Singular sets of a class of locally conformally flat manifolds

Abstract

We look at complete, locally conformally flat (lcf) metrics defined on a domain $\Omega \subset S^n$. There is a lot of information about the singular set $\partial \Omega$ contained in the positivity of ${\sigma }_k$, and, in particular, we obtain a bound for the Hausdorff dimension of $\partial \Omega$ in relation to Schoen and Yau's work [18] for the scalar curvature. On the other hand, since some locally conformally flat manifolds can be embedded into $S^n$, this dimension bound implies several topological corollaries.