Dirac and Plateau billiards in domains with corners

Misha Gromov

Open Mathematics (2014)

  • Volume: 12, Issue: 8, page 1109-1156
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Misha Gromov. "Dirac and Plateau billiards in domains with corners." Open Mathematics 12.8 (2014): 1109-1156. <http://eudml.org/doc/269152>.

@article{MishaGromov2014,
abstract = {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.},
author = {Misha Gromov},
journal = {Open Mathematics},
keywords = {Scalar curvature; Dirac operator; Plateau problem; Reflection groups; scalar curvature; domains with corners; Dirac; billiards; dihedrally extremal; Plateau-hedron; dihedrally rigid; Plateau traps; Reifenberg flatness; Plateau m-web},
language = {eng},
number = {8},
pages = {1109-1156},
title = {Dirac and Plateau billiards in domains with corners},
url = {http://eudml.org/doc/269152},
volume = {12},
year = {2014},
}

TY - JOUR
AU - Misha Gromov
TI - Dirac and Plateau billiards in domains with corners
JO - Open Mathematics
PY - 2014
VL - 12
IS - 8
SP - 1109
EP - 1156
AB - 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.
LA - eng
KW - Scalar curvature; Dirac operator; Plateau problem; Reflection groups; scalar curvature; domains with corners; Dirac; billiards; dihedrally extremal; Plateau-hedron; dihedrally rigid; Plateau traps; Reifenberg flatness; Plateau m-web
UR - http://eudml.org/doc/269152
ER -

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