Suppose that we have an $L$ bi-Lipschitz function $f:S\to U$, where $S$ is the square $S:=\{(x,y):|x|\leq 1,|y|\leq 1\}$ and $U$ is the upper hemisphere $U:=\{(x,y,z):x^2+y^2+z^2=1,z\geq 0\}$.
Questions:
(1) Is there a way to calculate the smallest bi-Lipschitz constant for which there exists such a function?
(2) More generally is there some condition (necessary and/or sufficient) we should impose on a compact surface $U$ so that there is a bi-Lipschitz map form $S$ to $U$?
(3) Can we say anything about the smallest bi-Lipschitz constant in this case assuming that we know the geometry of the surface?
Any references about where to learn more on bi-Lipschitz maps between metric spaces are welcome.