  This short note is a mostly expository article examining negatively curved
three-manifolds. We look at some rigidity properties related to isometric
embeddings into Minkowski space. We also review the Cross Curvature Flow (XCF)
as a tool to study the space of negatively curved metrics on hyperbolic
three-manifolds, the largest and least understood class of model geometries in
Thurston's Geometrisation. The relationship between integrability and
embedability yields interesting insights, and we show that solutions with fixed
Einstein volume are precisely the integrable solutions, answering a question
posed by Chow and Hamilton when they introduced the XCF.
