L p -pinching and the geometry of compact Riemannian manifolds

We prove a Harnack-type inequality inf|S|/sup|S|>1?ε(W, M, V) satisfied by the sections of a Riemannian vector bundleW lying in the kernel of a Schrödinger operator ∨*∨+V underL ^p -pinching assumptions on the potentialV and derive various topological and geometric consequences. For instance, we prove a fibration theorem which gives a classification of almost non-negatively curved compact manifolds by the first Betti number. In the case of almost non-positively curved compact manifolds, we prove that the minimal volume must vanish whenever the isometry group is not finite and give conditions implying that it is abelian.