Twisted entropy functional for three-manifolds and flows induced on the boundary

Considering a twisted version of the gravitational Chern–Simons action for three-manifolds as a Perelman entropy functional, a generalization of the Cotton flow for the metric, in co-evolution with torsion is developed. In the case of manifolds with boundary, there exists an entropy functional induced on it, and hence metric and torsional flows can be defined. The integrability of these boundary flows is studied in detail. In a particular case, the solutions of the two-dimensional logarithmic diffusion equation determine completely the dynamics of the fields in co-evolution. In this scheme of three-manifolds with evolving boundaries, the orthogonality of the twisted flows to the Yamabe-like flows is established; additionally, the evolution of the holonomy groups of the three-manifold and of its boundary is studied, showing in some cases an oscillatory behavior around orthogonal groups. In this approach, the twisted canonical geometries (TCG) are defined as manifolds with a vanishing contribution of the torsion to the curvature, which will be generated fully by the metric; these geometries evolve to manifolds with curvature generated by both metric and torsion, and cannot smoothly evolve into manifolds with a vanishing torsion.