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

Negatively Curved Three-Manifolds, Hyperbolic Metrics, Isometric Embeddings In Minkowski Space And The Cross Curvature Flow