Abstract:
Given two dimensional Riemannian manifolds $M,N$, I will present a sharp lower bound on the elastic energy (distortion) of embeddings $f:M \to N$, in terms of the areas' discrepancy of $M,N$.

We shall see that the minimizing maps attaining this bound go through a phase transition when the ratio of areas is $1/4$: The homotheties are the unique energy minimizers when the ratio $\frac{\operatorname{Vol}(N)}{\operatorname{Vol}(M)} \ge 1/4$, and they cease being minimizers when $\frac{\operatorname{Vol}(N)}{\operatorname{Vol}(M)} $ gets below $1/4$.

I will describe explicit minimizers in the non-trivial regime $\frac{\operatorname{Vol}(N)}{\operatorname{Vol}(M)} < 1/4$ when $M,N$ are disks, and give a proof sketch of the lower bound. If time permits, I will discuss the stability of minimizers.