Formal theory of cornered asymptotically hyperbolic Einstein metrics
This paper makes a formal study of asymptotically hyperbolic Einstein metrics given, as conformal infinity, a conformal manifold with boundary. The space on which such an Einstein metric exists thus has a finite boundary in addition to the usual infinite boundary and a corner where the two meet. On...
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| Format | Journal Article |
| Language | English |
| Published |
08.08.2017
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| Subjects | |
| Online Access | Get full text |
| DOI | 10.48550/arxiv.1708.02390 |
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| Summary: | This paper makes a formal study of asymptotically hyperbolic Einstein metrics
given, as conformal infinity, a conformal manifold with boundary. The space on
which such an Einstein metric exists thus has a finite boundary in addition to
the usual infinite boundary and a corner where the two meet. On the finite
boundary a constant mean curvature umbilic condition is imposed. 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. A model linear problem is then studied: a formal
expansion at the corner is derived for eigenfunctions of the scalar Laplacian
subject to certain boundary conditions. In doing so, scalar ODEs are studied
that are of relevance for a broader class of boundary value problems and also
for the Einstein problem. Next, unique formal existence at the corner, up to
order at least equal to the boundary dimension, of Einstein metrics in a
cornered asymptotically hyperbolic normal form which are 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 order,
which defines a conformal hypersurface invariant. |
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| DOI: | 10.48550/arxiv.1708.02390 |