Conformal Invariants Associated to a Measure

In this note, we study some conformal invariants of a Riemannian manifold ($M^{n}$, g) equipped with a smooth measure m. In particular, we show that there is a natural definition of the Ricci and scalar curvatures associated to such a space, both of which are conformally invariant. We also adapt the...

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Bibliographic Details
Published in:	Proceedings of the National Academy of Sciences - PNAS Vol. 103; no. 8; pp. 2535 - 2540
Main Authors:	Chang, Sun-Yung A., Gursky, Matthew J., Yang, Paul
Format:	Journal Article
Language:	English
Published:	United States National Academy of Sciences 21-02-2006 National Acad Sciences
Subjects:	Average linear density Curvature Integers Laplacians Mathematics Physical Sciences Preprints Riemann manifold Scalars Tensors Topological manifolds Vector fields
Online Access:	Get full text
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Summary:	In this note, we study some conformal invariants of a Riemannian manifold ($M^{n}$, g) equipped with a smooth measure m. In particular, we show that there is a natural definition of the Ricci and scalar curvatures associated to such a space, both of which are conformally invariant. We also adapt the methods of Fefferman and Graham [Fefferman, C. & Graham, C. R. (1985) Astérisque, Numero Hors Serie, 95-116] and Graham, Jenne, Mason, and Sparling [Graham, C. R., Jenne, R., Mason, L. J., & Sparling, G. A. J. (1992) J. London Math. Soc. 46, 557-565] to construct families of conformally covariant operators defined on these spaces. Certain variational problems in this setting are considered, including a generalization of the Einstein-Hilbert action.
Bibliography:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 23 Communicated by Charles L. Fefferman, Princeton University, Princeton, NJ, December 15, 2005 Author contributions: S.-Y.A.C., M.J.G., and P.Y. designed research; S.-Y.A.C., M.J.G., and P.Y. performed research; S.-Y.A.C., M.J.G., and P.Y. contributed new reagents/analytic tools; and S.-Y.A.C., M.J.G., and P.Y. wrote the paper.
ISSN:	0027-8424 1091-6490
DOI:	10.1073/pnas.0510814103