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Asymptotically symmetric Einstein metrics. Transl. from the French by Stephen S. Wilson. (English) Zbl 1112.53001
SMF/AMS Texts and Monographs 13. Providence, RI: American Mathematical Society (AMS); Paris: Société Mathématique de France, (ISBN 0-8218-3166-6/pbk). v, 105 p. (2006).
This beautiful book is a translation of [Métriques d’Einstein asymptotiquement symétriques. Astérisque 265 (2000; Zbl 0967.53030)]. The aim of this book is to study the following problem: given a conformal Carnot-Carathéodory metric \([\gamma]\) on the boundary \({\mathbb S}^{n-1}\) of the hyperbolic space \({\mathbb K}H^m\) with \(\mathbb K=\mathbb C,\mathbb H\) or \(\mathbb O\), study the nonlinear Dirichlet problem (i) \(\text{Ric}^g=-\lambda g\); (ii) the conformal infinity of \(g\) is \([\gamma]\).
Very shortly speaking, in this book, two situations are presented, in which the solution of the problem allows one to construct new Einstein metrics: the first solution uses analytic techniques; the second solution uses twistorial techniques. The reader should also compare the recently published book by John M. Lee [Mem. Am. Math. Soc. 864 (2006; Zbl 1112.53002)].

MSC:
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
53C28 Twistor methods in differential geometry
53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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