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

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)

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 tesselation 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]).

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.

References:

• [B] Bers, Riemann surfaces;

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

• [R] Rivin, Optimization in combinatorial geometry;

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

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

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)

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]).

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.

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;