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 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.
Department of Mathematics, Princeton University, Princeton, NJ 08540, USA. chang@math.princeton.edu
Sun-Yung A
SY
Chang
Matthew J
MJ
Gursky
Paul
P
Yang