Minimal Distortion Bending and Morphing of Compact Manifolds

TL;DR

This paper investigates minimal distortion transformations between compact smooth manifolds embedded in Euclidean space, introducing new cost functionals, existence results, and a review of minimal distortion morphing theory.

Contribution

It defines new distortion-based cost functionals, proves existence of their minima, and reviews the theory of minimal distortion morphing for compact manifolds.

Findings

Existence of minima for new distortion cost functionals

Definition and analysis of morphs between manifolds

Review of minimal distortion morphing theory

Abstract

Let $M$ and $N$ be compact smooth oriented Riemannian $n$-manifolds without boundary embedded in $\mathbb{R}^{n+1}$. Several problems about minimal distortion bending and morphing of $M$ to $N$ are posed. Cost functionals that measure distortion due to stretching or bending produced by a diffeomorphism $h:M \to N$ are defined, and new results on the existence of minima of these cost functionals are presented. In addition, the definition of a morph between two manifolds $M$ and $N$ is given, and the theory of minimal distortion morphing of compact manifolds is reviewed.