  We explore recent progress and open questions concerning local minima and
saddle points of the Cahn--Hilliard energy in $d\geq 2$ and the critical
parameter regime of large system size and mean value close to $-1$. We employ
the String Method of E, Ren, and Vanden-Eijnden -- a numerical algorithm for
computing transition pathways in complex systems -- in $d=2$ to gain additional
insight into the properties of the minima and saddle point. Motivated by the
numerical observations, we adapt a method of Caffarelli and Spruck to study
convexity of level sets in $d\geq 2$.
