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  1. 3 Open Mathematics

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  1. 3 Gromov M.

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  1. 2 2014

  2. 1 2012

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Dirac and Plateau billiards in domains with corners

100%
Gromov M.
Open Mathematics
|
2014
|
tom 12
|
nr 8
1109-1156
EN
Groping our way toward a theory of singular spaces with positive scalar curvatures we look at the Dirac operator and a generalized Plateau problem in Riemannian manifolds with corners. Using these, we prove that the set of C 2-smooth Riemannian metrics g on a smooth manifold X, such that scalg(x) ≥ κ(x), is closed under C 0-limits of Riemannian metrics for all continuous functions κ on X. Apart from that our progress is limited but we formulate many conjectures. All along, we emphasize geometry, rather than topology of manifolds with their scalar curvatures bounded from below.
2
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Hilbert volume in metric spaces. Part 1

100%
Gromov M.
Open Mathematics
|
2012
|
tom 10
|
nr 2
371-400
EN
We introduce a notion of Hilbertian n-volume in metric spaces with Besicovitch-type inequalities built-in into the definitions. The present Part 1 of the article is, for the most part, dedicated to the reformulation of known results in our terms with proofs being reduced to (almost) pure tautologies. If there is any novelty in the paper, this is in forging certain terminology which, ultimately, may turn useful in an Alexandrov kind of approach to singular spaces with positive scalar curvature [Gromov M., Plateau-hedra, Scalar Curvature and Dirac Billiards, in preparation].
3
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Plateau-Stein manifolds

100%
Gromov M.
Open Mathematics
|
2014
|
tom 12
|
nr 7
923-951
EN
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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