  In this paper we study the rigidity of infinite volume 3-manifolds with
sectional curvature $-b^2\le K\le -1$ and finitely generated fundamental group.
In-particular, we generalize the Sullivan's quasi-conformal rigidity for
finitely generated fundamental group with empty dissipative set to negative
variable curvature 3-manifolds. We also generalize the rigidity of
Hamenst\"{a}dt or more recently Besson-Courtois-Gallot, to 3-manifolds with
infinite volume and geometrically infinite fundamental group.
