Soit une variété à bord de dimension trois, le théorème de Cartan-Hadamard implique qu’il existe des obstacles à remplir l’intérieur d’une variété avec une métrique complète de courbure négative. Dans cet article, nous montrons que toute variété à bord de dimension trois peut être remplie conformément avec une métrique complète satisfaisant une condition de pincement : on suppose que le rapport entre la plus grande courbure sectionnelle et la valeur absolue de la courbure scalaire est bornée par une constante (petite). Cette condition signifie que la courbure est “presque négative” dans un sens invariant d’échelle.
Given a three-dimensional manifold with boundary, the Cartan-Hadamard theorem implies that there are obstructions to filling the interior of the manifold with a complete metric of negative curvature. In this paper, we show that any three-dimensional manifold with boundary can be filled conformally with a complete metric satisfying a pinching condition: given any small constant, the ratio of the largest sectional curvature to (the absolute value of) the scalar curvature is less than this constant. This condition roughly means that the curvature is “almost negative”, in a scale-invariant sense.
@article{AIF_2010__60_7_2421_0,
author = {Gursky, Matthew and Streets, Jeffrey and Warren, Micah},
title = {Conformally bending three-manifolds with boundary},
journal = {Annales de l'Institut Fourier},
volume = {60},
year = {2010},
pages = {2421-2447},
doi = {10.5802/aif.2613},
zbl = {1239.53047},
mrnumber = {2849268},
language = {en},
url = {http://dml.mathdoc.fr/item/AIF_2010__60_7_2421_0}
}
Gursky, Matthew; Streets, Jeffrey; Warren, Micah. Conformally bending three-manifolds with boundary. Annales de l'Institut Fourier, Tome 60 (2010) pp. 2421-2447. doi : 10.5802/aif.2613. http://gdmltest.u-ga.fr/item/AIF_2010__60_7_2421_0/
[1] Complete conformal metrics with negative scalar curvature in compact Riemannian manifolds, Duke Math. J., Tome 56 (1988) no. 2, pp. 395-398 | Article | MR 932852 | Zbl 0645.53023
[2] Courbure presque négative en dimension , Compositio Math., Tome 63 (1987) no. 2, pp. 223-236 | Numdam | MR 906372 | Zbl 0642.53047
[3] On the almost negatively curved -sphere, Geometry and analysis on manifolds (Katata/Kyoto, 1987), Springer, Berlin (Lecture Notes in Math.) Tome 1339 (1988), pp. 78-85 | MR 961474 | Zbl 0651.53032
[4] Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math., Tome 35 (1982) no. 3, pp. 333-363 | Article | MR 649348 | Zbl 0469.35022
[5] Complete conformal metrics of negative Ricci curvature on manifolds with boundary (to appear in Calc. Var.)
[6] Boundedly inhomogeneous elliptic and parabolic equations in a domain, Izv. Akad. Nauk SSSR Ser. Mat., Tome 47 (1983) no. 1, pp. 75-108 | MR 688919 | Zbl 0578.35024
[7] Partial differential equations invariant under conformal or projective transformations, Contributions to analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York (1974), pp. 245-272 | MR 358078 | Zbl 0298.35018
[8] Negative bending of open manifolds, J. Differential Geom., Tome 40 (1994) no. 3, pp. 461-474 | MR 1305978 | Zbl 0840.53023