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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.

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  • $\begingroup$ There exists an (almost) canonical metric on a Riemann surface: it is of constant curvature $1,-1$ or $0$. It is unique except the case of zero curvture when it is unique up to a non-zero constant multiple. $\endgroup$ Sep 8, 2019 at 13:27
  • $\begingroup$ The absolute value used by Jost is just a bad notation. What he really means here is the open metric ball $B(z_0,R_0)=\{z\in \Sigma_1: d(z, z_0)<R_0\}$, where $d$ is the Riemannian distance function induced by the given background (complete) Riemannian metric on $\Sigma_1$. $\endgroup$
    – Misha
    Sep 8, 2019 at 14:03
  • $\begingroup$ @Misha But in that case I am even more confused, because the following paragraphs all make use of the preceding lemmas, which are done only in the case of a domain in $\mathbb C$. If what you say is indeed what he means, I don't think those lemmas can be applied without any justification. $\endgroup$
    – trisct
    Sep 8, 2019 at 14:06
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    $\begingroup$ Another way to interpret what he says is to notice that Jost (again, sloppily) identifies coordinate neighborhoods on a surface with their images in ${\mathbb C}$. Then you can think of the absolute value as the one in the complex plane. (This gives you a collection of locally defined metrics on the surface.) When restricted to compactly contained sub-neighborhoods, the background Riemannian metric I mentioned, is Lipschitz-equivalent to the metric defined by the absolute value on the complex plane. Being local, the definition of Lipschitz notions does not depend on which approach you follow. $\endgroup$
    – Misha
    Sep 8, 2019 at 14:19
  • $\begingroup$ @Misha This actually makes sense. Thanks very much. And can you kindly point me to some reference that makes this precise? $\endgroup$
    – trisct
    Sep 8, 2019 at 14:22

1 Answer 1

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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).

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  • $\begingroup$ Does this mean in practice, we can choose any metric on $\Sigma_1$ to define the class of Lipschitz functions? If so, what's with the "absolute value", which does not seem to define a metric on the whole surface. And is there any other reference that makes this precise? Thanks a lot! $\endgroup$
    – trisct
    Sep 8, 2019 at 7:36

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