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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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Topological tools for the prescribed scalar curvature problem on S n

75%
Abuzaid D., Ben Mahmoud R., Chtioui H., Rigane A.
Open Mathematics
|
2014
|
tom 12
|
nr 12
1829-1839
EN
In this paper, we consider the problem of the existence of conformal metrics with prescribed scalar curvature on the standard sphere S n, n ≥ 3. We give new existence and multiplicity results based on a new Euler-Hopf formula type. Our argument also has the advantage of extending well known results due to Y. Li [16].
3
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Nontrivial examples of coupled equations for Kähler metrics and Yang-Mills connections

63%
Keller J., Tønnesen-Friedman C.
Open Mathematics
|
2012
|
tom 10
|
nr 5
1673-1687
EN
We provide nontrivial examples of solutions to the system of coupled equations introduced by M. García-Fernández for the uniformization problem of a triple (M; L; E), where E is a holomorphic vector bundle over a polarized complex manifold (M, L), generalizing the notions of both constant scalar curvature Kähler metric and Hermitian-Einstein metric.
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