| In this talk, I will present a Price type inequality for harmonic forms on manifolds without conjugate points and negative Ricci curvature. The techniques employed in the proof work particularly well for manifolds of non-positive sectional curvature, and in this case one can prove a strengthen result. Equipped with these Price type inequalities, I then study the asymptotic behavior of Betti numbers along infinite towers of regular coverings. Finally, I will discuss the case of compact real and complex hyperbolic manifolds in more details. Joint work with M. Stern (Duke University). |