Fredholm operators and Einstein metrics on conformally compact manifolds
Conformal geometry (in particular in dimension 4) and corresponding geometric partial differential equations have been studied carefully in the last few decades. New tools developed recently by Ch. Fefferman, R. Graham, M. Eastwood, R. Gover and T. Bailey have allowed a much better understanding of conformal invariants (including a complete classification in particular cases). A principal tool here is the ambient metric construction of Fefferman and Graham. The so-called AdS-CFT correspondence recently emerging from string theory is based on the notion of conformal infinity for a (pseudo-) Riemannian metric developed originally by R. Penrose. It caused a dramatic rise of interest in these problems both in mathematics and mathematical physics. The problem treated in the book is to find, for a compact manifold M with a given conformal structure c on its boundary ∂M, a complete asymptotically hyperbolic Einstein metric g on the interior of M such that c coincides with the conformal class of g on ∂M.
The book contains a systematic and self-contained description of the perturbation version of the problem (where we are looking for solutions of the problem for c near to a given c0, for which the solution exists). An important fact is that the solution has an optimal Hölder regularity up to the boundary (for n even). Related results (with less boundary regularity but also for the quaternionic and octonionic cases) were described recently in the book by O. Biquard (see EMS Newsletter 65, page 55). Methods used in the book are based on general sharp Fredholm and isomorphism theorems for geometric (degenerate) linear elliptic operators.