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**Minimal surfaces from circle patterns: geometry from combinatorics.**
*(English)*
Zbl 1122.53003

In this paper the authors fashion discretization of isothermic surfaces (including the minimal ones) from combinatorial properties of curvature line patterns and circles and spheres via the Koebe polyhedron. These discretizations are based on quadrilateral meshes formed by the curvature lines and they respect the conformal properties of the surfaces involved. A surface in \(\mathbb R^3\) is called isothermic if it admits a conformal curvature line parametrization away from the umbilic points. A discrete analog of this, a discrete isothermic surface, is a function \(f: V(\mathcal D) \to \mathbb R^3\) from the set of vertices of a quad-graph \(\mathcal D\) (a mesh of quadrilaterals, every interior vertex of which has an even number of edges) such that for every face of \(\mathcal D,\) with its vertices \(v_1, v_2, v_3, v_4\) counted cyclically, their images form a conformal square in \(\mathbb R^3\) (meaning their cross-ratio is \(-1\)). Further, a discrete S-isothermic surface is defined, which is, roughly, a polyhedral surface such that the faces have inscribed circles, each pair of neighbouring circles touch the common edge at the same point. It can be thought of as composed of touching spheres and circles with spheres and circles intersecting orthogonally.

As is well known, for an isothermic immersion \(f: D \to \mathbb R^3\) its Christoffel transform (or the dual isothermic surface) is another isothermic immersion \(f^*: D \to \mathbb R^3\) defined by

\[ f^*_x = f_x /| | f_x| | ^2 , \; f^*_y = f_y /| | f_y| | ^2 . \]

The Christoffel transform actually characterizes isothermic surfaces. Likewise, the discrete version of this transform defined on the set of vertices of some discrete isothermic surface \(f: V (\mathcal D) \to \mathbb R^3\) is defined (with appropriate assignment of \(\pm\) signs to the edges) by \(\Delta f^* = \pm\, \Delta f / | | \Delta f| | ^2,\) where \(\Delta f\) denotes the difference of neighboring vertices. The Christoffel transform plays an important role in the authors’ considerations.

From any discrete S-isothermic surface one gets a discrete isothermic surface by doing, so called, central extensions, i.e., by adding the centers of the spheres and circles. Christoffel’s duality transform preserves the class of discrete S-isothermic surfaces. Koebe’s polyhedron is a special convex polyhedron whose edges are tangent to the 2-sphere \(S^2.\) Koebe polyhedra give rise to special discrete S-isothermic surfaces which come from circle packings in \(S^2\) according to Koebe’s theorem [P. Koebe, Kontaktprobleme der konformen Abbildung, Ber. Sächs. Akad. Wiss. Leipzig Math.-phys. Kl. 88, 141–164 (1936; Zbl 0017.21701 and JFM 62.1217.04)].

Discrete minimal surfaces are accordingly defined as a subclass of S-isothermic discrete surfaces and it is shown that an S-isothermic surface is a discrete minimal surface if and only if the dual S-isothermic surface corresponds to a Koebe polyhedron. A descrete version of the Weierstrass representation is given which describes the S-isothermic minimal surface obtained by projecting the pattern stereographically to the sphere and dualizing corresponding Koebe polyhedron. It is also shown that the classical notion of the associated family of minimal surfaces (i.e., the 1-parameter family of isometric deformations of a given surface preserving the Gauss map) finds its counterpart in the discrete setting.

The authors suggest a general method of constructing discrete minimal surfaces from differentiable ones by looking at the image of curvature lines under the Gauss map, which gives a cell decomposition of \(S^2\) that gives rise to an orthogonal circle pattern. From this a Koebe polyhedron is constructed giving rise to a discrete minimal surface. Following this program the authors give examples of discretizations of Enneper’s surface, the catenoid, the Schwarz P-surface and the Scherk’s tower. Finally, the authors prove the convergence of discrete minimal S-isothermic surfaces to smooth minimal surfaces.

As is well known, for an isothermic immersion \(f: D \to \mathbb R^3\) its Christoffel transform (or the dual isothermic surface) is another isothermic immersion \(f^*: D \to \mathbb R^3\) defined by

\[ f^*_x = f_x /| | f_x| | ^2 , \; f^*_y = f_y /| | f_y| | ^2 . \]

The Christoffel transform actually characterizes isothermic surfaces. Likewise, the discrete version of this transform defined on the set of vertices of some discrete isothermic surface \(f: V (\mathcal D) \to \mathbb R^3\) is defined (with appropriate assignment of \(\pm\) signs to the edges) by \(\Delta f^* = \pm\, \Delta f / | | \Delta f| | ^2,\) where \(\Delta f\) denotes the difference of neighboring vertices. The Christoffel transform plays an important role in the authors’ considerations.

From any discrete S-isothermic surface one gets a discrete isothermic surface by doing, so called, central extensions, i.e., by adding the centers of the spheres and circles. Christoffel’s duality transform preserves the class of discrete S-isothermic surfaces. Koebe’s polyhedron is a special convex polyhedron whose edges are tangent to the 2-sphere \(S^2.\) Koebe polyhedra give rise to special discrete S-isothermic surfaces which come from circle packings in \(S^2\) according to Koebe’s theorem [P. Koebe, Kontaktprobleme der konformen Abbildung, Ber. Sächs. Akad. Wiss. Leipzig Math.-phys. Kl. 88, 141–164 (1936; Zbl 0017.21701 and JFM 62.1217.04)].

Discrete minimal surfaces are accordingly defined as a subclass of S-isothermic discrete surfaces and it is shown that an S-isothermic surface is a discrete minimal surface if and only if the dual S-isothermic surface corresponds to a Koebe polyhedron. A descrete version of the Weierstrass representation is given which describes the S-isothermic minimal surface obtained by projecting the pattern stereographically to the sphere and dualizing corresponding Koebe polyhedron. It is also shown that the classical notion of the associated family of minimal surfaces (i.e., the 1-parameter family of isometric deformations of a given surface preserving the Gauss map) finds its counterpart in the discrete setting.

The authors suggest a general method of constructing discrete minimal surfaces from differentiable ones by looking at the image of curvature lines under the Gauss map, which gives a cell decomposition of \(S^2\) that gives rise to an orthogonal circle pattern. From this a Koebe polyhedron is constructed giving rise to a discrete minimal surface. Following this program the authors give examples of discretizations of Enneper’s surface, the catenoid, the Schwarz P-surface and the Scherk’s tower. Finally, the authors prove the convergence of discrete minimal S-isothermic surfaces to smooth minimal surfaces.

Reviewer: Ivko Dimitric (Uniontown)

### MSC:

53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |

53A05 | Surfaces in Euclidean and related spaces |

52C99 | Discrete geometry |