## A discrete uniformization theorem for polyhedral surfaces.(English)Zbl 1396.30008

Let $$S$$ be a closed orientable surface and let $$V \subset S$$ be a non-empty finite set. A PL-metric $$d$$ on $$(S,V)$$ is a flat metric on $$S$$ whose cone points are contained in $$V$$. The discrete curvature of $$d$$ is the map $$K:V \to (-\infty, 2\pi)$$ defined as $$K(v)=2\pi-$$ cone angle at $$v$$. Examples of this situation are coming from the gluing of a finite collection of Euclidean triangles or by holomorphic quadratic forms on Riemann surfaces. In the paper under review, the notion for a PL-metric $$d'$$ on $$(S,V)$$ to be conformal to $$d$$ is provided, this involving the concept of Delaunay triangulations. Then main result is, for given $$K^{*}:V \to (-\infty, 2\pi)$$, such that $$\sum_{v \in V} K^{*}(v)=2 \pi \chi(S)$$, there exists a discrete conformal metric $$d'$$ whose discrete curvature is $$K^{*}$$. It is also shown that such a $$d'$$ can be obtained using a finite dimensional variational principle. The particular constant case $$K^{*}(v)=2 \pi \chi(S)/|V|$$ is a discrete version of the uniformization theorem.

### MSC:

 30F10 Compact Riemann surfaces and uniformization 30G25 Discrete analytic functions 52B70 Polyhedral manifolds 57Q15 Triangulating manifolds 58E30 Variational principles in infinite-dimensional spaces
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