Conformally bending three-manifolds with boundary
Annales de l'Institut Fourier, Tome 60 (2010) no. 7, pp. 2421-2447.

Soit M 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.

DOI : 10.5802/aif.2613
Classification : 53C20, 35J65
Keywords: Almost negative curvature, conformal filling, fully nonlinear equations
Mots clés : courbure presque négative, géométrie conforme, EDP non linéaire
Gursky, Matthew 1 ; Streets, Jeffrey 2 ; Warren, Micah 2

1 University of Notre Dame Department of Mathematics Notre Dame, IN 46556 (USA)
2 Princeton University Fine Hall Princeton, NJ 08544 (USA)
@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},
     pages = {2421--2447},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {60},
     number = {7},
     year = {2010},
     doi = {10.5802/aif.2613},
     zbl = {1239.53047},
     mrnumber = {2849268},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2613/}
}
TY  - JOUR
AU  - Gursky, Matthew
AU  - Streets, Jeffrey
AU  - Warren, Micah
TI  - Conformally bending three-manifolds with boundary
JO  - Annales de l'Institut Fourier
PY  - 2010
SP  - 2421
EP  - 2447
VL  - 60
IS  - 7
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2613/
DO  - 10.5802/aif.2613
LA  - en
ID  - AIF_2010__60_7_2421_0
ER  - 
%0 Journal Article
%A Gursky, Matthew
%A Streets, Jeffrey
%A Warren, Micah
%T Conformally bending three-manifolds with boundary
%J Annales de l'Institut Fourier
%D 2010
%P 2421-2447
%V 60
%N 7
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.2613/
%R 10.5802/aif.2613
%G en
%F 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) no. 7, pp. 2421-2447. doi : 10.5802/aif.2613. http://archive.numdam.org/articles/10.5802/aif.2613/

[1] Aviles, Patricio; McOwen, Robert C. Complete conformal metrics with negative scalar curvature in compact Riemannian manifolds, Duke Math. J., Volume 56 (1988) no. 2, pp. 395-398 | DOI | MR | Zbl

[2] Bavard, C. Courbure presque négative en dimension 3, Compositio Math., Volume 63 (1987) no. 2, pp. 223-236 | Numdam | MR | Zbl

[3] Buser, Peter; Gromoll, Detlef On the almost negatively curved 3-sphere, Geometry and analysis on manifolds (Katata/Kyoto, 1987) (Lecture Notes in Math.), Volume 1339, Springer, Berlin, 1988, pp. 78-85 | MR | Zbl

[4] Evans, Lawrence C. Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math., Volume 35 (1982) no. 3, pp. 333-363 | DOI | MR | Zbl

[5] Gursky, Matthew; Streets, Jeffrey; Warren, Micah Complete conformal metrics of negative Ricci curvature on manifolds with boundary (to appear in Calc. Var.)

[6] Krylov, N. V. Boundedly inhomogeneous elliptic and parabolic equations in a domain, Izv. Akad. Nauk SSSR Ser. Mat., Volume 47 (1983) no. 1, pp. 75-108 | MR | Zbl

[7] Loewner, Charles; Nirenberg, Louis 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 | Zbl

[8] Lohkamp, Joachim Negative bending of open manifolds, J. Differential Geom., Volume 40 (1994) no. 3, pp. 461-474 | MR | Zbl

Cité par Sources :