Geometria

Macroscopic band width inequalities
Daniel Räde (Augsburg)
Dia: 2 / 4 / 2020
Hora: 12:00
Lloc: Dpt Matematiques, Seminari C3B (C3B/158)

Web: Laboratori d'Interaccions entre Geometria, Àlgebra i Topologia

Resum: If $M^{n-1}$ is a closed smooth manifold and $g$ is a smooth Riemannian metric on $V:=M\times[0,1]$, we call $(V,g)$ a Riemannian band over $M$. The width of the Riemannian band $(V,g$ is defined to be the distance between the two boundary components.
In a recent paper M. Gromov conjectured that if $M$ does not admit a metric with positive scalar curvature and if we assume that $Sc(V,g)\geq\sigma>0$, then $width(V,g)$ is bounded from above by a sharp constant only depending on the dimension $n$ and $\sigma$. He proved this conjecture for several classes of manifolds including the torus $T^{n-1}$.
In this talk we want to discuss some results regarding band width estimates under a different condition on the metric. Instead of a lower scalar curvature bound we assume that unit balls in $(V,g)$ or in the universal cover $(\tilde{V},\tilde{g})$ have very small volume. This is based on L. Guth's notion of macroscopic scalar curvature and relates to systolic geometry.