We will explain geometric situations where a lower bound on the scalar curvature of a Riemannian manifold leads to quantitative distance estimates and rigidity results. The study of these has been prompted by several conjectures of Gromov from the recent years. Intuitively, these results can be seen as analogues for scalar curvature of comparison geometry statements such as the Bonnet-Myers theorem for Ricci curvature. However, unlike classical comparison geometry involving stronger curvature conditions, such results for scalar curvature typically rely on an additional topological assumption such as the non-existence of positive scalar curvature metrics on certain submanifolds. Along the way we will thus also provide a brief introduction to obstructions to the existence of positive scalar curvature metrics on closed manifolds.
This video was produced by the University of Münster, as part of the workshop Mathematics Münster Mid-term Conference.
