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Consider a unit disk with marked points $z_i$, $i=1, \dots , n$ on its boundary. Let us call this surface $X$. As it is well known, the disk can be equipped with an hyperbolic metric and is then called Poincaré disk. Consider the convex geodesic hull of the points $z_i$, i.e. the subset of the disk surrounded by the geodesics $\gamma_i$ joining consecutive pair points $(z_i, z_i+1)$, together with the geodesics and the points $z_i$ themselves. Let us call this surface $Y$.

Is there a biholomorphic map $\phi : X \to Y$ such that the points $\phi(z_i) = z_i$? In other words, are the surfaces with marked points $(X; z_i)$ and $(Y; z_i)$ isomorphic as Riemann surfaces?

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The answer is yes, in the following cases: when $n=3$, and when the points are at the vertices of a regular polygon inscribed in the unit circle. The reasons are evident.

For generic points the answer is no when $n\geq 4$. However I cannot give a simple proof of this at the moment, except by a numerical computation, or by asymptotics. This kind of computations and asymptotics are performed for example in these papers (for 4 points):

Gaven J. Martin, Random Lattices, Punctured Tori and the Teichmüller distribution, arXiv:1807.11127,

A. Eremenko, On the hyperbolic metric of the complement of a rectangular lattice, arXiv:1110.2696.

By asymptotics I mean that the points are the vertices of a rectangle inscribed in the circle, and the aspect ratio of this rectangle tends to $0$.

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  • $\begingroup$ Interesting. Let me step back a little bit then. Clearly we can associate the surface Y to the surface X, is this procedure invertible in a unique way? That is, given a disk with punctures Y is there a unique choice of points on the boundary on the unit disk (up to biholomorphisms of the disk) such that Y arise as the convex hull of those points? $\endgroup$
    – giulio bullsaver
    Commented Jul 22, 2019 at 17:37
  • $\begingroup$ This map is certainly surjective, by topological reasons, but injectivity I don't know how to prove, though it is plausible. (If I unnderstood you correctly. You use some weird terminology, calling marking boundary points "punctures" etc.) $\endgroup$
    – Alexandre Eremenko
    Commented Jul 23, 2019 at 17:47
  • $\begingroup$ oh yes apologies. My interest is to study the moduli space of the disk with marked points (sometimes called punctures) using the Fenchel-Nielsen coordinates. I noted that I can compute explicitly these coordinates for Y (since I have its metric by restricting the Poincaré metric of the disk to Y) and I was wondering if they were those of X as well (of which I cannot compute explicitly the metric). I obtained a simple and nice formula expressing the coordinates of Y in terms of the points (punctures/marked points) of X. $\endgroup$
    – giulio bullsaver
    Commented Jul 24, 2019 at 5:13
  • $\begingroup$ The word "puncture" means a hole. Marked points INSIDE the Riemann surface are called punctures. But marked points on the boundary are not punctures. Anyway, the "topological reasons" mentioned in the previous comment is the following theorem: If you have a continuous map of a simplex into itself, sending each face (of any dimension) to itself, then this map is surjective. $\endgroup$
    – Alexandre Eremenko
    Commented Jul 24, 2019 at 5:47
  • $\begingroup$ Thank you for the notational clarification, I was not aware of it. Regarding the topological reasons, I am not sure I understood it. Are you describing the map from X to Y? In my first comment I meant a different thing: whether the map from the "space of surfaces X" to the "space of surfaces Y" is 1-1. Alternatively, can I think of every disk with marked points Y to be biholomorphic to the geodesical convex hull of some points on the unit disk? $\endgroup$
    – giulio bullsaver
    Commented Jul 24, 2019 at 7:30

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