REGULARITY OF THE BELTRAMI EQUATION AND 1-QUASICONFORMAL EMBEDDINGS OF SURFACES IN R-3

A striking result in quasiconformal mapping theory states that if D is a domain in R-n (with n >= 3) and f : D -> R-n an embedding, then f is 1-QC if and only if f is a M <spacing diaeresis> obius 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 : R-n -> R-m from R-n into a higher dimensional space R-m (with m > n). In this paper we focus on smooth embeddings of planar domains into R-3. In particular, we show that a C1+alpha-smooth surface is 1-QC equivalent to a planar domain. We also show that a topological sphere that is C1+alpha-diffeomorphic to the standard sphere S-2 is also 1-QC equivalent to 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.