Abstract
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 language | English (US) |
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Pages (from-to) | 923-951 |
Number of pages | 29 |
Journal | Central European Journal of Mathematics |
Volume | 12 |
Issue number | 7 |
DOIs | |
State | Published - Jul 2014 |
Keywords
- Geometric measure theory
- Riemannian manifolds
- Stein Manifolds
ASJC Scopus subject areas
- General Mathematics