Suppose you have a riemannian surface $(\Sigma,g)$, and an open simply-connected set $U \subset \Sigma$. You know that you can find isothermal coordinates - that is a map $\varphi : U \rightarrow D$ ($D \subset \mathbb{C}$ stands for the unit disk) for which the metric reads $g = e^{2u(z)} |dz|^2$. $(U,g)$ is then isometric to $(D,d_g)$.

The question is the following : can one compare the distance $d_g$ on $D$, and the Euclidean distance $d_{euc}$ ? By comparing the two distances, I mean one of the following two inequalities : $C_1 {|z-z'|}^{C_2} \leq d_g(z,z') \leq C_3 {|z-z'|}^{C_4}$. The answer is "no" in general (with no difficulty you can find a function $u$ on the disc $D$ such that no such bound is possible).

So a more precise question is : given the open domain $U \subset \Sigma$, can one find a map $\varphi$ such that there exists such a bound ?

So the question can be thought as "can we find isothermal coordinates in which we control the metric" (in the sense stated above). Such things are classical (in any dimension) with harmonic coordinates : if one controls the curvature (in some sense...) and the injectivity radius, one can find a nice (harmonic) coordinate system in which the metric is controlled (in some sense again...). I'm wondering if this kind of things can exists for isothermal coordinates.


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You notice there is no comparaison in general. We can think to a shrinking sphere for instance. But if you assume that the distance between to point stay fixed, after some rescaling for instance. Then it suffies to control the L^2 norm of the second fondamental form if your surface is immersed. See theorem I.1 of https://people.math.ethz.ch/~riviere/papers/willmoreglobal.pdf


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