Critical metrics of the $L^{2}$-norm of the scalar curvature
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- by Giovanni Catino PDF
- Proc. Amer. Math. Soc. 142 (2014), 3981-3986 Request permission
Abstract:
In this paper we investigate complete critical metrics of the $L^{2}$-norm of the scalar curvature. We prove that any complete critical metric with positive scalar curvature has constant scalar curvature, and we characterize critical metrics with nonnegative scalar curvature in dimensions three and four.References
- Michael T. Anderson, Extrema of curvature functionals on the space of metrics on $3$-manifolds, Calc. Var. Partial Differential Equations 5 (1997), no. 3, 199–269. MR 1438146, DOI 10.1007/s005260050066
- Michael T. Anderson, Extrema of curvature functionals on the space of metrics on 3-manifolds. II, Calc. Var. Partial Differential Equations 12 (2001), no. 1, 1–58. MR 1808106, DOI 10.1007/s005260000043
- Arthur L. Besse, Einstein manifolds, Classics in Mathematics, Springer-Verlag, Berlin, 2008. Reprint of the 1987 edition. MR 2371700
- Guofang Wei and Will Wylie, Comparison geometry for the Bakry-Emery Ricci tensor, J. Differential Geom. 83 (2009), no. 2, 377–405. MR 2577473, DOI 10.4310/jdg/1261495336
- Shing Tung Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975), 201–228. MR 431040, DOI 10.1002/cpa.3160280203
Additional Information
- Giovanni Catino
- Affiliation: Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy
- MR Author ID: 887335
- Email: giovanni.catino@polimi.it
- Received by editor(s): December 27, 2012
- Published electronically: July 28, 2014
- Additional Notes: The author is partially supported by the Italian project FIRB–IDEAS “Analysis and Beyond”
- Communicated by: Lei Ni
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 142 (2014), 3981-3986
- MSC (2010): Primary 53C24, 53C25
- DOI: https://doi.org/10.1090/S0002-9939-2014-12238-6
- MathSciNet review: 3251738