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

Asaf Shachar (The Hebrew University of Jerusalem)
Non-Euclidean elasticity: Embedding surfaces with minimal distortion

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 $M,N$.

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
$\frac{\operatorname{Vol}(N)}{\operatorname{Vol}(M)}$ less than $1/4$ when $M,N$
are disks, and give a proof sketch of the lower bound. If time
permits, I will discuss the stability of minimizers.