Approximation of conformal mappings by circle patterns and discrete minimal surfaces

To a rhombic embedding of a planar graph with quadrilateral faces and vertices colored black and white there is an associated isoradial circle pattern C 1 with centers of circles at white vertices and radii equal to the edge length. Let C 2 be another circle pattern such that the rhombi correspond to kites of intersecting circles with the same intersection angles. We consider the mapping g C which maps the centers of circles and the intersection points to the corresponding points and which is an affine map on the rhombi. Let g be a locally injective holomorphic function. We specify the circle pattern C 2 by prescribing the radii or the angles on the boundary corresponding to values of log g. We show that g C approximates g and its first derivative uniformly on compact subsets and that a suitably normalized sequence converges to g if the radii of C 1 converge to 0. In particular, we study the case that C 1 is a quasicrystallic circle pattern, that is the number of different edge directions of the rhombic embedding is bounded by a fixed constant (for the whole sequence). For a class of such circle patterns we prove the convergence of discrete partial derivatives of arbitrary order to the corresponding continuous derivatives of g. For this purpose we use a discrete version of Hölder's inequality and a discrete regularity lemma for solutions of elliptic differential equations. Furthermore, we consider the special case of regular circle patterns with the combina-torics of the square grid and two (different) intersection angles, which correspond to the two different edge directions. We show the uniqueness of the embedded infinite circle pattern (up to similarities) and prove an estimation for the quotients of radii of neighboring circles of such an (finite) circle pattern with error of order 1/combinatorial distance of the circle to the boundary. We also carry this result over to certain classes of quasicrystallic circle patterns. In addition, we study the Z γ-circle patterns with the combinatorics of the square grid and regular intersection angles for γ ∈ (0, 2). We prove the uniqueness (up to scaling) of such embedded circle patterns which cover a corresponding sector of the plane, subject to some conditions on the intersection angles and γ. Similar results are also shown for some classes of quasicrystallic Z γ-circle patterns. For the case of orthogonal circle patterns with the …