I am not sure what Schoen and Yau had in mind. Any open contractible $n$-manifold that is simply-connected at infinity is homeomorphic to $\mathbb R^n$. This is due to Stallings  if $n\ge 5$ and
to Guilbault  if $n=4$, while the case $n=3$ follows
from the result of Husch and Price  and the non-existence of fake $3$-cells
by Perelman's solution of the Poincaré conjecture.
 J.~Stallings, The piecewise-linear structure of Euclidean space, Proc. Cambridge Philos. Soc. 58 (1962), 481--488
 C.~R. Guilbault, An open collar theorem for $4$-manifolds, Trans. Amer. Math. Soc. 331 (1992), no.~1, 227--245.
 L.~S. Husch and T.~M. Price, Finding a boundary for a $3$-manifold, Ann. of Math. 91 (1970), 223--235.
Since the question is about dimension $3$ let me give more details. By Perelman there is no fake $3$-cells. Now Theorem 2 of  implies that any contractible $3$-manifold that is simply-connected at infinity is homeomorphic to the interior of a compact $3$-manifold $N$ (in fact the result in  gives a general criterion of when one can attach a boundary to a $3$-manifold). Now $N$ is a contractible $3$-manifold whose boundary is a $2$-sphere, and again invoking Perelman we conclude that $N$ is a $3$-disk, up to homeomorphism. Finally, by Moise homeomorphic $3$-manifolds are diffeomorphic.