Asaf Shachar - Non-Euclidean elasticity: Embedding surfaces with minimal distortion

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

The minimizing maps attaining this bound go through a phase transition when the ratio of areas is 1/41/41/4: The homotheties are the unique energy minimizers when the ratio Vol⁡(N)Vol⁡(M)≥1/4\frac{\operatorname{Vol}(N)}{\operatorname{Vol}(M)} \ge 1/4Vol(M)Vol(N)​≥1/4, and they cease being minimizers when Vol⁡(N)Vol⁡(M)\frac{\operatorname{Vol}(N)}{\operatorname{Vol}(M)} Vol(M)Vol(N)​ gets below 1/41/41/4.

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