This is closed related to the question asked here. I wonder if there is any progress on Problem 13 from the "Problem Section" in Schoen and Yau, page 281, problem 13, which asks: Let $M_1$ and $M_2$ be compact Einstein manifolds with negative curvature. Suppose that $\pi_1(M_1)=\pi_1(M_2)$ and $\dim M_1\geq 3$. Is $M_1$ isometric to $M_2$?