7.3 Comparison with the linear theory of discrete harmonic functions
Discrete minimal surfaces
Minimal surfaces and solutions to Plateau’s problem have been studied for a long time and constitute still interesting (model) problems. Expositions of the theory of minimal surfaces can be found in standard textbooks, as for example [30, 56]. To explicitly construct minimal surfaces is an remaining challenge. Numerical approximations of minimal surfaces have been proposed and studied by several authors.
In this chapter, we first show how to apply the theory ofS-isothermic discrete minimal surfaces to the construction of examples, yielding in particular some triply-periodic dis-crete minimal surfaces and examples of solutions to Plateau’s problem. See also [23] for a shortened version. In Section 8.3 we apply the convergence results of Chapter 7 and prove C∞-convergence ofS-isothermic discrete minimal surfaces to smooth analogs. Finally, we compare this result with existing convergence results for other construction methods of discrete minimal surfaces in Section 8.4.
8.1 Construction ofS-isothermic discrete minimal surfaces with special boundary conditions
In the following we explain some details of how to apply the general constructions scheme forS-isothermic discrete minimal surfaces of Appendix A to examples with boundary condi-tions specified below. We focus on the first step (finding the combinatorial parametrization and boundary conditions) of the algorithm, as this constitutes the main ingredient for the remaining steps (construction of the corresponding spherical circle pattern, the Koebe polyhedron, and finally the discrete minimal surface by dualization).
8.1.1 Boundary conditions
Consider the family of all smooth minimal surfaces with boundary whose boundary curve can be divided into finitely many pieces of finite positive length. Furthermore, each of this boundary arcs
either (i) lies within a plane which intersects the surface orthogonally. Then this boundary arc is a curvature line and the surface may be continued smoothly across the plane by reflection in this plane. Moreover the image of this boundary arc under the Gauss map is (a part of) a great circle on the unit sphere.
or (ii) lies on a straight line. Then this boundary arc is an asymptotic line and the surface may be continued smoothly across this straight line by 180◦-rotation about it. Again the image of this boundary arc under the Gauss map is (a part of) a great circle on the unit sphere.
The implications in (i) and (ii) are well-known properties of smooth minimal surfaces and are called Schwarz’s reflection principles; cf. for example [30, Section 4.8].
Like a smooth minimal surface, an S-isothermic discrete minimal surface may also be continued by reflection in the boundary planes or by 180◦-rotation about straight boundary lines. This is due to the translation of the boundary conditions for the discrete minimal
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surfaces into angle conditions for boundary circles. In particular, the boundary circles of the spherical circle pattern intersect the corresponding great circle orthogonally.
To construct the examples, we always assume, that we know all boundary planes and straight lines and their intersection angles (and lengths if necessary). As indicated in Appendix A, to construct a discrete minimal surface, the first step consists in determining the combinatorics of the curvature lines under the Gauss map. To cope with this task, we first try to reduce the problem by symmetry. This is especially helpful when dealing with highly symmetric triply periodic minimal surfaces.
8.1.2 Reduction of symmetries
To simplify the construction of a discrete minimal surface, we only consider afundamental pieceof the smooth (and the discrete) minimal surface. This is a piece of the surface which is bounded by planar curvature lines and/or straight asymptotic lines (like the whole surface itself) with the following two properties.
(i) The surface is obtained from the fundamental piece by successive reflection/rotation in its boundary planes/lines and the new obtained boundary planes/lines.
(ii) There is no piece strictly contained in the fundamental piece which has property (i).
In general, the fundamental piece is not unique. In the remainder of this section, we assume that we are dealing with a fundamental piece.
8.1.3 Combinatorics of curvature lines
Knowing the boundary conditions, the main step consists in finding the ’right’ combinatorial data for the circle pattern. Thus we only draw general combinatorial pictures of curvature lines and indicate only sometimes intersection angles between boundary pieces. An illus-trating example of the following algorithm is given in Figures 8.1 and 8.2 corresponding to the image of a fundamental piece of Schwarz’s H surface under the Gauss map.
Given a fundamental piece of a smooth minimal surface, we first determine the image of the boundary pieces under the Gauss map. By our assumption explained in Section 8.1.1, each of these curves is mapped to a part of a great circle on the unit sphereS2. Then we extract the following ingredients for the construction (see Figure 8.1 for an example):
• a combinatorial picture of the image of the boundary pieces under the Gauss map,
• the angles between different boundary curves onS2,
• and eventually the lengths of the image of the boundary pieces under the Gauss map, if the angles do not uniquely determine the image of the boundary curve onS2 (up to rotations of the sphere). This case is more difficult; see Remark 8.1.
π α
curvature line asymptotic line umbilic point
angles of the boundary configuration onS2: q= right angle,α=π/3
Figure 8.1: A combinatorial picture of the boundary conditions on S2 of a fundamental region of Schwarz’s H surface.
curvature line asymptotic line umbilic point
m
n
Figure 8.2: A combinatorial conformal parametrization of the fundamental region of Schwarz’s H surface.
Given the combinatorial picture of the boundary, the remaining task consists in finding a combinatorial parametrization of the bounded domain corresponding to the curvature line parametrization of a smooth minimal surface. This implies in particular:
(1) Umbilic and singular points are taken from the smooth surface, but only their combi-natorial locations matter. The smooth surface also determines the number of curvature lines meeting at these points. For interior umbilic or singular points, we additionally have to take care how many times the interior region is covered by the Gauss map. As this case does not occur in any of the examples presented in Section 8.2, we assume for simplicity that all umbilic or singular points lie on the boundary.
(2) If a boundary curvature line and a boundary asymptotic line intersect at an angle of 3π/4, there is another curvature line meeting at this point. This property is a consequence of the conformality of the parametrization.
(3) The curvature lines of the combinatorial parametrization divide the domain into com-binatorial squares. The only exceptions are comcom-binatorial triangles formed by two curvature lines and by an asymptotic line on the boundary.
Hence in order to find the combinatorial curvature line parametrization, first determine all umbilic and singular points and all regular boundary points with 3π/4-angles. Then continue the additional curvature line(s) meeting at these points. In this way, the combi-natorial domain is divided into finitely many subdomains such that conditions (2) and (3) hold, see Figure 8.2 (left). If there are several combinatorially different possibilities to obtain such a subdivision, we have to choose the one with the same combinatorics as the corresponding curvature lines of the given smooth minimal surface. More precisely, the discrete curvature lines are chosen as an approximation of the smooth (infinite) curvature line pattern.
Given this coarse subdivision of the combinatorial domain, the parametrization of the subdomains is obvious. The only additional conditions occur at common boundaries of two subdomains, where the number of crossing curvature lines has to be equal on both sides.
Hence after this step, we know the maximum number of free integer parameters correspond-ing to the number of different combinatorial types of curvature lines whose numbers may be chosen independently. These integer parameters of the discrete minimal surface corre-spond to smooth parameters of the continuous minimal surface such as scaling or quotients of lengths (for example of the boundary curves). The number of free integer parameters is greater or equal to the number of smooth parameters of the corresponding continuous minimal surface.
Remark 8.1.Note that the free parameters of the combinatorial curvature line parametri-zation take integer values and the number of (quadrilateral) subdomains obtained by the subdivision process is finite. Also, all combinatorial curvature lines are closed modulo the boundary. In general these properties do not hold for curvature lines of smooth minimal
Figure 8.3: Gergonne’s surface. Copper plate engraving from H. A. Schwarz [68].
surfaces. Furthermore, there may be dependencies between the numbers of curvature lines of different types which have to be approximated by the choice of the free integer parameters of the combinatorial curvature line parametrization. These three aspects affect the different appearances of the smooth minimal surfaces and its discrete minimal analog.
In particular, for the of a smooth minimal surface whose curvature lines are closed mod-ulo the boundary and whose combinatorial curvature line parametrization has only one free parameter (corresponding to overall scaling), the discrete minimal analog will have exactly the same boundary planes/straight lines (up to translation and scaling). This is useful to create discrete minimal analogs to triply-periodic minimal surfaces. In general, we obtain discrete minimal approximations of such surfaces in the sense that the discrete surface does not exactly close up after suitable reflection/rotation in its boundary planes/straight edges.