On a Conformal Gauss-Bonnet-Chern inequality for LCF manifolds and related topics

Abstract

In this paper, we prove the following two results: First, we study a class of conformally invariant operators $P$ and their related conformally invariant curvatures $Q$ on even-dimensional Riemannian manifolds. When the manifold is locally conformally flat(LCF) and compact without boundary, $Q$-curvature is naturally related to the integrand in the classical Gauss-Bonnet-Chern formula, i.e., the Pfaffian curvature. For a class of even-dimensional complete LCF manifolds with integrable $Q$% -curvature, we establish a Gauss-Bonnet-Chern inequality. Second, a finiteness theorem for certain classes of complete LCF four-fold with integrable Pfaffian curvature is also proven. This is an extension of the classical results of Cohn-Vossen and Huber in dimension two. It also can be viewed as a fully non-linear analogue of results of Chang-Qing-Yang in dimension four.