  E. Calabi and J. Cao showed that a closed geodesic of least length in a
two-sphere with nonnegative curvature is always simple. Using min-max theory,
we prove that for some higher dimensions, this result holds without assumptions
on the curvature. More precisely, in a closed $(n+1)$-manifold with $2 \leq n
\leq 6$, a least area closed minimal hypersurface exists and any such
hypersurface is embedded. As an application, we give a short proof of the fact
that if a closed three-manifold $M$ has scalar curvature at least $6$ and is
not isometric to the round three-sphere, then $M$ contains an embedded closed
minimal surface of area less than $4\pi$. This confirms a conjecture of F. C.
Marques and A. Neves.
