Embedding surfaces inside small domains with minimal distortion

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