Regularity of the Beltrami equation and $1$-quasiconformal embeddings of surfaces in $\mathbb{R}^{3}$

A striking result in quasiconformal mapping theory states that if $D$ is a domain in $\mathbb{R} ^{n}$ (with $n\geq 3$) and $f: D\rightarrow \mathbb{R} ^{n}$ an embedding, then $f$ is $1$-QC if and only if $f$ is a Möbius transformation. This result has profound impact in quasiconformal analysis and differential geometry. This project reflects part of our effort to extend this type of rigidity results to embeddings $f: \mathbb{R} ^{n}\rightarrow \mathbb{R} ^{m}$ from $\mathbb{R} ^{n}$ into a higher dimensional space $\mathbb{R} ^{m}$ (with $m>n$). In this paper we focus on smooth embeddings of planar domains into $\mathbb{R} ^{3}$. In particular, we show that a $C^{1+\alpha}$-smooth surface is $1$-QC equivalent to a planar domain. We also show that a topological sphere that is $C^{1+\alpha }$-diffeomorphic to the standard sphere $\mathbb{S}^{2}$ is also $1$-QC equivalent to $\mathbb{S}^{2}$. Along the way, a regularity result is established for solutions of the Beltrami equation with degenerate coefficient, which is used in this paper and has its own interest.