Cornered Asymptotically Hyperbolic Metrics

Thesis (Ph.D.)--University of Washington, 2016-08

This thesis considers asymptotically hyperbolic manifolds that have a finite boundary in addition to the usual infinite boundary – cornered asymptotically hyperbolic manifolds. A theorem of Cartan-Hadamard type near infinity for the normal exponential map on the finite boundary is proved, and this is used to construct a corner normal form, analogous to the usual asymptotically hyperbolic normal form, and suitable for studying questions near the corner. Formal expansion at the corner of a boundary value problem for the scalar Laplacian is then studied in the special case that the finite boundary makes constant angle \pi/2 with the infinite boundary, and a formal existence and uniqueness result is proved. The thesis then takes up the study of cornered asymptotically hyperbolic Einstein metrics, with a constant mean curvature umbilic boundary condition imposed at the finite boundary. First, recent work of Nozaki, Takayanagi, and Ugajin is generalized and extended showing that such metrics cannot have smooth compactifications for generic corners embedded in the infinite boundary. Then unique formal existence at the corner, up to order equal to the boundary dimension, of Einstein metrics in normal form and polyhomogeneous in polar coordinates is demonstrated for arbitrary smooth conformal infinity. Finally it is shown that, in the special case that the finite boundary is taken to be totally geodesic, there is an obstruction to existence beyond this dimension, which defines a conformal hypersurface invariant.