Deformation Minimal Bending of Compact Manifolds: Case of Simple Closed Curves

The problem of minimal distortion bending of smooth compact embedded connected Riemannian $n$-manifolds $M$ and $N$ without boundary is made precise by defining a deformation energy functional $\Phi$ on the set of diffeomorphisms $\diff(M,N)$. We derive the Euler-Lagrange equation for $\Phi$ and determine smooth minimizers of $\Phi$ in case $M$ and $N$ are simple closed curves.