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Volume growth and finite topological type. (English) Zbl 0809.53048
Greene, Robert (ed.) et al., Differential geometry. Part 3: Riemannian geometry. Proceedings of a summer research institute, held at the University of California, Los Angeles, CA, USA, July 8-28, 1990. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 54, Part 3, 539-549 (1993).
A complete Riemannian manifold $$M$$ is said to have weak bounded geometry if it satisfies $K = \inf K_ M > -\infty, \quad v = \inf \text{vol } B(x,1) > 0.$ The main result is the following Theorem. Let $$M$$ be complete with weak bounded geometry and nonnegative $$k$$-th Ricci curvature outside a geodesic ball centered at $$p$$, for some $$2 \leq k \leq n - 1$$. There is a constant $$c$$ such that if $\varlimsup_{r \to\infty} {\text{vol }B(p,r)\over r^{1+1/ (k + 1)}} < c,$ then there exists a compact set $$C$$ such that $$M \setminus C$$ contains no critical points of $$Lp$$. In particular, $$M$$ has finite topological type. The authors point out that the examples by J.-P. Shah and D.-G. Yang [J. Differ. Geom. 29, No. 1, 95-103 (1989; Zbl 0654.53046)]are nonnegatively Ricci curved complete manifolds having infinite topological type. Their diameter growth and volume growth are estimated.
For the entire collection see [Zbl 0773.00024].

##### MSC:
 53C20 Global Riemannian geometry, including pinching
##### Keywords:
weak bounded geometry; diameter growth; volume growth
Zbl 0654.53046