Total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below. III. (English) Zbl 1246.53055

Kondo, Kei; Tanaka, Minoru

J. Math. Soc. Japan 64, No. 1, 185-200 (2012).

Recall that a von Mangoldt surface of revolution is a complete surface of revolution \(\widetilde M \) homeomorphic to \(\mathbb R^2\) with non-increasing Gaussian curvature along each meridian and that the Busemann function on a complete non-compact connected Riemannian manifold \(M\) is a function of a ray \(\gamma\) in \(M\) given by \(F_\gamma(x) = \lim_{t\to\infty}\{t-d(x, \gamma(t))\}\), and a function \(\varphi: M\to \mathbb R\) is called an exhaustion, if \(\varphi^{-1}(-\infty, a]\) is compact for all \(a\in \mathbb R\). In the paper [Math. Ann. 351, No. 2, 251–266 (2011; Zbl 1243.53072)], the authors showed that all Busemann functions on a complete non-compact connected Riemannian manifold \(M\) which is not less curved than a von Mangoldt surface of revolution \(\widetilde{M}\) are exhaustions, if the total curvature of \(\widetilde{M}\) is greater than \(\pi\) and the purpose of the paper under review is to generalize this theorem to a manifold M which is not less curved than a certain more general surface of revolution.

Reviewer: Witold Mozgawa (Lublin)

MSC:

53C20	Global Riemannian geometry, including pinching
53C21	Methods of global Riemannian geometry, including PDE methods; curvature restrictions

Keywords:

Busemann function; radial curvature; total curvature; von Mangoldt surface; exhaustion

Citations:

Zbl 1243.53072

Software:

Loki

References:

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