Deformation minimal bending of compact manifolds: case of simple closed curves

Abstract

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