Two Kazdan–Warner-type identities for the renormalized volume coefficients and the Gauss–Bonnet curvatures of a Riemannian metric

Bin Guo, Zheng-Chao Han and Haizhong Li

Vol. 251 (2011), No. 2, 257–268

DOI: 10.2140/pjm.2011.251.257

Abstract

We prove two Kazdan–Warner-type identities involving the renormalized volume coefficients v^(2k) of a Riemannian manifold (Mⁿ,g), the Gauss–Bonnet curvature G_2r, and a conformal Killing vector field on (Mⁿ,g). In the case when the Riemannian manifold is locally conformally flat, we find

and our results reduce to earlier ones established by Viaclovsky in 2000 and the second author in 2006.

Keywords

renormalized volume coefficients, v^(2k) curvature, conformal transformation, locally conformally flat, σ_k curvature, Gauss–Bonnet curvatures, Kazdan–Warner

Mathematical Subject Classification 2000

Primary: 53C20

Secondary: 53A30

Milestones

Received: 18 May 2010

Accepted: 1 June 2010

Published: 3 June 2011