The dual superconformal surface 4 1 0 M. Dajczer and T. Vlachos 2 n a J Abstract 7 It is shown that a superconformal surface with arbitrary codimension in flat ] G Euclideanspacehasa(necessarily unique)dualsuperconformalsurfaceifandonly if the surface is S-Willmore, the latter a well-known necessary condition to allow D a dual as shown by Ma [12]. Duality means that both surfaces envelope the same . h central sphere congruence and are conformal with the induced metric. Our main t a resultisthatthedualsurfacetoasuperconformalsurfacecaneasilybedescribedin m parametric form in terms of a parametrization of the latter. Moreover, it is shown [ that the starting surface is conformally equivalent, up to stereographic projection 1 in the nonflat case, to a minimal surface in a space form (hence, S-Willmore) if v and only if either the dual degenerates to a point (flat case) or the two surfaces 1 9 are conformally equivalent (nonflat case). 2 1 A surface f: M2 → Rn+2 in Euclidean space with codimension n ≥ 2 is called . 1 superconformal if at any point the ellipse of curvature is a nondegenerate circle. Recall 0 4 that the ellipse of curvature at p ∈ M2 is the ellipse in the normal space N M of f at f 1 p given by : v E(p) = {α (X,X) : X ∈ T M and |X| = 1}, f p i X where α denotes the second fundamental form of f with values in the normal bundle; r f a see [10] and references therein for several facts on this concept whose study started almost a century ago due to the work of Moore and Wilson [14], [15]. Superconformality is invariant under conformal transformations since the property of E(p) being a circle is invariant under conformal changes of the metric of the ambient space. Hence, the results in this paper belong to the realm of conformal (Moebius) geometry of surfaces and can also be stated in terms of surfaces in a space form. It was shown by Rouxel [16] that superconformal surfaces in codimension two always arise in pairs f,f˜: M2 → R4 of dual surfaces that induce conformal metrics on M2 and envelop a common central sphere congruence. Recall that the central sphere congruence (or mean curvature sphere congruence) of an Euclidean surface with any codimension is the family of two-dimensional spheres that are tangent to the surface and have the same mean curvature vector as the surface at the point of tangency. The concept of central sphere congruence (called the conformal Gauss map in a different context by Bryant 1 [5]) is central in conformal geometry and was extensively studied since the turn of the last century, fundamentally due to the work of Thomsen [17] and Blaschke [1]; see [11] for a detailed discussion of this subject. Rouxel also discovered that the surface of centers of the central sphere congruence is a minimal surface of R4. If f is free of minimal points, the surface of centers is the locus of centers of the spheres in the congruence, thus parametrically described by the map g: M2 → Rn+2 given by 1 g = f + H |H|2 where H denotes the mean curvature vector field of f. In this paper, we consider superconformal surfaces in Euclidean space in arbitrary codimension. To no surprise, the case of codimension two is rather special and this has much to do with the minimality of the surface of centers. In fact, this property and the classical Weierstrass representation of minimal surfaces allowed Dajczer and Tojeiro [7] to provide a complete local parametric representation of all superconformal surfaces in R4. Moreover, they showed that the dual to a superconformal surface in codimension two reduces to a point if and only if the surface is conformally equivalent, i.e., congruent by a conformal diffeomorphism of R4, to a holomorphic curve in C2. By a dual to a surface f: M2 → Rn+2 we mean an immersion f˜: M2 → Rn+2 that induces a conformal metric and possess a common central sphere congruence, that is, at each point of M2 the sphere in the their centrals sphere congruences is the same. In fact, for convenience we allow the dual to reduce to a single point. For a locally conformally substantial superconformal surface in codimension higher than two that carries a dual superconformal surface, it turns out that the surface of centers is never minimal. A surface being locally conformally substantial means that the image under f of any open subset of M2 is not contained in a proper affine subspace or a sphere in the ambient space Rn+2. This and the fact that in higher codimension superconformality is not longer such a strong assumption, make unlikely the goal to obtain, a complete parametric classification as in [7]. Nevertheless, it seems natural to expect for some class of superconformal surfaces the existence of a dual surface similar to the case considered by Rouxel. In fact, this turns out to be the case for the superconformal surfaces that are S-Willmore. The concept of S-Willmore was introduced by Ejiri [9] as a special class of Willmore surfaces. Ma [12] showed that being S-Willmore is the condition for a surface to have a dual that, in fact, is unique. For a complex coordinate z = y + iy associate to local 1 2 isothermal coordinates superconformality means that the complex line bundle spanned by α (∂ ,∂ ) is isotropic and S-Willmore that it is holomorphic with respect to the f z z normal connection. It is well-known [9] that minimal surfaces in space forms are the basic examples of S-Willmore surfaces. Hence, the “trivial” examples of superconformal S-Willmore sur- faces in Euclidean space are the ones conformally equivalent to minimal superconformal 2 surfaces in Euclidean space and the images under stereographic projection of the same class of surfaces in the sphere or hyperbolic space. Euclidean minimal superconformal surfaces are called 1-isotropic and admit a Weierstrass type representation given in [4] based on results in [3]. In the spherical case, this class of surfaces has been studied in different contexts, see [2], [13] and [18]. There are plenty of “non-trivial” examples of superconformal S-Willmore surfaces in Euclidean space. For instance, the image under stereographic projection of any super Willmore surface in an even dimensional sphere is a superconformal S-Willmore surface. The class of super Willmore surfaces was introduced and classified by Ejiri [9] in terms of isotropic holomorphic curves in complex projective spaces. Note that in conformal geometry we may assume, at least locally, that the mean curvature of a surface never vanishes by composing with a conformal diffeomorphism. Theorem 1. Let f: M2 → Rn+2, n ≥ 3, be a regular locally conformally substantial superconformal surface. Then f has a dual superconformal surface if and only if it is S-Willmore. Moreover, the dual surface can be parametrized as 2 f˜= f + (H)Λ, |H|2 where Λ is the normal subbundle of rank n −2 of the surface of centers perpendicular to the plane subbundle of the first normal bundle Nf of f orthogonal to the mean cur- 1 vature vector and (H)Λ denotes taking the Λ-component. Furthermore, up to conformal equivalence, we have the following cases: (i) The dual reduces to a single point if and only if f is a minimal surface. (ii) The dual is obtained by composing f with an inversion and a reflection with respect to its center if and only if f is the image under stereographic projection of a minimal surface in the sphere Sn+2. (iii) The dual is obtained by composing f with an inversion if and only if f is the image under stereographic projection of a minimal surface in the hyperbolic space Hn+2. The necessity of the surface being S-Willmore in the theorem is due to Ma [12] as already mentioned. A submanifold being regular (or nicely curved) means that the first normal spaces, i.e., the normal subspaces spanned by the second fundamental form, have constant dimension and thus form a subbundle of the normal bundle. Notice that any isometric immersion is regular along the connected components of an open dense subset of the manifold, hence in local submanifold theory, as is the case of this paper, regularity is just a minor technical assumption. Finally, we mention that part (i) is known (see Remark on p. 339 of [9]) but we were not able to find a proof. 3 Any superconformal surface in codimension two is S-Willmore, thus there is no need of such requirement in that case. The codimension three case is still quite special as shown by the following result. Theorem 2. Any superconformal Willmore surface f: M2 → R5 is S-Willmore. The paper concludes with a proof of the main result in [7] by means of the approach we developed here. 1 Preliminaries Inthis section, we first recall some basic properties of theellipse of curvature ofa surface and then briefly discuss the notions of superconformal and S-Willmore surface. Let f: M2 → Rn+2, n ≥ 2, stand for an isometric immersion of a two-dimensional Riemannian manifold into Euclidean space. Denote by α : TM × TM → N M its f f second fundamental form taking values in the normal bundle. Given an orthonormal basis {X ,X } of the tangent space T M at p ∈ M2, denote 1 2 p α = α (X ,X ), 1 ≤ i,j ≤ 2. Then, for any unit vector v = cosθX +sinθX we have ij f i j 1 2 α (v,v) = H +cos2θξ +sin2θξ , (1) f 1 2 where ξ = 1(α −α ), ξ = α and H = 1(α +α ) is the mean curvature vector 1 2 11 22 2 12 2 11 22 of f at p. Thus, when v goes once around the unit tangent circle, the vector α (v,v) f goes twice around the ellipse of curvature E(p) of f at p centered at H. Clearly E(p) degenerates into a line segment or a point if and only ξ and ξ are linearly dependent, 1 2 that is, at points where the normal curvature tensor R⊥ vanishes. It follows from (1) that E(p) is a circle if and only if for some (and hence any) orthonormal basis of T M p it holds that hα ,α −α i = 0 and |α −α | = 2|α |. 12 11 22 11 22 12 The complexified tangent bundle TM ⊗C is decomposed into the eigenspaces of the complex structure J, denoted by T′M and T′′M, corresponding to the eigenvalues i and −i. The complex structure of M2 is determined by the orientation and the induced metric. The second fundamental form can be complex linearly extended to TM ⊗ C with values in the complexified vector bundle N M ⊗C and then decomposed into its f (p,q)-components, p+q = 2, which are tensor products of p many 1-forms vanishing on T′′M and q many 1-forms vanishing on T′M. Taking local isothermal coordinates {y ,y } and z = y + iy , we have that the 1 2 1 2 surface f is superconformal if and only if the (2,0)-part of the second fundamental form is isotropic, or equivalently, if the complex line bundle α (∂ ,∂ ) is isotropic. A surface f z z 4 f: M2 → Rn+2 is called S-Willmore [9], [12] when the complex line bundle α (∂ ,∂ ) is f z z parallel in the normal bundle, that is, if ∇⊥α (∂ ,∂ ) is parallel to α (∂ ,∂ ). ∂z¯ f z z f z z It is well-known that any S-Willmore surface is always Willmore [9] but the converse isnottrue(cf.[8])unlessthesubstantial codimensionisn = 2. AsurfacebeingWillmore or S-Willmore is invariant under conformal diffeomorphisms of Euclidean space. Recall that a surface f: M2 → Rn+2 is called Willmore [9] if its mean curvature vector field H satisfies the Willmore surface equation obtained as the Euler-Lagrange equation of the Willmore functional, namely, if ∆⊥H −2|H|2H +Σ2 hH,α iα = 0 (2) i,j=1 ij ij where ∆⊥ is the Laplacian in N M and X ,X is an orthonormal frame. f 1 2 Using the Codazzi equation, it follows that 2 ∇⊥H = ∇⊥α (∂ ,∂ ), ∂z ρ2 ∂z¯ f z z where ds2 = ρ2|dz2| is the induced metric. Thus, the surface is S-Willmore if and only if ∇⊥H is parallel to α (∂ ,∂ ) or, equivalently, if ∂z f z z ∇⊥H is parallel to α (V,V) (3) V f for any V ∈ T′M. 2 The proofs We proceed with the proofs of the results stated in the introduction. We caution that several arguments contain simple but long computations denominated straightforward that may be only sketched. In the sequel we denote by f: M2 → Rn+2, n ≥ 2, a regular locally substantial superconformal surface. The latter assumption is that the image under f of any open subset of M2 is not contained in a proper affine subspace of the ambient space. Recall that regular means that the first normal spaces have constant dimension and thus form a subbundle of the normal bundle. The first normal space Nf of f at p ∈ M2 is the 1 normal subspace spanned by the second fundamental form, i.e., Nf(p) = span{α (X,Y) : X,Y ∈ T M}. 1 f p Under the above assumptions, it is easy to see that second fundamental form of the surface has the shape λ +µ 0 λ µ A = 1 , A = 2 and A = λI ξ1 (cid:18) 0 λ −µ (cid:19) ξ2 (cid:18)µ λ (cid:19) δ 1 2 5 with respect to orthonormal frames {X ,X } of the tangent bundle and {ξ ,ξ ,δ} of 1 2 1 2 the first normal subbundle Nf. Thus the mean curvature vector field of f is 1 H = λ ξ +λ ξ +λδ. 1 1 2 2 Notice that we cannot have that µ = 0 on an open subset of M2 since, otherwise, f would be totally umbilical along that set and this contradicts being substantial. In particular, the special case dimNf = 2 (in particular, if n = 2) can only occur if λ = 0. 1 From the Codazzi equations for ξ ,ξ and δ we obtain, respectively, that 1 2 X (λ )−X (µ) = −2µΓ −µγ +λ γ −λψ , X (λ )+X (µ) = 2µΓ −µγ +λ γ −λψ , 1 1 1 2 2 2 1 11 2 1 2 1 1 2 2 21 X (µ)−X (λ ) = 2µΓ +γ (λ +µ)+λψ , X (µ)−X (λ ) = 2µΓ +γ (λ −µ)+λψ , 1 2 2 2 2 1 22 2 1 2 1 1 1 12 and X (λ) = ψ (λ −µ)+λ ψ −µψ , X (λ) = ψ (λ +µ)+λ ψ −µψ , (4) 1 11 1 2 12 22 2 21 1 2 22 12 where we used the following notations: Γ = h∇ X ,X i, i 6= j, γ = h∇⊥ ξ ,ξ i and ψ = h∇⊥ δ,ξ i, i,j = 1,2. i Xi i j i Xi 1 2 ij Xi j The first four equations yield X (λ )−X (λ ) = λ γ +λ γ +λ(ψ −ψ ), 1 1 2 2 1 2 2 1 22 11 X (λ )+X (λ ) = −λ γ +λ γ −λ(ψ +ψ ). 2 1 1 2 1 1 2 2 21 12 Setting X (λ ) = λ γ −λψ −µa , X (λ ) = −λ γ −λψ −µa , (5) 1 1 2 1 11 1 1 2 1 1 12 2 for some smooth functions a ,a , we obtain that 1 2 X (λ ) = λ γ −λψ +µa , X (λ ) = −λ γ −λψ −µa , (6) 2 1 2 2 21 2 2 2 1 2 22 1 X (µ) = µ(2Γ +γ −a ), X (µ) = µ(2Γ −γ −a ). (7) 1 2 2 1 2 1 1 2 The Codazzi equation for any η ∈ (Nf)⊥ is equivalent to 1 h∇⊥ η,Hi = µ(h∇⊥ η,ξ i+h∇⊥ η,ξ i), h∇⊥ η,Hi = µ(h∇⊥ η,ξ i−h∇⊥ η,ξ i). X1 X1 1 X2 2 X2 X1 2 X2 1 Let {η } denote an orthonormal frame of (Nf)⊥ and set α 1≤α≤n−3 1 ψα = h∇⊥ η ,ξ i, 1 ≤ i,j ≤ 2. ij Xi α j If dimNf = 2 then 1 ≤ α ≤ n−2. In the sequel, we work the case dimNf = 3, but 1 1 most of the computations hold if dimNf = 2. For simplicity, we denote 1 ψ = ψ +ψ , ψ = ψ −ψ and ψα = ψα +ψα , ψα = ψα −ψα . 1 11 22 2 21 12 1 11 22 2 21 12 6 It follows using (4) and (5) to (7) that ∇⊥ H = −µ(a ξ +a ξ +ψ δ+Σ ψαη ), ∇⊥ H = µ(a ξ −a ξ +ψ δ+Σ ψαη ). (8) X1 1 1 2 2 1 α 1 α X2 2 1 1 2 2 α 2 α The locus of the centers of the central sphere congruence given by the map g: M2 → Rn+2 defined as g = f +r2H, where r = 1/|H|, satisfies g Z = f (I −r2A )Z +r2∇⊥H +Z(r2)H (9) ∗ ∗ H Z where |H|2+λ µ λ µ A = 1 2 . H (cid:18) λ µ |H|2−λ µ(cid:19) 2 1 Using that A2 = 2|H|2A −(|H|4−µ2θ)I H H where θ = λ2 +λ2 = |H|2−λ2, it follows that 1 2 hg Z,g Yi = r4µ2θhZ,Yi+r4h∇⊥H,∇⊥Hi. ∗ ∗ Z Y Thus f and g are conformal ⇐⇒ |∇⊥ H| = |∇⊥ H| and h∇⊥ H,∇⊥ Hi = 0. (10) X1 X2 X1 X2 Proposition 3. The following facts are equivalent: (i) The immersion f is S-Willmore. (ii) The immersions f and g are conformal and ∇⊥H ⊂ Nf. 1 (iii) ∇⊥H ⊂ Im (α −h , iH). f (iv) ψ = ψ = 0 and ψα = ψα = 0, 1 ≤ α ≤ n−3. 1 2 1 2 Proof: On one hand, α (X −iX ,X −iX ) = 2µ(ξ −iξ ). f 1 2 1 2 1 2 On the other hand, we have from (8) that 1 ∇⊥ H = −(a +ia )(ξ −iξ )−(ψ +iψ )δ −Σ (ψα +iψα)η , µ X1−iX2 1 2 1 2 1 2 α 1 2 α and it follows from (3) that (i) and (iv) are equivalent. From (8) we see that the right hand side of (10) is equivalent to ψ2 +Σ (ψα)2 = ψ2 +Σ (ψα)2 and ψ ψ +Σ ψαψα = 0, (11) 1 α 1 2 α 2 1 2 α 1 2 and the remaining of the argument follows easily from (8) to (11). 7 Corollary 4. If f is S-Willmore then (Nf)⊥ ⊂ N M. 1 g Proof: We have from (9) that hg Z,η i = 0, 1 ≤ α ≤ n−3, for any Z ∈ TM. ∗ α We now prove the second result stated in the introduction. Proof of Theorem 2: The Ricci equation hR⊥(X ,X )H,ξ i = h[A ,A ]X ,X i, j = 1,2, 1 2 j H ξj 1 2 together with (7) and (8) yield for j = 1 that X (a )+X (a )−2a a +a Γ +a Γ +ψ ψ +ψ ψ = 2µλ (12) 1 2 2 1 1 2 1 1 2 2 11 2 21 1 2 and for j = 2 that −X (a )+X (a )+a2 −a2 −a Γ +a Γ +ψ ψ +ψ ψ = −2µλ . (13) 1 1 2 2 1 2 1 2 2 1 12 2 22 1 1 On the other hand, h∆⊥H,ξ i = X h∇⊥ H,ξ i+X h∇⊥ H,ξ i−h∇⊥ H,∇⊥ ξ i−h∇⊥ H,∇⊥ ξ i j 1 X1 j 2 X2 j X1 X1 j X2 X2 j − Γ h∇⊥ H,ξ i−Γ h∇⊥ H,ξ i. 1 X2 j 2 X1 j Using (7), (8), (12) and (13) we easily obtain 1 1 h∆⊥H,ξ i = −2λ µ−ψ2 +ψ2 and h∆⊥H,ξ i = −2λ µ+2ψ ψ . µ 1 1 1 2 µ 2 2 1 2 Also, Σ2 hα (X ,X ),Hiα (X ,X ) = 2|H|2H +2µ2(λ ξ +λ ξ ). i,j=1 f i j f i j 1 1 2 2 Now, we have from (2) that f is Willmore if and only if ψ = 0 = ψ , and the result 1 2 follows from Proposition 3. Proposition 5. Let f: M2 → Rn+2 be a substantial superconformal S-Willmore surface with dimNf = 2. If n ≥ 3 then f is minimal. 1 Proof: The same proof given in Proposition 3 that parts (i) and (iv) are equivalent still holds if dimNf = 2. Thus ψα = 0 = ψα for 1 ≤ α ≤ n−2. On the other hand, the 1 1 2 Codazzi equation for η is α ψα A X +ψα A X = ψα A X +ψα A X . 11 ξ1 2 12 ξ2 2 21 ξ1 1 22 ξ2 1 We obtain that λ ψα −λ ψα = 0 and λ ψα +λ ψα = 0. 2 11 1 12 1 11 2 12 But θ 6= 0 would give ψα = 0, which is not possible. Thus f is minimal. ij 8 Proposition 6. Let f: M2 → Rn+2 be a superconformal S-Willmore surface with θ = 0 and dimNf = 3. Then f is minimal inside a sphere in Rn+2. 1 Proof: From (5) and (6) we obtain ψ = 0 = a = a . Then (4) implies that |H| is ij 1 2 constant. Since H 6= 0, then (8) and Proposition 3 show that the umbilical direction H is parallel in the normal connection. Remark 7. It follows from (9) that the locus of the centers of the central sphere congruence of a non-minimal surface is a point if and only if the surface is minimal in a sphere. In the sequel, we also assume that f is S-Willmore with θ 6= 0 6= λ everywhere. Set 1 h = r2(λ ξ −λ ξ ), h = r2H − δ and h = η , 3 ≤ j ≤ n−1. 1 2 1 1 2 2 j j−2 λ Thus Nf is spanned by orthogonal vectors 1 r2θ Nf = span{h ,h ,H} where |h |2 = r4θ and |h |2 = . 1 1 2 1 2 λ2 Lemma 8. The following equations hold: h = g ◦J +ω h +ω h +Σ ω h , (14) 1∗ ∗ 11 1 12 2 α 1α α h = g +ω h +ω h +Σ ω h , (15) 2∗ ∗ 21 1 22 2 α 2α α 1 1 h = − ω h − ω h +Σ ω h , where (16) α∗ |h |2 1α 1 |h |2 2α 2 β αβ β 1 2 λ λr2 1 ω = − d(λr2), ω = (Jgrad(λ/r2))∗, ω = − (Jgradλ)∗, 11 θr2 12 θ 21 λr2θ 1 1 ω = − dλ, ω = r2(A ω −B ω ), ω = − (C ω +D ω ), 22 λr2θ 1α α 1 α 2 2α λ α 1 α 2 ω = h∇⊥h ,h i αβ α β where ω = X∗, i = 1,2, Z∗ denotes the 1-form dual to Z ∈ TM. Also, i i C = h∇⊥ δ,h i, D = h∇⊥ δ,h i, A = λ ψα −λ ψα and B = λ ψα +λ ψα . α X1 α α X2 α α 1 12 2 11 α 1 11 2 12 Proof: A straightforward computation of the derivatives in the ambient space yields ∇¯ (λ ξ −λ ξ ) = µf (−λ X +λ X )−(µa +ψ λ)ξ +(µa +ψ λ)ξ X1 2 1 1 2 ∗ 2 1 1 2 2 12 1 1 11 2 + X (λ)δ +Σ A h , 2 α α α ¯ ∇ (λ ξ −λ ξ ) = µf (λ X +λ X )−(µa −ψ λ)ξ −(µa −ψ λ)ξ X2 2 1 1 2 ∗ 1 1 2 2 1 11 1 2 12 2 − X (λ)δ −Σ B h . 1 α α α 9 Another straightforward computation using (8), (9) and that X (1/r2) = −2µ(a λ +a λ ), X (1/r2) = 2µ(a λ −a λ ) (17) 1 1 1 2 2 2 2 1 1 2 gives λ λr2 h X = g X − X (λr2)h − X (λ/r2)h +r2Σ A h , 1∗ 1 ∗ 2 θ 1 1 θ 2 2 α α α λ λr2 h X = −g X − X (λr2)h + X (λ/r2)h −r2Σ B h . 1∗ 2 ∗ 1 θ 2 1 θ 1 2 α α α Similarly, we have 1 1 h X = g X + (X (λ)h −X (λ)h )− Σ C h , 2∗ 1 ∗ 1 λr2θ 2 1 1 2 λ α α α 1 1 h X = g X − (X (λ)h +X (λ)h )− Σ D h , 2∗ 2 ∗ 2 λr2θ 1 1 2 2 λ α α α and (14) and (15) follow. The third equation is just the Weingarten formula. We decompose h and h into its tangent and normal components to g, namely, 1 2 h = g Y +η, h = g Z +ξ. (18) 1 ∗ 2 ∗ Lemma 9. It holds that Y = Jgrad ̺ and Z = −grad ̺, (19) g g where ̺ = r2/2 and J denotes a complex structure in TM. Proof: Let u be the conformal factor between the metrics induced by g and f on M2, that is, h, i = uh, i . From (8), we have g f ∇⊥ H = −µ(a ξ +a ξ ) and ∇⊥ H = µ(a ξ −a ξ ). (20) X1 1 1 2 2 X2 2 1 1 2 We obtain using (9), (17) and (20) that 1 g Y = g (hh ,g X iX +hh ,g X iX ) ∗ ∗ 1 ∗ 1 1 1 ∗ 2 2 u µr4 = g ((a λ −a λ )X +(a λ +a λ )X ) ∗ 2 1 1 2 1 1 1 2 2 2 u r4 = g (X (1/r2)X −X (1/r2)X ) ∗ 2 1 1 2 2u r4 = − g Jgrad (1/r2) 2u ∗ f 1 = g Jgrad ̺. u ∗ f The computation of g Z is similar. ∗ 10