Bounds on the first Betti number - an approach via Schatten norm estimates on semigroup differences

TL;DR

This paper introduces a novel method to estimate the first Betti number of compact Riemannian manifolds using Schatten norm estimates on semigroup differences, avoiding the need for ultracontractivity assumptions.

Contribution

It provides new bounds on the first Betti number based on integral norms of the Ricci tensor, utilizing the Birman-Schwinger principle without ultracontractivity requirements.

Findings

Derived explicit bounds on the first Betti number.

Established estimates depend on integral norms of Ricci curvature.

Applied Schatten norm techniques to 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.