Prescription de courbure des feuilles des laminations : retour sur un théorème de Candel

Alvarez, Sébastien; Smith, Graham

Annales de l'Institut Fourier, Tome 71 (2021) no. 6, pp. 2549-2593.

Dans cet article, nous revenons sur un célèbre théorème de Candel que nous renforçons en prouvant qu’étant donnée une lamination compacte par surfaces hyperboliques, toute fonction négative lisse dans les feuilles et transversalement continue est la fonction courbure d’une unique métrique laminée dans la classe conforme correspondante. Nous interprétons ce fait comme la continuité de solutions de certaines EDP elliptiques dans une topologie, dite de Cheeger–Gromov, sur l’espace des variétés riemanniennes complètes pointées.

In the present paper, we revisit a famous theorem by Candel that we generalize by proving that given a compact lamination by hyperbolic surfaces, every negative function smooth inside the leaves and transversally continuous is the curvature function of a unique laminated metric in the corresponding conformal class. We give an interpretation of this result as a continuity result about the solutions of some elliptic PDEs in the so called Cheeger–Gromov topology on the space of complete pointed riemannian manifolds.

Classification : 57R30, 53C21, 30F10, 30F45
Mot clés : Laminations par surfaces hyperboliques, Prescription de courbure
Keywords: lamination by hyperbolic surfaces, prescrired curvature

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