Recent insights on the Uniqueness Problem of Diffeomorphisms determined by Prescribed Jacobian Determinant and Curl

TL;DR

This paper investigates whether the Jacobian determinant and curl uniquely determine a diffeomorphism, proposing a conjecture, experimental testing strategies, and partial theoretical proof related to the problem.

Contribution

It introduces a conjecture on the uniqueness of diffeomorphisms determined by Jacobian and curl, and provides a partial proof and experimental approach to test it.

Findings

Counterexamples, if any, must satisfy specific properties.

An experimental strategy to test the conjecture is formulated.

A partial proof shows near-identity diffeomorphisms are unique.

Abstract

Variational Principle (VP) forms diffeomorphisms with prescribed Jacobian determinant (JD) and curl. Examples demonstrate that, (i) JD alone can not uniquely determine a diffeomorphism without curl; and (ii) the solutions by VP seem to satisfy properties of a Lie group. Hence, it is conjectured that a unique diffeomorphism can be assured by its JD and curl (Uniqueness Conjecture). In this paper, (1) an observation based on VP is derived that a counter example to the Conjecture, if exists, should satisfy a particular property; (2) from the observation, an experimental strategy is formulated to numerically test whether a given diffeomorphism is a valid counter example to the conjecture; (3) a proof of an intermediate step to the conjecture is provided and referred to as the semi-general case, which argues that, given two diffeomorphisms, $\pmb{\phi}$ and $\pmb{\psi}$, if they are close to the identity map, $\pmb{id}$, then $\pmb{\phi}$ is identical $\pmb{\psi}$.