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 [1] if $n\ge 5$ and
to Guilbault [2] if $n=4$, while the case $n=3$ follows
from the result of Husch and Price [3] and the non-existence of fake $3$-cells
by Perelman's solution of the Poincaré conjecture.

[1] J.~Stallings, *The piecewise-linear structure of Euclidean space*, Proc. Cambridge Philos. Soc. **58** (1962), 481--488

[2] C.~R. Guilbault, *An open collar theorem for $4$-manifolds*, Trans. Amer. Math. Soc. **331** (1992), no.~1, 227--245.

[3] 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 [3] 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 [3] 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.

The piecewise-linear structure of Euclidean space. $\endgroup$