Sharp distortion growth for bilipschitz extension of planar maps
  This note addresses the quantitative aspect of the bilipschitz extension problem. The main result states that any bilipschitz embedding of $\mathbb R$ into $\mathbb R^2$ can be extended to a bilipschitz self-map of $\mathbb R^2$ with a linear bound on the distortion. 
Leonid V. Kovalev
math.CV math.MG