  We introduce a geometric evolution equation for 3-manifolds with sectional
curvature of one sign which is in some sense dual to the Ricci flow. On a
closed 3-manifold with negative sectional curvature, we establish short time
existence and a pair of monotonicity formulas for solutions to the flow. One of
these formulas shows that, provided the solution exists for all time, the
metric approaches hyperbolic in an integral sense. Long time existence is still
an open problem.

Differential Geometry;Geometric Topology