a complete Morse-theoretic analysis of a convex function can be obtained even though the function might not have C 2 regularity. In the authors' previous work [5] (see also their preliminary announcement in the last section of [17]), topological versions of the present paper's differentiable structure Theorems were established; the present paper's emphasis is thus specifically on differentiable structure. To some extent, the results on differentiable structure could be deduced from the topological versions combined with general results on differential topology.…”

Convex functions on Alexandrov surfaces

“…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.…”

“…A closed convex set A in a Riemannian manifold admits a tubular neighborhood U such that the distance function to A is C 1 -differentiable on U \A (see [6], Proposition 1.2). However, this is not true in our case, because the cut locus Cut(∂(X a )) to ∂(X a ) may have a sequence of points converging to a point on ∂(X a ).…”

“…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.…”