Quasiconformal Rigidity of Negatively Curved Three Manifolds

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.