  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.
