Euclidean surfaces with conical singularities and cusped hyperbolic surfaces

Question

Let $S$ be a compact orientable surface endowed with a singular euclidean metric $g$, with $n$ conical singularities $x_1,\ldots,x_n$.

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Construction 1: it is well-known that the conformal class $[g]$ of the metric makes of $S$ a Riemann surface, denoted by $X$. Then to the pair $(S,g)$, one can associate the $n$-punctured Riemann surface $(X,\{x_i\}_{ i=1}^n)$ (cf. [B] of [T] for a modern reference)

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There is another way to associate a punctured Riemann surface to $(S,g)$:

Construction 2: the Delaunay tesselation ${\cal D }(S,g)$ associated to $g$ is a particular polyhedral tessellation of $S$ by $g$-euclidean cyclic polygons with set of vertices equal to $V=\{x_1,\ldots,x_n\}$. It is canonically attached to $g$ hence makes of $S$ a euclidean polyhedral surface.

Let's assume (to simplify) that ${\cal D }(S,g)$ is a triangulation of $S$: any 2-face $T$ of ${\cal D }(S,g)$ is a euclidean triangle inscribed in a circle $C_T$. Considering the latter as the boundary of Klein's model of the hyperbolic disc, one can see $T$ as an ideal hyperbolic triangle. Gluing the 2-faces of ${\cal D}(S,g)$ with these ideal hyperbolic structures, one obtains a complete hyperbolic structure on $S\setminus \{x_i\}$ with cusps at the initial singularities $x_1,\ldots,x_n$. Let's denote by $X'$ this hyperbolic surface.

It is classical that $(X',\{x_i\}_{ i=1}^n)$ can be viewed as a $n$-punctured Riemann surface.

(this construction, using shear coordinates to glue the 2-faces of the Delaunay triangulation is due to Rivin; the equivalent construction described above using Klein's model of the hyperbolic disc is taken from [BPS]).

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In [R,Sect. 7], it is said that Construction 1 and Construction 2 lead to distinct Riemann surfaces, i.e. the two $n$-punctured Riemann surfaces $(X,\{x_i\})$ and $(X',\{x_i\})$ are distinct in general (a `remark' attributed to C.T. McMullen).

• Do I understand things properly?
• If yes, I would be interested by some details on this fact.
• Some (concrete?) examples would be welcome.
• Some references too (if any).

Thanks in advance for any help.

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References:

• [B] Bers, Riemann surfaces;

• [BPS] Bobenko, Pinkall & Springborn, Discrete conformal maps and ideal hyperbolic polyhedra;

• [R] Rivin, Combinatorial optimization in geometry;

• [T] Troyanov, Les surfaces euclidiennes à singularités coniques;

Answer 1

Your description is a little too abstract: it is easier to think of this geometrically [in a special case].

Fact (Rivin, 1991): Every complete finite area hyperbolic metric on a sphere with punctures can be realized in a unique way as the induced metric on a convex ideal polyhedron $P$ in $\mathbb{H}^3.$ Such a metric has a conformal class, but the sphere at infinity with the vertices of $P$ marked also defines a conformal class, and we thus have a map (which is easily seen to be a piecewise analytic homeomorphism) from the moduli space of such classes to itself. The obvious question (which is what your question is about) is: is this map the identity? Symmetric metrics (regular ideal simplex, equatorial square) are fixed points, but the map is not the identity. McMullen's observation was simply the following: look one dimension lower, and consider ideal quadrilaterals in $\mathbb{H}^2.$ Again, you have the four points on the circle at infinity, and their convex hull defining two conformal quadrilaterals, but as two of the vertices converge to (say) $-1$ and two others converge to (say) $1$ it is not hard to see that the moduli of the two quadrilaterals blow up at different rates (one is the log of the other), so the map is not the identity in that case, and since this is a section of the higher dimensional case (when the simplex is flat), the higher dimensional map is not the identity either.

The question of what the map is, exactly, is very interesting and completely open even for four-times-punctured spheres (although there is some related work, in completely different language, due to Zograf and Tachtajian).