  Motivated by problems on apparent horizons in general relativity, we prove
the following theorem on minimal surfaces: Let $g$ be a metric on the
three-sphere $S^3$ satisfying $Ric(g) \geq 2 g$. If the volume of $(S^3, g)$ is
no less than one half of the volume of the standard unit sphere, then there are
no closed minimal surfaces in the asymptotically flat manifold $(S^3 \setminus
\{P \}, G^4 g)$. Here $G$ is the Green's function of the conformal Laplacian of
$(S^3, g)$ at an arbitrary point $P$. We also give an example of $(S^3, g)$
with $Ric(g) > 0$ where $(S^3 \setminus \{P \}, G^4 g)$ does have closed
minimal surfaces.
