Ricci flow on contact manifolds

This paper is devoted to Ricci flow on contact manifolds. We define the contact curvature flow and establish a short time existence. Meanwhile, we study a contact Ricci soliton and prove that every solution of the unnormalized contact curvature flow is a selfsimilar solution corresponding to a contact Ricci soliton which is a steady soliton. Finally we show that a time dependent family of contact Einstein, Sasakian, $\mathrm K$-contact, or $\eta$-Einstein $1$-forms $\eta_t$ is a solution of the normalized contact curvature flow if it is a conformal variation of an initial $1$-form $\eta_0$.