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Metric Inequalities with Scalar Curvature

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Abstract

We establish several inequalities for manifolds with positive scalar curvature and, more generally, for the scalar curvature bounded from below. In so far as geometry is concerned these inequalities appear as generalisations of the classical bounds on the distances between conjugates points in surfaces with positive sectional curvatures. The techniques of our proofs is based on the Schoen–Yau descent method via minimal hypersurfaces, while the overall logic of our arguments is inspired by and closely related to the torus splitting argument in Novikov’s proof of the topological invariance of the rational Pontryagin classes.

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Authors and Affiliations

  1. Institut des Hautes Études Scientifiques, 35 Route de Chartres, 91440, Bures-sur-Yvette, France

    Misha Gromov

  2. Courant Institute of Mathematical Sciences - NYU, 251 Mercer Street, New York, NY, USA

    Misha Gromov

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  1. Misha Gromov
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Gromov, M. Metric Inequalities with Scalar Curvature. Geom. Funct. Anal. 28, 645–726 (2018). https://doi.org/10.1007/s00039-018-0453-z

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