Finite total $Q$-curvature on a locally conformally flat manifold

In this talk, we will discuss locally conformally flat manifolds with finite total curvature.

We prove that for such a manifold, the integral of the $Q$-curvature
equals an integral multiple of a dimensional constant. This
shows a new aspect of the $Q$-curvature on noncompact complete
manifolds. It provides further evidence that $Q$-curvature controls
geometry as the Gaussian curvature does in two dimension on locally conformally flat manifolds

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