  The Cahn-Hilliard energy landscape on the torus is explored in the critical
regime of large system size and mean value close to $-1$. Existence and
properties of a "droplet-shaped" local energy minimizer are established. A
standard mountain pass argument leads to the existence of a saddle point whose
energy is equal to the energy barrier, for which a quantitative bound is
deduced. In addition, finer properties of the local minimizer and appropriately
defined constrained minimizers are deduced. The proofs employ the
$\Gamma$-limit (identified in a previous work), quantitative isoperimetric
inequalities, variational arguments, and Steiner symmetrization.
