Geometric and Functional Analysis

, Volume 28, Issue 3, pp 645–726 | Cite as

Metric Inequalities with Scalar Curvature

  • Misha GromovEmail author


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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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut des Hautes Études ScientifiquesBures-sur-YvetteFrance
  2. 2.Courant Institute of Mathematical Sciences - NYUNew YorkUSA

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