Preliminary Version SCATTERING THEORY FOR CONFORMALLY COMPACT METRICS WITH VARIABLE CURVATURE AT INFINITY

Scattering Theory for Conformally Compact Metrics with Variable Curvature at Infinity

Inverse Scattering Results for Manifolds Hyperbolic Near Infinity

We study the inverse resonance problem for conformally compact manifolds which are hyperbolic outside a compact set. Our results include compactness of isoresonant metrics in dimension two and of isophasal negatively curved metrics in dimension three. In dimensions four or higher we prove topological finiteness theorems under the negative curvature assumption.

Unique Continuation Results for Ricci Curvature and Applications

Unique continuation results are proved for metrics with prescribed Ricci curvature in the setting of bounded metrics on compact manifolds with boundary, and in the setting of complete conformally compact metrics on such manifolds. Related to this issue, an isometry extension property is proved: continuous groups of isometries at conformal infinity extend into the bulk of any complete conformall...

Einstein Metrics with Prescribed Conformal Infinity on 4-manifolds

This paper considers the existence of conformally compact Einstein metrics on 4manifolds. A reasonably complete understanding is obtained for the existence of such metrics with prescribed conformal infinity, when the conformal infinity is of positive scalar curvature. We find in particular that general solvability depends on the topology of the filling manifold. The obstruction to extending the...

Q-curvature and Poincaré Metrics

This article presents a new definition of Branson’s Q-curvature in even dimensional conformal geometry. The Q-curvature is a generalization of the scalar curvature in dimension 2: it satisfies an analogous transformation law under conformal rescalings of the metric and on conformally flat manifolds its integral is a multiple of the Euler characteristic. Our approach is motivated by the recent w...