Boundary value problems on planar graphs and flat surfaces with integer cone singularities, I: The Dirichlet problem. (English) Zbl 1296.53077
J. Reine Angew. Math. 670, 65-92 (2012); addendum ibid. 713, 247-250 (2016).
Summary: Consider a planar, bounded, \(m\)-connected region \(\Omega \), and let \(\partial \Omega \) be its boundary. Let \(\mathcal T\) be a cellular decomposition of \(\Omega \cup \partial \Omega \), where each 2-cell is either a triangle or a quadrilateral. From these data and a conductance function we construct a canonical pair \(( S, f)\) where \(S\) is a genus \(( m - 1)\) singular flat surface tiled by rectangles and \(f\) is an energy preserving mapping from \(\mathcal T^{(1)}\) onto \(S\). By a singular flat surface, we will mean a surface which carries a metric structure locally modeled on the Euclidean plane, except at a finite number of points. These points have cone singularities, and the cone angle is allowed to take any positive value (see for instance [M. Troyanov, IRMA Lect. Math. Theor. Phys. 11, 507–540 (2007; Zbl 1127.32009)] for an excellent survey). Our realization may be considered as a discrete uniformization of planar bounded regions.
MSC:
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
| 32Q30 | Uniformization of complex manifolds |
| 57M50 | General geometric structures on low-dimensional manifolds |
Citations:
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