Bounds on the First Betti Number: An Approach via Schatten Norm Estimates on Semigroup Differences

Abstract

We derive new estimates for the first Betti number of compact Riemannian manifolds. Our approach relies on the Birman–Schwinger principle and Schatten norm estimates for semigroup differences. In contrast to previous works we do not require any a priori ultracontractivity estimates and we provide bounds which explicitly depend on suitable integral norms of the Ricci tensor.