Complete manifolds with positive spectrum
Let \(M\) be a complete Riemannian manifold of dimension \(n\geq 3\). Let \(\text{Ric}_{M}\) be the Ricci curvature of \(M\) and \(\lambda_{1}(M)\) the lower bound of the spectrum of the Laplacian on \(M\). Consider the following two conditions on \(M\): (1) \(\text{Ric}_{M}\geq -\frac{(n-1)}{n-2}\lambda_{1}(M)\), (2) \(\lambda_{1}(M)>0\). In the paper under review, the authors study the complete Riemannian manifolds that satisfy conditions (1) and (2). In particular, they show that if a complete Riemannian manifold of dimension \(n\geq 3\) satisfies (1) and (2), then either \(M\) has only one infinite volume end, or it is a warped product \(\mathbb R\times N\), where \(N\) is some compact manifold with Ricci curvature bounded from below by \(-(n-2)\). They also prove that if the curvature condition (1) is satisfied only at infinity with a strict inequality, then the manifold must have finitely many ends with infinite volume.
scientific article; zbMATH DE number 1782697