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**A characterization of spherical polyhedral surfaces.**
*(English)*
Zbl 1127.52032

A spherical angle structure on a closed triangulated surface \(S\) assigns angles \(x(e_i,f)\in(0,\pi)\) to the edges \(e_i\) of triangles \(f\) in the triangulation \(T\), in such a way that \(f\) with these inner angles may be realized as a spherical triangle. The edge invariant \(D\) of the spherical angle structure maps each edge \(e\in T\) to the sum \(D_x(e)=x(e,f)+x(e,f')\) of angles in the incident triangles.

Given a function \(D:E\to(0,\pi)\) defined on the set \(E\) of edges of a triangulated closed surface, a theorem by R. Guo [Proc. Am. Math. Soc. 135, No. 9, 3005–3011 (2007; Zbl 1129.57020)] provides the following set of necessary and sufficient linear inequalities for the existence of a spherical angle structure having \(D\) as an edge invariant: \[ \pi | X|< \sum_{e\in E(X)} D(e) \] for all subsets \(X\) of triangles in the triangulation; each sum runs over the set \(E(X)\) of edges in \(X\).

In the present paper, the author proves that if \(D\) satisfies these conditions, then there exists a unique spherical polyhedron metric \(l\) on \(S\), that is, a map \(l:E\to(0,\pi)\) such that the images under \(l\) of the three edges of each face in the triangulation can occur as edge lengths of a spherical triangle; this fixes the surface \(S\) up to isometry.

The main proof technique is to define the capacity function of triangles in the triangulation, and to show that it is strictly convex on the space of all spherical angle structures, with critical points exactly the spherical polyhedron metrics. (The bulk of the work is devoted to the analysis of degenerate cases.) Since strictly convex functions have at most one critical point, the theorem follows.

The capacity of a triangle has a straightforward geometric interpretation, related to volumes in hyperbolic space: any spherical, hyperbolic or Euclidean triangle in the boundary sphere of hyperbolic three-space is bounded by three circles. Its capacity is then, up to multiplicative or additive constants, the hyperbolic volume of the convex hull of the intersection points of these three circles.

Given a function \(D:E\to(0,\pi)\) defined on the set \(E\) of edges of a triangulated closed surface, a theorem by R. Guo [Proc. Am. Math. Soc. 135, No. 9, 3005–3011 (2007; Zbl 1129.57020)] provides the following set of necessary and sufficient linear inequalities for the existence of a spherical angle structure having \(D\) as an edge invariant: \[ \pi | X|< \sum_{e\in E(X)} D(e) \] for all subsets \(X\) of triangles in the triangulation; each sum runs over the set \(E(X)\) of edges in \(X\).

In the present paper, the author proves that if \(D\) satisfies these conditions, then there exists a unique spherical polyhedron metric \(l\) on \(S\), that is, a map \(l:E\to(0,\pi)\) such that the images under \(l\) of the three edges of each face in the triangulation can occur as edge lengths of a spherical triangle; this fixes the surface \(S\) up to isometry.

The main proof technique is to define the capacity function of triangles in the triangulation, and to show that it is strictly convex on the space of all spherical angle structures, with critical points exactly the spherical polyhedron metrics. (The bulk of the work is devoted to the analysis of degenerate cases.) Since strictly convex functions have at most one critical point, the theorem follows.

The capacity of a triangle has a straightforward geometric interpretation, related to volumes in hyperbolic space: any spherical, hyperbolic or Euclidean triangle in the boundary sphere of hyperbolic three-space is bounded by three circles. Its capacity is then, up to multiplicative or additive constants, the hyperbolic volume of the convex hull of the intersection points of these three circles.

Reviewer: Julian Pfeifle (Barcelona)

### MSC:

52C26 | Circle packings and discrete conformal geometry |

30F10 | Compact Riemann surfaces and uniformization |

57M50 | General geometric structures on low-dimensional manifolds |