Complete manifolds with positive spectrum. II.
The main result of the paper under review is the following one: Let \(M\) be a complete \(n\)-dimensional manifold with \(n\geq 4\). Let Ric\(_{M}\) be the Ricci curvature of \(M\) and \(\lambda_{1}(M)\) the largest lower bound of the spectrum of the Laplacian on \(M\) satisfying: \[ \text{Ric}_{M}\geq -(n-1)\text{ and } \lambda_{1}(M)\geq\frac{(n-1)^{2}}{4}. \] Then either: (1) \(M\) has only one end; or (2) \(M=\mathbb R\times\mathbb N\) with the warped product metric \[ ds^{2}_{M}=dt^{2}+\exp(2t) ds^{2}_{N}, \] where \(N\) is a compact manifold with nonnegative Ricci curvature. [For part I of the paper see the authors, ibid. 58, No. 3, 501--534 (2001; Zbl 1032.58016).]
scientific article; zbMATH DE number 2171913