Conformal invariants associated to a measure: conformally covariant operators

Sun-Yung A. Chang, Matthew J. Gursky and Paul Yang

Vol. 253 (2011), No. 1, 37–56

DOI: 10.2140/pjm.2011.253.37

Abstract

We study Riemannian manifolds (Mⁿ,g) equipped with a smooth measure m. We show that the construction of conformally covariant operators of Graham–Jenne–Mason–Sparling can be adapted to this setting. As a byproduct, we define a family of scalar curvatures, one of which corresponds to Perelman’s scalar-curvature function. We also study the variational problem naturally associated to these curvature/operator pairs.

Keywords

conformally invariant operators, metric measure spaces

Mathematical Subject Classification 2000

Primary: 53A30

Secondary: 58J05

Milestones

Received: 18 June 2009

Accepted: 4 February 2010

Published: 28 November 2011