Open Access

Daniel Räde

• A band is a connected compact manifold X together with a decomposition ∂X = ∂−X t ∂+X where ∂±X are non-empty unions of boundary components. If X is equipped with a Riemannian metric, the pair (X, g) is called a Riemannian band and the width of (X, g) is defined to be the distance between ∂−X and ∂+X with respect to g. Following Gromov’s seminal work on metric inequalities with scalar curvature, the study of Riemannian bands with lower curvature bounds has been an active field of research in recent years, which led to several breakthroughs on longstanding open problems in positive scalar curvature geometry and to a better understanding of the positive mass theorem in general relativity In the first part of this thesis we combine ideas of Gromov and Cecchini-Zeidler and use the variational calculus surrounding so called µ-bubbles to establish a scalar and mean curvature comparison principle for Riemannian bands with the property that no closed embedded hypersurface which separatesA band is a connected compact manifold X together with a decomposition ∂X = ∂−X t ∂+X where ∂±X are non-empty unions of boundary components. If X is equipped with a Riemannian metric, the pair (X, g) is called a Riemannian band and the width of (X, g) is defined to be the distance between ∂−X and ∂+X with respect to g. Following Gromov’s seminal work on metric inequalities with scalar curvature, the study of Riemannian bands with lower curvature bounds has been an active field of research in recent years, which led to several breakthroughs on longstanding open problems in positive scalar curvature geometry and to a better understanding of the positive mass theorem in general relativity In the first part of this thesis we combine ideas of Gromov and Cecchini-Zeidler and use the variational calculus surrounding so called µ-bubbles to establish a scalar and mean curvature comparison principle for Riemannian bands with the property that no closed embedded hypersurface which separates the two ends of the band admits a metric of positive scalar curvature. The model spaces we use for this comparison are warped product over scalar flat manifolds with log-concave warping functions. We employ ideas from surgery and bordism theory to deduce that, if Y is a closed orientable manifold which does not admit a metric of positive scalar curvature, dim(Y ) 6= 4 and Xn≤7 = Y ×[−1, 1], the width of X with respect to any Riemannian metric with scalar curvature ≥ n(n − 1) is bounded from above by 2π n. This solves, up to dimension 7, a conjecture due to Gromov in the orientable case. Furthermore, we adapt and extend our methods to show that, if Y is as before and Mn≤7 = Y × R, then M does not admit a metric of positive scalar curvature. This solves, up to dimension 7 a conjecture due to Rosenberg and Stolz in the orientable case. In the second part of this thesis we explore how these results transfer to the setting where the lower scalar curvature bound is replaced by a lower bound on the macroscopic scalar curvature of a Riemannian band. This curvature condition amounts to an upper bound on the volumes of all unit balls in the universal cover of the band. We introduce a new class of orientable manifolds we call filling enlargeable and prove: If Y is filling enlargeable, Xn = Y × [−1, 1] and g is a Riemannian metric on X with the property that the volumes of all unit balls in the universal cover of (X, g) are bounded from above by a small dimensional constant εn, then width(X, g) ≤ 1. Finally, we establish that whether or not a closed orientable manifold is filling enlargeable or not depends on the image of the fundamental class under the classifying map of the universal cover.…