In this talk, we define the torsion flow, a CR analogue of the Ricci flow. We first show that the torsion flow has short time existence for suitable initial conditions in a closed strictly pseudoconvex CR 3-manifold. Secondly, we give some examples for the long-time solution of the torsion flow. Finally, we derive monotonicity formulas for CR Perlman-type entropy functionals. As an application, we classify the torsion breathers. This is
  a jointed work with Otto van Koert and Chin-Tung Wu.