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The Algebraic Surfaces of the Enneper Family of Maximal Surfaces in Three Dimensional Minkowski Space

The Algebraic Surfaces of the Enneper Family of Maximal Surfaces in Three Dimensional Minkowski... axioms Article The Algebraic Surfaces of the Enneper Family of Maximal Surfaces in Three Dimensional Minkowski Space Erhan Güler Department of Mathematics, Faculty of Sciences, Kutlubey Campus, Bartın University, Bartın 74100, Turkey; eguler@bartin.edu.tr; Tel.: +90-378-5011000 (ext. 2275) Abstract: We consider the Enneper family of real maximal surfaces via Weierstrass data (1, z ) for z 2 C, m 2 Z . We obtain the irreducible surfaces of the family in the three dimensional Minkowski 2,1 2 space E . Moreover, we propose that the family has degree (2m + 1) (resp., class 2m(2m + 1)) in the cartesian coordinates x, y, z (resp., in the inhomogeneous tangential coordinates a, b, c). Keywords: Weierstrass representation; Enneper maximal surface; algebraic surface; degree; class MSC: primary 53A35; secondary 53C42, 65D18 1. Introduction A minimal surface is a surface of vanishing mean curvature in three dimensional Euclidean space E . There are many classical and modern minimal surfaces in the literature. See Darboux [1,2], Dierkes [3], Fomenko and Tuzhilin [4], Gray, Salamon, and Abbena [5], Nitsche [6], Osserman [7], Spivak [8] for some books, Lie [9], Schwarz [10], Small [11,12], and Weierstrass [13,14] for some papers related to minimal surfaces in Euclidean geometry. Lie [9] studied the algebraic minimal surfaces and gave a table classifying these Citation: Güler, E. The Algebraic surfaces. See also Enneper [15], Güler [16], Nitsche [6], and Ribaucour [17] for details. Surfaces of the Enneper Family of Weierstrass [13] revealed a representation for minimal surfaces in three dimensional Maximal Surfaces in Three Euclidean space E . Almost one hundred years later, Kobayashi [18] gave an analogous Dimensional Minkowski Space. Weierstrass-type representation for conformal spacelike surfaces with mean curvature 2,1 Axioms 2022, 11, 4. https://doi.org/ identically 0, called maximal surfaces, in three dimensional Minkowski space E . 10.3390/axioms11010004 In this paper, we consider the Enneper family of maximal surfaces E for positive integers m  1 by using Weierstrass data (1, z ) for z 2 C, and then show that these Academic Editor: Anna Maria Fino 2,1 surfaces are algebraic in E . See Güler [16] for a Euclidean case of Enneper ’s algebraic Received: 3 December 2021 minimal surfaces family. Accepted: 20 December 2021 In Section 2, we give this family of real maximal surfaces in (r, q) and (u, v) coordinates Published: 22 December 2021 2,1 by using Weierstrass representation in E . In Section 3, we find irreducible algebraic equations defining surfaces E (u, v) in terms of running coordinates x, y, z, and a, b, c, and Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in also compute degrees and classes of E (u, v). Finally, we summarize all findings in tables published maps and institutional affil- in the last section, then give some open problems. iations. 2. Family of Enneper Maximal Surfaces n,1 t Let E : = fx = (x , , x , x ) jx 2 Rg,h,i be the (n + 1)-dimensional Lorentz– 1 n 0 i Minkowski (for short, Minkowski) space with Lorentzian metric hx, yi = x y + + 1 1 Copyright: © 2021 by the authors. x y x y . n n 0 0 Licensee MDPI, Basel, Switzerland. n,1 A vector x 2 E is called space-like if hx, xi > 0, time-like if hx, xi < 0, and light-like This article is an open access article n,1 if x 6= 0 and x, x = 0. A surface in E is called space-like (resp. time-like, light-like) if the h i distributed under the terms and induced metric on the tangent planes is a Riemannian (resp. Lorentzian, degenerate) metric. conditions of the Creative Commons 2,1 Now, let E be three dimensional Minkowski space with Lorentzian metric h. , .i = Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ x y + x y x y . We identify x and x without further comment. 1 1 2 2 3 3 4.0/). Axioms 2022, 11, 4. https://doi.org/10.3390/axioms11010004 https://www.mdpi.com/journal/axioms Axioms 2022, 11, 4 2 of 12 LetU be an open subset of C. A maximal curve is an analytic function J : U ! C such D E 0 0 0 ¶J 0 0 that hJ (z), J (z)i = 0, where z 2 U , and J := . In addition, if J , J = jJ j 6= 0, then ¶z J is a regular maximal curve. We then have maximal surfaces in the associated family of a maximal curve, given by the following Weierstrass representation theorem for ZMC (zero mean curvature) surfaces, or maximal surfaces. Kobayashi [18] found a Weierstrass type representation for space-like conformal 2,1 maximal surfaces in E : Theorem 1. Let g(w) be a meromorphic function and let f (w) be a holomorphic function, f g is analytic, defined on a simply connected open subset U  C such that f (w) does not vanish on U except at the poles of g(w). Then, 2 2 x(u, v) = Re f 1 + g , i f 1 g ,2 f g dw, (z = u + iv) (1) is a space-like conformal immersion with mean curvature identically 0 (i.e., space-like conformal maximal surface). Conversely, any spacelike conformal maximal surface can be described in this manner. Next, we give some facts about Weierstrass data, and a maximal curve to construct some maximal surfaces. Definition 1. A pair of a meromorphic function g and a holomorphic function f , ( f , g) is called Weierstrass data for a maximal surface. Lemma 1. The curve of Enneper of order m: 2m+1 2m+1 m+1 z z 2z # (z) = z + , i z , (2) 2m + 1 2m + 1 m + 1 is a maximal curve, z 2 Cf0g, i = 1, m 6= 1,1/2. 2,1 Therefore, we have h# , # i = 0 by using (2). Hence, in E , the Enneper maximal m m surface is given by E (u, v) = Re # (z)dz, (3) m m where z = u + iv. Im # (z)dz gives the adjoint minimal surface E (u, v) of the surface E (u, v) in (3). Then, we get the following: Corollary 1. The Weierstrass data (1, z ) of (3) is a representation of the Enneper maximal surface, where integer m  1. iq Considering the findings above with z = re , we get the following Enneper family of maximal surfaces: 0 1 1 2m+1 r cos(q) + r cos[(2m + 1)q] 2m+1 1 2m+1 @ A E (r, q) = r sin(q) + r sin[(2m + 1)q] (4) 2m+1 m+1 r cos[(m + 1)q] m+1 where m 6= 1,1/2. See Figure 1 for Enneper maximal surfaces E ,E ,E in r, q ( ) 1 2 3 coordinates. Axioms 2022, 11, 4 3 of 12 Figure 1. Enneper maximal surfaces, and its top views (Left): E (r, q), (Middle): E (r, q), 1 2 (Right): E (r, q). Hence, using the binomial formula, we obtain more clear representation of E (u, v) in (3): 2m+1 2m + 1 1 2m+1k x(u, v) = Re u + iv + å u (iv) , 2m+1 k=0 2m+1 2m + 1 i 2m+1k y(u, v) = Re iu v + å u (iv) , (5) 2m+1 k=0 m+1 m + 1 2 m+1k z(u, v) = Re å u (iv) . m+1 k=0 We study surfaceE (u, v) in (u, v) coordinates for m = 1, 2, . . . , 5, taking z = u + iv at Cartesian coordinates x, y, z, and also in inhomogeneous tangential coordinates a, b, c, by using Weierstrass representation equation. Next, we give a theorem about maximality of surface E (u, v) (see Figure 1, Left): Theorem 2. The surface 0 1 0 1 1 3 2 u uv + u x(u, v) 1 3 2 @ A @ A E (u, v) = = y(u, v) (6) v + u v v 2 2 u + v z(u, v) 2,1 which has Weierstrass data (1, z), is an Enneper maximal surface in E . Proof. The coefficients of the first fundamental form of the surface E (u, v) (E , for short) 1 1 are given by E = (l 1) = G, F = 0, 2 2 where l = u + v . That is, conformality holds. Then, the Gauss map e (u, v) of E is as 1 1 follows 2u 2v l + 1 e = , , , (7) l 1 l 1 l 1 Axioms 2022, 11, 4 4 of 12 where l 6= 1. The coefficients of the second fundamental form of E are given by 2(3l + 1) L = = N, l 1 M = 0. Then, we obtain the following mean curvature and the Gaussian curvature of the surface E : H = 0, 4(3l + 1) K = , (l 1) EN+G L2F M LN M respectively. Here, H = hs, si , K = hs, si , where hs, si = 1. Hence, 2 2 2(EGF ) EGF the Enneper surface is maximal surface with positive Gaussian curvature. Therefore, we obtain the following parametric equations of the higher order maximal Enneper surfaces E (u, v) = (x(u, v), y(u, v), z(u, v)) (see Figure 2 Middle for E , and Figure 2 Right for E .): 0 1 1 5 3 2 4 u 2u v + uv + u 1 5 2 3 4 @ A E u, v = , (8) ( ) v 2u v + u v v 2 3 2 u + 2uv 0 1 1 7 5 2 3 4 6 u 3u v + 5u v uv + u 1 7 2 5 4 3 6 @ A E (u, v) = , (9) v + 3u v 5u v + u v v 1 4 2 2 1 4 u + 3u v v 2 2 0 1 1 9 7 2 5 4 23 3 6 8 u 4u v + 14u v u v + uv + u 9 3 1 9 2 7 4 5 23 6 3 8 @ A E (u, v) = v 4u v + 14u v u v + u v v , (10) 9 3 5 3 2 4 u + 4u v 2uv 0 1 11 9 2 7 4 5 6 3 8 10 u 5u v + 30u v 42u v + 15u v + uv + u 11 2 9 4 7 6 5 8 3 10 @ A E (u, v) = v + 5u v 30u v + 42u v 15u v + u v v . (11) 1 1 6 4 2 2 4 6 u + 5u v 5u v + v 3 3 Figure 2. Enneper maximal surfaces (Left): E (u, v), (Middle): E (u, v), (Right): E (u, v). 1 2 3 Axioms 2022, 11, 4 5 of 12 3. Degree and Class of Enneper Maximal Surfaces In this section, using some elimination techniques, we derive the irreducible algebraic surface equation, degree and class of Enneper maximal surfaces familyE (u, v) for integers 2,1 1  m  5 in three dimensional Minkowski space E . Let us see some basic notions of the surfaces. Definition 2. The set of roots of a polynomial Q(x, y, z) = 0 gives an algebraic surface equation. An algebraic surface s is said to be of degree d when d = deg(s). Definition 3. At a point (u, v) on a surface s(u, v) = (x(u, v), y(u, v), z(u, v)), the tangent pl ane is given by X x + Yy Zz + P = 0, (12) where e = (X(u, v), Y(u, v), Z(u, v)) is the Gauss map, and P = P(u, v). Then, in inhomogeneous tangential coordinates a, b, c, we have the following surface: b s(u, v) = (a, b, c) = (X/P, Y/P, Z/P). (13) Therefore, we can obtain an algebraic equation Q(a, b, c) = 0 of b s(u, v) in inhomoge- neous tangential coordinates. Definition 4. The maximum degree of the algebraic equation Q(a, b, c) = 0 of b s(u, v) in inhomo- geneous tangential coordinates gives the cl ass of b s(u, v). See [6], for details of a Euclidean case. Hence, we obtain the following findings for degrees and classes of Enneper maximal surfaces that we use: 3.1. Degree We compute the irreducible algebraic surface equation Q (x, y, z) = 0 (see Figure 3, Left) of Enneper ’s maximal surface E u, v in (6) by using some elimination techniques. ( ) We find the following algebraic equation: 9 2 6 2 6 4 3 2 2 3 2 5 Q (x, y, z) = 64z + 432x z 432y z 1215x z 6318x y z + 3888x z 4 3 2 5 7 6 4 2 4 2 1215y z + 3888y z 1152z + 729x 2187x y 4374x z (14) 2 4 2 4 6 4 2 2 4 4 +2187x y + 6480x z 729y + 4374y z 6480y z + 729x z 2 2 2 3 4 2 3 5 1458x y z 3888x z + 729y z 3888y z + 5184z . Then, its degree number is 9. Next, we continue our computations to find Q (x, y, z) = 0 for integers m = 2, 3. We compute the following irreducible algebraic surface equations Q (x, y, z) = 0 (see Figure 3, Middle) and Q (x, y, z) = 0 (see Figure 3, Right) of the surfaces E (u, v) and 3 2 E (u, v), respectively, 25 3 20 Q (x, y, z) = 847 288 609 443z 4358 480 501 250x z 2 20 6 15 +13 075 441 503 750xy z 131 157 978 046 875x z 4 2 15 474 186 536 015 625x y z + 107 other lower degree terms, Axioms 2022, 11, 4 6 of 12 Q (x, y, z) = 2475 880 078 570 760 549 798 248 448z 4 42 +5079 604 062 565 768 134 821 675 008x z 2 2 42 30 477 624 375 394 608 808 930 050 048x y z 4 42 +5079 604 062 565 768 134 821 675 008y z 8 35 633 850 350 654 216 217 766 624 493 568x z +446 other lower degree terms. Therefore, Q (x, y, z) = 0 are the algebraic maximal surfaces of the surfaces E (u, v), m m where m = 2, 3, and they have degree numbers 25 and 49, respectively. Figure 3. Enneper algebraic maximal surfaces (Left): Q (x, y, z) = 0, (Middle): Q (x, y, z) = 0, 1 2 (Right): Q (x, y, z) = 0. 3.2. Class Now, we introduce the class of the surfaces E (u, v) for integers 1  m  4. The case m = 5, marked with “*” presented in tables of Section 4. Computing the irreducible algebraic surface equations Q (a, b, c) = 0, we obtain the Gauss maps e (u, v) (see Figure 4 m m for e , e , e ) for integers 1  m  5 of the surfaces E (u, v), and we also generalize them 2 3 m as follows: u v l + 1 e = 2 ,2 , , l 1 l 1 l 1 2 2 2 u v 2uv l + 1 e = 2 ,2 , , (15) 2 2 2 l 1 l 1 l 1 3 2 2 3 3 u 3uv 3u v v l + 1 e = 2 ,2 , , (16) 3 3 3 l 1 l 1 l 1 4 2 2 4 3 3 4 u 6u v + v 4u v 4uv l + 1 e = 2 ,2 , , (17) 4 4 4 l 1 l 1 l 1 5 3 2 4 4 2 3 5 5 u 10u v + 5uv 5u v 10u v + v l + 1 e = 2 ,2 , , (18) 5 5 5 l 1 l 1 l 1 m m Re z Im z z + 1 ( ) ( ) j j e = 2 ,2 , , (z = u + iv, jzj = l). (19) m m m jzj 1 jzj 1 jzj 1 2 2 (l3)(u +v ) Using (6), (7), (12) and (13), with P (u, v) = , we get the following surface 3(l1) E (u, v) (see Figure 5, Left) in inhomogeneous tangential coordinates: 6u 6v 3(l + 1) a = , b = , c = . 2 2 2 2 2 2 (u + v )(l 3) (u + v )(l 3) (u + v )(l 3) Axioms 2022, 11, 4 7 of 12 2 2 where l = u + v , l 6= 3, u, v 6= 0. Therefore, we compute Enneper ’s irreducible algebraic ˆ b maximal surface equation Q (a, b, c) = 0 (see Figure 6, Left) of the surface E (u, v): 1 1 6 4 2 4 2 2 4 2 2 2 6 4 2 Q (a, b, c) = 4a 4a b 3a c 4a b + 6a b c + 4b 3b c 4 2 3 4 2 3 4 2 2 4 18a c + 12a c + 18b c 12b c + 9a + 18a b + 9b . So, Enneper ’s maximal surface E (u, v) in (6) has class number 6. Figure 4. Top views of the Gauss maps of the surfacesE (u, v) (Left): e (u, v), (Middle): e (u, v), m=1,2,3 1 2 (Right): e (u, v). Next, we continue our computations to find Q for integers 2, 3, 4. To find the class of surface E (u, v) (see Figure 5, Middle), we use (9), (12), (13) and (16). Calculating 3 2 2 4 u 3uv (l 5) ( ) P (u, v) = , we get the following surface E inhomogeneous tangential 2 2 15(l 1) coordinates: 2 2 2 15 u v 15uv 15 l + 1 a = , b = , c = , 3 2 2 3 2 2 3 2 2 2(u 3uv )(l 5) (u 3uv )(l 5) 4(u 3uv )(l 5) 2 2 2 where l = u + v , l 6= 5, u, v 6= 0. In the inhomogeneous tangential coordinates a, b, c, we find the following irreducible algebraic surface equation Q (a, b, c) = 0 (see Figure 6, Middle) of the surface E (u, v): 16 4 14 6 14 4 2 Q (a, b, c) = 2176 782 336a b + 5804 752 896a b 4837 294 080a b c 12 8 12 6 2 +2902 376 448a b 8062 156 800a b c + 120 other lower degree terms. ˆ b Hence, Q (a, b, c) = 0 is the algebraic surface of the surfaceE (u, v), and Enneper ’s maximal 2 2 surface E (u, v) in (8) has class number 20. Using similar ways, we compute the irreducible algebraic surface equation Q (a, b, c) = 0 (see Figure 6, Right) of surfaceE (u, v) (see Figure 5, Right) as follows: 42 40 2 Q (a, b, c) = 26 623 333 280 885 243 904a 718 829 998 583 901 585 408a b 40 2 38 4 104 829 374 793 485 647 872a c + 6868 819 986 468 392 927 232a b 38 2 2 +2935 222 494 217 598 140 416a b c + 774 other lower degree terms. Axioms 2022, 11, 4 8 of 12 ˆ b Q (a, b, c) = 0 is the algebraic surface of the surface E (u, v), and Enneper ’s maximal 3 3 surface E (u, v) in (9) has class number 42. We also compute the following irreducible ˆ b algebraic surface equation Q (a, b, c) = 0 of the surface E (u, v): 4 4 64 8 Q (a, b, c) = 42949672960000000000000000000000000000000000000000a b 62 8 2 247390116249600000000000000000000000000000000000000a b c 60 12 962072674304000000000000000000000000000000000000000a b 60 10 2 +247390116249600000000000000000000000000000000000000a b c 60 8 4 +667953313873920000000000000000000000000000000000000a b c +2604 other lower degree terms. ˆ b Q (a, b, c) = 0 is an algebraic surface of E (u, v), and Enneper ’s maximal surface E (u, v) 4 4 4 in (10) has class number 72. Figure 5. Enneper maximal surfaces in inhomogeneous tangential coordinates (Left): E (u, v), b b (Middle): E (u, v), (Right): E (u, v). 2 3 We obtain the following functions P (u, v), where 1  i  6, 2 2 3 2 2 (u v )(l3) 4(u 3uv )(l 5) P (u, v) = , P (u, v) = , 1 2 2 3(l1) 15(l 1) 4 2 2 4 3 5 3 2 4 4 3 u 6u v +v (l 7) 8 u 10u v +5uv l 9 ( ) ( )( ) P (u, v) = , P (u, v) = , 3 4 3 4 14(l 1) 45(l 1) 6 4 2 2 4 6 5 7 5 2 3 4 6 6 5(u 15u v +15u v v )(l 11) 12(u 21u v +35u v 7uv )(l 13) P (u, v) = , P (u, v) = . 5 5 6 6 33(l 1) 91(l 1) We generalize the above functions, and give the following results: Corollary 2. The functions P for integers m, are given by m1 h i 2k1 2(2k 1) l (2k + 1) 2k P =  Re z , 2k1 2k1 (k + 1)(2k + 1) l 1 h i 2k 4k l (2k + 1) 2k+1 P = Re z , 2k 2k (2k + 1)(4k + 1) l 1 Axioms 2022, 11, 4 9 of 12 where integers k  1, z = u + iv and z = l. j j b b So far, we find surfaces E and E . By using E E , e e , and also (12), (13), we 1 2 3 5 3 5 obtain the following surfaces: E (u, v) = a, b, c : ( ) 0 1 2u b @ A E = 2v , 2 2 (u v )(l 3) l + 1 0 1 2 2 2 u v @ A E = 4uv , 3 2 2 4(u 3uv )(l 5) l + 1 0  1 3 2 2 u 3uv 2 3 b @ A E = 2 u v v , 4 2 2 4 3 3 u 6u v + v (l 7) ( ) l + 1 0  1 4 2 2 4 2 u 6u v + v 3 3 b @ A E = 2 4u v 4uv , 5 3 2 4 4 8(u 10u v + 5uv )(l 9) l + 1 0 1 5 3 2 4 2 u 10u v + 5uv 4 2 3 5 b @ A E = 2 5u v 10u v + v . 6 4 2 2 4 6 5 5(u 15u v + 15u v v )(l 11) l + 1 We also generalize the above functions, and find the following results: Corollary 3. The surfaces S (u, v) for integers m, are given by m1 0 1 0 1 2k1 2Re z B C k(4k 1) b B C @ A 2k1 E (u, v) =    = b , 2k1 2Im z @ A 2k1 2k (2k 1) l (2k + 1) Re z 2k1 jzj + 1 0 1 0 1 2k 2Re z B C 2k + 1 4k + 1 ( )( ) b B C @ A 2k E (u, v) =    = b , 2k 2Im z @ A 2k 2k+1 4k l (4k + 1) Re z 2k jzj + 1 where integers k  1, z = u + iv and jzj = l. 2,1 Corollary 4. In E , the relations between the Enneper maximal surface E (u, v) in the inho- m1 mogeneous tangential coordinates and the Gauss map e (u, v) of the Enneper maximal surface m1 E (u, v) in the cartesian coordinates are given by m1 2k1 k(4k 1) l 1 E (u, v) = e (u, v), 2k1 2k1 2k1 2k (2k 1) l (2k + 1) Re z 2k (2k + 1)(4k + 1) l 1 E (u, v) = e (u, v), 2k 2k 2k 2k+1 4k l (2k + 1) Re z where integers k  1, z = u + iv and z = l. j j Axioms 2022, 11, 4 10 of 12 Figure 6. Enneper ’s algebraic maximal surfaces in inhomogeneous tangential coordinates ˆ ˆ ˆ (Left): Q (a, b, c) = 0, (Middle): Q (a, b, c) = 0, (Right): Q (a, b, c) = 0. 1 2 3 4. Conclusions To reveal the irreducible algebraic surface equations of the Enneper maximal surfaces 2,1 E (u, v) in E , we have tried a series of standard techniques in elimination theory: only Sylvester by hand for Q (x, y, z) = 0, and then projective (Macaulay) and sparse multivari- ate resultants implemented in the Maple software [19] package multires for Q (x, y, z) = 0 and Q (a, b, c) = 0. Maple’s native implicitization command Implicitize, and implicitization based on Maple’s native implementation of the Groebner Basis. For the latter, we implemented in Maple the method in [20] (Chapter 3, p. 128). Under reasonable time, we only succeed for m = 1, 2 in all above methods. For m = 3, the successful method we have tried was to compute the equation defining the elimination ideal using the Groebner Basis package FGb of Faugère in [21]. The time required to output the irreducible algebraic surface equations Q (x, y, z) = 0 (resp. Q (a, b, c) = 0) for integers 1  m  3 and polynomials defining the elimination ideal was under reasonable seconds determined by the following Table 1 (resp. Table 2). For the degree (resp. class) of the irreducible algebraic surface equation Q (x, y, z) = 0 ˆ b (resp. Q (a, b, c) = 0) of the surface E u, v (resp. E (u, v)), marked with “” in Table 1 ( ) 5 4 5 (resp. Table 2), was rejected (i.e., “out of memory”) by Maple 17 on a laptop Pentium Core i5-4310M 2.00 GHz, 4 GB RAM, with the time given in CPU seconds. Hence, we propose the following: Proposition 1. For integers m  1, degree number of the irreducible algebraic surfaces Q (x, y, z) = 0 in the Cartesian coordinates is of (2m + 1) , and class number of irreducible algebraic surfaces Q (a, b, c) = 0 in inhomogeneous tangential coordinates is of 2m(2m + 1) of the (1, z )-type real Enneper maximal surfacesE u, v . ( ) Open Problems Here, we give some problems that we could not find the answers in this paper: Problem 1. Find the irreducible Enneper algebraic maximal surface eq. Q (x, y, z) = 0 in the m4 cartesian coordinates by using the parametric equation of the Enneper maximal surface E u, v . ( ) m4 Problem 2. Find the irreducible Enneper algebraic maximal surface eq. Q (a, b, c) = 0 in the m5 inhomogeneous tangential coordinates by using the parametric equation of the Enneper maximal surface E (u, v). m5 Finally, we give all findings in Tables 1 and 2. Axioms 2022, 11, 4 11 of 12 Table 1. Results for the Enneper algebraic maximal surfaces Q (x, y, z) = 0. Algebraic Degree Number Gröbner FGb Surface of Surface of Terms Time (s) Time (s) Q 9 23 0.266 0.041 Q 25 112 321.953 0.835 Q 49 451  266.854 Q 81 . . . . . . . . . . . . . . . (2m + 1) Table 2. Results for the Enneper algebraic maximal surfaces Q (a, b, c) = 0. Algebraic Class Number Gröbner FGb Surface of Surface of Terms Time (s) Time (s) Q 6 14 0.94 0.030 Q 20 125 61.152 0.114 Q 42 779  125.904 Q 72 2609  1306.718 . . . . . . . . . . . . . . . Q 2m(2m + 1) Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Acknowledgments: The author is grateful to the referees for their suggestions, which improved the presentation of the paper. Conflicts of Interest: The author declares no conflict of interest. References 1. Darboux, G. 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J. für die reine und Angew. Math. (Crelle’s J.) 1875, 80, 280–300. 11. Small, A.J. Minimal surfaces in R and algebraic curves. Differ. Geom. Appl. 1992, 2, 369–384. [CrossRef] Axioms 2022, 11, 4 12 of 12 3 4 12. Small, A.J. Linear structures on the collections of minimal surfaces in R and R . Ann. Glob. Anal. Geom. 1994, 12, 97–101. [CrossRef] 13. Weierstrass, K. Untersuchungen über die Flächen, deren Mittlere Krümmung überall Gleich Null ist; Akademie der Wissenschaften zu Berlin: Berlin, Germany, 1866; pp. 612–625. 14. Weierstrass, K. Über die analytische Darstellbarkeit Sogenannter Willkürlicher Functionen einer reellen Veränderlichen; Akademie der Wissenschaften zu Berlin: Berlin, Germany, 1885; pp. 633–639, 789–805. 15. Enneper, A. Untersuchungen über einige Punkte aus der allgemeinen Theorie der Flächen. Math. Ann. 1870, 2, 58–623. [CrossRef] 16. Güler, E. Family of Enneper minimal surfaces. Mathematics 2018, 6, 281. [CrossRef] 17. Ribaucour, A. Etude des elassoides ou surfaces a courbure moyenne nulle. Mem. Cour. et Mem. Sav. Etr. Acad. R. Sci. Belg. Bruxelles 1882, 44, 236. 18. Kobayashi, O. Maximal surfaces in the 3-dimensional Minkowski space L . Tokyo J. Math. 1983, 6, 297–309. [CrossRef] 19. Maple Software, (Version 17); Waterloo Maple Inc.: Waterloo, ON, Canada. Available online: https://www.maplesoft.com/ (accessed on 10 October 2021). 20. Cox, D.; Little, J.; O’Shea, D. Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3rd ed.; Undergraduate Texts in Mathematics; Springer: New York, NY, USA, 2007. 21. Faugère, J.C. FGb: A library for computing Gröbner bases. In Proceedings of the Third International Congress Conference on Mathematical Software (ICMS’10), Kobe, Japan, 13–17 September 2010; Springer: Berlin/Heidelberg, Germany, 2010; pp. 84–87. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Axioms Multidisciplinary Digital Publishing Institute

The Algebraic Surfaces of the Enneper Family of Maximal Surfaces in Three Dimensional Minkowski Space

Güler, Erhan
Axioms , Volume 11 (1) – Dec 22, 2021
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axioms Article The Algebraic Surfaces of the Enneper Family of Maximal Surfaces in Three Dimensional Minkowski Space Erhan Güler Department of Mathematics, Faculty of Sciences, Kutlubey Campus, Bartın University, Bartın 74100, Turkey; eguler@bartin.edu.tr; Tel.: +90-378-5011000 (ext. 2275) Abstract: We consider the Enneper family of real maximal surfaces via Weierstrass data (1, z ) for z 2 C, m 2 Z . We obtain the irreducible surfaces of the family in the three dimensional Minkowski 2,1 2 space E . Moreover, we propose that the family has degree (2m + 1) (resp., class 2m(2m + 1)) in the cartesian coordinates x, y, z (resp., in the inhomogeneous tangential coordinates a, b, c). Keywords: Weierstrass representation; Enneper maximal surface; algebraic surface; degree; class MSC: primary 53A35; secondary 53C42, 65D18 1. Introduction A minimal surface is a surface of vanishing mean curvature in three dimensional Euclidean space E . There are many classical and modern minimal surfaces in the literature. See Darboux [1,2], Dierkes [3], Fomenko and Tuzhilin [4], Gray, Salamon, and Abbena [5], Nitsche [6], Osserman [7], Spivak [8] for some books, Lie [9], Schwarz [10], Small [11,12], and Weierstrass [13,14] for some papers related to minimal surfaces in Euclidean geometry. Lie [9] studied the algebraic minimal surfaces and gave a table classifying these Citation: Güler, E. The Algebraic surfaces. See also Enneper [15], Güler [16], Nitsche [6], and Ribaucour [17] for details. Surfaces of the Enneper Family of Weierstrass [13] revealed a representation for minimal surfaces in three dimensional Maximal Surfaces in Three Euclidean space E . Almost one hundred years later, Kobayashi [18] gave an analogous Dimensional Minkowski Space. Weierstrass-type representation for conformal spacelike surfaces with mean curvature 2,1 Axioms 2022, 11, 4. https://doi.org/ identically 0, called maximal surfaces, in three dimensional Minkowski space E . 10.3390/axioms11010004 In this paper, we consider the Enneper family of maximal surfaces E for positive integers m  1 by using Weierstrass data (1, z ) for z 2 C, and then show that these Academic Editor: Anna Maria Fino 2,1 surfaces are algebraic in E . See Güler [16] for a Euclidean case of Enneper ’s algebraic Received: 3 December 2021 minimal surfaces family. Accepted: 20 December 2021 In Section 2, we give this family of real maximal surfaces in (r, q) and (u, v) coordinates Published: 22 December 2021 2,1 by using Weierstrass representation in E . In Section 3, we find irreducible algebraic equations defining surfaces E (u, v) in terms of running coordinates x, y, z, and a, b, c, and Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in also compute degrees and classes of E (u, v). Finally, we summarize all findings in tables published maps and institutional affil- in the last section, then give some open problems. iations. 2. Family of Enneper Maximal Surfaces n,1 t Let E : = fx = (x , , x , x ) jx 2 Rg,h,i be the (n + 1)-dimensional Lorentz– 1 n 0 i Minkowski (for short, Minkowski) space with Lorentzian metric hx, yi = x y + + 1 1 Copyright: © 2021 by the authors. x y x y . n n 0 0 Licensee MDPI, Basel, Switzerland. n,1 A vector x 2 E is called space-like if hx, xi > 0, time-like if hx, xi < 0, and light-like This article is an open access article n,1 if x 6= 0 and x, x = 0. A surface in E is called space-like (resp. time-like, light-like) if the h i distributed under the terms and induced metric on the tangent planes is a Riemannian (resp. Lorentzian, degenerate) metric. conditions of the Creative Commons 2,1 Now, let E be three dimensional Minkowski space with Lorentzian metric h. , .i = Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ x y + x y x y . We identify x and x without further comment. 1 1 2 2 3 3 4.0/). Axioms 2022, 11, 4. https://doi.org/10.3390/axioms11010004 https://www.mdpi.com/journal/axioms Axioms 2022, 11, 4 2 of 12 LetU be an open subset of C. A maximal curve is an analytic function J : U ! C such D E 0 0 0 ¶J 0 0 that hJ (z), J (z)i = 0, where z 2 U , and J := . In addition, if J , J = jJ j 6= 0, then ¶z J is a regular maximal curve. We then have maximal surfaces in the associated family of a maximal curve, given by the following Weierstrass representation theorem for ZMC (zero mean curvature) surfaces, or maximal surfaces. Kobayashi [18] found a Weierstrass type representation for space-like conformal 2,1 maximal surfaces in E : Theorem 1. Let g(w) be a meromorphic function and let f (w) be a holomorphic function, f g is analytic, defined on a simply connected open subset U  C such that f (w) does not vanish on U except at the poles of g(w). Then, 2 2 x(u, v) = Re f 1 + g , i f 1 g ,2 f g dw, (z = u + iv) (1) is a space-like conformal immersion with mean curvature identically 0 (i.e., space-like conformal maximal surface). Conversely, any spacelike conformal maximal surface can be described in this manner. Next, we give some facts about Weierstrass data, and a maximal curve to construct some maximal surfaces. Definition 1. A pair of a meromorphic function g and a holomorphic function f , ( f , g) is called Weierstrass data for a maximal surface. Lemma 1. The curve of Enneper of order m: 2m+1 2m+1 m+1 z z 2z # (z) = z + , i z , (2) 2m + 1 2m + 1 m + 1 is a maximal curve, z 2 Cf0g, i = 1, m 6= 1,1/2. 2,1 Therefore, we have h# , # i = 0 by using (2). Hence, in E , the Enneper maximal m m surface is given by E (u, v) = Re # (z)dz, (3) m m where z = u + iv. Im # (z)dz gives the adjoint minimal surface E (u, v) of the surface E (u, v) in (3). Then, we get the following: Corollary 1. The Weierstrass data (1, z ) of (3) is a representation of the Enneper maximal surface, where integer m  1. iq Considering the findings above with z = re , we get the following Enneper family of maximal surfaces: 0 1 1 2m+1 r cos(q) + r cos[(2m + 1)q] 2m+1 1 2m+1 @ A E (r, q) = r sin(q) + r sin[(2m + 1)q] (4) 2m+1 m+1 r cos[(m + 1)q] m+1 where m 6= 1,1/2. See Figure 1 for Enneper maximal surfaces E ,E ,E in r, q ( ) 1 2 3 coordinates. Axioms 2022, 11, 4 3 of 12 Figure 1. Enneper maximal surfaces, and its top views (Left): E (r, q), (Middle): E (r, q), 1 2 (Right): E (r, q). Hence, using the binomial formula, we obtain more clear representation of E (u, v) in (3): 2m+1 2m + 1 1 2m+1k x(u, v) = Re u + iv + å u (iv) , 2m+1 k=0 2m+1 2m + 1 i 2m+1k y(u, v) = Re iu v + å u (iv) , (5) 2m+1 k=0 m+1 m + 1 2 m+1k z(u, v) = Re å u (iv) . m+1 k=0 We study surfaceE (u, v) in (u, v) coordinates for m = 1, 2, . . . , 5, taking z = u + iv at Cartesian coordinates x, y, z, and also in inhomogeneous tangential coordinates a, b, c, by using Weierstrass representation equation. Next, we give a theorem about maximality of surface E (u, v) (see Figure 1, Left): Theorem 2. The surface 0 1 0 1 1 3 2 u uv + u x(u, v) 1 3 2 @ A @ A E (u, v) = = y(u, v) (6) v + u v v 2 2 u + v z(u, v) 2,1 which has Weierstrass data (1, z), is an Enneper maximal surface in E . Proof. The coefficients of the first fundamental form of the surface E (u, v) (E , for short) 1 1 are given by E = (l 1) = G, F = 0, 2 2 where l = u + v . That is, conformality holds. Then, the Gauss map e (u, v) of E is as 1 1 follows 2u 2v l + 1 e = , , , (7) l 1 l 1 l 1 Axioms 2022, 11, 4 4 of 12 where l 6= 1. The coefficients of the second fundamental form of E are given by 2(3l + 1) L = = N, l 1 M = 0. Then, we obtain the following mean curvature and the Gaussian curvature of the surface E : H = 0, 4(3l + 1) K = , (l 1) EN+G L2F M LN M respectively. Here, H = hs, si , K = hs, si , where hs, si = 1. Hence, 2 2 2(EGF ) EGF the Enneper surface is maximal surface with positive Gaussian curvature. Therefore, we obtain the following parametric equations of the higher order maximal Enneper surfaces E (u, v) = (x(u, v), y(u, v), z(u, v)) (see Figure 2 Middle for E , and Figure 2 Right for E .): 0 1 1 5 3 2 4 u 2u v + uv + u 1 5 2 3 4 @ A E u, v = , (8) ( ) v 2u v + u v v 2 3 2 u + 2uv 0 1 1 7 5 2 3 4 6 u 3u v + 5u v uv + u 1 7 2 5 4 3 6 @ A E (u, v) = , (9) v + 3u v 5u v + u v v 1 4 2 2 1 4 u + 3u v v 2 2 0 1 1 9 7 2 5 4 23 3 6 8 u 4u v + 14u v u v + uv + u 9 3 1 9 2 7 4 5 23 6 3 8 @ A E (u, v) = v 4u v + 14u v u v + u v v , (10) 9 3 5 3 2 4 u + 4u v 2uv 0 1 11 9 2 7 4 5 6 3 8 10 u 5u v + 30u v 42u v + 15u v + uv + u 11 2 9 4 7 6 5 8 3 10 @ A E (u, v) = v + 5u v 30u v + 42u v 15u v + u v v . (11) 1 1 6 4 2 2 4 6 u + 5u v 5u v + v 3 3 Figure 2. Enneper maximal surfaces (Left): E (u, v), (Middle): E (u, v), (Right): E (u, v). 1 2 3 Axioms 2022, 11, 4 5 of 12 3. Degree and Class of Enneper Maximal Surfaces In this section, using some elimination techniques, we derive the irreducible algebraic surface equation, degree and class of Enneper maximal surfaces familyE (u, v) for integers 2,1 1  m  5 in three dimensional Minkowski space E . Let us see some basic notions of the surfaces. Definition 2. The set of roots of a polynomial Q(x, y, z) = 0 gives an algebraic surface equation. An algebraic surface s is said to be of degree d when d = deg(s). Definition 3. At a point (u, v) on a surface s(u, v) = (x(u, v), y(u, v), z(u, v)), the tangent pl ane is given by X x + Yy Zz + P = 0, (12) where e = (X(u, v), Y(u, v), Z(u, v)) is the Gauss map, and P = P(u, v). Then, in inhomogeneous tangential coordinates a, b, c, we have the following surface: b s(u, v) = (a, b, c) = (X/P, Y/P, Z/P). (13) Therefore, we can obtain an algebraic equation Q(a, b, c) = 0 of b s(u, v) in inhomoge- neous tangential coordinates. Definition 4. The maximum degree of the algebraic equation Q(a, b, c) = 0 of b s(u, v) in inhomo- geneous tangential coordinates gives the cl ass of b s(u, v). See [6], for details of a Euclidean case. Hence, we obtain the following findings for degrees and classes of Enneper maximal surfaces that we use: 3.1. Degree We compute the irreducible algebraic surface equation Q (x, y, z) = 0 (see Figure 3, Left) of Enneper ’s maximal surface E u, v in (6) by using some elimination techniques. ( ) We find the following algebraic equation: 9 2 6 2 6 4 3 2 2 3 2 5 Q (x, y, z) = 64z + 432x z 432y z 1215x z 6318x y z + 3888x z 4 3 2 5 7 6 4 2 4 2 1215y z + 3888y z 1152z + 729x 2187x y 4374x z (14) 2 4 2 4 6 4 2 2 4 4 +2187x y + 6480x z 729y + 4374y z 6480y z + 729x z 2 2 2 3 4 2 3 5 1458x y z 3888x z + 729y z 3888y z + 5184z . Then, its degree number is 9. Next, we continue our computations to find Q (x, y, z) = 0 for integers m = 2, 3. We compute the following irreducible algebraic surface equations Q (x, y, z) = 0 (see Figure 3, Middle) and Q (x, y, z) = 0 (see Figure 3, Right) of the surfaces E (u, v) and 3 2 E (u, v), respectively, 25 3 20 Q (x, y, z) = 847 288 609 443z 4358 480 501 250x z 2 20 6 15 +13 075 441 503 750xy z 131 157 978 046 875x z 4 2 15 474 186 536 015 625x y z + 107 other lower degree terms, Axioms 2022, 11, 4 6 of 12 Q (x, y, z) = 2475 880 078 570 760 549 798 248 448z 4 42 +5079 604 062 565 768 134 821 675 008x z 2 2 42 30 477 624 375 394 608 808 930 050 048x y z 4 42 +5079 604 062 565 768 134 821 675 008y z 8 35 633 850 350 654 216 217 766 624 493 568x z +446 other lower degree terms. Therefore, Q (x, y, z) = 0 are the algebraic maximal surfaces of the surfaces E (u, v), m m where m = 2, 3, and they have degree numbers 25 and 49, respectively. Figure 3. Enneper algebraic maximal surfaces (Left): Q (x, y, z) = 0, (Middle): Q (x, y, z) = 0, 1 2 (Right): Q (x, y, z) = 0. 3.2. Class Now, we introduce the class of the surfaces E (u, v) for integers 1  m  4. The case m = 5, marked with “*” presented in tables of Section 4. Computing the irreducible algebraic surface equations Q (a, b, c) = 0, we obtain the Gauss maps e (u, v) (see Figure 4 m m for e , e , e ) for integers 1  m  5 of the surfaces E (u, v), and we also generalize them 2 3 m as follows: u v l + 1 e = 2 ,2 , , l 1 l 1 l 1 2 2 2 u v 2uv l + 1 e = 2 ,2 , , (15) 2 2 2 l 1 l 1 l 1 3 2 2 3 3 u 3uv 3u v v l + 1 e = 2 ,2 , , (16) 3 3 3 l 1 l 1 l 1 4 2 2 4 3 3 4 u 6u v + v 4u v 4uv l + 1 e = 2 ,2 , , (17) 4 4 4 l 1 l 1 l 1 5 3 2 4 4 2 3 5 5 u 10u v + 5uv 5u v 10u v + v l + 1 e = 2 ,2 , , (18) 5 5 5 l 1 l 1 l 1 m m Re z Im z z + 1 ( ) ( ) j j e = 2 ,2 , , (z = u + iv, jzj = l). (19) m m m jzj 1 jzj 1 jzj 1 2 2 (l3)(u +v ) Using (6), (7), (12) and (13), with P (u, v) = , we get the following surface 3(l1) E (u, v) (see Figure 5, Left) in inhomogeneous tangential coordinates: 6u 6v 3(l + 1) a = , b = , c = . 2 2 2 2 2 2 (u + v )(l 3) (u + v )(l 3) (u + v )(l 3) Axioms 2022, 11, 4 7 of 12 2 2 where l = u + v , l 6= 3, u, v 6= 0. Therefore, we compute Enneper ’s irreducible algebraic ˆ b maximal surface equation Q (a, b, c) = 0 (see Figure 6, Left) of the surface E (u, v): 1 1 6 4 2 4 2 2 4 2 2 2 6 4 2 Q (a, b, c) = 4a 4a b 3a c 4a b + 6a b c + 4b 3b c 4 2 3 4 2 3 4 2 2 4 18a c + 12a c + 18b c 12b c + 9a + 18a b + 9b . So, Enneper ’s maximal surface E (u, v) in (6) has class number 6. Figure 4. Top views of the Gauss maps of the surfacesE (u, v) (Left): e (u, v), (Middle): e (u, v), m=1,2,3 1 2 (Right): e (u, v). Next, we continue our computations to find Q for integers 2, 3, 4. To find the class of surface E (u, v) (see Figure 5, Middle), we use (9), (12), (13) and (16). Calculating 3 2 2 4 u 3uv (l 5) ( ) P (u, v) = , we get the following surface E inhomogeneous tangential 2 2 15(l 1) coordinates: 2 2 2 15 u v 15uv 15 l + 1 a = , b = , c = , 3 2 2 3 2 2 3 2 2 2(u 3uv )(l 5) (u 3uv )(l 5) 4(u 3uv )(l 5) 2 2 2 where l = u + v , l 6= 5, u, v 6= 0. In the inhomogeneous tangential coordinates a, b, c, we find the following irreducible algebraic surface equation Q (a, b, c) = 0 (see Figure 6, Middle) of the surface E (u, v): 16 4 14 6 14 4 2 Q (a, b, c) = 2176 782 336a b + 5804 752 896a b 4837 294 080a b c 12 8 12 6 2 +2902 376 448a b 8062 156 800a b c + 120 other lower degree terms. ˆ b Hence, Q (a, b, c) = 0 is the algebraic surface of the surfaceE (u, v), and Enneper ’s maximal 2 2 surface E (u, v) in (8) has class number 20. Using similar ways, we compute the irreducible algebraic surface equation Q (a, b, c) = 0 (see Figure 6, Right) of surfaceE (u, v) (see Figure 5, Right) as follows: 42 40 2 Q (a, b, c) = 26 623 333 280 885 243 904a 718 829 998 583 901 585 408a b 40 2 38 4 104 829 374 793 485 647 872a c + 6868 819 986 468 392 927 232a b 38 2 2 +2935 222 494 217 598 140 416a b c + 774 other lower degree terms. Axioms 2022, 11, 4 8 of 12 ˆ b Q (a, b, c) = 0 is the algebraic surface of the surface E (u, v), and Enneper ’s maximal 3 3 surface E (u, v) in (9) has class number 42. We also compute the following irreducible ˆ b algebraic surface equation Q (a, b, c) = 0 of the surface E (u, v): 4 4 64 8 Q (a, b, c) = 42949672960000000000000000000000000000000000000000a b 62 8 2 247390116249600000000000000000000000000000000000000a b c 60 12 962072674304000000000000000000000000000000000000000a b 60 10 2 +247390116249600000000000000000000000000000000000000a b c 60 8 4 +667953313873920000000000000000000000000000000000000a b c +2604 other lower degree terms. ˆ b Q (a, b, c) = 0 is an algebraic surface of E (u, v), and Enneper ’s maximal surface E (u, v) 4 4 4 in (10) has class number 72. Figure 5. Enneper maximal surfaces in inhomogeneous tangential coordinates (Left): E (u, v), b b (Middle): E (u, v), (Right): E (u, v). 2 3 We obtain the following functions P (u, v), where 1  i  6, 2 2 3 2 2 (u v )(l3) 4(u 3uv )(l 5) P (u, v) = , P (u, v) = , 1 2 2 3(l1) 15(l 1) 4 2 2 4 3 5 3 2 4 4 3 u 6u v +v (l 7) 8 u 10u v +5uv l 9 ( ) ( )( ) P (u, v) = , P (u, v) = , 3 4 3 4 14(l 1) 45(l 1) 6 4 2 2 4 6 5 7 5 2 3 4 6 6 5(u 15u v +15u v v )(l 11) 12(u 21u v +35u v 7uv )(l 13) P (u, v) = , P (u, v) = . 5 5 6 6 33(l 1) 91(l 1) We generalize the above functions, and give the following results: Corollary 2. The functions P for integers m, are given by m1 h i 2k1 2(2k 1) l (2k + 1) 2k P =  Re z , 2k1 2k1 (k + 1)(2k + 1) l 1 h i 2k 4k l (2k + 1) 2k+1 P = Re z , 2k 2k (2k + 1)(4k + 1) l 1 Axioms 2022, 11, 4 9 of 12 where integers k  1, z = u + iv and z = l. j j b b So far, we find surfaces E and E . By using E E , e e , and also (12), (13), we 1 2 3 5 3 5 obtain the following surfaces: E (u, v) = a, b, c : ( ) 0 1 2u b @ A E = 2v , 2 2 (u v )(l 3) l + 1 0 1 2 2 2 u v @ A E = 4uv , 3 2 2 4(u 3uv )(l 5) l + 1 0  1 3 2 2 u 3uv 2 3 b @ A E = 2 u v v , 4 2 2 4 3 3 u 6u v + v (l 7) ( ) l + 1 0  1 4 2 2 4 2 u 6u v + v 3 3 b @ A E = 2 4u v 4uv , 5 3 2 4 4 8(u 10u v + 5uv )(l 9) l + 1 0 1 5 3 2 4 2 u 10u v + 5uv 4 2 3 5 b @ A E = 2 5u v 10u v + v . 6 4 2 2 4 6 5 5(u 15u v + 15u v v )(l 11) l + 1 We also generalize the above functions, and find the following results: Corollary 3. The surfaces S (u, v) for integers m, are given by m1 0 1 0 1 2k1 2Re z B C k(4k 1) b B C @ A 2k1 E (u, v) =    = b , 2k1 2Im z @ A 2k1 2k (2k 1) l (2k + 1) Re z 2k1 jzj + 1 0 1 0 1 2k 2Re z B C 2k + 1 4k + 1 ( )( ) b B C @ A 2k E (u, v) =    = b , 2k 2Im z @ A 2k 2k+1 4k l (4k + 1) Re z 2k jzj + 1 where integers k  1, z = u + iv and jzj = l. 2,1 Corollary 4. In E , the relations between the Enneper maximal surface E (u, v) in the inho- m1 mogeneous tangential coordinates and the Gauss map e (u, v) of the Enneper maximal surface m1 E (u, v) in the cartesian coordinates are given by m1 2k1 k(4k 1) l 1 E (u, v) = e (u, v), 2k1 2k1 2k1 2k (2k 1) l (2k + 1) Re z 2k (2k + 1)(4k + 1) l 1 E (u, v) = e (u, v), 2k 2k 2k 2k+1 4k l (2k + 1) Re z where integers k  1, z = u + iv and z = l. j j Axioms 2022, 11, 4 10 of 12 Figure 6. Enneper ’s algebraic maximal surfaces in inhomogeneous tangential coordinates ˆ ˆ ˆ (Left): Q (a, b, c) = 0, (Middle): Q (a, b, c) = 0, (Right): Q (a, b, c) = 0. 1 2 3 4. Conclusions To reveal the irreducible algebraic surface equations of the Enneper maximal surfaces 2,1 E (u, v) in E , we have tried a series of standard techniques in elimination theory: only Sylvester by hand for Q (x, y, z) = 0, and then projective (Macaulay) and sparse multivari- ate resultants implemented in the Maple software [19] package multires for Q (x, y, z) = 0 and Q (a, b, c) = 0. Maple’s native implicitization command Implicitize, and implicitization based on Maple’s native implementation of the Groebner Basis. For the latter, we implemented in Maple the method in [20] (Chapter 3, p. 128). Under reasonable time, we only succeed for m = 1, 2 in all above methods. For m = 3, the successful method we have tried was to compute the equation defining the elimination ideal using the Groebner Basis package FGb of Faugère in [21]. The time required to output the irreducible algebraic surface equations Q (x, y, z) = 0 (resp. Q (a, b, c) = 0) for integers 1  m  3 and polynomials defining the elimination ideal was under reasonable seconds determined by the following Table 1 (resp. Table 2). For the degree (resp. class) of the irreducible algebraic surface equation Q (x, y, z) = 0 ˆ b (resp. Q (a, b, c) = 0) of the surface E u, v (resp. E (u, v)), marked with “” in Table 1 ( ) 5 4 5 (resp. Table 2), was rejected (i.e., “out of memory”) by Maple 17 on a laptop Pentium Core i5-4310M 2.00 GHz, 4 GB RAM, with the time given in CPU seconds. Hence, we propose the following: Proposition 1. For integers m  1, degree number of the irreducible algebraic surfaces Q (x, y, z) = 0 in the Cartesian coordinates is of (2m + 1) , and class number of irreducible algebraic surfaces Q (a, b, c) = 0 in inhomogeneous tangential coordinates is of 2m(2m + 1) of the (1, z )-type real Enneper maximal surfacesE u, v . ( ) Open Problems Here, we give some problems that we could not find the answers in this paper: Problem 1. Find the irreducible Enneper algebraic maximal surface eq. Q (x, y, z) = 0 in the m4 cartesian coordinates by using the parametric equation of the Enneper maximal surface E u, v . ( ) m4 Problem 2. Find the irreducible Enneper algebraic maximal surface eq. Q (a, b, c) = 0 in the m5 inhomogeneous tangential coordinates by using the parametric equation of the Enneper maximal surface E (u, v). m5 Finally, we give all findings in Tables 1 and 2. Axioms 2022, 11, 4 11 of 12 Table 1. Results for the Enneper algebraic maximal surfaces Q (x, y, z) = 0. Algebraic Degree Number Gröbner FGb Surface of Surface of Terms Time (s) Time (s) Q 9 23 0.266 0.041 Q 25 112 321.953 0.835 Q 49 451  266.854 Q 81 . . . . . . . . . . . . . . . (2m + 1) Table 2. Results for the Enneper algebraic maximal surfaces Q (a, b, c) = 0. Algebraic Class Number Gröbner FGb Surface of Surface of Terms Time (s) Time (s) Q 6 14 0.94 0.030 Q 20 125 61.152 0.114 Q 42 779  125.904 Q 72 2609  1306.718 . . . . . . . . . . . . . . . Q 2m(2m + 1) Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Acknowledgments: The author is grateful to the referees for their suggestions, which improved the presentation of the paper. Conflicts of Interest: The author declares no conflict of interest. References 1. Darboux, G. 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Etude des elassoides ou surfaces a courbure moyenne nulle. Mem. Cour. et Mem. Sav. Etr. Acad. R. Sci. Belg. Bruxelles 1882, 44, 236. 18. Kobayashi, O. Maximal surfaces in the 3-dimensional Minkowski space L . Tokyo J. Math. 1983, 6, 297–309. [CrossRef] 19. Maple Software, (Version 17); Waterloo Maple Inc.: Waterloo, ON, Canada. Available online: https://www.maplesoft.com/ (accessed on 10 October 2021). 20. Cox, D.; Little, J.; O’Shea, D. Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3rd ed.; Undergraduate Texts in Mathematics; Springer: New York, NY, USA, 2007. 21. Faugère, J.C. FGb: A library for computing Gröbner bases. In Proceedings of the Third International Congress Conference on Mathematical Software (ICMS’10), Kobe, Japan, 13–17 September 2010; Springer: Berlin/Heidelberg, Germany, 2010; pp. 84–87.

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Axioms – Multidisciplinary Digital Publishing Institute

Published: Dec 22, 2021

Keywords: Weierstrass representation; Enneper maximal surface; algebraic surface; degree; class

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