Minimal Distortion Morphs Generated by Time-Dependent Vector Fields

TL;DR

This paper introduces measures for the distortion caused by morphs between Riemannian manifolds, proves the existence of minimal distortion morphs generated by time-dependent vector fields, and establishes the existence of energy minimizers within this framework.

Contribution

It defines distortion energies for morphs generated by time-dependent vector fields and proves the existence of minimal distortion morphs and energy minimizers.

Findings

Existence of minimal distortion morphs between manifolds.

Definition of bending and morphing distortion energies.

Existence of energy minimizers in Sobolev space of vector fields.

Abstract

A morph between two Riemannian $n$-manifolds is an isotopy between them together with the set of all intermediate manifolds equipped with Riemannian metrics. We propose measures of the distortion produced by some classes of morphs and diffeomorphisms between two isotopic Riemannian $n$-manifolds and, with respect to these classes, prove the existence of minimal distortion morphs and diffeomorphisms. In particular, we consider the class of time-dependent vector fields (on an open subset $\Omega$ of $ \R^{n+1}$ in which the manifolds are embedded) that generate morphs between two manifolds $M$ and $N$ via an evolution equation, define the bending and the morphing distortion energies for these morphs, and prove the existence of minimizers of the corresponding functionals in the set of time-dependent vector fields that generate morphs between $M$ and $N$ and are $L^2$ functions from $[0,1]$ to the Sobolev space $W^{k,2}_0(\Omega,\R^{n+1})$.