C^∞-Convergence of Circle Patterns to Minimal Surfaces

Abstract

Abstract. Given a smooth minimal surface F : Ω → R 3 defined on a simply connected region Ω in the complex plane C, there is a regular SG circle pattern Q ǫ Ω . By the Weierstrass representation of F and the existence theorem of SG circle patterns, there exists an associated SG circle pattern P ǫ Ω in C with the combinatoric of Q ǫ Ω . Based on the relationship between the circle pattern P ǫ Ω and the corresponding discrete minimal surfaceThe theory of discrete differential geometry is presently emerging on the border of differential and discrete geometry, which studies geometric shapes with a finite number of elements (polyhedra) and aims at a development of discrete equivalents of the geometric notions and methods of surface theory (see . A smooth geometric shape (such as surface) appears then as a limit of the refinement of the discretization. One of the central problems of discrete differential geometry is to find proper discrete analogues of special classes of surfaces, such as minimal, constant mean curvature, isothermic, etc. In [2], a new discrete model was introduced to investigate conformal discretizations of minimal surface, i.e., the analogous discrete minimal surfaces consisting of touching spheres, and of circles which intersect the spheres orthogonally in their points of touch. It is proved that the discrete minimal surfaces converge to the smooth ones. The advantages of the discretizations are that they respect conformal properties of surfaces, possess a maximum principle, etc. Here, we are concerned with the C ∞ -convergence of discrete minimal surfaces given in terms of circles

Approximation of Conformal Mappings Using Conformally Equivalent Triangular Lattices

“…Also, there is a link to integrable structures via isoradial circle patterns. The approximation of conformal maps using circle patterns has been studied in [4,5,12,15,18].…”

Approximation of quasiconformal mappings by SG circle patterns

“…Next, circle patterns were used to approximate conformal mappings (see [6][7][8][9]). It was also proved that circle patterns converge to minimal surfaces (see [10,11]). Current development of research related to circle patterns can be found in [12][13][14][15].…”