On the structure of locally convex filtrations on complete manifolds (Q1117474)

The author extends structure theorems on complete Riemannian manifolds M which carry a convex function [\textit{V. Bangert}, Arch. Math. 31, 163-170 (1978; Zbl 0372.53021), \textit{R. E. Greene} and \textit{K. Shiohama}, Invent. Math. 63, 129-157 (1981; Zbl 0468.53033); Ann. Sci. Ec. Norm. Supér., IV. Sér. 14, 357-367 (1981; Zbl 0488.57012)] to the case where the function f: \(M\to {\mathbb{R}}\) has only locally convex sublevels. Thus, also compact manifolds are included. If the function is locally nonconstant and regular (i.e. has no branch points of infinite order), it is called a regular convex filtration. In this situation, it is shown that \(M\setminus H^*_ f\) is homeomorphic to the normal bundle of a submanifold. Here, \(H^*_ f\) denotes union of level components of f intersecting the closure of the local maximum set \(H_ f\) of f. In case that there are no branch points at all (f is then called ``nice''), the author establishes a classification of all compact manifolds into four types up to diffeomorphisms. Finally, for noncompact surfaces of finite topological type not homeomorphic to \({\mathbb{R}}^ 2\), the absolute value of the Euler characteristic is identified with the minimal number of connected components of \(H_ f\) when f runs over all nice filtrations for all metrics admitting such filtrations. Also, ``strict'' nice filtrations are considered.