# Band width estimates via the Dirac operator

## TL;DR

This paper proves a conjecture of Gromov relating scalar curvature bounds to the width of manifolds with non-vanishing Dirac index, establishing new obstructions to positive scalar curvature and introducing the concept of KO-width.

## Contribution

It verifies Gromov's conjecture for manifolds with non-vanishing Dirac index and introduces KO-width as an obstruction to positive scalar curvature.

## Key findings

- Bound on boundary distance in manifolds with scalar curvature ≥ σ
- Quadratic decay estimate for scalar curvature in certain manifolds
- Infinite KO-width obstructs positive scalar curvature

## Abstract

Let $M$ be a closed connected spin manifold such that its spinor Dirac operator has non-vanishing (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 $C_n = \sqrt{(n-1)/{n}} \cdot C$ 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 high-dimensional 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.