REGULARITY OF THE BELTRAMI EQUATION AND 1-QUASICONFORMAL EMBEDDINGS OF SURFACES IN R3

Abstract. A striking result in quasiconformal mapping theory states that if D is a domain in Rn (with n ≥ 3) and f: D → Rn 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 re-flects part of our effort to extend this type of rigidity results to embeddings f: Rn → Rm from Rn into a higher dimensional space Rm (with m> n). In this paper we focus on smooth embeddings of planar domains into R3. In particular, we show that a C1+α-smooth surface is 1-QC equivalent to a planar domain. We also show that a topological sphere that is C1+α-diffeomorphic to the standard sphere S2 is also 1-QC equivalent to S2. Along the way, a regular-ity result is established for solutions of the Beltrami equation with degenerate coefficient, which is used in this paper and has its own interest. 1