The multivalued inverse of a quasiregular map is a quasiregular curve
Elefterios Soultanis
TL;DR
The paper constructs a natural multivalued inverse for a proper quasiregular map $f: \\Omega\\to \\mathbb{R}^n$ of finite degree $d$, taking values in Almgren's space $\\mathcal{A}_d(\\mathbb{R}^n)$. It proves that this inverse $\\accentset{\\leftarrow}{f}$ is an $\\omega$-quasiregular curve with $\\omega=\\operatorname{tr}(\\operatorname{vol}_{\\mathbb{R}^n})$ and belongs to $W_{loc}^{1,n}$, and that the induced map onto its image $\\Omega_f$ is geometrically quasiconformal; the image space $\\Omega_f$ is $n$-rectifiable and upper $n$-Ahlfors-regular and supports an infinitesimal $n$-Poincaré inequality. These results connect the theories of quasiregular maps, quasiregular curves, and metric-space quasiconformal theory, and are underpinned by a general pull-back theory for differential forms on $\\mathcal{A}_d(\\mathbb{R}^n)$ for multi-valued Sobolev maps. The framework also yields higher integrability results for the generalized inverse and offers potential new directions, including bi-Lipschitz parametrizations of $\\Omega_f$ via Almgren embedding $\\mathcal{A}_d(\\mathbb{R}^n)\\hookrightarrow \\mathbb{R}^N$.
Abstract
We observe that a proper quasiregular map $f:Ω\to \mathbb{R}^n$ of finite degree $d$ has a natural multivalued inverse $F:f(Ω)\to \mathcal{A}_d(\mathbb{R}^n)$ taking values in the space $\mathcal{A}_d(\mathbb{R}^n)$ of unordered $d$-tuples in $\mathbb{R}^n$, and show that the multivalued inverse is a quasiregular curve (in the sense of Pankka) with respect to a natural $n$-form on $\mathcal{A}_d(\mathbb{R}^n)$ (suitably interpreted). We moreover show that $F$ is a geometrically quasiconformal map onto its image $Ω_f=F(f(Ω))$, and that the Hausdorff $n$-measure on $Ω_f$ is upper Ahlfors regular. These results demonstrate a new connection between quasiregular maps, quasiregular curves, and quasiconformal theory in metric spaces. To establish them, we develop a theory of pull-backs of differential forms on $\mathcal{A}_d(\mathbb{R}^n)$ by multi-valued Sobolev maps, which may be of independent interest.