  In hep-th/9910245, Witten and Yau consider the AdS/CFT correspondence in the
context of a Riemannian Einstein manifold $M^{n+1}$ of negative Ricci curvature
which admits a conformal compactification with conformal boundary $N^n$. They
prove that if the conformal class of the boundary contains a metric of positive
scalar curvature, then $M$ and $N$ have several desirable properties: (1) $N$
is connected, (2) the $n$th homology of the compactified $M$ vanishes, and (3)
the fundamental group of $M$ is "bounded by" that of $N$. Here it is shown that
all of these results extend to the case where the conformal class of the
boundary contains a metric of nonnegative scalar curvature. (The case of zero
scalar curvature is of interest as it is borderline for the stability of the
theory.) The proof method used here is different from, and in some sense dual
to, that used by Witten and Yau. While their method involves minimizing the
co-dimension one brane action on $M$, and requires the machinery of geometric
measure theory, the main arguments presented here use only geodesic geometry.
