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Pairings, duality, amenability and bounded cohomology

Jacek Brodzki, Graham A. Niblo, Nick J. Wright (2012)

Journal of the European Mathematical Society

We give a new perspective on the homological characterizations of amenability given by Johnson & Ringrose in the context of bounded cohomology and by Block & Weinberger in the context of uniformly finite homology. We examine the interaction between their theories and explain the relationship between these characterizations. We apply these ideas to give a new proof of non-vanishing for the bounded cohomology of a free group.

Plateau-Stein manifolds

Misha Gromov (2014)

Open Mathematics

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.

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