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

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

2004-03-13

arxiv.org/abs/math/0403221v2

Mathematics

Differential Geometry

26 pages, 0 figure

Scientific paper

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.

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