$C_0$-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 $0\mathrm{\ne }X\in T_{1,0}(M)$. 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.