Assume that $\gamma$ is a rectifiable Jordan curve in the complex plane of length $2\pi$. Then there exists a Riemann isometry $f$ between $\gamma$ and the unit circle $T$. My question is, does this isometry provides the minimal constant of bi-Lipschitz mappings (w.r.t. euclidean metric) between $\gamma$ and $T$ (provided that the last set is non-empty).
The formulation of the question could be better; I had to guess what could you had in mind and likely I made it wrong.
Imagine a curve which pass very close to itself say at the points $A$ and $B$, but both arcs from $A$ to $B$ are nearly round and both have length near $\pi$. If you map it isometrically, the distortion is very bad for $A$ and $B$ and you can decrease distortion by shrinking the image of one of the arcs.