Almost-Schur lemma

Almost-Schur lemma

Schur’s lemma states that every Einstein manifold of dimension n  ≥ 3 has constant scalar curvature. In this short note we ask to what extent the scalar curvature is constant if the traceless Ricci tensor is assumed to be small rather than identically zero. In particular, we provide an optimal L 2 e...

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Veröffentlicht in:Calculus of variations and partial differential equations 2012-03, Vol.43 (3-4), p.347-354
Hauptverfasser: Lellis, Camillo De, Topping, Peter M.
Format: Artikel
Sprache:eng
Schlagworte:
Analysis
Calculus of variations
Calculus of Variations and Optimal Control
Optimization
Control
Curvature
Estimates
Manifolds
Mathematical analysis
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Optimization
Partial differential equations
Scalars
Systems Theory
Tensors
Theoretical
Topological manifolds
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Beschreibung
Zusammenfassung:Schur’s lemma states that every Einstein manifold of dimension n  ≥ 3 has constant scalar curvature. In this short note we ask to what extent the scalar curvature is constant if the traceless Ricci tensor is assumed to be small rather than identically zero. In particular, we provide an optimal L 2 estimate under suitable assumptions and show that these assumptions cannot be removed.
ISSN:0944-2669
1432-0835
DOI:10.1007/s00526-011-0413-z