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

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.

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.

Received:
Accepted:
DOI: 10.5802/aif.2613
Classification: 53C20,  35J65
Keywords: Almost negative curvature, conformal filling, fully nonlinear equations
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)
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Gursky, Matthew; Streets, Jeffrey; Warren, Micah. Conformally bending three-manifolds with boundary. Annales de l'Institut Fourier, Volume 60 (2010) no. 7, pp. 2421-2447. doi : 10.5802/aif.2613. https://aif.centre-mersenne.org/articles/10.5802/aif.2613/

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