Volume and macroscopic scalar curvature
Differential Geometry
2021-11-09 v2 Geometric Topology
Abstract
We prove the macroscopic cousins of three conjectures: 1) a conjectural bound of the simplicial volume of a Riemannian manifold in the presence of a lower scalar curvature bound, 2) the conjecture that rationally essential manifolds do not admit metrics of positive scalar curvature, 3) a conjectural bound of -Betti numbers of aspherical Riemannian manifolds in the presence of a lower scalar curvature bound. The macroscopic cousin is the statement one obtains by replacing a lower scalar curvature bound by an upper bound on the volumes of -balls in the universal cover.
Cite
@article{arxiv.2012.08999,
title = {Volume and macroscopic scalar curvature},
author = {Sabine Braun and Roman Sauer},
journal= {arXiv preprint arXiv:2012.08999},
year = {2021}
}
Comments
48 pages; added a statement about integral foliated simplicial volume in the introduction and made minor corrections; to be published in GAFA