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.
Keywords: Almost negative curvature, conformal filling, fully nonlinear equations
Mots clés : courbure presque négative, géométrie conforme, EDP non linéaire
@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://www.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://www.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://www.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://www.numdam.org/articles/10.5802/aif.2613/
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