Critical metrics of the 𝐿²-norm of the scalar curvature

Catino, Giovanni

doi:10.1090/S0002-9939-2014-12238-6

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