Band width estimates via the Dirac operator

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 "$\mathcal{K}\mathcal{O}$-width" of a closed manifold and deduce that infinite $\mathcal{K}\mathcal{O}$-width is an obstruction to positive scalar curvature.

24 pages, 2 figures; v2: minor additions and improvements; v3: minor corrections and slightly improved estimates. To appear in J. Differential Geom

References in corpus (5)

• The topology of positive scalar curvature
• Enlargeability and index theory: Infinite covers
• The positive scalar curvature cobordism category
• The Gromov-Lawson codimension 2 obstruction to positive scalar curvature and the C*-index
• Slant products on the Higson-Roe exact sequence

Cited by in corpus (10)

• Width, Largeness and Index Theory
• Scalar and mean curvature comparison via the Dirac operator
• Nonnegative scalar curvature on manifolds with at least two ends
• A quantitative relative index theorem and Gromov's conjectures on positive scalar curvature
• Quantitative K-theory, positive scalar curvature, and band width
• A proof of Gromov's cube inequality on scalar curvature
• Decay of scalar curvature on uniformly contractible manifolds with finite asymptotic dimension
• The positive mass theorem and distance estimates in the spin setting
• Macroscopic band width inequalities
• On Gromov's dihedral extremality and rigidity conjectures