Construction de métriques d’Einstein à partir de transformations biconformes
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 15 (2006) no. 3, p. 553-588

We give a new method for constructing Einstein metrics as follows. Given a harmonic morphism ϕ:(M,g)(N,h), we deform the metric g biconformally in such a way as to preserve harmonicity. The condition that the new metric be Einstein determines a first order system in terms of the scaling factors of the deformation. By choosing our initial metric g conveniently, and with assumptions on the scaling factors, this system corresponds to a dynamical system. In such cases we are able to establish local existence of solutions. We describe some explicit cases which correspond to Einstein metrics in dimension 4.

L’objectif de cet article est de proposer une nouvelle méthode de construction de métriques d’Einstein. Le procédé consiste à considérer un morphisme harmonique ϕ:(M,g)(N,h) ; on déforme ensuite biconformément la métrique g en g ˜, en conservant l’harmonicité, ce qui simplifie le calcul de la courbure de Ricci. L’équation Ric ˜=Cg ˜ se traduit alors en un système différentiel en termes des paramètres de la déformation. On montre d’abord l’existence de solutions par un procédé dynamique. Puis, on résout ce système dans des exemples en dimension 4, exhibant ainsi des métriques d’Einstein.

@article{AFST_2006_6_15_3_553_0,
     author = {Danielo, Laurent},
     title = {Construction de m\'etriques d'Einstein \`a partir de transformations biconformes},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {6e s{\'e}rie, 15},
     number = {3},
     year = {2006},
     pages = {553-588},
     doi = {10.5802/afst.1129},
     mrnumber = {2246414},
     zbl = {1127.53037},
     language = {fr},
     url = {http://www.numdam.org/item/AFST_2006_6_15_3_553_0}
}
Danielo, Laurent. Construction de métriques d’Einstein à partir de transformations biconformes. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 15 (2006) no. 3, pp. 553-588. doi : 10.5802/afst.1129. http://www.numdam.org/item/AFST_2006_6_15_3_553_0/

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