On locally conformally flat manifolds with finite total $Q$-curvature

Abstract

In this paper, we focus our study on 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 $c_n$, where $c_n$ 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 dimension.