Negatively pinched $3$-manifolds admit hyperbolic metrics

We show that any compact 3-manifold carrying a metric with sufficiently pinched negative Ricci curvature admits a hyperbolic metric. This proof is a corrected version of the proof first suggested by Maung Min-Oo. The key insight in this new proof is that the error in Min-Oo's paper does not occur if the type $(4,0)$ curvature is considered instead of the type $(3,1)$ curvature.