Plateau-Stein manifolds

Misha Gromov

Research output: Contribution to journalArticlepeer-review


We study/construct (proper and non-proper) Morse functions f on complete Riemannian manifolds X such that the hypersurfaces f(x) = t for all -∞ < t < +∞ have positive mean curvatures at all non-critical points x ∈ X of f. We show, for instance, that if X admits no such (not necessarily proper) function, then it contains a (possibly, singular) complete (possibly, compact) minimal hypersurface of finite volume.

Original languageEnglish (US)
Pages (from-to)923-951
Number of pages29
JournalCentral European Journal of Mathematics
Issue number7
StatePublished - Jul 2014


  • Geometric measure theory
  • Riemannian manifolds
  • Stein Manifolds

ASJC Scopus subject areas

  • General Mathematics


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