  Given two-dimensional Riemannian manifolds $\mathcal{M},\mathcal{N}$, we
prove a lower bound on the distortion of embeddings $\mathcal{M} \to
\mathcal{N}$, in terms of the areas' discrepancy
$V_{\mathcal{N}}/V_{\mathcal{M}}$, for a certain class of distortion
functionals. For $V_{\mathcal{N}}/V_{\mathcal{M}} \ge 1/4$, homotheties,
provided they exist, are the unique energy minimizing maps attaining the bound,
while for $V_{\mathcal{N}}/V_{\mathcal{M}} \le 1/4$, there are non-homothetic
minimizers. We characterize the maps attaining the bound, and construct
explicit non-homothetic minimizers between disks. We then prove stability
results for the two regimes. We end by analyzing other families of distortion
functionals. In particular we characterize a family of functionals where no
phase transition in the minimizers occurs; homotheties are the energy
minimizers for all values of $V_{\mathcal{N}}/V_{\mathcal{M}}$, provided they
exist.
