A volume comparison theorem for manifolds with asymptotically nonnegative curvature and its applications.

*(English)*Zbl 0953.53027From the introduction: The original motivation for this paper is to investigate the ends of manifolds of asymptotically nonnegative Ricci curvature. We note that a bound on the number of ends is a consequence of a general volume comparison theorem for concentric metric balls at an arbitrary point. For manifolds of nonnegative Ricci curvature, this comparison theorem holds and is known as the Bishop-Gromov volume comparison theorem. For manifolds with asymptotically nonnegative Ricci curvature, we prove the following: Theorem 2.1. Let \((M^n,g)\) be a complete open Riemannian manifold with \(\text{Ric}(x)\geq -(n- 1)\lambda(d_{P_0}(x))\), and \(\int^\infty_0 t\lambda(t) dt= b_0< \infty\). Then for any subset \(\Gamma\) of the unit sphere at \(P_0\), \({\text{vol}(B^\Gamma_{P_0}(R))\over \text{vol}(B^\Gamma_{P_0}(r))}\leq e^{(n- 1)b_0}\cdot \left({R\over r}\right)^n\) if \(R\geq r> 0\).

We remark that Theorem 2.1 is a statement only about balls with center at the base point (take \(\Gamma= S^{n-1}\)). It does not hold for balls centered at other points (the constant in front of \(\left({R\over r}\right)^n\) depends on the center). Thus we can not conclude a bound on the number of ends from Theorem 2.1.

The difficulty of working with manifolds of asymptotically nonnegative curvature (sectional or Ricci) is caused by the special role played by the base point (for example, the covering manifold with the pulled back metric is not asymptotically nonnegative). Essentially, we can only work with geodesics eminating from \(P_0\). This limits to a great extent the kind of results we can obtain. It is easily seen that without further conditions, the topological and metric structures at infinity are the only things we can hope to control. However, in §3, as an application of Theorem 2.1, will prove the following topological rigidity result which controls the global topology of such manifolds. In the following, we denote by \(\omega_n\) the volume of the \(n\)-dimensional unit ball.

Theorem 3.1. For any \(\delta> 0\), there is a constant \(\varepsilon= \varepsilon(n,\delta)\) such that if a complete noncompact manifold \((M^n,g)\) satisfies \(K(x)\geq -\lambda(d_{P_0}(x))\), \(\text{Vol}(B_{P_0}(r))\geq \left({1\over 2}+\delta\right) \omega_nr^n\), and \(\int^\infty_0 t\lambda(t)\leq \varepsilon\), then \(d_{P_0}\) has no critical point. In particular, \(M\) is diffeomorphic to \(\mathbb{R}^n\).

We remark that Theorem 2.1 is a statement only about balls with center at the base point (take \(\Gamma= S^{n-1}\)). It does not hold for balls centered at other points (the constant in front of \(\left({R\over r}\right)^n\) depends on the center). Thus we can not conclude a bound on the number of ends from Theorem 2.1.

The difficulty of working with manifolds of asymptotically nonnegative curvature (sectional or Ricci) is caused by the special role played by the base point (for example, the covering manifold with the pulled back metric is not asymptotically nonnegative). Essentially, we can only work with geodesics eminating from \(P_0\). This limits to a great extent the kind of results we can obtain. It is easily seen that without further conditions, the topological and metric structures at infinity are the only things we can hope to control. However, in §3, as an application of Theorem 2.1, will prove the following topological rigidity result which controls the global topology of such manifolds. In the following, we denote by \(\omega_n\) the volume of the \(n\)-dimensional unit ball.

Theorem 3.1. For any \(\delta> 0\), there is a constant \(\varepsilon= \varepsilon(n,\delta)\) such that if a complete noncompact manifold \((M^n,g)\) satisfies \(K(x)\geq -\lambda(d_{P_0}(x))\), \(\text{Vol}(B_{P_0}(r))\geq \left({1\over 2}+\delta\right) \omega_nr^n\), and \(\int^\infty_0 t\lambda(t)\leq \varepsilon\), then \(d_{P_0}\) has no critical point. In particular, \(M\) is diffeomorphic to \(\mathbb{R}^n\).

##### MSC:

53C20 | Global Riemannian geometry, including pinching |