Isoperimetric rigidity and distributions of 1-Lipschitz functions

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Abstract

We prove that if a geodesic metric measure space satisfies a comparison condition for isoperimetric profile and if the observable variance is maximal, then the space is foliated by minimal geodesics, where the observable variance is defined to be the supremum of the variance of 1-Lipschitz functions on the space. Our result can be considered as a variant of Cheeger-Gromoll's splitting theorem and also of Cheng's maximal diameter theorem. As an application, we obtain a new isometric splitting theorem for a complete weighted Riemannian manifold with a positive Bakry-Émery Ricci curvature.

Original languageEnglish
Pages (from-to)1198-1233
Number of pages36
JournalAdvances in Mathematics
Volume349
DOIs
Publication statusPublished - 2019 Jun 20

Keywords

  • Concentration of measure
  • Isoperimetric profile
  • Lipschitz function
  • Metric measure space
  • Observable variance

ASJC Scopus subject areas

  • Mathematics(all)

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