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Scalar and mean curvature comparison via the Dirac operator

Simone Cecchini and Rudolf Zeidler

Geometry & Topology 28 (2024) 1167–1212

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. These include optimal estimates for Riemannian bands and for the long neck problem. In the case of bands over manifolds of nonvanishing  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.

scalar curvature, mean curvature, Dirac operator, Callias operator, comparison geometry, rigidity
Mathematical Subject Classification
Primary: 53C21, 53C24
Secondary: 53C23, 53C27, 58J20
Received: 28 July 2021
Revised: 14 March 2022
Accepted: 10 April 2022
Published: 10 May 2024
Proposed: John Lott
Seconded: Bruce Kleiner, Tobias H Colding
Simone Cecchini
Department of Mathematics
Texas A&M University
College Station, TX
United States
Rudolf Zeidler
Mathematical Institute
University of Münster

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