Super-conformal surfaces associated with null complex holomorphic curves

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1 Bull. London Math. Soc. 41 (2009) C 2009 London Mathematical Society doi: /blms/bdp005 Super-conformal surfaces associated with null complex holomorphic curves Katsuhiro Moriya Abstract A correspondence from a null complex holomorphic curve in four-dimensional complex Euclidean space to a super-conformal surface in four-dimensional Euclidean space is defined by the quaternionic theory of surfaces. As an application, a transformation of super-conformal surfaces is defined. 1. Introduction The construction of surfaces is an important and interesting problem. A transformation of a surface is a way to construct a new surface from a given one. The integrable system method is a powerful method for defining transformations of surfaces. The theory of quaternionic holomorphic vector bundles [2, 4, 7] is influenced by the theory of integrable systems and is also a powerful method for defining transformations of surfaces. Burstall, Ferus, Leschke, Pedit, and Pinkall [2] defined the Bäcklund transforms and the Darboux transforms for Willmore surfaces in a four-dimensional sphere in terms of quaternionic holomorphic vector bundles. Generalizing this, Bohle [1] defined the Bäcklund transforms for conformal maps from a Riemann surface into a four-dimensional sphere, and Leschke and Pedit [5] defined the Bäcklund transformations for quaternionic holomorphic curves. It was shown in [2] that a super-conformal surface in four-dimensional Euclidean space R 4 is a Euclidean realization of a twistor projection of a complex holomorphic curve in threedimensional complex projective space CP 3. An arbitrary choice of two Euclidean realizations defines a pair of super-conformal surfaces. Then we can consider one super-conformal surface as a transformation of another super-conformal surface. Since the relation between a complex holomorphic curve in CP 3 and a super-conformal surface is unclear in this correspondence, it is an interesting problem to define a more explicit correspondence between a complex holomorphic curve and a super-conformal surface. In this paper, we define two correspondences from a null complex holomorphic curve in fourdimensional complex Euclidean space C 4 to a super-conformal surface in R 4 by the quaternionic theory of surfaces. The real part and the imaginary part of a null complex holomorphic curve in C 4 are minimal surfaces in R 4. They have a common left normal vector and a common right normal vector. Our correspondences are obtained by a simple calculation using these minimal surfaces, their left normal vector, and their right normal vector. The composition of these two correspondences defines a transformation between super-conformal surfaces in R 4. These correspondences are an application of a transformation that we define between conformal immersions from a Riemann surface M to R 4. This transformation preserves the right normal vector of a conformal immersion. Received 1 June 2007; revised 21 October 2008; published online 4 March Mathematics Subject Classification 53C42 (primary), 53A10 (secondary). Partly supported by the Grant-in-Aid for Young Scientists (B) no , The Ministry of Education, Culture, Sports, Science and Technology, Japan.

2 328 KATSUHIRO MORIYA Remark 1. different way. Our theorem was proved independently by Dajczer and Tojeiro [3] in a 2. Surfaces in terms of quaternions We review the quaternionic theory of surfaces [2, 7]. The set of quaternions H is the unitary real algebra generated by the symbols i, j, andk with relations i 2 = j 2 = k 2 = 1, ij = ji = k, jk = kj = i, ki = ik = j. For a = a 0 + a 1 i + a 2 j + a 3 k H with a 0, a 1, a 2,anda 3 R, we denote by â = a 0 a 1 i a 2 j a 3 k its quaternionic conjugation. We identify H with R 4 by the identification of a 0 + a 1 i + a 2 j + a 3 k H with (a 0,a 1,a 2,a 3 ) R 4. We identify the set of complex numbers C with the unitary real algebra generated by the symbol i with the relation i 2 = 1. Let M be a Riemann surface with complex structure J. We denote by Ω 1 (M) the set of one-forms on M. We define an operator :Ω 1 (M) Ω 1 (M) by ω = ω J for every ω Ω 1 (M). An immersion F : M H is said to be conformal if the induced metric on M by F from H is compatible to the complex structure of M. For an immersion F : M H, the following three conditions are equivalent (see [2, Definition 2, Remark 2]). (1) The immersion F is conformal. (2) There exists a unique smooth map N : M H such that (df )=N(dF ). (3) There exists a unique smooth map R : M H such that (df )=(df )( R). The map N is called the left normal vector of F and the map R is called the right normal vector of F. By the definition of a left normal vector and a right normal vector, we have N 2 = R 2 = 1. Hence N(dN) = (dn)n, R(dR) = (dr)r. Let F : M H be a conformal immersion with left normal vector N and right normal vector R. For the mean curvature vector H : M H of F, the equations 2Ĥ(dF )= (dr)+r(dr), 2(dF )Ĥ = (dn)+n(dn) are satisfied [2, Proposition 8]. A conformal immersion F : M H is called a minimal surface if its mean curvature vanishes everywhere. A conformal immersion F is a minimal surface if and only if d (df ) = 0. In terms of a left normal vector N and a right normal vector R, a conformal immersion F is a minimal surface if and only if (dn) = N(dN) =(dn)n, or, equivalently, (dr) = R(dR) =(dr)r. A conformal immersion F : M H is called a super-conformal surface if its curvature ellipse is a circle (see [2]). A conformal immersion F : M H is a super-conformal surface [2] ifand only if one of the following equations holds for the left normal vector N or the right normal vector R of F : (dn) =N(dN) = (dn)n, (2.1) (dr) =R(dR) = (dr)r. (2.2) The left normal vector N and the right normal vector R of a minimal surface define (branched) conformal immersions F N = N : M H and F R = R : M H with

3 SUPER-CONFORMAL SURFACES 329 (df N )= N(dF N )=(df N )N and (df R )= R(dF R )=(df R )R. In particular, F N and F R are super-conformal. Similarly, the left normal vector N with (2.1) of a super-conformal surface defines a (branched) minimal immersion F N = N : M H with (df N )=N(dF N )= (df N )N, and the right normal vector R with (2.2) of a super-conformal surface defines a (branched) minimal immersion F R = R : M H with (df R )=R(dF R )= (df R )R.Asuperconformal surface is a Euclidean realization of a twistor projection of a complex holomorphic curve in the three-dimensional complex projective space [2, Theorem 5]. Let F : M H and G : M H be conformal immersions with their left normal vector N such that F Gλ for every λ H. Then there exists a nowhere-vanishing conformal (branched) immersion ψ 0 : M H such that (df )=(dg)ψ 0 (see [5, 7]). Then (dg)ψ 0 is called a Weierstrass representation of F in Bohle [1]. Similarly, if F : M H and G : M H are conformal immersions with their right normal vector R such that F λg for every λ H, then there exists a nowhere-vanishing conformal (branched) immersion ψ 1 : M H such that (df )=ψ 1 (dg). Then ψ 1 (dg) is also called a Weierstrass representation of F. Let F : M H be a conformal immersion with right normal vector R F and let G : M H be a conformal immersion with left normal vector N G. Then (df ) (dg) = 0 if and only if R F = N G [2, Proposition 16]. The map G was called a forward Bäcklund transformation of F and the map F is called a backward Bäcklund transformation of G in Bohle [1]. We see that a Bäcklund transformation of a minimal surface is a super-conformal surface and a forward Bäcklund transformation or a backward Bäcklund transformation of a super-conformal surface is a minimal surface, depending on whether (2.1) or(2.2) holds. 3. Transformations of surfaces We make an observation on Bäcklund transformations and Weierstrass representations. We call a pair (F, G) of nowhere-vanishing conformal immersions F : M H and G : M H a Bäcklund pair if (df ) (dg) = 0. Since d(fg)=(df )G + F (dg), there exists a conformal immersion from M to H with its Weierstrass representation (df )G if and only if there exists a conformal immersion from M to H with Weierstrass representation F (dg). We say that a Bäcklund pair (F, G) isexact if there exists a conformal immersion with Weierstrass representation (df )G. We assume that (F 0,G 0 )isanexactbäcklund pair, that N F0 is the left normal vector of F 0, and that R G0 is the right normal vector of G 0. Then there exist conformal immersions F 1 : M H and G 1 : M H with Weierstrass representations (df 0 )G 0 and F 0 (dg 0 ), respectively. We see that F 1 = F 0 G 0 G 1 up to constants. Indeed, d(f 0 G 0 G 1 )=(df 0 )G 0 + F 0 (dg 0 ) (dg 1 )=(df 0 )G 0 = df 1, d(f 0 G 0 F 1 )=(df 0 )G 0 + F 0 (dg 0 ) (df 1 )=F 0 (dg 0 )=dg 1. Because (F 0,G 0 )isabäcklund pair if and only if (Ĝ0, ˆF 0 )isabäcklund pair, we focus on G 0 and G 1. Then G 1 is considered as a transformation of G 0 preserving the right normal vector as follows. Lemma 1. Let G 0 : M H be a nowhere-vanishing conformal immersion with left normal vector N G0 and right normal vector R G0. We assume that there exists a nowhere-vanishing conformal immersion F 0 : M H such that (F 0,G 0 ) is an exact Bäcklund pair. Let F 1 : M H be a conformal immersion such that (df 0 )G 0 is its Weierstrass representation. Then G 1 = F 0 G 0 F 1 is a conformal immersion with left normal vector F 0 N G0 F0 1 and right normal vector R G0.

4 330 KATSUHIRO MORIYA The properties of G 0 that only depend on its right normal vector are preserved under this transformation. For example, this transformation is a transformation between minimal surfaces, between super-conformal surfaces, between Lagrangian surfaces (see [6]), and between Hamiltonian-minimal Lagrangian surfaces (see [6]). The Willmore energy is preserved under these transformations (see [7]). However, this transformation may not be a transformation between Willmore surfaces since it does not transform non-conformal variations. Hence this transformation is a transformation between constrained Willmore surfaces (see [5]). 4. Null complex holomorphic curves and super-conformal surfaces Applying Lemma 1, we define a correspondence from a null complex holomorphic immersion in C 4 to a super-conformal surface in R 4. We consider the complexification C R H of H as C 4. Let =2 1 (d i d) and = 2 1 (d + i d). Then a smooth map φ : M C 4 with φ = 0 is a complex holomorphic map. For a complex holomorphic immersion φ = F 0 + if 1 with smooth maps F 0 : M H and F 1 : M H, the map F 0 is a conformal immersion with left normal vector N and right normal vector R if and only if the map F 1 is a conformal immersion with left normal vector N and right normal vector R. Indeed, an immersion φ is complex holomorphic if and only if (df 0 )= (df 1 ). Then F 0 and F 1 are minimal surfaces since d (df 0 )= d(df 1 )=0 and d (df 1 )=d(df 0 ) = 0. Hence φ is a null complex holomorphic immersion, that is, 3 m=0 ( φ m)( φ m )=0 for φ = φ 0 + φ 1 i + φ 2 j + φ 3 k with smooth maps φ m : M C (m = 0, 1, 2, 3). Our correspondence is defined in the following theorem. Theorem 1. Let F 0 + if 1 : M C 4 be a nowhere-vanishing null complex holomorphic immersion such that the minimal surfaces F 0 : M H and F 1 : M H are nowhere vanishing and have both left normal vector N and right normal vector R.IfN and R are immersions, then G 0 = NF 0 F 1 : M H and G 1 = F 0 R F 1 : M H are super-conformal surfaces such that (dg 0 )= N(dG 0 )=(dg 0 )F0 1 NF 0 and that (dg 1 )= F 0 RF0 1 (dg 1 )=(dg 1 )R. Proof. Let F 0 + if 1 : M C 4, N : M H, andr : M H be as above. Then (df 0 )= (df 0 )( R) = (df 1 ). Hence (F 0,R)isanexactBäcklund pair and (df 0 )R is a Weierstrass representation of F 1. Since F 0 is a minimal surface, we have (dr) = R(dR) =(dr)r. Then G 1 is a conformal immersion with left normal vector F 0 RF0 1 and right normal vector R by Lemma 1. Since d( R) ( R)d( R) = 0, the map G 1 is a super-conformal surface. The rest of the proof is similar. The above theorem yields a transformation on super-conformal surfaces. We give a direct proof of it as follows. Corollary 1. Let G 0 : M H be a super-conformal surface with left normal vector N G0 such that (dn G0 ) N G0 (dn G0 )=0.IfN G0 is an immersion and a smooth map F 0 : M H defined by (dg 0 )=(dn G0 )F 0 is a conformal immersion such that its right normal vector R F0 is an immersion, then a smooth map G 1 defined by G 1 = N G0 F 0 F 0 R F0 G 0 is a superconformal surface with right normal vector R F0 such that (dr F0 )+R F0 (dr F0 )=0. Proof. Let G 0, N G0, F 0,andR F0 be as above. Since 0 = d(dg 0 )=(dn G0 ) (df 0 ), the pair (N G0,F 0 )isanexactbäcklund pair and (df 0 )= N G0 (df 0 ). Hence F 0 is a minimal surface

5 SUPER-CONFORMAL SURFACES 331 and it is nowhere vanishing by its definition. Then (dg 1 )=(dn G0 )F 0 + N G0 (df 0 ) (df 0 )R F0 F 0 (dr F0 ) (dg 0 )= F 0 (dr F0 ). Since R F0 is an immersion, the map G 1 is a conformal immersion with right normal vector R F0. Then G 1 is a super-conformal surface with (dr F0 )+R F0 (dr F0 )=0. Defining a smooth map F 1 by F 1 = G 0 N G0 F 0, the smooth map F 0 + if 1 : M C 4 is a nowhere-vanishing null complex holomorphic immersion and N G0 is the left normal vector of F 0. The right normal vector of G 0 is F0 1 N G0 F 0 and the left normal vector of G 1 is F 0 R F0 F0 1. The rigid motion of G 0 is αg 0 β + γ with quaternions α, β, andγ such that αˆα =1andβ ˆβ = 1. Then G 1 is a rigid motion of G 0 if and only if F0 1 N G0 F 0 = βr F0 β 1 and F 0 R F0 F0 1 = αn G0 α 1. Hence G 1 is not a rigid motion of G 0 generally. Acknowledgement. reference [3]. The author is grateful to the referee for drawing his attention to the References 1. C. Bohle, Möbius invariant flows of tori in S 4, Doctoral Thesis, Technischen Universität Berlin, Berlin, F. E. Burstall, D. Ferus, K. Leschke, F. Pedit and U. Pinkall, Conformal geometry of surfaces in S 4 and quaternions, Lecture Notes in Mathematics 1772 (Springer, Berlin, 2002). 3. M. Dajczer and R. Tojeiro, All superconformal surfaces in R 4 in terms of minimal surfaces, Math. Z. (4) 261 (2009). 4. D. Ferus, K. Leschke, F. Pedit and U. Pinkall, Quaternionic holomorphic geometry: Plücker formula, Dirac eigenvalue estimates and energy estimates of harmonic 2-tori, Invent. Math. (3) 146 (2001) K. Leschke and F. Pedit, Bäcklund transforms of conformal maps into the 4-sphere, PDEs, submanifolds and affine differential geometry, (ed. B. Opozda, U. Simon and M. Wiehe), Banach Center Publications 69 (Polish Academy Sciences, Warsaw, 2005) K. Moriya, The denominators of Lagrangian surfaces in complex Euclidean plane, Ann. Global Anal. Geom. (1) 34 (2008) F. Pedit and U. Pinkall, Quaternionic analysis on Riemann surfaces and differential geometry, Proceedings of the International Congress of Mathematicians (Universität Bielefeld, Bielefeld, 1998), Doc. Math. Extra Vol. II (1998) (electronic). Katsuhiro Moriya Institute of Mathematics University of Tsukuba Tennodai, Tsukuba-shi Ibaraki-ken Japan moriya@math tsukuba ac jp