Asymptotically Symmetric Einstein Metrics

The relation between the Riemannian structure of a hyperbolic manifold and the induced conformal structure on its boundary has been intensively studied recently, in particular in connection with the so called AdS/CFT correspondence discovered in theoretical physics. Basic examples of such correspondence are given by suitable homogeneous spaces (and their boundaries) but the whole scheme works as well in a curved situation (where Einstein metrics are just asymptotically symmetric). The correspondence studied in this book is modelled on hyperbolic spaces over basic fields (real, complex, quaternionic and octonionic in dimension two) and their boundaries. Basic homogeneous examples show that the induced conformal metric on the boundary is a Carnot-Carathéodory metric (which is defined, with the exception of the real case, only on a suitable sub-bundle of the tangent bundle). The main problem addressed in the book is a Dirichlet problem for a nonlinear system of partial differential equations (the Einstein equation with some additional constraints), the boundary data being given by the conformal class of a chosen Carnot-Carathéodory metric. The book contains results both on global solutions and local solutions near infinity. The book brings together new and important results in a modern field of mathematics.

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