Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 035, 26 pages      arXiv:2306.04015      https://doi.org/10.3842/SIGMA.2024.035
Contribution to the Special Issue on Differential Geometry Inspired by Mathematical Physics in honor of Jean-Pierre Bourguignon for his 75th birthday

Scalar Curvature Rigidity of Warped Product Metrics

Christian Bär a, Simon Brendle b, Bernhard Hanke c and Yipeng Wang b
a) Institut für Mathematik, Universität Potsdam, 14476 Potsdam, Germany
b) Department of Mathematics, Columbia University, New York NY 10027, USA
c) Institut für Mathematik, Universität Augsburg, 86135 Augsburg, Germany

Received June 09, 2023, in final form April 08, 2024; Published online April 18, 2024

Abstract
We show scalar-mean curvature rigidity of warped products of round spheres of dimension at least 2 over compact intervals equipped with strictly log-concave warping functions. This generalizes earlier results of Cecchini-Zeidler to all dimensions. Moreover, we show scalar curvature rigidity of round spheres of dimension at least 3 with two antipodal points removed. This resolves a problem in Gromov's ''Four Lectures'' in all dimensions. Our arguments are based on spin geometry.

Key words: scalar curvature; warped product; bandwidth estimate; Llarull's theorem; holographic index theorem.

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