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MINIMAL SURFACES THAT GENERALIZE the ENNEPER's SURFACE 1. Introduction

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MINIMAL SURFACES THAT GENERALIZE the ENNEPER's SURFACE 1. Introduction Novi Sad J. Math. Vol. 40, No. 2, 2010, 17-22 MINIMAL SURFACES THAT GENERALIZE THE ENNEPER'S SURFACE Dan Dumitru1 Abstract. In this article we give the construction of some minimal surfaces in the Euclidean spaces starting from the Enneper's surface. AMS Mathematics Subject Classi¯cation (2000): 53A05 Key words and phrases: Enneper's surface, minimal surface, curvature 1. Introduction We start with the well-known Enneper's surface which is de¯ned in the 2 3 u3 3-dimensional Cartesian by the application x : R ! R ; x(u; v) = (u ¡ 3 + 2 v3 2 2 2 uv ; v¡ 3 +vu ; u ¡v ): The point (u; v) is mapped to (f(u; v); g(u; v); h(u; v)), which is a point on Enneper's surface [2]. This surface has self-intersections and it is one of the ¯rst examples of minimal surface. The fact that Enneper's surface is minimal is a consequence of the following theorem: u3 2 Theorem 1.1. [1] Consider the Enneper's surface x(u; v) = (u ¡ 3 + uv ; v ¡ v3 2 2 2 3 + vu ; u ¡ v ): Then: a) The coe±cients of the ¯rst fundamental form are (1.1) E = G = (1 + u2 + v2)2;F = 0: b) The coe±cients of the second fundamental form are (1.2) e = 2; g = ¡2; f = 0: c) The principal curvatures are 2 2 (1.3) k = ; k = ¡ : 1 (1 + u2 + v2)2 2 (1 + u2 + v2)2 Eg+eG¡2F f It follows that the mean curvature H = EG¡F 2 is zero, which means the Enneper's surface is minimal. If we look more carefully to the formula of the mean curvature ,H , of the E(e+g)¡2f¢0 Enneper's surface we observe that H = EG¡F 2 = 0; because e + g = 0: Starting with this observation we want to ¯nd out new examples of minimal surface in R3; for which the ¯rst and the second fundamental forms are in some sense similar to the ¯rst and the second fundamental forms of the Enneper's surface. That means if we consider a surface in R3;' : R2 ! R3 such that the ¯rst fundamental form satis¯es E = G, F = 0 and the second fundamental form satis¯es e + g = 0, with f arbitrary, this surface will be minimal. 1Spiru Haret University of Bucharest, Faculty of Mathematics and Computer Science, 13 Ghica Str., Bucharest, Romania, email: [email protected] 18 Dan Dumitru 2. Main results We consider an arbitrary surface in R3;' : R2 ! R3; where the compo- nents of the surface are '(u; v) = (f(u; v); g(u; v); h(u; v)); f; g; h : R2 ! R: 0 0 0 0 0 0 0 0 00 00 00 00 We denote by 'u := (fu; gu; hu);'v := (fv; gv; hv);'u2 := (fu2 ; gu2 ; hu2 ); 00 00 00 00 00 00 00 00 'v2 := (fv2 ; gv2 ; hv2 );'uv := (fuv; guv; huv): We want to ¯nd ' di®erent from the Enneper's surface such that the ¯rst fundamental form of ' satis¯es E0 = G0, F0 = 0 and the second fundamental form of ' satis¯es e0 + g0 = 0: We know that the coe±cients of the ¯rst fundamental form are given by the following formulas: 0 2 0 2 0 2 0 2 E0 = j'uj = (fu) + (gu) + (hu) ; 0 2 0 2 0 2 0 2 (2.1) G0 = j'vj = (fv) + (gv) + (hv) ; 0 0 0 0 0 0 0 0 F0 =< 'u;'v >= fufv + gugv + huhv: 0 2 0 2 0 2 0 2 0 2 0 2 So, E0 = G0 implies (f ) + (g ) + (h ) = (f ) + (g ) + (h ) and 0 0 0 0 u 0 0 u u v v v F0 = 0 implies fufv + gugv + huhv = 0: 0 0 'u£'v If we consider N = 0 0 the unit normal vector to the surface then the j'u£'v j coe±cients of the second fundamental form are given by the formulas: 00 e0 =< 'u2 ; N >; 00 (2.2) g0 =< 'v2 ; N >; 00 f0 =< 'uv; N > : 00 00 So, e0 + g0 =< 'u2 + 'v2 ; N > and a su±cient condition that e0 + g0 = 0 is 00 00 that 'u2 + 'v2 = (0; 0; 0); which implies 4f = 4g = 4h = 0; where 4 is the Laplace operator. Hence, because we are looking for new surfaces which satisfy E0 = G0; F0 = 0; and e0 + g0 = 0 it is su±cient to look for solution of the following system 4f = 4g = 4h = 0 (that means f; g; h are harmonic); 0 0 0 0 0 0 (2.3) fufv + gugv + huhv = 0; 0 2 0 2 0 2 0 2 0 2 0 2 (fu) + (gu) + (hu) = (fv) + (gv) + (hv) : Remark 2.1. Every solution of the system (2.3) will give us a minimal surface in R3: In general, it is very di±cult to give the general solution of the above system, but we are interested in some solution of it, to give examples of minimal surfaces. We keep in mind that we want to generalize the Enneper's surface, so we will consider that f; g; h satisfy the following conditions, because the components of the Enneper's surface satisfy these conditions: 0 0 0 0 00 0 00 0 (2.4) fv = gu; fu + gv = 2; fv2 = hu; fuv = ¡hv: Minimal surfaces that generalize the Enneper's surface 19 Remark 2.2. We only have to consider f to be harmonic, because from (2.4) it follows that g and h are also harmonic. Thus, from (2.4) the second equation of the system (2.3) becomes: 0 0 0 0 00 00 0 = fufv + fv gv + fv2 (¡fuv) 0 0 0 00 00 = fv(fu + gv) ¡ fv2 fuv (2.5) 0 00 00 = 2fv ¡ fv2 fuv ) 0 00 00 2fv ¡ fv2 fuv = 0; and the third equation becomes: 0 2 0 2 0 2 0 2 0 2 0 2 (fu) + (gu) + (hu) = (fv) + (gv) + (hv) ) 0 2 0 2 00 2 0 2 0 2 00 2 (fu) + (fv) + (fv2 ) = (fv) + (2 ¡ fu) + (¡fuv) ) (2.6) 0 2 0 2 00 2 0 2 0 2 0 00 2 (fu) + (fv) + (fv2 ) = (fu) + (fv) ¡ 4fu + 4 + (fuv) ) 0 00 2 00 2 4 ¡ 4fu ¡ (fv2 ) + (fuv) = 0: Now we reduced our system (2.3) to a particular one which depends only on f; and that is: 4f = 0 (that means f is harmonic), 0 00 00 (2.7) 2fv ¡ fv2 fuv = 0, 0 00 2 00 2 4 ¡ 4fu ¡ (fv2 ) + (fuv) = 0: We observe that one particular solution of the second equation of the system 0 2 (2.7) is fv = (au+b)( a v+c); where a 6= 0; b; c 2 R. We will see that this solution will lead to some generalization of the Enneper's surface. 1 2 So, we have that f(u; v) = (au + b)( a v + cv + d) + Ã(u); where à : R ! R is some function. Hence 0 0 1 2 fu = à (u) + a( a v + cv + d); (2.8) 00 2 b fv2 = a (au + b) = 2u + 2 a ; 00 2 fuv = a( a v + c) = 2v + ac; and if we introduce them in the third equation of the system (2.7) we obtain 0 00 2 00 2 b 2 2 4fu = 4 ¡ (fv2 ) + (fuv) = 4 ¡ (2u + 2 a ) + (2v + ac) ) (2.9) 0 2 b b2 2 a2c2 fu = 1 ¡ u ¡ 2 a u ¡ a2 + v + acv + 4 : 0 0 2 But fu = à (u) + v + acv + ad; and then 0 2 b b2 a2c2 à (u) = 1 ¡ u ¡ 2 a u ¡ a2 + 4 ¡ ad ) u3 b 2 b2 a2c2 Ã(u) = u ¡ 3 ¡ a u + (¡ a2 + 4 ¡ ad)u + e ) (2.10) u3 2 f(u; v) = (u ¡ 3 + uv ) b 2 b 2 b2 a2c2 +[¡ a u + acuv + a v + (¡ a2 + 4 )u + bcv + bd + e]; where a 6= 0; b; c; d; e 2 R: 20 Dan Dumitru Using (2.4) we obtain that 0 0 2 fv = gu = (au + b)( a v + c) ) a 2 2 g(u; v) = ( 2 u + bu + k)( a v + c) + θ(v); where θ : R ! R is a function. But, 0 0 gv = 2 ¡ fu 2 b b2 2 a2c2 = 2 ¡ (1 ¡ u ¡ 2 a u ¡ a2 + v + acv + 4 ) 0 2 b 2k = θ (v) + u + 2 a u + a ) 0 2 b2 a2c2 2k (2.11) θ (v) = ¡v ¡ acv + (1 + a2 ¡ 4 ¡ a ) ) v3 ac 2 b2 a2c2 2k θ(v) = ¡ 3 ¡ 2 v + (1 + a2 ¡ 4 ¡ a )v + l ) v3 2 ac 2 b g(u; v) = (v ¡ 3 + vu ) + [¡ 2 v + 2 a uv ac 2 b2 a2c2 + 2 u + ( a2 ¡ 4 )v + bcu + kc + l]; where k; l 2 R. 0 00 b 2 b Using again (2.4) we get that hu = fv2 = 2u+2 a ) h(u; v) = u +2 a u+Á(v), 0 00 where Á : R ! R is a function and hv = ¡fuv = ¡(2v + ac). So, we have that 0 Á (v) = ¡2v ¡ ac ) Á(v) = ¡v2 ¡ acv + j ) (2.12) 2 2 b h(u; v) = (u ¡ v ) + 2 a u ¡ acv + j; with j 2 R: With loss of generality we can consider the parameters l = k = j = e = d = 0; and obtain the following minimal surfaces ' : R2 ! R3;'(u; v) = (f(u; v); g(u; v); h(u; v)); where f; g; h are given by: u3 2 f(u; v) = (u ¡ 3 + uv )+ b 2 b 2 b2 a2c2 [¡ a u + acuv + a v + (¡ a2 + 4 )u + bcv]; (2.13) v3 2 g(u; v) = (v ¡ 3 + vu )+ ac 2 b ac 2 b2 a2c2 [¡ 2 v + 2 a uv + 2 u + ( a2 ¡ 4 )v + bcu]; 2 2 b h(u; v) = (u ¡ v ) + 2 a u ¡ acv; where a 6= 0, b; c 2 R: b If we denote ® = a and ¯ = ac then ®¯ = bc and we can consider a family of minimal surfaces which are only indexed by two real parameters ®; ¯.
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