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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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