On algebraic minimal surfaces Boris Odehnal December 21, 2016 Abstract 1 Introduction Minimal surfaces have been studied from many different points of view. Boundary We give an overiew on various constructions value problems, uniqueness results, stabil- of algebraic minimal surfaces in Euclidean ity, and topological problems related to three-space. Especially low degree exam- minimal surfaces have been and are stil top- ples shall be studied. For that purpose, we ics for investigations. There are only a use the different representations given by few results on algebraic minimal surfaces. Weierstraß including the so-called Björ- Most of them were published in the second ling formula. An old result by Lie dealing half of the nine-teenth century, i.e., more with the evolutes of space curves can also or less in the beginning of modern differen- be used to construct minimal surfaces with tial geometry. Only a few publications by rational parametrizations. We describe a Lie [30] and Weierstraß [50] give gen- one-parameter family of rational minimal eral results on the generation and the prop- surfaces which touch orthogonal hyperbolic erties of algebraic minimal surfaces. This paraboloids along their curves of constant may be due to the fact that computer al- Gaussian curvature. Furthermore, we find gebra systems were not available and clas- a new class of algebraic and even rationally sical algebraic geometry gained less atten- parametrizable minimal surfaces and call tion at that time. Many of the compu- them cycloidal minimal surfaces. tations are hard work even nowadays and synthetic reasoning is somewhat uncertain. Keywords: minimal surface, algebraic Besides some general work on minimal sur- surface, rational parametrization, poly- faces like [5, 8, 43, 44], there were some iso- nomial parametrization, meromorphic lated results on algebraic minimal surfaces function, isotropic curve, Weierstraß- concerned with special tasks: minimal sur- representation, Björling formula, evolute of faces on certain scrolls [22, 35, 47, 49, 53], a spacecurve, curve of constant slope. minimal surfaces related to congruences of lines [25, 28, 34, 38] minimal surfaces with MSC 2010: 53A10, 53A99, 53C42, 49Q05, a given geodesic [23], minimal surfaces of a 14J26, 14Mxx. certain degree, class, or genus (whether real 1 or not) [1, 10, 11, 19, 20, 21, 31, 41, 42, 48], Theorem 1.2. Let a minimal surface be minimal surfaces touching surfaces along tangent to a cylinder . If is algebraic,M Z M special curves [22], minimal surfaces show- then the orthogonal cross-section c of is ing special symmetries [14, 15, 16, 17], or the evolute of an algebraic curve. If c isZ the minimal surface which allow isometries to evolute of a transcendental curve, then M special classes of surfaces [4, 6, 18, 52]. is also transcendental. The famous algebraic minimal surface by Rib- Enneper which is of degree 9 and class However, according to a theorem by aucour Enneper 6 attracted intensive investigation. Con- , ’s surface, like many sequently, researchers have found different other minimal surfaces, appears as the cen- generations of this surface: as the envelope tral envelope of isotropic congruences of of the planes of symmetry of all points on lines, see [25, 28, 34, 38, 45]. the pair of focal parabolas Among the real algebraic minimal sur- faces, Enneper’s surface has lowest pos- 4 2 2 1 p1(u)=( 3 u, 0, 3 u 3 ), sible degree 9. But there are algebraic min- 4 1 −2 2 p2(v)=(0, v, v ) imal surfaces that can be found in [12, 13, 3 3 − 3 21, 30] which are of degree 3 and 4 having or as the unique minimal surface (22) the equations through the rational curve 1 : (x iy)4 + 3(x2 + y2 + z2)=0 γ(t)= t t3, t2, 0 G − − 3 and having γ’s normals for its surface normals. Since γ is planar, the surface normals of the : 2(x iy)3 6i(x iy)z 3(x+iy)=0 uniquely defined minimal surface form a de- L − − − − velopable surface (to be precise, a plane), with respect to a properly chosen Cartesian and thus, γ is a planar geodesic on En- coordinate system. The surfaces and G L neper’s minimal surface. The plane of γ is have no real equation (polynomial equation a plane of symmetry of Enneper’s surface. with real coefficients exclusively) and do not This is a manifestation of a more general carry a single real point. result by Henneberg, see [21, 24, 30, 33]: is usually called Geiser’s surface and G Lie Geiser L Theorem 1.1. A minimal surface car- is named after . ’s minimal sur- ries a planar and not straight curveMc as a face is a minimal surface of revolution with geodesic. If is algebraic, then the invo- an isotropic axis. Obviously, it is of degree M lutes of c have to be algebraic or c is the 4 and some computation tells us that the equation of its dual surface ⋆, i.e., the sur- evolute of a planar algebraic curve. G face of its tangent planes has the equation We shall make use of this fact later in ⋆ 2 4 2 2 2 3 Sec. 7 when we construct cycloidal minimal : 9w0(w1 iw2) (w1 +w2 +w3) =0 surfaces. G − − A further result due to Henneberg (see which is, therefore, of degree 6, and thus, [21, 24, 30, 33]) is the following is of class 6. G 2 Whereas Lie’s surface is of degree 3 and the parametrization also of class 3 since the implicit equation of the dual surface ⋆ reads c3uS3v 3cuSv L − f(u, v)= s3uS3v + 3suSv (1) ⋆ 2 2 2 3 : 27w0(w2+iw1) +9i(w1+w2)w3 4iw3 =0. 3c uC v L − 2 2 Geiser’s surface meets the ideal plane in is an example for that, since the implicit the same ideal line as Lie’s surface does. equation of its dual surface equals The ideal line x iy =0 is a 4-fold line on 2 2 2 − u0(u1 + u2) + and a 3-fold line on . It is remarkable (2) G L +u (u2 u2)(3u2 +3u2 +2u2)=0. that complex (non-real) algebraic minimal 3 1 − 2 1 2 3 surfaces have been undergoing detailed in- The algebraic degree of Henneberg’s sur- vestigations, see, e.g., [1, 10, 12, 13, 48]. face equals 15. Enneper’s surface is the In [30], Lie gives a result dealing with the only known example of a minimal surface ideal curves of algebraic minimal surfaces: where the degree and class sum up to 15: the degree equals 9 (cf. (23)), the class Theorem 1.3. The intersection of an al- equals 6 (cf. (24)). gebraic minimal surface with the plane at Lie gives also results on the class of an infinity consists of finitely many lines. algebraic minimal surface: Some of the ideal lines on a minimal Theorem 1.5. The class of an orientable surface may have higher multiplicities and algebraic minimal surface is alwas even. pairs of complex conjugate lines can also Henneberg occcur. ’s surface is of class 5 and For the coordinatization of ideal points non-orientable. The rational minimal and lines we refer to Sec. 2. Möbius strip given in [35] is of class 15. The results on degrees, ranks, and classes In Sec. 2, we introduce coordinates and of real algebraic minimals surfaces differ define all necessary abbreviations. Then, from the results on complex algebraic min- the different parametrization techniques for imal surfaces. For real algebraic minimal minimal surfaces are collected. Proofs for surfaces we have (see [30]) these can be found in most of the standard monographs on minimal surfaces or differ- Theorem 1.4. The sum of the degree and ential geometry such as [2, 33, 46]. Sec. 3 class of a real algebraic minimal surface is is dedicated to Enneper’s surface and its at least 15. natural generalizations. In Sec. 4, Bour’s minimal surfaces gain attention. We show The two aforementioned examples of different ways to find these minimal sur- complex minimal surfaces obviously show a faces and give estimates on the algebraic different behaviour. degrees of these surfaces. Then, in Sec. It is well-kown (cf. [30, 33]) that 5 is the 5, Richmond’s surface appears as one in lowest possible class of a real algebraic min- a one-parameter family. Sec. 6 gives ad- imal surface. Henneberg’s surface with ditional and apparently new results on a 3 well-known kind of minimal surface tangent 816, and 969 terms provided that no special to a hyperbolic paraboloid. Sec. 7 deals coordinate system is chosen and that the with an apparently new class of minimal equations are expanded in full length. surfaces. The fact that cycloids (cycloidal curves with cusps) have rational normals and are algebraic as well as their evolutes 2 Prerequisites and involutes are (see [32, 51, 55, 56]), al- lows us to construct a family of algebraic Since we are dealing with minimal surfaces minimal surfaces that admit even rational in the Euclidean three-space, Cartesian co- parametrizations. We debunk their rela- ordinates (x, y, z) are sufficient. Vectors tions to curves of constant slope on quadrics and matrices are written in bold characters. of revolution. The canonical innerproduct of two vectors u, v R3 is denoted by u, v . The Eu- The reasons for the interest in algebraic ∈ h i clidean length v of a vector v is then and, especially in rational minimal sur- k k faces are manifold: Rational parametriza- given by v = v, v . The induced crossproductk k of two vectorsh i u, v R3 is the tions can be converted into a geometrically p ∈ vector u v R3.