# Deformations of Bi-conformal Energy and a new Characterization of Quasiconformality

## TL;DR

This paper extends quasiconformal mappings to hyperelastic deformations with bi-conformal energy, introduces a new characterization of quasiconformality based on modulus of continuity, and explores the differences in continuity behavior between these deformations and classical quasiconformal maps.

## Contribution

It develops the concept of hyperelastic deformations of bi-conformal energy, providing a new way to characterize quasiconformality and analyzing their continuity properties.

## Key findings

- Quasiconformal maps exhibit a gain/loss property in their modulus of continuity between a map and its inverse.
- Deformations of bi-conformal energy can have the same optimal modulus of continuity as their inverse, unlike quasiconformal maps.
- Examples suggest potential applications in nonlinear elasticity and materials science.

## Abstract

The concept of hyperelastic deformations of bi-conformal energy is developed as an extension of quasiconformality. These are homeomorphisms $h:X \to Y$ between domains $ X, Y \subset \mathbb R^n$ of the Sobolev class $W^{1,n}_{loc} (X, Y)$ whose inverse $f =h^{-1}:Y \to X$ also belongs to $W^{1,n}_{loc}(Y, X)$. Thus the paper opens new topics in Geometric Function Theory with connections to mathematical models of Nonlinear Elasticity. In seeking differences and similarities with quasiconformal mappings we examine closely the modulus of continuity of deformations of bi-conformal energy. This leads us to a new characterization of quasiconformality. Specifically, it is observed that quasiconformal mappings behave locally at every point like radial stretchings. Without going into detail, if a quasiconformal map $h$ admits a function $\phi$ as its optimal modulus of continuity at a point $x_0$, then $f = h^{-1}$ admits the inverse function $\psi = \phi^{-1}$ as its modulus of continuity at $y_0 = h(x_0)$. That is to say; a poor continuity of $h$ at a given point $x_0$ is always compensated by a better continuity of $f$ at $y_0$, and vice versa. Such a gain/loss property, seemingly overlooked by many authors, is actually characteristic of quasiconformal mappings. It turns out that the elastic deformations of bi-conformal energy are very different in this respect. Unexpectedly, such a map may have the same optimal modulus of continuity as its inverse deformation. In line with Hooke's Law, when trying to restore the original shape of the body (by the inverse transformation) the modulus of continuity may neither be improved nor become worse. However, examples to confirm this phenomenon are far from being obvious. We eventually hope that our examples will gain an interest in the materials science, particularly in mathematical models of hyperelasticity.