Almost everywhere conformal maps

Question

Let $d\geq 2$. A map $f:\mathbb{R}^d\to\mathbb{R}^d$ is called conformal at a point $x\in\mathbb{R}^d$ if $Df(x)=\lambda U$, where $\lambda>0$ and $U$ is an orthogonal map (that definition includes orientation reversing maps which to exclude we have to assume that $U$ has positive determinant). Holomorphic maps in the complex plane are examples of maps that are conformal everywhere (except at their critical points). In higher dimensions now Liouville's theorem says that a map $f:G\to\mathbb{R}^d$ that is conformal in a domain $G\subset \mathbb{R}^d$ is equal with a Mobius transformation on that domain.

My question is: Are there any examples of maps defined in some domain in $\mathbb{R}^d$, $d> 2$ that are conformal Lebesgue almost everywhere in the domain of definition but are not Mobius transformations? Is that even possible?

Answer 1

The answer is yes. See Theorem 5.2 in this paper

Answer 2

I will try here to give an answer to my question in case anyone has the same question as me in the future.

So a conformal map $f:G\to\mathbb{R}^d$, $G$ domain, is one that satisfies the equation \begin{equation}\tag{1}Df^T(x)Df(x)=J_f I_d ,\end{equation} where $J_f$ is the jacobian and $I_d$ the identity matrix (this is equivalent with the definition I gave in the OP). In fact we can look for weak solutions to that PDE so our map does not even have to be differentiable. So assume that our map belongs to some Sobolev space $W^{1,p}_{loc}$. Here is what Liouville's theorem roughly says now:

Any map that satisfies equation (1) a.e. on some domain $G$ and belongs to some Sobolev space $W^{1,p}_{loc}(G)$ with $p\geq d/2$ in case $d$ is even and $p\geq d-\epsilon$ (some $\epsilon$ whose precise value is still not known) in case $d$ is odd, is in fact a Mobius transformation on $G$.

On the other hand, according to that paper that Malik pointed out there are examples of maps belonging in some Sobolev space $W^{1,p}_{loc}(G)$ with $p<d/2$ that satisfy equation (1) a.e. (so they are conformal a.e.) but are not Mobius transformations in $G$. So those maps show that the answer to my question is yes!