Exponential mixing of frame flows for three dimensional manifolds of quarter-pinched negative curvature

For a compact three-dimensional smooth Riemannian manifold of strictly 1/4-pinched negative sectional curvature, we establish exponential mixing of the frame flow with respect to the normalized volume. More generally this result extends to a class of torus extensions of Anosov flows, subject to assumptions on the Brin transitivity group and the smoothness of the stable subbundle. Our approach is based on a simplified dynamical model for studying the extension flow, constructed via a Young tower of the underlying Anosov flow. Exponential mixing is then obtained through a strengthened Dolgopyat type estimate on the corresponding transfer operators.