  We define a new geometric flow, which we shall call the $K$-flow, on
3-dimensional Riemannian manifolds; and study the behavior of Thurston's model
geometries under this flow both analytically and numerically. As an example, we
show that an initially arbitrarily deformed homogeneous 3-sphere flows into a
round 3-sphere and shrinks to a point in the unnormalized flow; or stays as a
round 3-sphere in the volume normalized flow. The $K$-flow equation arises as
the gradient flow of a specific purely quadratic action functional that has
appeared as the quadratic part of New Massive Gravity in physics; and a decade
earlier in the mathematics literature, as a new variational characterization of
three-dimensional space forms. We show the short-time existence of the $K$-flow
using a DeTurck-type argument.
