Super-conformal surfaces associated with null complex holomorphic curves

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1 Super-conformal surfaces associated with null complex holomorphic curves Katsuhiro Moriya June 26, 2007 Abstract We define a correspondence from a null complex holomorphic curve in four-dimensional complex Euclidean space to a super-conformal surface in four-dimensional Euclidean space by the quaternionic theory of surfaces. As an application, we define a transformation of superconformal surfaces. 1 Introduction Construction of surfaces is an important and an interesting problem. A transformation of a surface is a way to construct a new surface from given one. Integrable system method is powerful to define transformations of surfaces. Theory of quaternionic holomorphic vector bundles [2], [3], and [6] is influenced by the theory of integrable systems and powerful to define transformations of surfaces, too. Burstall, Ferus, Leschke, Pedit, and Pinkall [2] defined the Bäcklund transforms and the Darboux transforms for Willmore surfaces in the four-dimensional sphere in terms of quaternionic holomorphic vector bundles. Generalizing it, Bohle [1] defined the Bäcklund transforms for conformal maps from a Riemann surface into the four-dimensional sphere and Leschke and Pedit [4] defined the Bäcklund transformations for quaternionic holomorphic curves. Partly supported by the Grant-in-Aid for Young Scientists (B), The Ministry of Education, Culture, Sports, Science and Technology, Japan. 1

2 It is showed 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 three-dimensional complex projective space CP 3. An arbitrary choice of two Euclidean realization define a pair of super-conformal surfaces. Then we can consider one super-conformal surface as a transformation of another super-conformal surface. Since the other relation between a complex holomorphic curve in CP 3 and a super-conformal surface except twistor projection is unclear in this correspondence, it is interesting problem to define 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 four-dimensional complex Euclidean space C 4 to a superconformal 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 simple calculation using these minimal surfaces, their left normal vector, and their right normal vector. Composition of these two correspondences define a transformation between super-conformal surfaces in R 4. These correspondences are an application of a transformation we define between conformal immersions from a Riemann surface M to R 4. This transformation preserves the right normal vector of a conformal immersion. 2 Surfaces in terms of quaternions We review the quaternionic theory of surfaces [2] and [6]. The set of quaternions H is the unitary real algebra generated by the symbols i, j, and k 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, and a 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. Let M be a Riemann surface with its 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) 2

3 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 Definition 2 and Remark 2 in [2]): 1. An 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 its left normal vector N and its 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 (Proposition 8 in [2]). 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]). This is equivalent to that the equation (dn) = N(dN) = (dn)n 3

4 or (dr) = R(dR) = (dr)r is satisfied (see [2]). Hence the left normal vector and the right normal vector of a minimal surface are super-conformal surfaces and the left normal vector or the right normal vector of a super-conformal surface are superconformal surface around every immersed point. A super-conformal surface is a Euclidean realization of a twistor projection of a complex holomorphic curve in the three-dimensional complex projective space (Theorem 5 in [2]). Let F : M H and G: M H are conformal immersions with their left normal vector N. Then there exists a nowhere vanishing conformal immersion ψ 0 : M H such that (df ) = (dg)ψ 0 (see [6] and [4]). Then (dg)ψ 0 is called a Weierstrass representation of F in Bohle [1]. Similarly, if F and G: M H are conformal immersions with their right normal vector R, then there exists a nowhere vanishing conformal immersion ψ 1 : M H such that (df ) = ψ 1 (dg). Then ψ 1 (dg) is called a Weierstrass representation of F, too. Let F : M H be a conformal immersion with its right normal vector R F and G: M H a conformal immersion with its left normal vector N G. Then (df ) (dg) = 0 if and only if R F = N G (Proposition 16 in [2]). The map G is 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. 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(f G) = (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 its Weierstrass representation F (dg). We say that a Bäcklund pair (F, G) is exact if there exists a conformal immersion with its Weierstrass representation (df )G, 4

5 We assume that (F 0, G 0 ) is an exact Bäcklund 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 their Weierstrass representations F 0 (dg 0 ) and (df 0 )G 0 respectively. We see that F 1 = F 0 G 0 G 1 and G 1 = F 0 G 0 F 1 up to constants. Indeed, d(f 0 G 0 G 1 ) = (df 0 )G 0 + F 0 (dg 0 ) (df 1 ) = (df 0 )G 0, d(f 0 G 0 F 1 ) = (df 0 )G 0 + F 0 (dg 0 ) (dg 1 ) = F 0 (dg 0 ). Because (F 0, G 0 ) is a Bäcklund pair if and only if (Ĝ0, ˆF 0 ) is, 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 its left normal vector N G0 and its 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 its left normal vector F 0 N G0 F0 1 and its right normal vector R G0. The properties of G 0 which depends on its right normal vector only are preserved under this transformation. For example, this transformation is a transformation between minimal surfaces, between super-conformal surfaces, between Lagrangian surfaces (see [5]), and between Hamiltonian-minimal Lagrangian surfaces (see [5]). The Willmore energy is preserved under these transformations (see [6]). However, this transformation may not be a transformation between Willmore surfaces since it does not transform nonconformal variations. Hence this transformation is a transformation between constrained Willmore surfaces (see [2]). 4 Null complex holomorphic curves and superconformal surfaces Applying the above lemma, 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 5

6 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 its left normal vector N and its right normal vector R if and only if the map F 1 is a conformal immersion with its left normal vector N and its 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 as follows: Theorem 1. Let F 0 + if 1 : M C 4 be a nowhere vanishing null complex holomorphic immersion such that minimal surfaces F 0 : M H and F 1 : M H are nowhere vanishing and have their left normal vector N and their right normal vector R. If N 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 R, and R: M R as above. Then (df 0 ) = (df 0 )( R) = (df 1 ). Hence (F 0, R) is an exact Bäcklund 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 its left normal vector F 0 RF0 1 and its right normal vector R by the above lemma. Since d( R) ( R)d( R) = 0, a map G 1 is a super-conformal surface. The rest of the proof is similar. Starting from a super-conformal surface for another super-conformal surface in the above theorem, we obtain a transformation of a super-conformal surface. We give a direct proof of it as follows. Corollary 1. Let G 0 : M H be a super-conformal surface with its left normal vector N G0 such that (dn G0 ) N G0 (dn G0 ) = 0. If N 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 super-conformal surface with its right normal vector R F0 such that (dr F0 )+R F0 (dr F0 ) = 0. 6

7 Proof. Let G 0, N G0, F 0, and R F0 be as above. Since 0 = d(dg 0 ) = (dn G0 ) (df 0 ), the pair (N G0, F 0 ) is an exact Bäcklund pair and (df 0 ) = N G0 (df 0 ). Hence F 0 is a minimal surface 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 its 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, and γ such that αˆα = 1 and β 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. References [1] C. Bohle, Möbius invariant flows of tori in S 4, Dissertation, Technischen Universität Berlin, [2] 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, vol. 1772, Springer-Verlag, Berlin, [3] 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. 146 (2001), no. 3, [4] K. Leschke, and F. Pedit, Bäcklund transforms of conformal maps into the 4-sphere, PDEs, submanifolds and affine differential geometry, Banach Center Publ. 69, , Polish Acad. Sci., [5] K. Moriya, The denominators of Lagrangian surfaces in complex Euclidean plane, Mathematical Research Note , University of Tsukuba,

8 [6] F. Pedit and U. Pinkall, Quaternionic analysis on Riemann surfaces and differential geometry, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), no. Extra Vol. II, 1998, pp (electronic). Katsuhiro Moriya Institute of Mathematics, University of Tsukuba, Tennodai, Tsukuba-shi, Ibaraki-ken, Japan 8