Complete surfaces with negative extrinsic curvature

N. V. Efimov [Efi64] proved that there is no complete, smooth surface in R with uniformly negative curvature. We extend this to isometric immersions in a 3-manifold with pinched curvature: if M has sectional curvature between two constants K2 and K3, then there exists K1 < min(K2, 0) such that M contains no smooth, complete immersed surface with curvature below K1. Optimal values of K1 are determined. This results rests on a phenomenon of propagations for degenerations of solutions of hyperbolic Monge-Ampère equations. Résumé N. V. Efimov [Efi64] a montré qu’il n’existe pas de surface complète à courbure uniformément négative dans R. On étend ce résultat aux immersions isométriques dans les 3-variétés à courbure pincée: si M a sa courbure sectionnelle comprise entre deux constantes K2 et K3, alors il existe une constante K1 < min(K2, 0) telle que M ne contient pas de surface immergée complète et régulière à courbure inférieure à K1. Des valeurs optimales de K1 sont déterminées. Ce résultat repose sur un phénomène de propagation pour les dégénérescences de solutions d’équations de Monge-Ampère hyperboliques. AMS classifications: 53C45, 58G16, 35L55. Key-words: isometric, immersion, surface, Monge-Ampère, hyperbolic. Hilbert [Hil01] proved that there is no smooth isometric immersion of the hyperbolic plane H into the Euclidean 3-space R. This was extended by Efimov, who replaced H by any complete surface with uniformly negative curvature: Theorem 0.1 (N. V. Efimov [Efi64]). Let (Σ, σ) be a smooth, complete Riemannian surface with curvature K ≤ −1. Then (Σ, σ) has no C isometric immersion into R. This result was proved using some subtle geometric constructions, strongly based on the Euclidean structure of the target space. More details can be found in [Efi68a], [Klo72] or in [BS92, Roz92], and some extensions and related results in [Efi68b, Efi62, Efi66]. It seems rather natural to try to extend Hilbert’s result further by replacing also R by a Riemannian manifold. This was started in [Sch99], where the target space can be a Riemannian or Lorentzian 3dimensional space-form. The present paper treats the case where it is a Riemannian manifold with pinched curvature. Theorem 0.2. Let (M,μ) be a complete Riemannian 3-manifold, with sectional curvature KM between two constants K2 ≤ K3. Let (Σ, σ) be a complete Riemannian surface, with curvature KΣ ≤ K1, with K1 < 0, K1 < K2 ≤ K3, and: • either K3 ≥ 0 and (K3 −K2) < 16|K1|(K2 −K1) ; Mathématiques, UMR 8628 du CNRS, Bât. 425, Université Paris-Sud, F-91405 Orsay Cedex, France ; currently: FIM, ETHZ, Rämistr. 101, CH-8092 Zürich. [email protected]. http://www.math.u-psud.fr/~schlenker.