Orbifold Degeneration of Conformally Compact Einstein Metrics
Girao, Frederico Vale
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In his investigation of the Dirichlet problem for conformally compact Einstein metrics, Anderson showed that there are (at most) three possibilities for the behavior, under subsequences, of a sequence of conformally compact Einstein metrics, with controlled conformal infinities, on a four-manifold: convergence, orbifold degeneration, or cusp formation. Motivated by this result, we study the phenomenon of orbifold degeneration of a curve of conformally compact Einstein metrics. We start by presenting some background material. After this, we survey the known results concerning the Dirichlet problem, and we address some open questions regarding orbifold degeneration. We then analyze a concrete example of orbifold degeneration, namely, the Taub-bolt family of conformally compact Einstein metrics on the tangent bundle of the two-sphere, and we show that the orbifold Taub-bolt metric is nondegenerate, that is, the kernel of the Bianchi gauged Einstein operator is trivial for this metric. Finally, we obtain results related to a conjecture of Anderson about the boundary of the completion, in the pointed Gromov-Hausdorff topology, of the space of conformally compact Einstein metrics on a four-manifold. These last results give necessary conditions for orbifold degeneration to occur.