September 2022 Band width estimates via the Dirac operator
Rudolf Zeidler
Author Affiliations +
J. Differential Geom. 122(1): 155-183 (September 2022). DOI: 10.4310/jdg/1668186790


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 $\sigma \gt 0$, 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 $C \lt 8(1+\sqrt{2})$ 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.

Funding Statement

Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2044–390685587, Mathematics Münster: Dynamics–Geometry–Structure.


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Rudolf Zeidler. "Band width estimates via the Dirac operator." J. Differential Geom. 122 (1) 155 - 183, September 2022.


Received: 1 July 2019; Accepted: 3 February 2020; Published: September 2022
First available in Project Euclid: 11 November 2022

Digital Object Identifier: 10.4310/jdg/1668186790

Primary: 53C21
Secondary: 19K56 , 58J22

Rights: Copyright © 2022 Lehigh University


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Vol.122 • No. 1 • September 2022
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