Universität Augsburg
Institut für Mathematik

Let \rho denote the lowest eigenvalue of the Ricci tensor on a smooth complete Riemannian manifold. Pointwise lower bounds on \rho impose restrictions on the geometry and topology of a manifold. Motivated in part by questions arising from the study of stable minimal hypersurfaces, there has recently been interest in replacing such pointwise assumptions by weaker spectral ones, namely lower bounds for the first eigenvalue of operators of the form -\gamma\Delta + \rho, where \gamma > 0. Here \Delta is the usual Laplacian.

In this talk, I will discuss the following question: for which values of \gamma does the spectral condition -\gamma\Delta + \rho >= 0 impose the same topological obstructions as \rho >= 0? I will give a sharp answer for compact manifolds with mean-convex boundaries, and discuss what is known in the closed case. This is joint work with Y. Li and P. Sweeney Jr.