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].


53C20 Global Riemannian geometry, including pinching


Zbl 0654.53046