Band width estimates via the Dirac operator
Abstract
Let $M$ be a closed connected spin manifold such that its spinor Dirac operator has nonvanishing (Rosenberg) index. We prove that for any Riemannian metric on $V = M \times [1,1]$ with scalar curvature bounded below by $\sigma > 0$, the distance between the boundary components of $V$ is at most $C_n/\sqrt{\sigma}$, where $C_n = \sqrt{(n1)/{n}} \cdot C$ with $C < 8(1+\sqrt{2})$ being a universal constant. This verifies a conjecture of Gromov for such manifolds. In particular, our result applies to all highdimensional closed simply connected manifolds $M$ which do not admit a metric of positive scalar curvature. We also establish a quadratic decay estimate for the scalar curvature of complete metrics on manifolds, such as $M \times \mathbb{R}^2$, which contain $M$ as a codimension two submanifold in a suitable way. Furthermore, we introduce the "$\mathcal{KO}$width" of a closed manifold and deduce that infinite $\mathcal{KO}$width is an obstruction to positive scalar curvature.
 Publication:

arXiv eprints
 Pub Date:
 May 2019
 arXiv:
 arXiv:1905.08520
 Bibcode:
 2019arXiv190508520Z
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  KTheory and Homology;
 53C21 (Primary) 19K56;
 58J22 (Secondary)
 EPrint:
 24 pages, 2 figures