Regularity and expanding entropy for connection Ricci flow

We study a generalization of Ricci flow to connections with torsion. This system of equations in fact arises naturally in the study of nonlinear sigma models [T. Oliynyk, V. Suneeta, E. Woolgar, A gradient flow for worldsheet nonlinear sigma models, Nuclear Phys. B 739 (2006) 441–458]. The picture we give allows one to easily extend the proofs of derivative estimates and compactness of solutions to the case of a connection with torsion. We also examine gradient properties of this flow. Indeed it was shown in Oliynyk et al. (see the citation above) that the monotonicity of Perelman’s FF-functional extends to the case of a connection with torsion. We show that the expander entropy of Feldman, Ilmanen, and Ni [M. Feldman, T. Ilmanen, L. Ni, Entropy and reduced distance for Ricci expanders, arXiv:math/0405036] also extends to the connection Ricci flow.