Scalar and mean curvature comparison via the Dirac operator

Abstract

We use the Dirac operator technique to establish sharp distance estimates for compact spin manifolds under lower bounds on the scalar curvature in the interior and on the mean curvature of the boundary. In the situations we consider, we thereby give refined answers to questions on metric inequalities recently proposed by Gromov. This includes optimal estimates for Riemannian bands and for the long neck problem. In the case of bands over manifolds of non-vanishing $\hat{\mathrm{A}}$-genus, we establish a rigidity result stating that any band attaining the predicted upper bound is isometric to a particular warped product over some spin manifold admitting a parallel spinor. Furthermore, we establish scalar- and mean curvature extremality results for certain log-concave warped products. The latter includes annuli in all simply-connected space forms. On a technical level, our proofs are based on new spectral estimates for the Dirac operator augmented by a Lipschitz potential together with local boundary conditions.

48 pages, 2 figures; v2: optimality of long neck problem and improvements of exposition; v3: minor improvements and added two figures. To appear in Geometry & Topology

References in corpus (4)

• Generalized soap bubbles and the topology of manifolds with positive scalar curvature
• Width, Largeness and Index Theory
• No metrics with Positive Scalar Curvatures on Aspherical 5-Manifolds
• Rigidity results for complete manifolds with nonnegative scalar curvature

Cited by in corpus (7)

• A quantitative relative index theorem and Gromov's conjectures on positive scalar curvature
• The positive mass theorem and distance estimates in the spin setting
• Spectral flow of Callias operators, odd K-cowaist, and positive scalar curvature
• $\mathrm{K}$-cowaist on complete foliated manifolds
• On Gromov's dihedral extremality and rigidity conjectures
• The odd-dimensional long neck problem via spectral flow
• The degree condition in Llarull's theorem on scalar curvature rigidity