CONFORMALLY BENDING THREE-MANIFOLDS WITH BOUNDARY

Given a three-dimensional manifold with boundary, the Caftan-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.