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Scalar and mean curvature
comparison via the Dirac operator
Simone Cecchini and Rudolf Zeidler |
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Geometry & Topology 28 (2024) 1167–1212
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Abstract |
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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 –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. |
Keywords
scalar curvature, mean curvature, Dirac operator, Callias
operator, comparison geometry, rigidity
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Mathematical Subject Classification
Primary: 53C21, 53C24
Secondary: 53C23, 53C27, 58J20
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References |
Publication
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
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Authors |
| Simone Cecchini | |
| Department of Mathematics Texas A&M University College Station, TX United States |
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| https://simonececchini.org | |
| Rudolf Zeidler | |
| Mathematical Institute University of Münster Münster Germany |
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| https://www.rzeidler.eu | |
| © 2024 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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