$C_0$-positivity and a classification of closed three-dimensional CR torsion solitons

A closed CR 3-manifold is said to have $C_{0}$-positive pseudohermitian curvature if $(W+C_{0}Tor)(X,X)>0$ for any $0\neq X\in T_{1,0}(M)$. We discover an obstruction for a closed CR 3-manifold to possess $C_{0}$-positive pseudohermitian curvature. We classify closed three-dimensional CR Yamabe solitons according to $C_{0}$-positivity and $C_{0}$-negativity whenever $C_{0}=1$ and the potential function lies in the kernel of Paneitz operator. Moreover, we show that any closed three-dimensional CR torsion soliton must be the standard Sasakian space form. At last, we discuss the persistence of $C_{0}$-positivity along the CR torsion flow starting from a pseudo-Einstein contact form.