  Suppose that $N$ is a smooth manifold with a smooth Riemannian metric $g_0$,
and that $\Gamma$ is a smooth submanifold of $N$. This paper proves that for a
generic (in the sense of Baire category) smooth metric $g$ conformal to $g_0$,
if $F$ is any simple $g$-minimal immersion of a closed manifold into N, then
$F$ is transverse to $\Gamma$ and $F$ is self-transverse. The theorem remains
true with "transverse" and "self-transverse" replaced by "strongly transverse"
and "strongly self-transverse". The theorem also holds for hypersurfaces of
constant mean curvature or, more generally, of prescribed mean curvature.
  The paper also proves that for a generic ambient metric, every
$2$-dimensional surface (integral current or flat chain mod $2$) without
boundary that minimizes area in its homology class has support equal to a
smoothly embedded minimal surface.
