Holomorphic Legendrian curves in CP 3 and superminimal surfaces in S 4
aa r X i v : . [ m a t h . DG ] O c t HOLOMORPHIC LEGENDRIAN CURVES IN CP AND SUPERMINIMAL SURFACES IN S ANTONIO ALARC ´ON, FRANC FORSTNERI ˇC, AND FINNUR L ´ARUSSON
Abstract.
We obtain a Runge approximation theorem for holomorphic Legendrian curvesand immersions in the complex projective 3-space CP , both from open and compact Rie-mann surfaces, and we prove that the space of Legendrian immersions from an open Riemannsurface into CP is path connected. We also show that holomorphic Legendrian immersionsfrom Riemann surfaces of finite genus and at most countably many ends, none of which arepoint ends, satisfy the Calabi-Yau property. Coupled with the Runge approximation theo-rem, we infer that every open Riemann surface embeds into CP as a complete holomorphicLegendrian curve. Under the twistor projection π : CP → S onto the 4-sphere, immersedholomorphic Legendrian curves M → CP are in bijective correspondence with superminimalimmersions M → S of positive spin according to a result of Bryant. This gives as corol-laries the corresponding results on superminimal surfaces in S . In particular, superminimalimmersions into S satisfy the Runge approximation theorem and the Calabi-Yau property. Introduction
It is well known that the 3-dimensional complex projective space CP admits a uniquecomplex contact structure, that is to say, a completely noninvolutive holomorphic hyperplanesubbundle ξ of the tangent bundle T CP such that any other holomorphic contact bundleon CP is contactomorphic to ξ by an automorphism of CP (see C. LeBrun and S. Salamon[34, 33]). This contact structure is determined by the following homogeneous 1-form on C via the standard projection C \ { } → CP :(1.1) α = z dz − z dz + z dz − z dz . (See Sect. 2.) Uniqueness makes this contact structure fundamentally interesting. This wasamplified in 1982 when R. Bryant [18] discovered that R. Penrose’s twistor projection π : CP → S (a fibre bundle projection onto the 4-sphere whose fibres are projective lines)induces a bijective correspondence between immersed holomorphic Legendrian curves in CP and immersed superminimal surfaces of positive spin in S . (When speaking of the 4-sphere, wealways consider it endowed with the spherical metric induced by the Euclidean metric on theunit sphere S ⊂ R .) Furthermore, the contact bundle ξ on CP is the orthogonal complementof the vertical tangent bundle of π in the Fubini-Study metric, and the differential dπ maps ξ isometrically onto T S , so π maps Legendrian curves locally isometrically to superminimalsurfaces in S . The latter form an interesting subclass of the class of all minimal surfaces in S . Bryant proved in [18, Theorem F] that for any pair of meromorphic functions f, g on aRiemann surface M with g nonconstant, the map given in homogeneous coordinates by(1.2) B ( f, g ) = (cid:2) dg : f dg − gdf : gdg : df (cid:3) : M −→ CP Date : 28 October 2019.2010
Mathematics Subject Classification.
Primary 53D10. Secondary 32E30, 32H02, 53A10.
Key words and phrases.
Riemann surface, Legendrian curve, Runge approximation, superminimal surface. s a holomorphic Legendrian curve in CP . Using this formula, he showed that any compactRiemann surface M admits a holomorphic embedding into CP as a Legendrian curve (see[18, Theorem G]), and he inferred that any such M admits a conformal, generically injectiveimmersion M → S onto a superminimal surface in S (see [18, Corollary H]).In the present paper we go considerably further by treating Legendrian curves in CP andconformal superminimal surfaces in S parametrised not only by compact, but also by openand by compact bordered Riemann surfaces. In particular, we obtain what seem to be thefirst general existence and approximation results in the literature for complete noncompactsuperminimal surfaces in the 4-sphere. More about this below.Let us now describe the content of the paper.We begin by presenting in Sect. 2 a unified approach from first principles to a couple ofrepresentation formulas for Legendrian curves in CP , the one of Bryant (1.2) and anotherone adapted from the recent papers [12, 24]; see (2.7), (2.8). The relationship between them isgiven by (2.9). As pointed out in Remark 2.6, the optimal choice of a formula to use dependson the particular problem one wants to solve. Although each formula has a set of exceptionalcurves it does not cover, any given Legendrian curve is nonexceptional in some homogeneouscoordinates on CP (see Proposition 2.2). By choosing homogeneous coordinates on CP sothat the hyperplane at infinity intersects our Legendrian curve transversely, which is possibleby Bertini’s theorem, the pair of meromorphic functions determining the curve have onlysimple poles. This condition means that the curve is immersed near the poles.In Sect. 3 we use Bryant’s formula (1.2) to prove the Runge approximation theorem cou-pled with the Weierstrass interpolation theorem for holomorphic Legendrian curves in CP ,both from compact and open Riemann surfaces (see Theorem 3.2), as well as the correspond-ing result for holomorphic Legendrian immersions (see Theorem 3.4). For open Riemannsurfaces, we also have a Runge approximation theorem for Legendrian embeddings into CP (see Corollary 3.7), where by an embedding we mean an injective immersion.In Sect. 4 we use the second representation formula (2.8) to prove that the space of allLegendrian immersions M → CP from an arbitrary open Riemann surface is path connected;see Theorem 4.1. On the other hand, the space of Legendrian immersions is not connected if M is compact. It will split into components by degree, and perhaps further.These results imply that every formal Legendrian immersion from an open Riemann sur-face to CP can be deformed to a genuine holomorphic Legendrian immersion, unique upto homotopy (see Theorem 5.1). It remains an open problem whether the inclusion of thespace of holomorphic Legendrian immersions M → CP into the space of formal Legendrianimmersions satisfies the full parametric h-principle. For immersed Legendrian curves in C n +1 with its standard contact structure the parametric h-principle was proved in [24]; however,the technical problems that arise for Legendrian curves in projective spaces are considerable.In Sect. 6 we introduce an axiomatic approach to the Calabi-Yau problem which unifiesrecent results in this direction in various geometries. The motivation behind results of thistype is the
Calabi-Yau problem for minimal hypersurfaces , asking whether there exist com-plete bounded minimal hypersurfaces in R n for n ≥
3. This problem originates in Calabi’sconjecture from 1965 that such hypersurfaces do not exist (see [32, p. 170]). Although noth-ing seems known for n ≥
4, several constructions of complete bounded 2-dimensional minimalsurfaces in R n for any n ≥ nspired by S.-T. Yau’s [43] where he revisited Calabi’s conjecturesand proposed several questions concerning topology, complex structure, and boundary be-haviour of complete bounded minimal surfaces in R . A recent survey of this topic can befound in [8, Sect. 5.3]; see also the paper [9] where the Calabi-Yau theorem was establishedfor immersed minimal surfaces in R n , n ≥
3, from any open Riemann surface of finite genusand at most countably many ends, none of which are point ends.Recently, the Calabi-Yau phenomenon has been discovered in other geometries, and itis reasonable to expect that more examples will follow. This motivated us to formulate anaxiomatic approach by introducing the
Calabi-Yau property which a class of immersions intoa given Riemannian manifold N (or a class of manifolds) may or may not have; see Definition6.1 and Theorem 6.2. This property means that one can enlarge the intrinsic diameter ofan immersed manifold as much as desired by C small perturbations of the immersion inthe given class. Combining the Calabi-Yau property with Runge’s approximation propertyfor immersions of the given class into the manifold N (see Definition 6.8) gives completeimmersions of this class from all open admissible manifolds into N (see Theorem 6.9).As a particular case of interest, we discuss Legendrian immersions. It was proved in [7]that holomorphic Legendrian immersions from bordered Riemann surfaces into any complexcontact manifold with an arbitrary Riemannian metric enjoy the Calabi-Yau property. Weshow that one can at the same time interpolate the given map at finitely many points (seeCorollary 6.7). Coupled with the Runge approximation theorem for Legendrian embeddingsof open Riemann surfaces into CP (see Corollary 3.7) and Bryant’s Legendrian embeddingtheorem for compact Riemann surfaces [18, Theorem G], it follows that every Riemann surfaceembeds into CP as a complete holomorphic Legendrian curve (see Corollary 6.11).In Sect. 7 we apply our results to the study of superminimal surfaces in the 4-sphere, S ,endowed with the spherical metric. It follows in particular that the Runge approximationtheorem and the Weierstrass interpolation theorem hold for conformal superminimal immer-sions of Riemann surfaces (both open and closed) into S , and every open Riemann surface isthe conformal structure of a complete conformally immersed superminimal surface in S (seeCorollaries 7.3 and 7.4). Furthermore, any conformal superminimal immersion M → S froma compact bordered Riemann surface can be approximated as closely as desired uniformly on M by a continuous map M → S whose restriction to the interior of M is a complete confor-mal superminimal surface with Jordan boundary (see Theorem 7.5). The analogous result forminimal surfaces in R n , n ≥
3, with the Euclidean metric has been proved only recently in [4].Finally, for every open Riemann surface M , the spaces of conformal superminimal immersions M → S of positive or negative spin are path connected; see Corollary 7.7.Results of this paper concerning holomorphic Legendrian curves in CP can be generalisedto higher dimensional projective spaces CP n +1 with the unique holomorphic contact structuredetermined by the following holomorphic 1-form on C n +2 : α = n X j =0 z j dz j +1 − z j +1 dz j . Since CP n +1 is the twistor space of the quaternionic projective space HP n (see [34, p. 113]),this gives similar applications to superminimal surfaces in HP n for n >
1. The twistorprojection CP → HP = S considered above is the basic case of this correspondence for n = 1. We shall not give the details of this generalisation because this would considerablyenlarge the paper without providing any substantially new ideas or techniques. . Representation formulas for Legendrian curves in CP Let α be the homogeneous 1-form on C defined by (1.1). Its differential is the standardcomplex symplectic form on C . At each point z = ( z , z , z , z ) ∈ C \ { } , ker α ( z ) is acomplex hyperplane in T z C containing the radial vector P i =0 z i ∂∂z i . Let π : C \ { } → CP be the standard projection and [ z : z : z : z ] be the homogeneous coordinates on CP .Since α is homogeneous, there is a unique holomorphic hyperplane subbundle ξ ⊂ T CP defined by the condition (cid:8) v ∈ T z C : dπ z ( v ) ∈ ξ π ( z ) (cid:9) = ker α ( z ) , z ∈ C \ { } . It turns out that ξ is a holomorphic contact bundle on T CP , and the essentially unique one(see [34] or [33, Proposition 2.3]). The following lemma shows that the restriction of ξ to anyaffine chart C ⊂ CP is linearly contactomorphic to the standard contact structure on C . Lemma 2.1.
For every projective hyperplane CP ∼ = H ⊂ CP there are linear coordinates ( z ′ , z ′ , z ′ ) on C = CP \ H in which ξ is defined by the contact form (2.1) α = dz ′ + z ′ dz ′ − z ′ dz ′ . Note that every linear automorphism of C extends to a unique projective automorphismof CP . Hence, in the context of the lemma there exists φ ∈ Aut( CP ) such that φ ( H ) = H and φ ∗ ( ξ ) = ker α on C = CP \ H . Proof.
Due to symmetries of α (1.1) it suffices to consider hyperplanes H ⊂ CP of the form z = a z + a z + a z for some a , a , a ∈ C . The affine chart CP \ H = C is thendetermined by the affine hyperplaneΛ = { z = 1 + a z + a z + a z } ⊂ C . Note that ( z , z , z ) are affine coordinates on Λ, and the restriction of α to it is α = (1 + a z + a z ) dz − ( z + a z ) dz + ( z − a z ) dz . We introduce new linear coordinates on C by z ′ = z , z ′ = z − a z , z ′ = z + a z . Then, (1 + a z + a z ) dz = (1 + a z ′ + a z ′ ) dz , ( z + a z ) dz = z ′ ( dz ′ + a dz ) = z ′ dz ′ + a z ′ dz , ( z − a z ) dz = z ′ ( dz ′ − a dz ) = z ′ dz ′ − a z ′ dz , and hence α is given in these coordinates by (2.1). (cid:3) Lemma 2.1 shows that for any hyperplane H ⊂ CP , homogeneous coordinates on CP can be chosen such that H = { z = 0 } and the contact structure ξ is given on C = CP \ H as the kernel of the holomorphic contact form(2.2) α = dz + z dz − z dz , α ∧ dα = 2 dz ∧ dz ∧ dz = 0 . Globally on CP , α is a meromorphic 1-form with a second order pole along the hyperplane H = { z = 0 } . It can be viewed as a nowhere vanishing holomorphic contact 1-form on CP with values in the normal line bundle L = T CP /ξ of the contact structure. (See [34, Sect. 2]for the precise explanation.) Furthermore, ω = α ∧ dα is a holomorphic 3-form on CP with alues in the line bundle L , hence an element of H ( CP , K ⊗ L ) where K = Λ ( T ∗ CP ) isthe canonical bundle of CP . Being nowhere vanishing, ω defines a holomorphic trivialisationof K ⊗ L , so we infer that L ∼ = K − / = O CP (2). In other words, the dual bundle L ∗ = L − is the square of the universal bundle on CP .We also consider the contact form on C given by(2.3) β = dz + z dz , with β ∧ dβ = dz ∧ dz ∧ dz . The map ψ : C → C defined by(2.4) ψ ( z , z , z ) = (cid:16) z + z z , z , − z (cid:17) is a polynomial automorphism of C , and a simple calculation shows that ψ ∗ α = β . Itfollows that ψ maps β -Legendrian curves to α -Legendrian curves. Clearly, we can represent β -Legendrian curves in either of the following two forms:(2.5) z = f, z = − dfdg , z = g, (2.6) z = − Z hdg, z = h, z = g, where f, g, h are meromorphic functions on a given Riemann surface M . (The part of the curvecontained in C is the image of the complement M \ P of the polar set P of the respectivepair of functions ( f, g ) or ( h, g ).)In the first case (2.5), the pair of functions ( f, g ) is arbitrary subject only to the conditionthat g is nonconstant. The exceptional family of Legendrian lines with z = const. , z = const. cannot be represented in this way.In the second case (2.6), the pair ( h, g ) must be such that hdg is an exact meromorphic1-form, which therefore has a meromorphic primitive f = − R hdg determined up to an ad-ditive constant. We discuss this condition in Proposition 2.4. Conversely, assuming that g isnonconstant, we can express h in terms of f by h = − df /dg .Applying the automorphism ψ ∈ Aut( C ), given by (2.4), to β -Legendrian curves (2.5),(2.6) yields the following formulas for α -Legendrian curves in CP : B ( f, g ) = (cid:20) f − g dfdg : g : 12 dfdg (cid:21) = (cid:2) dg : f dg − gdf : gdg : df (cid:3) , (2.7) F ( h, g ) = (cid:20) hg − Z hdg : g : − h (cid:21) = (cid:20) Z gdh − hg g : − h (cid:21) . (2.8)Both formulas depend on the choice of homogeneous coordinates and are related by(2.9) B ( f, g ) = F ( h, g ) , where f = − Z hdg and h = − df /dg. The formula (2.7) was used by Bryant [18] to prove that every compact Riemann surface em-beds in CP as a holomorphic Legendrian curve. The second formula (2.8) has been exploitedin the study of Legendrian curves in C in the recent work [12].The family of exceptional β -Legendrian lines z = a = const. , z = 2 t ∈ C , z = b = const. is mapped by the automorphism ψ (2.4) to the family of exceptional α -Legendrian lines(2.10) [1 : a + bt : b : − t ] , t ∈ CP , a, b ∈ C , hich are not of the form B ( f, g ). On the other hand, every Legendrian curve intersectingthis affine chart equals F ( h, g ) for a unique pair of meromorphic functions ( h, g ) and a choiceof an additive constant determining the value of the integral R hdg at an initial point p ∈ M .We now show that every nonconstant Legendrian curve in CP is of the form B ( f, g ) and F ( h, g ) in some homogeneous coordinate system on CP . Proposition 2.2.
Let F : M → CP be a nonconstant holomorphic Legendrian curve froman open or compact Riemann surface M . (a) There are homogeneous coordinates on CP such that F = B ( f, g ) (see (2.7) ), where f and g are meromorphic functions on M with only simple poles. (b) There are homogeneous coordinates on CP such that F = F ( h, g ) (see (2.8) ), where h and g are meromorphic functions on M with only simple poles.Furthermore, every Legendrian curve M → CP given by (2.7) or (2.8) , with the functions f, g, h having only simple poles, is an immersion in a neighbourhood of the union of the polarsets of f and g (for (2.7) ), or h and g (for (2.8) ).Proof. Let F : M → CP be a nonconstant holomorphic Legendrian curve. In view of E.Bertini’s theorem (see e.g. [27, p. 150] or [31] and note that this is essentially an applicationof the transversality theorem), F intersects most complex hyperplanes H ⊂ CP transversely.Fix such H and choose homogeneous coordinates [ z : z : z : z ] on CP with H = { z = 0 } and such that the contact form on CP \ H = C is given by (2.2). The preimage F − ( H ) = { p ∈ M : F ( p ) ∈ H } is then a closed discrete subset of M . Hence, we can represent F ineither form (2.7), (2.8), the only exceptions being the family of projective lines (2.10) whichcannot be represented by Bryant’s formula (2.7). We shall deal with this issue later.Consider a point p ∈ F − ( H ). Choose a local holomorphic coordinate ζ on M with ζ ( p ) = 0. Write F = [1 : F : F : F ] and let k ∈ N be the maximal order of poles at p of thecomponents F , F , F . Multiplying by ζ k we obtain F ( ζ ) = (cid:2) ζ k : ζ k F ( ζ ) : ζ k F ( ζ ) : ζ k F ( ζ ) (cid:3) , where the functions ζ k F j ( ζ ), j ∈ { , , } , are regular at ζ = 0 and at least one of them isnonvanishing at ζ = 0. Looking at the map F in the corresponding affine chart { z j = 1 } , wesee that F is transverse to H at the point p if and only if the derivative dz /dζ is nonvanishingat ζ = 0, which holds if and only if k = 1. Inspection of the formulas for B ( f, g ) and F ( h, g )then shows that the functions f, g or h, g have at most simple poles at p . Conversely, theabove argument shows that the intersection of F with H is transverse at any simple pole ofthe functions f, g or h, g . In particular, F is an immersion near such points.It remains to show that the exceptional lines (2.10) become nonexceptional in anothercoordinate system. Consider the following coordinates on CP : z ′ = z , z ′ = z , z ′ = − z , z ′ = z . We have not changed H = { z = 0 } , so we are still in the same affine chart. In thesecoordinates, the form α (1.1) restricted to the affine chart { z = 1 } equals α = dz ′ + z ′ dz ′ − z ′ dz ′ , and the exceptional family of lines is given in the new coordinates by[1 : a + bt : t : b ] , t ∈ CP , a, b ∈ C . his curve equals B ( f, g ) with f ( t ) = a + 2 bt and g ( t ) = t . This shows that every nonconstantLegendrian curve in CP is of the form B ( f, g ) in some homogeneous coordinate system. (cid:3) There are Legendrian immersions (2.7), (2.8) given by functions f, h, g with higher orderpoles. However, this means that the hyperplane H determining the affine chart was not wellchosen, and a small deformation of it yields a representation by functions with simple poles.The following is an immediate corollary to Proposition 2.2. Corollary 2.3.
Let F = F ( h, g ) : M → CP be a holomorphic Legendrian curve of theform (2.8) with g, h having only simple poles. Then, F is an immersion if and only if ( h, g ) : M \ P → C is an immersion, where P = P ( h ) ∪ P ( g ) is the union of polar loci of h and g .Proof. By Proposition 2.2, F is an immersion if and only if its restriction M \ P → C is an immersion. This restriction is equivalent to the β -Legendrian curve (2.6) under theautomorphism ψ ∈ Aut( C ) given by (2.4). Obviously, the map (2.6) is an immersion if andonly if ( h, g ) : M \ P → C is an immersion. (cid:3) The precise conditions for a Legendrian map F = B ( f, g ) to be an immersion are morecomplicated. By Lemma 3.3, if g is an immersion, then B ( f, g ) is an immersion. See thediscussion preceding Theorem 3.4 for more information.Let us look more closely at the formula (2.8). The meromorphic 1-form hdg on M is exactif and only if R C hdg = 0 for every closed curve C in M which does not contain any poles of hdg . There are two types of curves to consider: those in a homology basis of M (they can bechosen in the complement of the polar set of hdg ), and small loops around the poles of hdg .The integral of hdg around a pole a equals 2 π i Res a ( hdg ). Let us record this observation. Proposition 2.4.
A pair of meromorphic functions ( h, g ) on a Riemann surface M deter-mines a Legendrian immersion F = F ( h, g ) : M → CP (2.8) if and only if the following twoconditions hold: (a) R C hdg = 0 for every closed curve in a basis of the homology group H ( M, Z ) , and (b) Res a ( hdg ) = 0 holds at every pole of hdg .If a is a simple pole of g or h , then condition (b) is equivalent to (2.11) c − ( h, a ) c ( g, a ) − c − ( g, a ) c ( h, a ) = Res a ( hdg ) = 0 , where c k ( h, a ) denotes the coefficient of the term ( z − a ) k in a Laurent series representationof h at a (so c − ( h, a ) = Res a h ). The situation is more complicated at poles of higher order. However, the case of first orderpoles is a generic one in view of Proposition 2.2.
Proof.
It remains to show that (2.11) holds at a simple pole a ∈ M of h or g . In a localholomorphic coordinate z on M , with z ( a ) = 0, we have that h ( z ) = c − ( h ) z + c ( h ) + c ( h ) z + · · · ,g ( z ) = c − ( g ) z + c ( g ) + c ( g ) z + · · · ,g ′ ( z ) = − c − ( g ) z + c ( g ) + · · · , rom which we easily infer thatRes ( hg ′ ) = c − ( h ) c ( g ) − c − ( g ) c ( h ) . This gives (2.11) and completes the proof. (cid:3)
Remark 2.5.
In particular, if a ∈ M is a simple pole of h while g is regular at a , we haveRes a ( hdg ) = Res a ( hg ′ ) = g ′ ( a )Res a h. Similarly, if a is a simple pole of g while h is regular at a , we haveRes a ( hdg ) = h ′ ( a )Res a g. Assuming that h and g have only simple poles and their polar sets are disjoint, we infer thatthe 1-form hdg has vanishing residues precisely when h has a critical point at each pole of g ,and g has a critical point at each pole of h . (cid:3) Remark 2.6.
An advantage of Bryant’s formula (2.7) over (2.8) is that it applies to any pair( f, g ) of meromorphic functions with g nonconstant. A disadvantage is that the Legendriancurve B ( f, g ) does not depend continuously on ( f, g ) near a common critical point of f and g (see Remark 3.9). This becomes a major drawback especially when trying to constructfamilies of Legendrian curves depending continuously on parameters. A similar difficulty wasencountered in [23] when studying holomorphic Legendrian curves in projectivised cotangentbundles. On the other hand, the Legendrian curve F ( h, g ) (2.8) depends continuously on thepair ( h, g ) for which hdg is an exact 1-form. (cid:3) Approximation and interpolation for Legendrian curves in CP In this section we prove the Runge approximation theorem with interpolation at finitelymany points for holomorphic Legendrian curves in CP (see Theorem 3.2) and holomorphicLegendrian immersions (see Theorem 3.4), both from compact and open Riemann surfaces.As a corollary, we obtain the interpolation theorem on a discrete set (see Corollary 3.6).We shall use the following version of Runge approximation theorem, proved by H. L.Royden [39] in 1967, which we state here for the reader’s convenience. In this theorem, thegiven function is allowed to have poles on the set where the approximation takes place. Theorem 3.1 (Royden [39]) . Let M be a compact Riemann surface and K = M be a com-pact subset of M . Let E consist of one point in each connected component of M \ K , f beholomorphic on a neighbourhood of K except for finitely many poles in K , and D be an ef-fective divisor with support in K . Given ǫ > , there is a meromorphic function F on M ,holomorphic on M \ E except at the poles of f , such that ( f − F ) ≥ D and | f − F | < ǫ on K . The condition ( f − F ) ≥ D in the theorem simply means that F agrees with f to order D ( x ) > x ∈ K of the finite support of the divisor D .Here is our first approximation theorem. Theorem 3.2.
Let M be a Riemann surface, open or compact, and let K be a compactsubset of M . Every holomorphic Legendrian map Φ from a neighbourhood of K to CP can beapproximated uniformly on K by holomorphic Legendrian maps M → CP . The approximantscan be taken to agree with Φ to any finite order at each point of any finite subset of K . We take a Riemann surface to be connected by definition, but the neighbourhood in thetheorem need not be connected. roof. First we note that the compact case of the theorem implies the open case. Indeed,if M is open, we exhaust M by smoothly bounded compact domains containing K and useinduction, applying the compact case of the theorem to a compactification of each domain.Hence, from now on, we assume that M is compact.Let Φ be a holomorphic Legendrian map from a neighbourhood V of K = M to CP . ByProposition 2.2, we may assume that Φ = B ( f, g ), where f and g are meromorphic on V and g is not constant on any connected component of V .Let B be the finite subset of K consisting of the poles of f , the poles of g , and the commoncritical points of f and g in K . We use Royden’s theorem to approximate f and g uniformlyon a neighbourhood of K by meromorphic functions f n and g n on M , respectively, such thatthe functions φ n = f n − f and ψ n = g n − g , which are holomorphic and go to zero uniformlyon a neighbourhood of K , vanish at each point of B to sufficiently high order N , independentof n , to be specified as the proof progresses.We claim that if N is sufficiently large, then the holomorphic Legendrian maps B ( f n , g n ) : M → CP converge to B ( f, g ) uniformly on K as n → ∞ .Near a point p of K \ B , with respect to a local coordinate z centred at p , B ( f, g ) = (cid:2) g ′ : f g ′ − f ′ g : gg ′ : f ′ (cid:3) . On a neighbourhood U of p with U ∩ B = ∅ , f n → f and g n → g uniformly, these functionsare holomorphic, and the same holds for their derivatives, so (cid:0) g ′ n , f n g ′ n − f ′ n g n , g n g ′ n , f ′ n (cid:1) −→ (cid:0) g ′ , f g ′ − f ′ g, gg ′ , f ′ (cid:1) uniformly on U as n → ∞ . Also, ( g ′ , f g ′ − f ′ g, gg ′ , f ′ ) = (0 , , ,
0) at every point of U , so B ( f n , g n ) → B ( f, g ) uniformly on U .Next, let p ∈ B . Then the lowest order m ∈ Z at p of the components g ′ , f g ′ − f ′ g , gg ′ , f ′ of B ( f, g ) is not zero. If N is large enough, then a component of B ( f, g ) of order m corresponds to a component of B ( f n , g n ) of lowest order, and that lowest order is also m .If we now multiply the components by z − m , then we are in the same situation as before andneed to show that (cid:0) z − m g ′ n , z − m ( f n g ′ n − f ′ n g n ) , z − m g n g ′ n , z − m f ′ n (cid:1) −→ (cid:0) z − m g ′ , z − m ( f g ′ − f ′ g ) , z − m gg ′ , z − m f ′ (cid:1) uniformly near p as n → ∞ . Note that each difference z − m g ′ n − z − m g ′ , z − m ( f n g ′ n − f ′ n g n ) − z − m ( f g ′ − f ′ g ) ,z − m g n g ′ n − z − m gg ′ , z − m f ′ n − z − m f ′ is a sum of terms of the form z − m times one of the functions φ ′ n , ψ ′ n , φ n ψ ′ n , φ ′ n ψ n , ψ n ψ ′ n , f ′ ψ n , f ψ ′ n , g ′ φ n , gφ ′ n , g ′ ψ n , gψ ′ n (perhaps with a factor of ). By the maximum principle, z − N φ n and z − N ψ n go to zerouniformly near p . Likewise, z − N +1 φ ′ n and z − N +1 ψ ′ n go to zero uniformly near p . Hence, if N is big enough, all those differences go to zero uniformly near every point p in the finite set B .Jet interpolation can be achieved by taking N large enough and, if necessary, addingfinitely many points to B . (cid:3) ext we adapt Theorem 3.2 to immersions. First we need to determine those meromorphicfunctions f and g for which B ( f, g ) is an immersion.First, if f is constant, then B ( f, g ) = [ dg : f dg : gdg : 0] = [1 : f : g : 0] is an immersionif and only if g is an immersion. Now suppose that f is not constant. In suitable localcoordinates centred at a point p in M and at the point g ( p ) in CP , write g ( x ) = x b , b = 0,and f ( x ) = x a h ( x ), where h is holomorphic near p and h (0) = 0. If a = 0, then B ( f, g ) = h bx b − : bx b − f ( x ) − x b f ′ ( x ) : bx b − : f ′ ( x ) i . The orders of the components at p are (cid:2) b − b − b − p f ′ ≥ (cid:3) . If a = 0, then B ( f, g ) = (cid:2) bx b − : bx b − x a h ( x ) − x b ( ax a − h ( x ) + x a h ′ ( x )) : bx b − : ( ax a − h ( x ) + x a h ′ ( x )) (cid:3) , so the orders of the components at p are[ b − c : 2 b − a − , where c = (cid:26) a + b − a = 2 b , a + b + ord p h ′ if a = 2 b .Now B ( f, g ) is regular at p if and only if the smallest and the second smallest order differby 1. It is easily checked that this condition is satisfied when g is regular at p , that is, when b = ±
1. Indeed, for b = 1, the orders are[0 : 0 : 1 : ≥
0] if a = 0,[0 : ≥ a = 2,[0 : a : 1 : a −
1] if a = 0 , b = −
1, the orders are[ − − − ≥
0] if a = 0,[ − ≥ − − −
3] if a = − − a − − a −
1] if a = 0 , − Lemma 3.3. If g is an immersion, then B ( f, g ) is an immersion. We see that when g is critical at p , that is, b ≥ b ≤ −
2, then B ( f, g ) is regular at p if, for example, a = b −
1. On the other hand, regularity of f may not be enough to ensureregularity of B ( f, g ), for example when a = 1 and b = 3.In fact, we see that if g is critical at p , then B ( f, g ) is regular at p if and only if the degreesof the first two terms in the Laurent series of f at p belong to a certain set of admissible pairsof integers that only depends on the order of g at p (and that is quite complicated to describeexplicitly). Theorem 3.4.
Let M be a Riemann surface, open or compact, and let K be a compact subsetof M . Every holomorphic Legendrian immersion Φ from a neighbourhood of K to CP canbe uniformly approximated on K by holomorphic Legendrian immersions M → CP . Theapproximants can be taken to agree with Φ to any given finite order at each point of any givenfinite subset of K . emark 3.5. We shall see in Corollary 6.11 (ii) that the approximating immersion M → CP in Theorem 3.4 can always be chosen complete , i.e., such that the pullback of any Riemannianmetric on CP by the immersion is a complete metric on M . (cid:3) Proof of Theorem 3.4.
As in the proof of Theorem 3.2, it suffices to take M to be compact.By Theorem 3.2, a Legendrian immersion φ from a neighbourhood U of K to CP can beapproximated on K by a Legendrian map B ( f , g ) : M → CP . If we approximate sufficientlywell on a compact neighbourhood of K in U , then B ( f , g ) will be an immersion on a neigh-bourhood V of K . On M \ V , g has finitely many critical points, at which B ( f , g ) may notbe regular.Let h be a meromorphic function on a neighbourhood of the disjoint union L of a compactneighbourhood K ′ of K and closed coordinate discs centred at each of the critical points of g in M \ K , such that h = f near K ′ , and h has order b − g of order b . As in the proof of Theorem 3.2, we can use Royden’s theorem to approximate h uniformlyon L by a meromorphic function f on M such that: • B ( f, g ) is as close as we wish to B ( f , g ) on K ′ , so in particular, B ( f, g ) is an im-mersion on a neighbourhood of K , • at each critical point of g in M \ K of order b , f has order b −
1, so B ( f, g ) is regularthere.Then B ( f, g ) : M → CP is a Legendrian immersion that uniformly approximates φ on K asclosely as desired. Finally, jet interpolation can be incorporated using Theorem 3.2. (cid:3) Corollary 3.6.
Let E be a closed discrete subset of a Riemann surface M . Every map E → CP can be extended to a holomorphic Legendrian immersion M → CP .Proof. For a compact Riemann surface M this is an immediate corollary of Theorem 3.4,applied to K being the union of small mutually disjoint discs around the points of E . Foran open Riemann surface we apply Theorem 3.4 inductively, interpolating at more and morepoints of the given discrete set as we go. (cid:3) Corollary 3.7.
Let M be an open Riemann surface. Every holomorphic Legendrian immer-sion M → CP can be approximated, uniformly on compact subsets of M , by holomorphicLegendrian embeddings M ֒ → CP .Proof. Let φ : M → CP be a holomorphic Legendrian immersion. Choose an exhaustion K ⊂ K ⊂ · · · of M by compact subsets without holes so that each K j is contained in theinterior of the next set K j +1 . By [7, Theorem 1.2], φ can be approximated arbitrarily closely,uniformly on K ◦ , by a holomorphic Legendrian embedding ψ : K ◦ → CP . By Theorem 3.4, ψ can be approximated uniformly on K by a holomorphic Legendrian immersion φ : M → CP .If the approximation is close enough then φ is injective on K . Repeating the same argument, φ can be approximated arbitrarily closely on K by a holomorphic Legendrian immersion φ : M → CP that is an embedding on K . Continuing in this way with sufficiently closeapproximations and passing to the limit, the corollary is proved. (cid:3) Problem 3.8.
Let M be a compact Riemann surface. Is it possible to approximate everyholomorphic Legendrian immersion M → CP by a holomorphic Legendrian embedding? n this connection, Bryant did prove in [18, Theorem G] that every compact Riemannsurface admits a holomorphic Legendrian embedding into CP , but his proof does not seemto provide an answer to the above problem, and we could not find one either. Remark 3.9.
We have a bijection ( f, g ) B ( f, g ) from the space of pairs ( f, g ) of mero-morphic functions on M with g nonconstant to the space of holomorphic Legendrian maps M → CP whose image does not lie in a plane of the form [ z : z ] = constant. As alreadynoted in Remark 2.6, this bijection is not continuous. (The analogous phenomenon for pro-jectivised cotangent bundles was observed in [23].) Take, for example, M = C , f ( x ) = x ,and g ǫ ( x ) = ( x + ǫ ) , ǫ ∈ C . Then B ( f, g ǫ )( x ) = (cid:2) x + ǫ : x ( x − ǫ ) : ( x + ǫ ) : x (cid:3) and in particular, B ( f, g )( x ) = (cid:2) x : x : 1 (cid:3) , so B ( f, g ǫ )(0) = [1 : 0 : ǫ : 0] for ǫ = 0,but B ( f, g )(0) = [1 : 0 : 0 : 1].The inverse bijection B − , however, is continuous. Indeed, we can retrieve g from B ( f, g )by postcomposing by the meromorphic function ψ : [ z : z : z : z ] z z , and retrieve f by postcomposing by φ : [ z : z : z : z ] z z + z z z , so B − ( h ) = ( φ ◦ h, ψ ◦ h ) for a holomorphic Legendrian map h : M → CP whose image doesnot lie in a plane of the form [ z : z ] = constant. To see that B − is continuous, note thatthe image of h will not lie in the indeterminacy locus of either φ or ψ , since both loci lie inthe plane where z = 0.As shown in the proof of Theorem 3.2, despite the failure of continuity of B , if ( f n , g n ) → ( f, g ) uniformly on a neighbourhood of a compact subset K of a Riemann surface and thefunctions f n − f and g n − g , which are holomorphic and go to zero uniformly on a neighbourhoodof K , vanish to sufficiently high order, independent of n , at each pole of f , pole of g , andcommon critical point of f and g in K , then the holomorphic Legendrian maps B ( f n , g n )converge to B ( f, g ) uniformly on K as n → ∞ . (cid:3) The space of Legendrian immersions M → CP is path connected In this section we prove the following result.
Theorem 4.1.
The space of holomorphic Legendrian immersions from an open Riemannsurface to CP is path connected in the compact-open topology. From a purely technical viewpoint, this may be the most difficult result in the paper.The fact that homogeneous coordinates on CP can be chosen such that the meromorphicfunctions, defining a given immersed holomorphic Legendrian curve, have only simple poles(see Proposition 2.2) is essential in our proof.We shall need the following parametric version of Weierstrass’s interpolation theorem forfinitely many points in an open Riemann surface. A similar result holds for a variable familyof infinite discrete sets of points, but this simple version suffices for our present application. emma 4.2. Let M be an open Riemann surface. Given maps a j : [0 , → M , j = 1 , . . . , k ,of class C r for some r ∈ { , , . . . , ∞ , ω } and integers n , . . . , n k ∈ N , there is a path ofholomorphic functions f t ∈ O ( M ) with C r dependence on t such that for every t ∈ [0 , and j = 1 , . . . , k , the function f t vanishes to order n j at a j ( t ) and has no other zeros.Proof. It suffices to prove the result for k = 1 and n = 1; the general case is then obtainedby taking for each j = 1 , . . . , k , a path of functions f j,t with a simple zero at a j ( t ) and noother zeros, and letting f t = Q kj =1 f n j j,t .Hence, let a : [0 , → M be a C r function. Assume first that a is real analytic. Then, a complexifies to a holomorphic map from an open simply connected neighbourhood D ⊂ C ofthe interval [0 , ⊂ R ⊂ C to M . Its graph Σ = { ( z, a ( z )) : z ∈ D } ⊂ D × M is a smoothcomplex hypersurface which defines a divisor on the Stein surface D × M . Since we clearlyhave H ( D × M, Z ) = 0, K. Oka’s solution of the second Cousin problem [36] shows that thisdivisor is defined by a holomorphic function f ∈ O ( D × M ) which vanishes to order 1 on Σand has no other zeros. The function f t = f ( t, · ) ∈ O ( M ) then has a simple zero at a ( t ) andno other zeros, and its dependence on t ∈ [0 ,
1] is real analytic.If a is of class C r but not real analytic, we proceed as follows. Choose a nowhere vanishingholomorphic vector field V on M and a relatively compact Runge domain M ⋐ M such that a ( t ) ∈ M for all t ∈ [0 , ǫ > φ s ( x ) of V exists for anyinitial point φ ( x ) = x ∈ M and for all s ∈ C with | s | < ǫ , and s φ s ( x ) maps the disc | s | < ǫ biholomorphically onto a neighbourhood U ( x ) ⊂ M of x . The diameter of theseneighbourhoods in a fixed metric on M is uniformly bounded from below for x ∈ M . Hence,approximating a : [0 , → M sufficiently closely by a real analytic map ˜ a : [0 , → M , wehave that ˜ a ( t ) = φ s ( t ) ( a ( t )) for a unique C r function s = s ( t ) with | s ( t ) | < ǫ for all t ∈ [0 , f t ∈ O ( M ) is a real analytic path of functions with simple zeros at ˜ a ( t ) for t ∈ [0 , f t = ˜ f t ◦ φ s ( t ) ∈ O ( M ) is a C r path of functions with simple zeros at a ( t ).To complete the proof, we approximate the path f t by a path of holomorphic functions on M without creating additional zeros. This is done inductively, exhausting M by an increasingsequence of Runge domains M ⊂ M ⊂ · · · ⊂ S ∞ j =1 M j = M and constructing a sequenceof C r paths f j,t ∈ O ( M j ) ( j ∈ Z + ) having simple zeros at a ( t ) and converging to a C r path f t ∈ O ( M ) with the same property. Note that f j +1 ,t = f j,t g j,t on M j , where g j,t is a C r path in O ( M j , C ∗ ). By the parametric Oka theorem for maps to C ∗ , we can approximate thepath g j,t by a C r path ˜ g j,t ∈ O ( M j +1 , C ∗ ). Replacing f j +1 ,t by f j,t ˜ g j,t gives a C r path in O ( M j +1 , C ∗ ) which approximates f j,t as closely as desired on a chosen compact subset of M j .Assuming that the approximations are close enough, the sequence f j,t converges as j → ∞ toa C r path f t ∈ O ( M ) solving the problem. (cid:3) Proof of Theorem 4.1.
Given a pair of Legendrian immersions F , F : M → CP , we mustfind a path of Legendrian immersions F t : M → CP , t ∈ [0 , F to F .Choose a hyperplane H ⊂ CP such that F and F are transverse to H . (Most hyperplanessatisfy this condition by Bertini’s theorem, cf. [31].) By Proposition 2.2, there are homoge-neous coordinates [ z : z : z : z ] on CP , with H = { z = 0 } , such that F j = F ( h j , g j )( j = 0 ,
1) where h j , g j are meromorphic functions on M with only simple poles satisfyingconditions (2.11). To prove the theorem, we shall find a path ( h t , g t ) ( t ∈ [0 , M connecting ( h , g ) to ( h , g ) and satisfying the followingconditions for every t ∈ [0 , t = 0 , i) h t and g t have only simple poles and the relations (2.11) hold.(ii) R C h t dg t = 0 for every closed curve C in a homology basis of M .(iii) Let P t = P ( h t ) ∪ P ( g t ) ⊂ M denote the union of the polar loci of h t and g t . Then themap ( h t , g t ) : M \ P t → C is an immersion.In light of (i), condition (iii) is clearly equivalent to(iv) ( h t , g t ) : M → ( CP ) is an immersion.The path of holomorphic Legendrian immersions F t = F ( h t , g t ) : M → CP , t ∈ [0 , h t , g t ) satisfying the stated condi-tions on some connected Runge domain D ⋐ M by a path satisfying the same conditions ona bigger Runge domain D ′ ⋐ M . To be precise, we exhaust M by an increasing sequence K ⊂ K ⊂ · · · ⊂ ∞ [ j =1 K j = M of compact smoothly bounded domains without holes such that K j ⊂ K ◦ j +1 for every j ∈ N .(The set K may be chosen as big as desired.) At every stage we shall approximate a givenfamily of solutions ( h jt , g jt ) on a neighbourhood of K j by one on a neighbourhood of K j +1 which has the same jets at a prescribed finite family of points in K j . Furthermore, we willensure that ( h jt , g jt ) agrees with the given pair ( h t , g t ) for t = 0 and t = 1.Since we shall be using partitions of unity on [0 , < r < / h t , g t ) = ( h , g ) for t ∈ [0 , r ], ( h t , g t ) = ( h , g ) for t ∈ [1 − r , . For t ∈ [0 , r ] ∪ [1 − r ,
1] let A t , B t , C t denote closed discrete subsets of M such that(a) A t is the set of poles of h t which are not poles of g t .(b) B t is the set of poles of g t which are not poles of h t .(c) C t is the set of common poles of g t and h t .Thus, the set P t := A t ∪ B t ∪ C t ⊂ M is the union of polar loci of h t and g t . (The reason for specifically distinguishing points in C t will become apparent shortly.) These sets do not depend on t in the indicated pair of intervals,but they will become t -dependent in subsequent steps when extending them to bigger sets ofparameter values t ∈ [0 , h t , g t ) of pairs ofmeromorphic functions on an open neighbourhood U = U of K in M such that conditions(a)–(c) above hold for all t ∈ [0 ,
1] at the points in P t ∩ U . This will be done in four steps.The neighbourhood U may shrink around K at every step without saying so each time.Fix once and for all a holomorphic immersion ζ : M → C (as provided by the theorem ofR. Gunning and R. Narasimhan [28]), so ζ provides a local holomorphic coordinate at everypoint of M . We can express any meromorphic function h in a neighbourhood of a point p ∈ M by a Laurent series in the local holomorphic coordinate z = ζ − ζ ( p ). We denote by c k ( h, p )the k -th coefficient of h at p in this series. Note that these coefficients are already defined forour functions h t , g t near t = 0 ,
1, they vanish for k < − tep 1. Choose a connected open neighbourhood D of K such that bD ∩ P = bD ∩ P = ∅ .For t ∈ [0 , r ] ∪ [1 − r , A t = A t ∩ D , B t = B t ∩ D , C t = C t ∩ D . We now extend each of these sets to all parameter values t ∈ [0 ,
1] (possibly adding morepoints to the already given sets) as follows. We connect a point a ∈ A to a point a ′ ∈ A by a smooth path a ( t ) ∈ D , t ∈ [0 , a ( t ) = a for t ∈ [0 , r ] and a ( t ) = a ′ for t ∈ [1 − r , t ∈ [0 , A and A if and only if they have the samecardinality. If A has more points than A , we choose a path a ( t ) starting at a point A without a matching pair in A such that a ( t ) exits D at some time t ∈ (0 , M \ D ) for the remaining values t ∈ ( t , A has more points than A , we do the same for points in A without matchingpairs in A , with t now running from t = 1 back to t = 0. (The parts of these paths lying in M \ D will be redefined in the next stage of the induction process.) Let A t ⊂ M , t ∈ [0 , A t for each t equals the bigger of the cardinalities of A ∩ D and A ∩ D , and in the process we mayhave added more points to these sets.We repeat the same procedure with the points in the families B and C , making sure thatthe resulting sets A t , B t , C t are pairwise disjoint for all t ∈ [0 , P t := A t ∪ B t ∪ C t . By the construction, the points in P t vary smoothly with t and their number does not dependon t . Hence, by Lemma 4.2 there is a smooth path of holomorphic functions f t ∈ O ( M ), t ∈ [0 , P t and nowhere else. Step 2.
Let D be the neighbourhood of K as in step 1. We extend the meromorphic jets of( h t , g t ) containing terms of orders − , , P t ∩ D for t ∈ [0 , r ] ∪ [1 − r , P t ∩ D , t ∈ [0 , h t dg t at the points in P t ∩ D .)Let us explain the details. We are interested in jets of the form ξ hp ( ζ ) = c h − ( p )( ζ − ζ ( p )) − + c h ( p ) + c h ( p )( ζ − ζ ( p )) , (4.2) ξ gp ( ζ ) = c g − ( p )( ζ − ζ ( p )) − + c g ( p ) + c g ( p )( ζ − ζ ( p )) . (4.3)For p ∈ P t ∩ D ( t ∈ [0 , r ] ∪ [1 − r , j = − , ,
1, we set c hj ( p ) = c j ( h t , p ) , c gj ( p ) = c j ( g t , p ) , where h t , g t are the initially given meromorphic functions. It is elementary to extend thecoefficients c hj , c gj to smooth functions c hj , c gj : P t ∩ D → C , t ∈ [0 , , j = − , , , satisfying the following conditions. • At points p ∈ A t ∩ D we have c h − ( p ) = 0 and c g ( p ) = 0. • At points p ∈ B t ∩ D we have c g − ( p ) = 0 and c h ( p ) = 0. • At points p ∈ C t ∩ D we have c h − ( p ) = 0, c g − ( p ) = 0, and c h − ( p ) c g ( p ) − c g − ( p ) c h ( p ) = 0 . hese are precisely conditions (2.11). There are no conditions on c h , c g . Remark 4.3.
The above conditions on points p ∈ C t allow nonzero values of c h ( p ) and c g ( p ),while those for points p ∈ A t force c g ( p ) = 0, and those for points p ∈ B t force c h ( p ) = 0.Hence, if a common pole of h t and g t would split into a pair of distinct poles of these functionsfor nearby values of t , the required conditions could not always be satisfied in a continuousway. For this reason, these three types of poles must remain distinct for all values of t . (cid:3) Step 3.
We shall find smooth paths h t , g t of meromorphic functions on a neighbourhood U ⊂ D of K having the jets constructed in step 2 at the points of P t ∩ U . (It will suffice touse paths of class C in the variable t ∈ [0 , t near 0 ,
1, and they will satisfy the following conditions for every p ∈ P t ∩ U and t ∈ [0 , c h ± ( p ) = c ± ( h t , p ) , c g ± ( p ) = c ± ( g t , p ) . These are Mittag-Leffler interpolation conditions at a variable family of points in the openRiemann surface M . This problem can be solved by using the ∂ -equation together withLemma 4.2. An important point is that a convex combination of solutions is again a solution,a fact which allows for the use of partitions of unity in the t -variable. A detail that one mustpay attention to is that the number of points in the sets P t ∩ D may vary with t . We nowexplain how to do this.Fix a point t ∈ (0 , t near t . (For t = 0 ,
1, we take the already given functions defined on all of M .) Choose a domain D ′ with K ⊂ D ′ ⊂ D such that P t ∩ bD ′ = ∅ (see (4.1)). Then, there is an open interval I t ⊂ [0 , t such that P t ∩ bD ′ = ∅ for all t ∈ I t . Hence, the number of points in the set P t ∩ D ′ = { p ( t ) , . . . , p k ( t ) } is independent of t ∈ I t . Choose small pairwise disjoint coordinate neighbourhoods U j ⊂ D ′ of the points p j ( t ), j = 1 , . . . , k , and for each j choose a smooth function χ j : M → [0 ,
1] whichequals 1 on a neighbourhood V j ⋐ U j of p j ( t ) and has support contained in U j . Shrinkingthe interval I t around t , we may assume that p j ( t ) ∈ V j for all t ∈ I t and j = 1 , . . . , k .Recall that the jet ξ hp is given by (4.2). We define˜ h t ( x ) = k X j =1 χ j ( x ) ξ hp j ( t ) ( ζ ( x ) − ζ ( p j ( t ))) , x ∈ M. The same expression, with ξ h replaced by ξ g , defines ˜ g t . Note that ˜ h t is a smooth functionon M \ P t whose restriction to V j agrees with ξ hp j ( t ) for every j = 1 , . . . , k ; the analogousstatement holds for ˜ g t . Since the number of points p j ( t ) ∈ P t ∩ D ′ is independent of t ∈ I t ,Lemma 4.2 furnishes a path of holomorphic functions f t ∈ O ( M ), t ∈ I t , vanishing to order2 at these points and nowhere else. We look for the desired path ( h t , g t ), t ∈ I t , in the form h t = ˜ h t − f t µ t , g t = ˜ g t − f t ν t , where µ t , ν t are paths of smooth functions to be found. The choice of f t implies that ( h t , g t )has the same jet with coefficients − , , h t , ˜ g t ) at the points in P t ∩ D ′ , and henceconditions (2.11) still hold for ( h t , g t ). he condition that h t is holomorphic (except at the poles in P t ∩ D ′ ) is0 = ∂h t = ∂ ˜ h t − f t ∂µ t ⇐⇒ ∂µ t = ∂ ˜ h t f t =: α t . Note that ∂ ˜ h t vanishes in V jt for each j = 1 , . . . , k , and since f t has zeros only on P t ∩ D ′ , α t is a smooth (0 , M depending smoothly on t ∈ I t . Hence, the equation ∂µ t = α t has a solution depending smoothly on t ∈ I t , and we get a desired path of functions h t asabove. The same procedure applies to g t . Remark 4.4 (On the parametric ∂ -equation) . There are several approaches in the literatureto solving the nonhomogeneous ∂ -equation by bounded linear operators, which therefore alsoapply in the parametric case. In the simple case at hand we have a 1-parameter family of ∂ -equations on a domain in an open Riemann surface. In this case, a solution operator –a Cauchy-Green-type operator similar to the one in the complex plane – has already beenconstructed by H. Behnke and K. Stein in 1949; see [16]. A discussion of this topic can befound in [20, Sect. 2]; see in particular Remark 1 in the cited paper. (cid:3) It remains to combine the partial solutions, obtained in this way on parameter subintervals I t ⊂ [0 , h t , g t ) ( t ∈ [0 , U of K . This is done byapplying a smooth partition of unity on [0 , g t is a nonconstantfunction for each t and ( h t , g t ) agrees with the initial pair ( h t , g t ) for t near 0 , Step 4.
We shall deform the path ( h t , g t ) ( t ∈ [0 , P t ∩ U for all t , and on U for t near0 and 1, to a path of immersions ( h t , g t ) : U → ( CP ) such that the 1-forms h t dg t havevanishing periods on a system of curves forming a homology basis of K . (The neighbourhood U is allowed to shrink around K .) Then, F t = F ( h t , g t ), t ∈ [0 , K into CP which agrees withthe given path for t near 0 and 1.The deformation will consist of two substeps. In the first one we shall obtain a pathof immersions U → ( CP ) , and in the second one we will modify it (through a path ofimmersions) to one satisfying also the period vanishing conditions.For substep 1 we consider paths ( h t , g t ) of the form(4.4) h t = h t + f t ξ t , g t = g t + f t η t , where f t ∈ O ( M ) is a path of holomorphic functions vanishing to the second order at thepoints of P t and nowhere else (such exists by Lemma 4.2) and ξ t , η t ∈ O ( U ) are paths ofholomorphic functions to be chosen. Note that every such map is already an immersion into( CP ) in small neighbourhoods of the points in P t ∩ U for all t . For a generic choice ofthe pair ξ t , η t ∈ O ( U ) near the zero function, the map ( h t , g t ) : U → ( CP ) is then animmersion by H. Whitney’s general position theorem [40]. Indeed, the domain of the map hasreal dimension 3 (including the t variable) and the maps are smooth, so the derivative withrespect to the variable x ∈ U of a generic map ( h t , g t ) of this kind misses the origin 0 ∈ C ,the latter being of real codimension 4.To simplify the notation, we denote the resulting path of immersions U → ( CP ) again( h t , g t ). We may assume that g t is nonconstant for each t . In substep 2 we keep g t fixed andconsider deformations of h t of the form(4.5) ˜ h t = h t + m t ξ t , t ∈ [0 , , here m t ∈ O ( U ) is a path of holomorphic functions vanishing to the second order at thepoints in P t ∩ U , and also at every critical point of g t in U which is not a pole of g t , while ξ t ∈ O ( U ) is a path of holomorphic functions. A suitable path of multipliers m t is given by m t = ( f t ) ( g ′ t ) , t ∈ [0 , , where f t ∈ O ( M ) is a path of holomorphic functions vanishing to the second order at thepoints of P t (such exists by Lemma 4.2) and g ′ t = dg t /dζ . (Here, ζ : M → C is an immersionchosen at the beginning of the proof.) Indeed, at any (simple) pole of g t the derivative g ′ t hasa second order pole, and since f t has a second order zero there, m t vanishes to order 6 − g t which is not a pole, the function ( g ′ t ) has a secondorder zero, and hence so does m t . We have that d ˜ h t = dh t + ξ t dm t + m t dξ t . At any critical point of g t not in P t , the second and the third term on the right hand sidevanish but dh t does not (since ( h t , g t ) is an immersion at such point), and hence d ˜ h t does notvanish either. This shows that any choice of path ξ t in (4.5) furnishes a path of immersions(˜ h t , g t ) : U → ( CP ) , and at the poles these functions have only changed for a second orderterm which does not affect the residues of h t dg t (these remain zero).It remains to choose the path ξ t ∈ O ( U ) in (4.5) such that the 1-form ˜ h t dg t has vanishingperiods on a homology basis of K . This can be done by the method in [12, Sect. 4]. Indeed,since we are deforming our maps only on the complement of the polar sets, we are dealingwith the standard contact form (2.3) on C and the results in the cited paper apply. Oneuses the convex integration method along with period dominating sprays and the parametricMergelyan approximation theorem. (A proof of the parametric Mergelyan approximationtheorem for maps to any complex manifold is spelled out in [22, Theorem 4.3].) Note thatthe problem is linear in ξ t , so we may use partitions of unity in the t variable. This reducesthe problem to small subintervals of [0 ,
1] where it is almost the same as the problem for asingle map treated in [12] (since the poles vary smoothly with t ). In particular, locally in t we can choose the homology basis of K in the complement of P t . (The parametric case forLegendrian immersions is treated in more detail in [24].)This completes the initial stage of the induction. Let us denote the resulting path ofmeromorphic functions, defined on a neighbourhood of K , again by ( h t , g t ). The constructionensures that ( h t , g t ) agrees with the initial family ( h t , g t ) near the endpoints of [0 , h t , g t ) ( t ∈ [0 , K which approximates ( h t , g t ) from stage 1 on K , it agrees withit near t = 0 ,
1, and it satisfies conditions (i)–(iii) (stated at the beginnning of the proof) onthe set K . This can be done by the essentially same procedure as in the initial stage, but usingalso the parametric Runge approximation theorem (a special case of the parametric Oka-Weiltheorem, see [21, Theorem 2.8.4]). In the first step, the sets A t , B t , C t must be defined sothat they agree with A t , B t , C t in a neighbourhood of K (this amounts to redefining the setsfrom stage 1 in the complement of K ). Let P t = A t ∪ B t ∪ C t ; so P t ∩ K = P t ∩ K for all t . In step 2, we extend the jets of ( h t , g t ) ( t ∈ [0 , P t ∩ K to P t ∩ K suchthat conditions (2.11) hold. When solving the ∂ -problem in step 3, we correct the solutionsby using the parametric Runge’s theorem so that they approximate those from stage 1 on K . Step 4 can be handled by the same tools, using period dominating sprays and parametricMergelyan approximation theorem in order to preserve the period vanishing conditions on K nd in addition fulfil those on the new curves in the period basis for K . The details aresimilar to those in the paper [24] and we leave them out.Proceeding in the same way, we obtain a sequence of solutions ( h mt , g mt ) ( t ∈ [0 , K m ( m ∈ N ) which approximates ( h m − t , g m − t ) on K m − and agrees with the initial data ( h t , g t )for t near 0 and 1. Assuming as we may that the approximations are close enough at everystep, the sequence ( h mt , g mt ) converges to a solution ( h t , g t ) on M as m → ∞ . (cid:3) The homotopy principle
Let M be an open Riemann surface. Let L formal ( M, CP ) be the space of formal holomor-phic Legendrian immersions from M to CP , that is, commuting squares T M φ / / (cid:15) (cid:15) ξ (cid:15) (cid:15) M f / / CP where ξ is the contact subbundle of T CP , φ is a monomorphism, and f is holomorphic. Inthis section we show that the inclusion L ( M, CP ) ֒ → L formal ( M, CP )induces a bijection of path components.There are very few results of this kind in the literature. The full parametric h-principleholds for Legendrian holomorphic immersions of M into C n +1 with the standard complexcontact structure (see [24]). Here, crucial use is made of the projection C n +1 → C n ,( x, y, z ) ( x, y ) (the standard contact form is dz + xdy ). For plain maps, not necessar-ily immersions, the h-principle is obvious.There are also results for projectivised cotangent bundles with the standard complexcontact structure (see [23]). The inclusion of the space of holomorphic Legendrian maps M → P T ∗ Z , where Z is a manifold with dim Z ≥
2, into the space of formal holomorphicLegendrian maps induces a surjection of path components. For closed holomorphic curvesthat are strong immersions, the inclusion induces a bijection of path components. And ifdim Z ≥
3, the inclusion also induces an epimorphism of fundamental groups, but this failsin general when dim Z = 2. Here, crucial use is made of the projection P T ∗ Z → Z and thefact that its fibres are Oka.Consider now formal holomorphic immersions of M into an Oka manifold Y , directed bya subbundle ξ of T Y . Trivialise
T M once and for all. Then a formal holomorphic immersion M → Y , directed by ξ , is nothing but a holomorphic map M → E , where E is the holomorphicfibre bundle over Y obtained from ξ by removing the zero section. The fibre of E is C k \ { } ,where k is the rank of ξ . Hence E is an Oka manifold, so the inclusion O ( M, E ) ֒ → C ( M, E ) isa weak equivalence. Determining the weak homotopy type of the space of formal holomorphicimmersions M → Y , directed by ξ , is thus reduced to a purely topological problem.The long exact sequence of homotopy groups · · · → π ( C k \ { } ) → π ( E ) → π ( Y ) → · · · shows that if Y is simply connected and k ≥
2, then E is simply connected, so O ( M, E ) is pathconnected. The basic h-principle follows, as long as there is at least one genuine holomorphicimmersion M → Y , directed by ξ . ow we return to holomorphic Legendrian immersions of M into CP . Here, of course, CP is simply connected and k = 2, so L formal ( M, CP ) is path connected. Also, by Theorem4.1, L ( M, CP ) is path connected and clearly nonempty (consider for example B (1 , g ) =[1 , , g, g : M → C is a holomorphic immersion, as provided by the theorem ofGunning and Narasimhan [28]). Thus we have the following h-principle. Theorem 5.1.
Every formal holomorphic Legendrian immersion from an open Riemannsurface to CP can be deformed to a genuine holomorphic Legendrian immersion, unique upto homotopy. Calabi-Yau property and complete immersions
In this and the following sections we discuss implications of our new results, as well asthose from some other recent papers, to the existence of complete Legendrian curves in CP and conformally immersed superminimal surfaces in the 4-sphere.We begin by discussing completeness of immersions on a formal level, with the aim ofconceptualising and unifying phenomena of this type in different geometries.Let ( N, g ) be a connected smooth Riemannian manifold of dimension n , possibly endowedwith some additional structure. For example, N could be a complex manifold, a complexcontact manifold, a manifold with a chosen subset of the tangent bundle, etc.Assume that M is a smooth manifold of dimension dim M < n . For every immersion F : M → N we have the induced Riemannian metric F ∗ g and distance function dist F on M .Assume now that M is either noncompact, or a compact manifold with nonempty boundary.For a fixed interior point p ∈ M we denote by R F ( M, p ) ∈ (0 , + ∞ ]the intrinsic radius of M , defined as the infimum of the lengths in the metric F ∗ g of alldivergent paths γ : [0 , → M with γ (0) = p . (The path is said to be divergent if thepoint γ ( t ) ∈ M leaves any compact subset of the interior of M when t → M is acompact manifold with boundary bM and F : M → N is an immersion, then R F ( M, p ) =dist F ( p , bM ) is simply the distance from p to bM in the metric F ∗ g . An immersion F is saidto be complete if F ∗ g is a complete metric on M ; if M is an open manifold, this is equivalentto R F ( M, p ) = + ∞ . Changing the base point clearly changes the intrinsic radius by anadditive constant which is irrelevant in our considerations. (We refer to M. do Carmo [19] foran introduction to Riemannian geometry where these concepts are explained in detail.)Consider a class of immersions F ( · , N ) from smooth manifolds M of dimension dim M 3, etc. Weshall say that M and N are admissible for the given class of immersions if the definition of he class makes sense for them; when writing F ∈ F ( M, N ) we tacitly assume that M and N are admissible for this class. For example, when considering holomorphic immersions, ourmanifolds must be complex, and for conformal immersions, they must be conformal manifolds.The smoothness class of manifolds and immersions may depend on the situation.We shall assume the following conditions on a class F ( · , N ) that we wish to consider.(a) If F ∈ F ( M, N ) and M ⊂ M is either an open domain or a compact smoothly boundeddomain, then F | M ∈ F ( M , N ). Conversely, if F : M → N is an immersion which is ofclass F in an open neighbourhood of every point of M , then F ∈ F ( M, N ).(b) If M is a compact admissible manifold with boundary, then F ( M, N ) is nonempty.(c) If a sequence F j ∈ F ( M, N ) converges in the C ( M, N ) topology to an immersion F : M → N , then F ∈ F ( M, N ).(d) (Interior estimates.) Let g be a Riemannian metric on M . Given F ∈ F ( M, N ), a pairof relatively compact domains M ⋐ M ⊂ M and a number ǫ > 0, there is δ > G ∈ F ( M , N ), we have that(6.1) max p ∈ M dist g ( F ( p ) , G ( p )) < δ = ⇒ max p ∈ M dist g ,g ( dF p , dG p ) < ǫ. Condition (a) says that immersions of class F are sections of a sheaf of immersions.Condition (b) is typically fulfilled by the existence of F -immersions M → N with valuesin a chart of N . Condition (c) says that F ( M, N ) is closed in the space of all immersions M → N in the C topology. Condition (d) means that the distance between F and G inthe C topology on the smaller domain M can be estimated by the distance between thetwo maps in the C topology (i.e., the uniform distance) on the bigger domain M . This isa typical elliptic type estimate which holds whenever our immersions are solutions of someelliptic PDE; in particular, it holds for harmonic and holomorphic maps.We have already mentioned the Calabi-Yau problem for minimal surfaces in the intro-ductory section. We now introduce the following key condition which lies behind all recentlyestablished Calabi-Yau-type theorems in various geometries. Definition 6.1 (Calabi-Yau property) . Assume that ( N, g ) is a Riemannian manifold and F ( · , N ) is a class of immersions into N satisfying conditions (a)–(d) above. The class F ( · , N )enjoys the Calabi-Yau property if the following holds true. Given a compact F -admissiblemanifold M with boundary bM , a point p ∈ M ◦ = M \ bM , an immersion F ∈ F ( M, N ),and numbers ǫ > λ > G ∈ F ( M, N ) such that(6.2) dist g ( G, F ) := max p ∈ M dist g ( G ( p ) , F ( p )) < ǫ and R G ( M, p ) > λ. The following result may be viewed as an abstract Calabi-Yau theorem . It is motivatedby the classical Calabi-Yau problem for minimal surfaces, and it summarises all recent resultson this subject in the literature (see Example 6.4). For the history of this problem, see thediscussion and references in [9, 8]. Theorem 6.2. Assume that ( N, g ) is a Riemannian manifold and F ( · , N ) is a class ofimmersions into N satisfying conditions (a)–(d) above and the Calabi-Yau property (seeDefinition 6.1). Let M be a compact F -admissible manifold with boundary. Then, every F ∈ F ( M, N ) can be approximated as closely as desired uniformly on M by a continuousmap F : M → N such that F | M ◦ : M ◦ = M \ bM → N is a complete immersion in F ( M ◦ , N ) .If in addition the immersion G in (6.2) can always be chosen injective on M or bM , then F can be chosen injective on M or bM , respectively. f in addition G in (6.2) can always be chosen to agree with F to a given finite order ateach point in a given finite subset of M ◦ , then F can also be so chosen.Furthermore, if M is a domain in M ◦ obtained by removing from M ◦ a countable familyof pairwise disjoint, compact, smoothly bounded domains D j , j ∈ N , then for every F ∈ F ( M, N ) and ǫ > , there exists a continuous map F : M → N such that dist g ( F, F | M ) < ǫ and F | M : M → N is a complete immersion in F ( M , N ) .Proof. The first statement is seen by following [4, proof of Theorem 1.1]. Indeed, the Calabi-Yau property allows one to construct a sequence of immersions F j ∈ F ( M, N ) ( j ∈ N ) whichconverges uniformly on M to a continuous map F : M → N and such that(6.3) lim j →∞ R F j ( M, p ) = + ∞ . Assuming as we may that the approximation of F j by F j +1 is sufficiently close in everystep, condition (d) on the class F (see in particular (6.1)) implies that the restrictions of F j to any relatively compact subset of M ◦ converge in the C topology to an immersion,and hence F | M ◦ ∈ F ( M ◦ , N ) in view of condition (a). Completeness of the limit immersion F | M ◦ : M ◦ → N follows from (6.3) in view of [9, Lemma 2.2] which shows that the intrinsicradius R F j ( M, p ) can decrease only a little under C -small deformations of the map. (Thisis obvious for small C deformations, but the point is that it also holds for C deformations.)Alternatively, one can apply the argument in [4, proof of Theorem 1.1], which controls frombelow the intrinsic radii of an increasing sequence of compact domains in M exhausting M ◦ ,using the fact that the convergence F j → F is in the C topology on each compact subsetof M ◦ . There it is also explained how to obtain injectivity of the limit map F on M or bM provided the immersions F j in the sequence can be chosen injective and the uniformapproximation is close enough at each step.The second statement is seen by [9, proof of Theorem 1.1] where this was proved forconformal minimal immersions from Riemann surfaces to flat Euclidean spaces R n . Fix a point p ∈ M and consider the decreasing sequence of domains M k = M ◦ \ S kj =1 D j , k ∈ N . Byusing the Calabi-Yau property (see in particular (6.2)) we construct a sequence of immersions F k ∈ F ( M k , N ), k = 1 , , . . . , converging uniformly on M = T k M k to a continuous map F : M → N and such that(6.4) lim k →∞ R F k ( M k , p ) = + ∞ . Assuming as we may that the convergence F k → F on M is fast enough, the interior estimates(6.1) ensure that the sequence F k converges in the C topology on any compact subset of M to an immersion, and hence F | M ∈ F ( M , N ) by condition (a). Finally, from (6.4) and [9,Lemma 2.2] it follows that F | M is a complete immersion. (cid:3) Remark 6.3. Since the immersions in Theorem 6.2 have ranges contained in a compactneighbourhood of the range of the initial immersion and any two metrics on N are comparableon a compact set, the immersions F ∈ F ( M ◦ , N ) and F ∈ F ( M , N ) in the above theoremcan be chosen complete in any given Riemannian metric on N . Example 6.4. The following classes of manifolds and immersions are known to enjoy theCalabi-Yau property (6.2), and hence the conclusion of Theorem 6.2 holds for them.(i) N = R n with n ≥ M is a compact conformal surface with boundary (or a compactbordered Riemann surface), and F ( M, R n ) is the space of conformal minimal (i.e., con-formal harmonic) immersions M → R n . See [4, Theorem 1.1] for the orientable case and 11, Theorem 6.6] for the nonorientable one. Injectivity on bM can be obtained for any n ≥ 3, and injectivity on M for any n ≥ N = C n , M is a compact bordered Riemann surface, and F ( M, C n ) is the space ofholomorphic (or null holomorphic for n ≥ 3) immersions; see [5]. In this case, injectivityon M can be obtained for any n ≥ 3, and injectivity on bM for any n ≥ N = C n +1 with the standard complex contact structure given by (2.2), M is a compactbordered Riemann surface, and F ( M, C n +1 ) is the space of Legendrian immersions ofclass C ( M, C n +1 ) which are holomorphic on M ◦ (see [12, Theorem 1.2 and Lemma6.5]). The limit map can be chosen injective on M .(iv) ( N, ξ ) is an arbitrary complex contact manifold, M is a compact smoothly boundeddomain in a Riemann surface f M , and F ( M, N ) is the space of holomorphic Legendrianimmersions from open neighbourhoods of M in f M to N . As in the previous case, thelimit map can be chosen injective on M . See [7, Theorem 1.3] where this is reduced to thecase N = C n +1 by using the existence of holomorphic Darboux neighbourhoods of anynoncompact immersed holomorphic Legendrian curve in a complex contact manifold.In all the examples listed above, the Calabi-Yau condition was obtained by finding ap-proximate solutions to the Riemann-Hilbert boundary value problem in the respective geom-etry, combined with the method of exposing boundary points of compact bordered Riemannsurfaces. This is usually the most difficult part of the work, intricately depending on theparticular properties of the given class of immersions. The Riemann-Hilbert method providesvery precise geometric control on the placement of the image manifold inside N , somethingwhich was impossible by the earlier methods used in the Calabi-Yau problem for minimalsurfaces. This allows us in particular to keep the source manifold M and its associated struc-tures (such as the conformal structure) unchanged. Sufficient conditions for the existence ofinjective immersions are obtained by proving a general position theorem for the given class ofimmersions, and this typically depends on the dimensions of the manifolds.The following is a challenging question. Problem 6.5. Let F ( · , N ) be the class of conformal minimal (that is, conformal harmonic)immersions from smooth conformal surfaces into a smooth Riemannian manifold ( N, g ) ofdimension at least 3. Does this class enjoy the Calabi-Yau property for every ( N, g )?We do not see any a priori reasons against this being true. Affirmative answers are knownonly when N is a flat Euclidean space (see Example 6.4 (i)), or the 4-sphere with the sphericalmetric and we are considering conformal supermininal immersions (see Theorem 7.5).In complex analytic problems of this type, the analogue of the Calabi-Yau problem iscalled Yang’s problem after P. Yang [42], who in 1977 asked about the existence of completebounded complex submanifolds in complex Euclidean spaces. There has been a recent surgeof activity on this problem which is briefly surveyed in [8, pp. 291–292]. In some of theseworks (see in particular [1, 6, 13, 14]), a weaker analogue of the Calabi-Yau property wasestablished by a different technique, using holomorphic automorphisms of complex Euclideanspaces to successively deform a given complex submanifold so that it avoids more and morepieces of a certain labyrinth, thereby increasing its intrinsic radius. The advantage of thismethod, when compared to the Riemann-Hilbert method, is that it preserves embeddedness,but the disadvantage is that one must cut away pieces of the source manifold to keep theimage bounded, so one loses the control of its complex structure (unless it is the disc). mmersions of types (i) and (ii) in Example 6.4 are known to satisfy the interpolation con-dition in Theorem 6.2. We now show that the classes (iii) and (iv) also satisfy this condition.The following is an extension of [7, Lemma 4.1]. Lemma 6.6. Let N be a complex contact manifold equipped with a Riemannian metric. Alsolet M be a compact bordered Riemann surface, E ⊂ M ◦ = M \ bM be a finite set, p ∈ M ◦ be a point, and F : M → N be a holomorphic Legendrian immersion. Given a number λ > (big), F can be approximated uniformly on M by holomorphic Legendrian immersions F : M → N satisfying the following conditions. (i) dist F ( p , bM ) > λ . (ii) F agrees with F to any given finite order at every point of E .Furthermore, if F | E : E → N is injective, then F : M → N can be chosen an embedding. We are assuming as we may that M is a smoothly bounded compact domain in an openRiemann surface f M , and by a holomorphic Legendrian immersion M → N we mean a holo-morphic Legendrian immersion defined on an unspecified neighbourhood of M in f M . Proof. The only novelty with respect to [7, Lemma 4.1] is condition (ii). When N = C n +1 with the Euclidean metric, the lemma coincides (except for condition (ii)) with [12, Lemma6.5] which holds true for any compact bordered Riemann surface (see the discussion at the be-ginning of [12, Sec. 6]). The interpolation condition at finitely many points is easily achievedby the techniques developed in [12]. It is the same technique which gives holomorphic im-mersions ( x, y ) : M → C n for which xdy = P nj =1 x j dy j is an exact 1-form on M ; any suchdefines a Legendrian immersion F = ( x, y, z ) : M → C n +1 with the last component functiongiven by z = − R xdy . To achieve the interpolation conditions, we arrange in addition thatthe immersion ( x, y ) has correct jets at points of the given finite set E (matching those ofthe initially given immersion to specified orders), and the integral of xdy has suitably pre-scribed values on a collection of arcs in M connecting a base point p ∈ M to the points in E . The last condition, which is arranged by the methods in [12, proof of Theorem 5.1], canbe used to ensure that the last component function z = − R xdy also has correct values at thepoints of E ; the jet intepolation condition for z at the points of E then follows immediatelyfrom those for ( x, y ). For the details in a similar setting, see the paper [3] where the authorsproved approximation results with jet interpolation for directed holomorphic immersions ofopen Riemann surfaces into complex Euclidean spaces.This proves the lemma for N = C n +1 . It is shown in [7, Theorem 1.1] that every complexcontact manifold N admits a holomorphic Darboux chart around any immersed noncompactholomorphic Legendrian curve. Using such charts, the general case of the lemma is obtainedby following word for word the proof of [7, Lemma 4.1], but applying the special case ofLemma 6.6 for N = C n +1 instead of [12, Lemma 6.5]. (cid:3) In view of Theorem 6.2, Lemma 6.6 implies the following Calabi-Yau type theorem forholomorphic Legendrian curves in any complex contact manifold. Except for the interpolationcondition, the statement for finite bordered Riemann surfaces coincides with [7, Theorem 1.3],while the part for surfaces with countably many boundary curves is new. Corollary 6.7 (Calabi-Yau theorem for Legendrian curves) . Holomorphic Legendrian immer-sions of compact bordered Riemann surfaces (possibly with countably many boundary curves)into any complex contact manifold N satisfy the conclusion of Theorem 6.2. n particular, if M is an open Riemann surface of finite genus and with at most countablymany ends, none of which are point ends, then M admits a complete injective holomorphicLegendrian immersion into any complex contact manifold. By the uniformisation theorem of X. He and O. Schramm [29], every open Riemann surfaceas in the second part of the above corollary is conformally equivalent to a domain in a compactRiemann surface with at most countably many closed geometric discs removed. Hence, it isof the kind as in the last statement in Theorem 6.2, so that result applies.We now introduce the Runge property for a class of immersions. Definition 6.8. A class of immersions F ( · , N ) enjoys the Runge property if every open F -admissible manifold M can be exhausted by a sequence M ⊂ M ⊂ · · · ⊂ S ∞ j =1 M j = M ofcompact smoothly bounded domains such that for every j ∈ N we have M j ⊂ M ◦ j +1 and every F ∈ F ( M j , N ) can be approximated in C ( M j , N ) by immersions in F ( M j +1 , N ).An exhaustion of M as in the above definition is called an F -Runge exhaustion . Forclasses of holomorphic or harmonic immersions from open Riemann surfaces, one typicallytries to establish Runge approximation on compact sets without holes in M . This holds fornull holomorphic immersions M → C n and more general directed holomorphic immersions(see [5, Corollary 2.7]), conformal minimal immersions into R n for any n ≥ n = 3 and [10, Theorem 5.3] for the general case), and holomorphic Legendrianimmersions into complex Euclidean or projective spaces with their standard complex contactstructures (see [12] for C n +1 and Sect. 3 of this paper for CP ). Theorem 6.9. Assume that ( N, g ) is a Riemannian manifold and F ( · , N ) is a class of im-mersions into N enjoying the Calabi-Yau property (see Definition 6.1) and the Runge property(see Definition 6.8). Then, every open F -admissible manifold M admits a complete immer-sion F : M → N in F ( M, N ) . The immersion F may be chosen an embedding provided theCalabi-Yau property and the Runge property hold for embeddings.Proof. Let ( M j ) j ∈ N be an F -Runge exhaustion on M (see Definition 6.1). By alternately usingthe Runge property and the Calabi-Yau property, we can find a sequence F j ∈ F ( M j , N ) suchthat for every j = 1 , , . . . , F j +1 | M j approximates F j as closely as desired in C ( M j , N ) andthe intrinsic diameter of M j +1 with respect to F j +1 is arbitrarily big. By doing this in the rightway, the sequence F j converges in C ( M k , N ) for each k to an immersion F : M → N whichis complete and belongs to F ( M, N ). If the Runge property and the Calabi-Yau propertyhold for embeddings, we can obtain a complete embedding M ֒ → N in F ( M, N ). (cid:3) Remark 6.10. (a) Since the Calabi-Yau property pertains to maps with ranges in a relativelycompact neighbourhood of the range of a given map, we see that the immersion F ∈ F ( M, N )in Theorem 6.9 can be chosen complete in any given metric on N .(b) In many cases of interest, it is possible to include the jet interpolation condition intothe Runge approximation property and thereby obtain a version of Theorem 6.9 with jetinterpolation on infinite closed discrete subsets of M . However, an axiomatic formulationwould demand further conditions on F -Runge exhaustions. (cid:3) Corollary 6.11 (Complete embedded Legendrian curves) . (i) Every Riemann surface ad-mits a complete injective holomorphic Legendrian immersion into CP . (ii) In the statement of the Runge Theorem 3.4 for Legendrian immersions of open Riemannsurfaces into CP , the approximating immersion can be chosen complete. iii) Every open Riemann surface admits a complete injective holomorphic Legendrian im-mersion into CP with (everywhere) dense image. (iv) If N is a connected complex contact manifold enjoying the Runge property with jet inter-polation on finite sets (see Def. 6.8) for holomorphic Legendrian immersions from openRiemann surfaces on any exhaustion by compact domains without holes, then every openRiemann surface admits a complete injective holomorphic Legendrian immersion into N with dense image.Proof. For a compact Riemann surface, (i) holds by Bryant’s theorem [18, Theorem G]. For anopen Riemann surface, it is seen by combining Theorem 3.4 (the Runge approximation theo-rem for holomorphic Legendrian immersions into CP ), Lemma 6.6 (the Calabi-Yau propertywith interpolation for Legendrian immersions), the general position theorem for holomorphicLegendrian immersions into an arbitrary complex contact manifold (see [7, Theorem 1.2]),and the proof of Theorem 6.9. The same argument gives part (ii).To obtain (iii), we apply the same proof but add also the interpolation condition at finitelymany points at every step of the inductive construction, adding more and more points of agiven closed discrete subset E in the open Riemann surface M as we go. In this way, wecan arrange that the resulting injective Legendrian immersion F : M → CP interpolates aprescribed injective map E → CP with dense image, and hence the curve F ( M ) is densein CP . (For the details, we refer to [2, Sect. 4.4] where the analogous result is proved forconformal minimal surfaces in R n ( n ≥ C n ( n ≥ C n +1 for any n ∈ N .)The same arguments apply in any complex contact manifold enjoying the Runge propertyfor holomorphic Legendrian immersions from open Riemann surfaces, thereby giving (iv). (cid:3) Superminimal surfaces in the four-dimensional sphere We begin by recalling the construction and basic properties of the Penrose twistor map π : CP → S ; see e.g. [37, 38]. We shall follow Bryant’s paper [18, Sect. 1], but the readermay also wish to consult Bolton and Woodward [17] and Wood [41].We let H denote the algebra of quaternions. An element of H can be written uniquely as q = x + i y + j u + k v = z + j w, z = x + i y ∈ C , w = u − i v ∈ C . Here, i , j , k are the quaternionic units. In this way we identify H with C and the quaternionicplane H = H × H with C . We write H ∗ = H \ { } . Consider the following diagram: C ∗ (cid:31) (cid:127) / / H ∗ = C ∗ φ (cid:15) (cid:15) CP (cid:31) (cid:127) / / CP π (cid:15) (cid:15) HP = S The map φ : C ∗ → CP is the standard quotient projection. The map π ◦ φ : H ∗ → HP associates to each quaternionic line H ⊂ H the corresponding point in the quaternionic 1-dimensional projective space HP , which is the 4-sphere. Each complex line Λ ⊂ C = H spans the unique quaternionic line H = Λ ⊕ j Λ ⊂ H , and the space of all complex lines withina given quaternionic line (which may be identified with C ) is clearly parameterised by CP . his observation defines the fibre bundle projection π : CP → S with fibre CP , called the twistor map or the twistor projection . It is not holomorphic, but its fibres are projective lines CP ⊂ CP . We endow CP with the Fubini-Study metric and S with the spherical metric.Among all minimal surfaces in S , there is a natural and important subclass consisting ofthe superminimal surfaces. The term was introduced in 1982 by Bryant [18], although super-minimal surfaces had been studied much earlier. In particular, Bryant mentions in his paperseveral works by E. Calabi and S. S. Chern from the period 1967–70, in which they exploitedthe fact that every minimal immersion of the 2-sphere into a higher-dimensional sphere issuperminimal. A minimal immersion from a Riemann surface M into S is superminimal ifa certain holomorphic quartic form on M vanishes identically [18, p. 466] (on the 2-sphereit always does). An equivalent, more geometric condition is that the curvature ellipse in thenormal plane to M at each of its points, which is determined by the second fundamentalform of M in S , is a circle centred at the origin. This characterisation is due to Friedrich(see [25, 26]). Superminimal immersions can also be characterised in terms of Legendrianimmersions via the twistor map, as we shall now describe.The properties of the twistor projection most relevant to our purposes are summarised inthe following theorem due to Bryant. The reference for part (1) is [18, Theorem A], whileparts (2) and (3) are the content of [18, Theorems B, B’, D]. When saying that X : M → S is a superminimal immersion, we shall always tacitly assume that X is conformal; parts (2)and (3) require this assumption. Theorem 7.1 (Bryant [18]) . Let ξ ⊂ T CP be the holomorphic contact bundle. (1) At every point p ∈ CP , the contact plane ξ p ⊂ T p CP is the orthogonal complement of ker( dπ p ) with respect to the Fubini-Study metric, and the differential dπ p : ξ p → T π ( p ) S is an isometry in the Fubini-Study metric on CP and the spherical metric on S . (2) If M is a Riemann surface and X : M → CP is a holomorphic or antiholomorphicLegendrian immersion, then π ◦ X : M → S is a superminimal immersion. (3) Conversely, every superminimal immersion M → S lifts to a unique holomorphic orantiholomorphic Legendrian immersion M → CP . Superminimal immersions M → S from a Riemann surface M which lift to holomorphicLegendrian immersions M → CP are said to have positive spin . We denote the space of suchimmersions by S + ( M, S ). Those lifting to antiholomorphic Legendrian immersions are saidto have negative spin . We denote the space of such immersions by S − ( M, S ). Remark 7.2. The main (and in fact the only) issue behind the spin is orientation. Orientationis irrelevant for a superminimal immersion Y : M → S , but it matters for its Legendrianlifting X : M → CP . Since Y is conformal, X is also conformal by Theorem 7.1 (1), andit parameterises a Legendrian surface S = X ( M ) ⊂ CP . Since any smooth Legendriansurface is necessarily a complex curve (see [7, Lemma 5.1]), a conformal parameterisation X : M → S is either holomorphic or antiholomorphic. Replacing the complex structureoperator J : T M → T M by its conjugate J , determined by the same conformal structurebut the opposite orientation on M , the spaces S + ( M, S ) and S − ( M, S ) interchange. Hence,there is no real loss of generality in focusing on superminimal surfaces of positive spin. (cid:3) Uniqueness of Legendrian lifting is intimately connected to the nonintegrability of thecontact structure ξ : given a superminimal surface at a point s ∈ S , there is a unique point p n the fibre π − ( s ) ⊂ CP such that the surface admits a local horizontal (Legendrian) liftingthrough p . Explicit formulas for the lifting can be found in [18, Sect. 1].By Theorem 7.1, postcomposition by π : CP → S defines a homeomorphism π ∗ : L ( M, CP ) −→ S + ( M, S )from the space of holomorphic Legendrian immersions M → CP to the space of superminimalimmersions M → S of positive spin, both endowed with the compact-open topology.We now give corollaries of the results on holomorphic Legendrian immersions into CP ,obtained in the previous sections. Corollary 7.3 (Runge approximation theorem for superminimal surfaces in S ) . Let M bea Riemann surface, open or compact, and let K be a compact subset of M . Every conformalsuperminimal immersion from a neighbourhood of K to S can be approximated uniformly on K by complete superminimal immersions Y : M → S . Furthermore, we may choose Y toagree with X to a given finite order at each point of a given finite subset of K .In particular, every Riemann surface immerses into the -sphere as a complete conformalsuperminimal surface.Proof. Let X : K → S be a superminimal immersion which may be assumed to be of positivespin (see Remark 7.2). Fix ǫ > 0, a finite set E ⊂ K , and an integer k ∈ N . By Theorem 7.1(3), X lifts to a holomorphic Legendrian immersion F : K → CP , i.e., X = π ◦ F . By Theorem3.4 (see also Remark 3.5) we can approximate F uniformly on K by complete holomorphicLegendrian immersions G : M → CP agreeing with F to order k at each point of E . (If M is compact, then every immersion from it is complete; the main point here concerns openRiemann surfaces.) The projection Y = π ◦ G : M → S is then a superminimal immersion(see Theorem 7.1 (3)) that approximates X on K and agrees with X to order k at each pointof E . Since G is complete and the twistor projection is an isometry on the contact subbundle(see Theorem 7.1), Y is complete as well. (cid:3) Similarly one proves the following interpolation theorem. Corollary 7.4 (Weierstrass interpolation theorem for superminimal surfaces in S ) . Let M be a Riemann surface, open or compact, and let E be a closed discrete subset of M . Everymap E → S extends to a complete superminimal immersion M → S . From the Calabi-Yau theorem for Legendrian immersions to CP (see Corollary 6.7) andthe fact that π is an isometry on the contact subbundle ξ ⊂ T CP we infer the following. Theorem 7.5 (Calabi-Yau theorem for conformal superminimal surfaces in S ) . If M is acompact bordered Riemann surface and X : M → S is a superminimal immersion (definedon a neighbourhood of M in some bigger Riemann surface), then X can be approximatedas closely as desired uniformly on M by a continuous map Y : M → S whose restrictionto the interior M ◦ is a complete, generically injective superminimal immersion, and whoserestriction to the boundary bM is a topological embedding. In particular, Y ( bM ) ⊂ S is aunion of pairwise disjoint Jordan curves.The analogous result holds for bordered surfaces with countably many boundary curves andwithout point ends (see the last part of Theorem 6.2 for the precise statement).Proof. This is seen by the same argument as in the proof of Corollary 7.3; however, wemust justify the statement that Y can be chosen generically injective on M and injective on M . To this end, it suffices to show that at every step of the inductive construction, theLegendrian immersion X j : M → CP can be chosen such that the superminimal immersion Y j := π ◦ X j : M → S is generically injective on M and injective on bM . If we approximatesufficiently closely at every step, then the limit map Y = lim j →∞ Y j : M → S will enjoy thesame properties. (For the details in a similar setting, see [4, proof of Theorem 1.1].)Let X : M → CP be a holomorphic Legendrian immersion. We claim that there is anarbitrarily C small holomorphic Legendrian perturbation X of X such that π ◦ X : M → S is generically injective and π ◦ X : bM → S is injective; this will complete the proof.Pick a point p ∈ S which does not lie on the surface π ◦ X ( M ) ⊂ S and choose Euclideancoordinates on S \ { p } = R . We associate to any map X : M → CP uniformly close to X the difference map δX : M × M → R defined by(7.1) δX ( x, x ′ ) = π ◦ X ( x ) − π ◦ X ( x ′ ) ∈ R , x, x ′ ∈ M. Since X is a Legendrian immersion, the map π ◦ X : M → S is an immersion by Theorem 7.1,and hence there is an open neighbourhood U ⊂ M × M of the diagonal ∆ := { ( x, x ) : x ∈ M } such that δX does not assume the value 0 ∈ R on U \ ∆. The same is then true for allmaps sufficiently close to X in C ( M, CP ). By the general position argument in [12, proofof Lemma 4.4], a generic holomorphic Legendrian immersion X : M → CP close to X in C ( M, CP ) is such that the difference map δX : M × M → R , and also its restriction δX : bM × bM → R , are transverse to the origin 0 ∈ R on M × M \ U and bM × bM \ U ,respectively. (The argument in [12, Lemma 4.4] is written for the standard contact structureon CP , but it applies in any complex contact manifold in view of the Darboux neighbourhoodtheorem [7, Theorem 1.1]. Compare with [7, proof of Theorem 1.2].) Assume that X is such.Since dim bM × bM = 2 < 4, it follows that δX does not assume the value 0 ∈ R on bM × bM \ ∆, which means that π ◦ X is injective on bM . Also, since dim M × M = 4,transversality of δX to 0 ∈ R on M × M \ U implies that ( δX ) − (0) ⊂ M × M consists ofthe diagonal ∆ together with at most finitely many points in M × M \ ∆. (cid:3) The following is an immediate consequence of Corollary 6.11 (iii) and Theorem 7.1. Corollary 7.6. Every open Riemann surface admits a complete superminimal immersion into S with dense image. Finally, Theorem 4.1 on path connectedness of the space of Legendrian immersions fromany open Riemann surface into CP immediately implies the following. Corollary 7.7. For every open Riemann surface M , the spaces S + ( M, S ) and S − ( M, S ) ofsuperminimal immersions M → S of positive resp. negative spin are path connected. Acknowledgements. A. Alarc´on is supported by the State Research Agency (SRA) and Eu-ropean Regional Development Fund (ERDF) via the grant no. MTM2017-89677-P, MICINN,Spain. F. Forstneriˇc is supported by the research program P1-0291 and the research grantJ1-9104 from ARRS, Republic of Slovenia. F. L´arusson is supported by Australian ResearchCouncil grant DP150103442. A part of the work on this paper was done while the second andthe third named authors were visiting the University of Granada in September 2019. Theywish to thank the university for the invitation and partial support. eferences [1] A. Alarc´on. Complete complex hypersurfaces in the ball come in foliations. ArXiv e-prints , Feb. 2018. https://arxiv.org/abs/1802.02004 .[2] A. Alarc´on and I. Castro-Infantes. Complete minimal surfaces densely lying in arbitrary domains of R n . Geom. Topol. , 22(1):571–590, 2018.[3] A. Alarc´on and I. Castro-Infantes. Interpolation by conformal minimal surfaces and directed holomorphiccurves. Anal. PDE , 12(2):561–604, 2019.[4] A. Alarc´on, B. Drinovec Drnovˇsek, F. Forstneriˇc, and F. J. L´opez. Every bordered Riemann surface is acomplete conformal minimal surface bounded by Jordan curves. Proc. Lond. Math. Soc. (3) , 111(4):851–886, 2015.[5] A. Alarc´on and F. Forstneriˇc. Null curves and directed immersions of open Riemann surfaces. Invent.Math. , 196(3):733–771, 2014.[6] A. Alarc´on and F. Forstneriˇc. A foliation of the ball by complete holomorphic discs. arXiv e-prints , May2019. https://arXiv:1905.09878 . Math. Z. , to appear.[7] A. Alarc´on and F. Forstneriˇc. Darboux charts around holomorphic Legendrian curves and applications. Int. Math. Res. Not. IMRN , (3):893–922, 2019.[8] A. Alarc´on and F. Forstneriˇc. New complex analytic methods in the theory of minimal surfaces: a survey. J. Aust. Math. Soc. , 106(3):287–341, 2019.[9] A. Alarc´on and F. Forstneriˇc. The Calabi-Yau problem for Riemann surfaces with finite genus and count-ably many ends. arXiv e-prints , Apr. 2019. https://arXiv:1904.08015 . Rev. Mat. Iberoam. , to appear.[10] A. Alarc´on, F. Forstneriˇc, and F. J. L´opez. Embedded minimal surfaces in R n . Math. Z. , 283(1-2):1–24,2016.[11] A. Alarc´on, F. Forstneriˇc, and F. J. L´opez. New complex analytic methods in the study of non-orientableminimal surfaces in R n . ArXiv e-prints , Mar. 2016. Memoirs Amer. Math. Soc. , to appear.[12] A. Alarc´on, F. Forstneriˇc, and F. J. L´opez. Holomorphic Legendrian curves. Compos. Math. , 153(9):1945–1986, 2017.[13] A. Alarc´on and J. Globevnik. Complete embedded complex curves in the ball of C can have any topology. Anal. PDE , 10(8):1987–1999, 2017.[14] A. Alarc´on, J. Globevnik, and F. J. L´opez. A construction of complete complex hypersurfaces in the ballwith control on the topology. J. Reine Angew. Math. , 751:289–308, 2019.[15] A. Alarc´on and F. J. L´opez. Minimal surfaces in R properly projecting into R . J. Differential Geom. ,90(3):351–381, 2012.[16] H. Behnke and K. Stein. Entwicklung analytischer Funktionen auf Riemannschen Fl¨achen. Math. Ann. ,120:430–461, 1949.[17] J. Bolton and L. M. Woodward. Higher singularities and the twistor fibration π : C P → S . Geom.Dedicata , 80(1-3):231–245, 2000.[18] R. L. Bryant. Conformal and minimal immersions of compact surfaces into the 4-sphere. J. DifferentialGeom. , 17(3):455–473, 1982.[19] M. P. do Carmo. Riemannian geometry . Mathematics: Theory & Applications. Birkh¨auser Boston, Inc.,Boston, MA, 1992. Translated from the second Portuguese edition by Francis Flaherty.[20] J. E. Fornæss, F. Forstneriˇc, and E. F. Wold. Holomorphic approximation: the legacy of Weierstrass,Runge, Oka-Weil, and Mergelyan. arXiv e-prints , Feb. 2018.[21] F. Forstneriˇc. Stein manifolds and holomorphic mappings (The homotopy principle in complex analysis) ,volume 56 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge . Springer, Cham, second edition,2017.[22] F. Forstneriˇc. Immersions of open Riemann surfaces into the Riemann sphere. arXiv e-prints , Oct. 2019. https://arxiv.org/abs/1910.06221 .[23] F. Forstneriˇc and F. L´arusson. Holomorphic Legendrian curves in projectivised cotangent bundles. arXive-prints , page arXiv:1809.09391, Sep. 2018.[24] F. Forstneriˇc and F. L´arusson. The Oka principle for holomorphic Legendrian curves in C n +1 . Math. Z. ,288(1-2):643–663, 2018.[25] T. Friedrich. On surfaces in four-spaces. Ann. Global Anal. Geom. , 2(3):257–287, 1984.[26] T. Friedrich. On superminimal surfaces. Arch. Math. (Brno) , 33(1-2):41–56, 1997. 27] M. Goresky and R. MacPherson. Stratified Morse theory , volume 14 of Ergebnisse der Mathematik undihrer Grenzgebiete, 3. Folge . Springer-Verlag, Berlin, 1988.[28] R. C. Gunning and R. Narasimhan. Immersion of open Riemann surfaces. Math. Ann. , 174:103–108, 1967.[29] Z.-X. He and O. Schramm. Fixed points, Koebe uniformization and circle packings. Ann. of Math. (2) ,137(2):369–406, 1993.[30] L. P. d. M. Jorge and F. Xavier. A complete minimal surface in R between two parallel planes. Ann. ofMath. (2) , 112(1):203–206, 1980.[31] S. L. Kleiman. The transversality of a general translate. Compositio Math. , 28:287–297, 1974.[32] S. Kobayashi and J. Eells Jr. Proceedings of the United States-Japan Seminar in Differential Geometry,Kyoto, Japan, 1965 . Nippon Hyoronsha Co., Ltd., Tokyo, 1966.[33] C. LeBrun. Fano manifolds, contact structures, and quaternionic geometry. Internat. J. Math. , 6(3):419–437, 1995.[34] C. LeBrun and S. Salamon. Strong rigidity of positive quaternion-K¨ahler manifolds. Invent. Math. ,118(1):109–132, 1994.[35] N. Nadirashvili. Hadamard’s and Calabi-Yau’s conjectures on negatively curved and minimal surfaces. Invent. Math. , 126(3):457–465, 1996.[36] K. Oka. Sur les fonctions analytiques de plusieurs variables. III. Deuxi`eme probl`eme de Cousin. J. Sci.Hiroshima Univ., Ser. A , 9:7–19, 1939.[37] R. Penrose. Twistor theory as an approach to fundamental physics. In Foundations of mathematics andphysics one century after Hilbert , pages 253–285. Springer, Cham, 2018.[38] R. Penrose and M. A. H. MacCallum. Twistor theory: an approach to the quantisation of fields andspace-time. Phys. Rep. , 6C(4):241–315, 1973.[39] H. L. Royden. Function theory on compact Riemann surfaces. J. Analyse Math. , 18:295–327, 1967.[40] H. Whitney. Differentiable manifolds. Ann. of Math. (2) , 37(3):645–680, 1936.[41] J. C. Wood. Twistor constructions for harmonic maps. In Differential geometry and differential equations(Shanghai, 1985) , volume 1255 of Lecture Notes in Math. , pages 130–159. Springer, Berlin, 1987.[42] P. Yang. Curvatures of complex submanifolds of C n . J. Differential Geom. , 12(4):499–511 (1978), 1977.[43] S.-T. Yau. Review of geometry and analysis. In Mathematics: frontiers and perspectives , pages 353–401.Amer. Math. Soc., Providence, RI, 2000. Antonio Alarc´onDepartamento de Geometr´ıa y Topolog´ıa e Instituto de Matem´aticas (IEMath-GR), Univer-sidad de Granada, Campus de Fuentenueva s/n, E–18071 Granada, Spaine-mail: [email protected] Franc ForstneriˇcFaculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI–1000 Ljubl-jana, SloveniaInstitute of Mathematics, Physics and Mechanics, Jadranska 19, SI–1000 Ljubljana, Sloveniae-mail: [email protected] Finnur L´arussonSchool of Mathematical Sciences, University of Adelaide, Adelaide SA 5005, Australiae-mail: [email protected]@adelaide.edu.au