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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.

Riesz potentials and amalgams

Michael Cowling, Stefano Meda, Roberta Pasquale (1999)

Annales de l'institut Fourier

Let ( M , d ) be a metric space, equipped with a Borel measure μ satisfying suitable compatibility conditions. An amalgam A p q ( M ) is a space which looks locally like L p ( M ) but globally like L q ( M ) . We consider the case where the measure μ ( B ( x , ρ ) of the ball B ( x , ρ ) with centre x and radius ρ behaves like a polynomial in ρ , and consider the mapping properties between amalgams of kernel operators where the kernel ker K ( x , y ) behaves like d ( x , y ) - a when d ( x , y ) 1 and like d ( x , y ) - b when d ( x , y ) 1 . As an application, we describe Hardy–Littlewood–Sobolev type regularity theorems...

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