This appears in the section 3.7 of the book *Compact Riemann Surfaces* by Jurgen Jost, right after Lemma 3.7.3. The exact words are



> 
> Now let $v:\Sigma\_1\to\Sigma\_2$ be a Lipschitz continuous map. Cover $\Sigma\_1$ by coordinate neighborhoods. Choose $R\_0<1$ so that, for every $z\_0\in\Sigma\_1$, a disc of the form
>  $$B(z\_0,R\_0)=\{z:|z-z\_0|<R\_0\}$$
>  is contained inside a coordinate neighborhood.
> 
> 
> 


Note that the only mention of $\Sigma\_1$ before the cited paragraph assumes $\Sigma\_1$ to be a compact surfaces, and judging from how these arguments are to be used later, $\Sigma\_1$ is undoubtedly a compact Riemann surface (with no given metric). 


My questions are:


(1) What is a Lipschitz continuous map from a surface in this context?


(2) What is the absolute value used in the representation of $B(z\_0,R\_0)$?


(3) It seem that the author is trying to identify (locally) $\Sigma\_1$ with its local coordinates. However, if we do so, two problems arise: first, since we can multiply a chart by a constant, we can modify the distance of two points (covered by the same chart, distance given by the absolute value) to any distance we want, and the Lipschitz bound is not well-defined; second, for the same reason above, there's no need to define $R\_0$.


I have searched this section over and again for what the absolute value on $\Sigma\_1$ is but I can't come up with a plausible explanation.


Any help is greatly appreciated.



What is a Lipschitz continuous map between Riemann surfaces in Jost's book Compact Riemann Surfaces?
While changing the Riemannian metric will change the constants, as you observe, on a compact manifold it will not change the class of Lipschitz continuous function, defined as usual as the set of functions that do not increase distances between points by more than a constant factor. This is a common regularity hypothesis, for example allowing to use the Rademacher theorem (Lipschitz continuous functions are differentiable almost everywhere). This class is to be thought of as a slight relaxation of $C^1$, and makes exactly as much sense even without specifying a metric. Another way to see this is to observe that smooth changes of coordinates preserve the Lipschitz modulus of continuity (up to a multiplicative constant, again).


