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

TL;DR

This paper investigates the minimal distortion bending problem for smooth compact manifolds, deriving conditions for optimal deformations, with explicit solutions for simple closed curves.

Contribution

It formulates a deformation energy functional for minimal distortion bending and finds smooth minimizers specifically for simple closed curves.

Findings

Derived the Euler-Lagrange equation for the deformation energy functional.

Identified smooth minimizers for the case of simple closed curves.

Provided a framework for minimal distortion bending of compact manifolds.

Abstract

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.