Calc. Var. (2017) 56:98
Calculus of Variations
On locally conformally ﬂat manifolds with ﬁnite total
· Yi Wang
Received: 3 May 2016 / Accepted: 4 May 2017
© Springer-Verlag Berlin Heidelberg 2017
Abstract In this paper, we study the ends of a locally conformally ﬂat complete manifold
with ﬁnite total Q-curvature. We prove that for such a manifold, the integral of the Q-
curvature equals an integral multiple of a dimensional constant c
is the integral
of the Q-curvature on the unit n-sphere. It provides further evidence that the Q-curvature on
a locally conformally ﬂat manifold controls geometry as the Gaussian curvature does in two
Mathematics Subject Classiﬁcation Primary 53A30 · Secondary 53C21
The Q-curvature arises naturally as a conformal invariant associated to the Paneitz operator.
When n = 4, the Paneitz operator is deﬁned as:
Communicated by A. Malchiodi.
Zhiqin Lu is partially supported by NSF grant DMS-1510232, and Yi Wang is partially supported by NSF
grants DMS-1547878 and DMS-1612015.
Department of Mathematics, University of California, Irvine, 410D Rowland Hall, Irvine, CA
Department of Mathematics, Johns Hopkins University, 404 Krieger Hall, 3400 N. Charles Street,
Baltimore, MD 21218, USA