Harmonic mappings and conformal minimal immersions of Riemann surfaces into RN

HARMONIC MAPPINGS AND CONFORMAL MINIMAL IMMERSIONS OF RIEMANN SURFACES INTO RN ANTONIO ALARC ´ ON†, ISABEL FERN ´ ANDEZ‡, AND FRANCISCO J. L ´ OPEZ† ABSTRACT. We prove that for any open Riemann surface N, natural number N≥3, non-constant harmonic map h:N → RN−2and holomorphic 2-form Hon N, there exists a weakly complete harmonic map X= (Xj)j=1,...,N:N → RNwith Hopf differential Hand (Xj)j=3,...,N=h. In particular, there exists a complete conformal minimal immersion Y= (Yj)j=1,...,N:N → RNsuch that (Yj)j=3,...,N=h. As some consequences of these results: •There exists complete full non-decomposable minimal surfaces with arbitrary conformal structure and whose generalized Gauss map is non-degenerate and fails to intersect Nhyperplanes of CPN−1in general position. •There exists complete non-proper embedded minimal surfaces in RN,∀N>3. 1. INTRODUCTION In this paper we use methods coming from the study of minimal surfaces to construct harmonic mappings of Riemann surfaces into RNwith prescribed geometry. A basic reference for this topic is, for instance, Klotz’s work [K]. Our main result states (see Corollary 4.5): Theorem A. For any open Riemann surface N,natural number N≥3, non-constant harmonic map h :N → RN−2and holomorphic 2-form Hon N,there exists a weakly complete harmonic map X = (Xj)j=1,...,N:N → RNwith Hopf differential Hand (Xj)j=3,...,N=h. Recall that the Hopf differential QXof a harmonic map X:N → RNis the holomorphic 2-form given by QX:=h∂zX,∂zXi, where ∂zmeans complex differential. By definition, Xis said to be weakly complete if ΓX:=|∂zX|2is a complete conformal Riemannian metric in N(see [K]). The fact that conformal minimal immersions are harmonic maps strongly influences the global theory of this kind of surfaces. It is well known that a harmonic immersion X:N → RNis minimal if and only if it is conformal, or equivalently, QX=0. Weakly completeness is equivalent to Riemannian completeness under minimality assumptions. The geometry of complete minimal surfaces in RN, specially those properties regarding the Gauss map, has been the object of extensive study over the last past decades (see for instance [O1,CO,C,F3,R]). In the recent paper [AFL], the authors constructed complete minimal surfaces in R3with arbitrarily prescribed conformal structure and non-constant third coordinate function (see also [AF]). As a consequence, any open Riemann surface admits a complete conformal minimal immersion in 2010 Mathematics Subject Classification. 53C43; 53C42, 30F15, 49Q05. Key words and phrases. Complete minimal surfaces, harmonic mappings of Riemann surfaces, Gauss map. †Research partially supported by MCYT-FEDER research project MTM2007-61775 and Junta de Andaluc´ıa Grant P09- FQM-5088. ‡Research partially supported by MCYT-FEDER research project MTM2007-64504 and Junta de Andaluc´ıa Grant P09- FQM-5088. 2 A. ALARC ´ ON, I. FERN ´ ANDEZ, AND F.J. L ´ OPEZ R3whose Gauss map omits two antipodal points of the unit sphere. Theorem A let us generalize these results to arbitrary higher dimensions (see Corollary 4.6): Theorem B. For any open Riemann surface N,natural number N≥3and non-constant harmonic map h :N → RN−2,there exists a complete conformal minimal immersion X= (Xj)j=1,...,N:N → RNwith (Xj)j=3,...,N=h. Under some compatibility conditions depending on the map h, the flux map of the immersion Xcan be also prescribed. Recall that the flux map of a conformal minimal immersion X:N → RN is given by pX(γ) = Im Rγ∂zXfor all γ∈ H1(N,Z). In particular, if his the real part of a holomorphic map H:N → CN−2, Theorem B provides a complete null holomorphic curve F= (Fj)j=1,...,N:N → CNsuch that (Fj)j=3,...,N=H. Likewise, Y= (Fj)j=2,...,N:N → CN−1is a complete holomorphic immersion whose last N−2 coordinates coincide with H. Theorem B also includes some information about the Gauss map of minimal surfaces in RN. Given a conformal minimal immersion X:N → RN, its generalized Gauss map GX:N → CPN−1,GX(P) = ∂zX(P), is holomorphic and takes values on the complex hyperquadric {∑N j=1w2 j= 0}. Chern and Osserman [C,CO] showed that if Xis complete then either X(N)is a plane or GX(N)intersects a dense set of complex hyperplanes. Even more, Ru [R] proved that if Xis complete and non-flat then GXcannot omit more than N(N+1)/2 hyperplanes in CPN−1lo- cated in general position (see also the works of Fujimoto [F2,F3] for a good setting). Under the non-degeneracy assumption on GX, this upper bound is sharp for some values of N, see [F4]. However, the number of exceptional hyperplanes strongly depends on the underlying conformal structure of the surface. Indeed, Ahlfors [A] proved that any holomorphic map G:C→CPN−1, N≥3, avoiding N+1 hyperplanes of CPN−1in general position is degenerate, that is to say, G(C)lies in a proper projective subspace of CPN−1(see [W, Chapter 5, §5] and [F1] for further generalizations). So, it is natural to wonder whether any open Riemann surface admits a complete conformal minimal immersion in RNwhose generalized Gauss map is non-degenerate and omits Nhyperplanes in general position. An affirmative answer to this question can be found in the following (see Corollary 4.8) Theorem C. Let Nbe an open Riemann surface, and let p:H1(N,Z)→RNbe a group morphism, N≥3. Then there exists a complete conformal full non-decomposable minimal immersion X : N → RNwith pX=pand whose generalized Gauss map is non-degenerate and omits N hyperplanes in general position. On the other hand, Theorem B leads to some interesting consequences regarding Calabi-Yau conjectures. The embedded Calabi-Yau problem for minimal surfaces asks for the existence of complete bounded embedded minimal surfaces in R3. Complete embedded minimal surfaces in R3with finite genus and countably many ends are proper in space [MPR,CM]. However, this result fails to be true for arbitrary higher dimensions. For instance, taking Nthe unit disc Din Cand h:D→R2the map h(z) = (Re(z), Im(z)), Theorem B generates complete non-proper embedded minimal discs in R4(so in RNfor all N≥4), see Corollary 4.7 for more details. The paper is laid out as follows. In Section 2 we introduce the necessary background and notations. In Section 3 we prove a basic approximation result by holomorphic 1-forms in open Riemann surfaces (Lemma 3.3), which is the key tool for proving our main results. Finally, in Section 4 we state and prove Theorems A, B and C. It is worth mentioning that all these theorems actually follows from the more general result Theorem 4.4 in Section 4. HARMONIC MAPPINGS AND MINIMAL IMMERSIONS IN RN3 ACKNOWLEDGMENTS The authors would like to thank Jos´e A. G´alvez for directing their attention to the paper [K]. 2. PRELIMINARIES Given a topological surface M,∂Mwill denote the one dimensional topological manifold determined by the boundary points of M. Given S⊂M, call by S◦and Sthe interior and the closure of Sin M, respectively. Open connected subsets of M−∂Mwill be called domains, and those proper topological subspaces of Mbeing surfaces with boundary are said to be regions. The surface Mis said to be open if it is non-compact and ∂M=∅. If Mis a Riemann surface, ∂zwill denote the global complex operator given by ∂z|U=∂ ∂wdw for any conformal chart (U,w)on M. Remark 2.1. Throughout this paper Nand Nwill denote a fixed but arbitrary open Riemann surface and natural number greater than or equal to three, respectively. Let Sdenote a subset of N,S6=N. We denote by F0(S)as the space of continuous functions f:S→Cwhich are holomorphic on an open neighborhood of Sin N. Likewise, F∗ 0(S)will denote the space of continuous functions f:S→Cbeing holomorphic on S◦. As usual, a 1-form θon Sis said to be of type (1, 0)if for any conformal chart (U,z)in N, θ|U∩S=h(z)dz for some function h:U∩S→C. We denote by Ω0(S)as the space of holomorphic 1-forms on an open neighborhood of Sin N. We call Ω∗ 0(S)as the space of 1-forms θof type (1, 0)on Ssuch that (θ|U)/dz ∈ F∗ 0(S∩U)for any conformal chart (U,z)on N. We denote by ℧0(S)as the space of holomorphic 2-forms on an open neigborhood of Sin N. Let Div(S)denote the free commutative group of divisors of Swith multiplicative notation. A divisor D∈Div(S)is said to be integral if D=∏n i=1Qni iand ni≥0 for all i. Given D1, D2∈Div(S), we write D1≥D2if and only if D1D−1 2is integral. For any f∈ F0(S)we denote by (f)its associated integral divisor of zeros in S. Likewise we define (θ)for any θ∈Ω0(S). In the sequel we will assume that Sis a compact subset of N. A compact Jordan arc in Nis said to be analytical if it is contained in an open analytical Jordan arc in N. By definition, a connected component Vof N −Sis said to be bounded if Vis compact, where Vis the closure of Vin N. Moreover, a subset K⊂ N is said to be Runge (in N) if N −K has no bounded components. Definition 2.2. A compact subset S⊂ N is said to be admissible if and only if (see Figure 2.1): •Sis Runge, •MS:=S◦consists of a finite collection of pairwise disjoint compact regions in Nwith C0 boundary, •CS:=S−MSconsists of a finite collection of pairwise disjoint analytical Jordan arcs, and •any component αof CSwith an endpoint P∈MSadmits an analytical extension βin N such that the unique component of β−αwith endpoint Plies in MS. Let Wbe a domain in N, and let Sbe either a compact region or an admissible subset in N. Wis said to be a tubular neighborhood of Sif S⊂Wand W−Sconsists of a finite collection of pairwise disjoint open annuli. In addition, if Wis a compact region isotopic to Wthen Wis said to be a compact tubular neighborhood of S. Here isotopic means that ∗:H1(W,Z)→ H1(W,Z)is an isomorphism, where :W→Wis the inclusion map. 4 A. ALARC ´ ON, I. FERN ´ ANDEZ, AND F.J. L ´ OPEZ FIGURE 2.1. An admissible subset. Let W⊂ N be a domain with S⊂W. We shall say that a function f∈ F∗ 0(S)can be uniformly approximated on Sby functions in F0(W)if there exists {fn}n∈N⊂ F0(W)such that {|fn−f|}n∈N→0 uniformly on S. A 1-form θ∈Ω∗ 0(S)can be uniformly approximated on S by 1-forms in Ω0(W)if there exists {θn}n∈N⊂Ω0(W)such that {θn−θ dz }n∈N→0 uniformly on S∩U, for any conformal closed disc (U,dz)on W. Given an admissible compact set S⊂W, a function f:S→Cn,n∈N, is said to be smooth if f|MSadmits a smooth extension f0to an open domain Vin Wcontaining MS, and for any component αof CSand any open analytical Jordan arc βin Wcontaining α,fadmits an smooth extension fβto βsatisfying that fβ|V∩β=f0|V∩β. Likewise, an 1-form θof type (1, 0)on Sis said to be smooth if for any closed conformal disc (U,z)on Nsuch that S∩Uis admissible, the function θ dz is smooth on S∩U. Given a smooth f∈ F∗ 0(S), we set d f ∈Ω∗ 0(S)as the smooth 1-form given by d f |MS=d(f|MS)and d f |α∩U= ( f◦α)′(x)dz|α∩U, where (U,z=x+iy)is a conformal chart on Wsuch that α∩U=z−1(R∩z(U)). Obviously, d f |α(t) = ( f◦α)′(t)dt for any component αof CS, where tis any smooth parameter along α. This definition makes sense also for smooth functions with poles in S◦. A smooth 1-form θ∈Ω∗ 0(S)is said to be exact if θ=d f for some smooth f∈ F∗ 0(S), or equivalently if Rγθ=0 for all γ∈ H1(S,Z). 2.1. Harmonic maps and minimal surfaces in RN.Given a non-constant harmonic map X= (Xj)j=1,...,N:N → RN, the holomorphic quadratic differential QX:=h∂zX,∂zXi= N ∑ j=1 (∂zXj)2 is said to be the Hopf differential of X. We also consider the conformal metric, possibly with isolated singularities, ΓX:=1 2 N ∑ j=1|∂zXj|2. It is clear that 2ΓXis greater than or equal to the Riemannian metric on N(possibly with singularities) induced by X. When Xis an immersion then ΓXis a Riemannian metric, and if in addition Xis complete then ΓXis complete as well [K]. However, the reciprocal does not hold in general. Definition 2.3. We say that a harmonic map X:N → RNis weakly complete (or complete in the sense of [K]) if ΓXis a complete metric in N. We also associate to Xthe group morphism pX:H1(N,Z)→RN,pX(γ) = Im Zγ∂zX. HARMONIC MAPPINGS AND MINIMAL IMMERSIONS IN RN5 Remark 2.4. If QX=0 and ΓXnever vanishes, then Xis a conformal minimal immersion, ΓXis the metric induced on Nby X, and pXis the flux map of X. If in addition Xis a conformal minimal immersion and we write ∂zXj=fjdζin terms of a local parameter ζon N,j=1, . . ., N, then the (generalized) Gauss map of Xis given by GX:N → CPN−1,GX(ζ) = [( fj(ζ))j=1,...,N], where [w]is the class of win CPN−1,∀w∈CN. It is well known that GXis a holomorphic map taking values in the complex quadric {[(wj)j=1,...,N]∈CPN−1|∑N j=1w2 j=0}. A set of hyperplanes in CPN−1is said to be in general position if each subset of khyperplanes, with k≤N−1, has an (N−1−k)-dimensional intersection. Definition 2.5 ([O2]).Let X:N → RNbe a conformal minimal immersion. •Xis said to be decomposable if, with respect to suitable rectangular coordinates in RN, one has ∑m k=1(∂zXk)2=0 for some m<N. •Xis said to be full if X(N)is contained in no hyperplane of RN. •The Gauss map GXis said to be degenerate if GX(N)lies in a hyperplane of CPN−1. When N=3, decomposable, non-full and degenerate are equivalent. However, if one passes to higher dimensions then no two of these conditions are equivalent (see [O2]). 3. THE APPROXIMATION LEMMA The next two lemmas are the key tools in the proof of the main result of this section (Lemma 3.3). They represent a slight generalization of Lemmas 2.4 and 2.5 in [AL]. From now on, ıdenotes the imaginary unit and the symbol 6≡ 0 means non-identically zero. Lemma 3.1. Let W ⊂ N be a domain with finite topology and S ⊂ N an admissible subset with S ⊂W. Consider f ∈ F∗ 0(S)∩F0(MS)with f |MS6≡ 0. Then f can be uniformly approximated on S by functions {fn}n∈Nin F0(W)satisfying that (fn) = (f|MS)on W.In particular, fnnever vanishes on W −MSfor all n. Proof. Let {Mk}k∈Nbe a sequence of compact tubular neighborhoods of MSin Wsuch that Mk⊂M◦ k−1for any k,∩k∈NMk=MSand fholomorphically extends (with the same name) to M1and has no zeros on M1−MS(take into account that f|MS6≡ 0). Choose Mkso that, in addition, the compact set Sk:=Mk∪CS⊂Wis admissible and α−M◦ kis a (non-empty) Jordan arc for any component αof CS. In particular, MSk=Mkand CSk=CS−M◦ k,k∈N. For any k∈Ntake gk∈ F∗ 0(Sk)∩F0(MSk)satisfying •gk|MSk=f|MSk, •gknever vanishes on Sk−S◦ k(recall that fhas no zeros on M1−MS), and •the sequence {gk|S}k∈Nuniformly converges to fon S. The construction of such functions is standard, we omit the details. Since gksatisfies the hypotheses of Lemma 2.4 in [AL], it can be uniformly approximated on Skby a sequence {gk,n}n∈N⊂ F0(W)with (gk,n) = (gk|MSk) = ( f|MS)on W, for any k. A standard diagonal argument concludes the proof.  Lemma 3.2. Let W ⊂ N be a domain with finite topology and S ⊂ N an admissible subset with S ⊂W. Consider θ∈Ω∗ 0(S)∩Ω0(MS)with θ|MS6≡ 0. 6 A. ALARC ´ ON, I. FERN ´ ANDEZ, AND F.J. L ´ OPEZ Then θcan be uniformly approximated on S by 1-forms {θn}n∈Nin Ω0(W)satisfying that (θn) = (θ|MS)on W.In particular, θnnever vanishes on W −MSfor all n. Proof. Let ϑbe a never vanishing 1-form in Ω0(W). Define f:=θ/ϑ∈ F∗ 0(S)∩F0(MS). By Lemma 3.1, fcan be uniformly approximated on Sby a sequence {fn}n∈Nin F0(W)satisfying that (fn) = ( f|MS)on Wfor all n. It suffices to set θn:=fnϑ,n∈N. Lemma 3.3. Let W ⊂ N be a domain with finite topology and S ⊂ N an admissible subset with S ⊂W. Let Θ∈℧0(W)and Φ= (φ1,φ2)be a smooth pair in Ω∗ 0(S)2∩Ω0(MS)2satisfying φ2 1+φ2 2=Θ|S and either of the following conditions: (A) φ1|MSand φ2|MSare linearly independent in Ω0(MS)and Θhas no zeros on CS. (B) Θ=0and φ1|MS6≡ 0. Then Φcan be uniformly approximated on S by a sequence {Φn= (φ1,n,φ2,n)}n∈N⊂Ω0(W)2 satisfying (a) φ2 1,n+φ2 2,n=Θ, (b) Φn−Φis exact on S,and (c) the zeros of Φnon W are those of Φon MS(in particular, Φnnever vanishes on W −MS). Proof. Assume (A) holds. Claim 3.4. Without loss of generality it can be assumed that φ1,φ2and dξnever vanish on CS,where ξ:=φ1/φ2. Proof. Assume for a moment that the conclusion of the lemma holds when φ1,φ2and dξnever vanish on CS. Take a sequence {Mk}k∈Nas in the proof of Lemma 3.1 such that Φholomorphically extends (with the same name) to M1, and φ1,φ2and dξnever vanish on M1−MS, for all n(take into account (A)). Recall that Sk:=Mk∪CS⊂W◦is an admissible set and CSk=CS−M◦ k,k∈N. Since Θnever vanishes on CS, which consists of a finite collection of pairwise disjoint analytical Jordan arcs, then we can find θ∈Ω0(CS)with θ2=Θ|CS. Consider fj:CS→C,fj=φj/θ, j=1, 2, and notice that f2 1+f2 2=1 and ξ|CS=f1/f2. Consider a sequence {(f1,k,f2,k)}k∈Nof pairs of smooth functions on CSksatisfying: i)f1,k,f2,kand d f1,knever vanish on CSk, ii)f2 1,k+f2 2,k=1, iii)the function gj,kgiven by gj,k|MSk=fj,gj,k|CSk=fj,k, lies in F∗ 0(Sk)and is smooth, j=1, 2, iv){fj,k}k∈Nuniformly converges to fjon CS,j=1, 2, and v)Ψk|S−Φis exact on S, where Ψk:= (gj,kθ)j=1,2 ∈Ω∗ 0(Sk)2∩Ω0(MSk)2. The construction of this data is standard, we omit the details. Write Ψk= (ψj,k)j=1,2 and ξk= ψ1,k/ψ2,k. From i),ii)and the definition of θfollow that ψ2 1,k+ψ2 2,k=Θand dξknever vanishes on CSk. Moreover, iv)gives that {Ψk|S}k∈Nuniformly converges to Φon S. By hypothesis, Lemma 3.3 holds for any Ψk, then there exists a sequence {Ψk,n}n∈Nuniformly converging to Ψkon Skand satisfying (a), (b) and (c) of Lemma 3.3 for Φ=Ψkand S=Sk. Using that {Ψk|S}k∈Nconverges to Φ, the zeros of Ψkin MSkare those of Φin MS,v), and a standard diagonal argument, we can obtain a sequence satisfying the conclusion of the lemma, proving the claim.  In the sequel we will assume that φ1,φ2and dξnever vanish on CS. HARMONIC MAPPINGS AND MINIMAL IMMERSIONS IN RN7 Label η=φ1−ıφ2∈Ω∗ 0(S)∩Ω0(MS)and observe that Θ/η=φ1+ıφ2∈Ω∗ 0(S)∩Ω0(MS). Notice that (Θ/η)|MS,η|MS6≡ 0, φ1=1 2η+Θ ηand φ2=ı 2η−Θ η. Let BSbe a homology basis of H1(S,Z)and label νas its cardinal number. Consider in F∗ 0(S) the maximum norm and the Fr´echet differentiable map P:F∗ 0(S)→C2ν,P(f) = Zcefη+e−fΘ η−2φ1,efη−e−fΘ η+2ıφ2c∈BS . Label A:F∗ 0(S)→C2νas the Fr´echet derivative of Pat 0. Claim 3.5. A|F0(W)is surjective. Proof. Reason by contradiction and assume that A(F0(W)) lies in a complex subspace U= {(xc,yc)c∈BS∈C2ν|∑c∈BSAcxc+Bcyc=0}, where Ac,Bc∈C,∀c∈ BS, and (3.1) ∑ c∈BS|Ac|+|Bc|6=0. Then, writing Γ1=∑c∈BSAccand Γ2=∑c∈BSBcc, we have (3.2) ZΓ1 fφ2+ıZΓ2 fφ1=0, ∀f∈ F0(W). Denote by Σ={f∈ F0(W)|(f)≥(φ2|MS)2}(recall that φ2never vanishes on CS). Then for any f∈Σthe function d f /φ2∈ F∗ 0(S)∩F0(MS), so it can be uniformly approximated on S by functions in F0(W). This fact is trivial when fis constant, otherwise use Lemma 3.1. Hence equation (3.2) applies and gives (3.3) 0 =ZΓ2 ξd f =ZΓ2 f dξ,∀f∈Σ, where we have used integration by parts (notice that fξ,ξd f and f dξare smooth). Suppose Γ26=0 and take [τ]∈ H1 hol(W)(the first holomorphic De Rham cohomology group of W) and g∈ F0(W)so that RΓ2τ6=0, the function f:= (τ+dg)/dξlies in F∗ 0(S)∩F0(MS) and (f|MS)≥(φ2|MS)2. The existence of such 1-form and function follows from well known arguments on Riemann surfaces theory (take into account that (A) implies dξ|MS6≡ 0). By Lemma 3.1, fcan be uniformly approximated on Sby functions in Σ, so equation (3.3) applies and shows that 0 =RΓ2f dξ=RΓ2(τ+dg) = RΓ2τ6=0, a contradiction. Therefore Γ2=0. Replacing (ξ,φ1,φ2,Γ1,Γ2)by (1/ξ,φ2,φ1,Γ2,Γ1)and using a symmetric argument, we can prove that Γ1=0. This contradicts (3.1) and concludes the proof.  Let {e1, . . . , e2ν}be a basis of C2ν, fix fi∈ A−1(ei)∩F0(W)for all i, and set Q:C2ν→C2νas the analytical map given by Q((zi)i=1,...,2ν) = P(∑ i=1,...,2ν zifi). By Claim 3.5 the differential dQ0of Qat 0 is an isomorphism, then there exists a closed Euclidean ball U⊂C2νcentered at the origin such that Q:U→ Q(U)is an analytical diffeomorphism. Furthermore, notice that 0 =Q(0)∈ Q(U)is an interior point of Q(U). Consider a sequence {θn}n∈N⊂Ω0(W)uniformly approximating ηon Sand with (θn) = (η|MS)for all n(recall that η|MS6≡ 0 and see Lemma 3.2). 8 A. ALARC ´ ON, I. FERN ´ ANDEZ, AND F.J. L ´ OPEZ Label Pn:F∗ 0(S)→C2νas the Fr´echet differentiable map given by Pn(f) = Zcefθn+e−fΘ θn−2φ1,efθn−e−fΘ θn +2ıφ2c∈BS ,∀n∈N. Call Qn:C2ν→C2νas the analytical map Qn((zi)i=1,...,2ν) = Pn(∑i=1,...,2νzifi)for all n∈N. Since {Qn}n∈N→ Q uniformly on compacts subsets of C2ν, without loss of generality we can suppose that Qn:U→ Qn(U)is an analytical diffeomorphism and 0 ∈ Qn(U)for all n. Label αn= (α1,n, . . . , α2ν,n)as the unique point in Usuch that Qn(αn) = 0 and note that {αn}n∈N→0. Set ηn:=e∑2ν i=1αi,nfiθn,φ1,n:=1 2ηn+Θ ηnand φ2,n:=ı 2ηn−Θ ηn,∀n∈N and let us check that the sequence {Φn= (φ1,n,φ2,n)}n∈Nsatisfies the conclusion of the lemma. Indeed, since (ηn) = (θn) = (η|MS)one has Θ/ηn∈Ω0(W)and so Φn∈Ω0(W)2. The convergence of {Φn}n∈Nto Φon Sfollows from the ones of {θn}n∈Nto ηand of {αn}n∈Nto 0. A straightforward computation gives (a). The fact that Qn(αn) = 0, n∈N, implies (b). Finally, (ηn) = (η|MS)for all nimplies (c). The proof of the lemma in case (B) goes as follows. Notice that Θ=0 is nothing but φ2=βφ1, where β∈ {ı,−ı}. As above, we can assume without loss of generality that φ1never vanishes on CS(we omit the details). Reasoning as in case (A), we can prove that ˆ A|F0(W):F0(W)→Cνis surjective, where ˆ Ais the Fr´echet derivative of ˆ P:F∗ 0(S)→Cν,ˆ P(f) = Rc(ef−1)φ1c∈ˆ BS , at 0. Take ˆ fi∈ˆ A−1(ˆ ei)∩F0(W)for all i, where ˆ BS={ˆ e1, . . . , ˆ eν}is a basis of Cν, and define ˆ Q:Cν→Cν by ˆ Q((zi)i=1,...,ν) = ˆ P(∑i=1,...,νziˆ fi). Now, consider a sequence {ˆ θn}n∈N⊂Ω0(W)that uniformly approximates φ1on Sand (ˆ θn) = (φ1|MS)for all n(as above, recall that φ1|MS6≡ 0 and see Lemma 3.2). Set ˆ Pn:F∗ 0(S)→Cνby ˆ Pn(f) = Rc(efˆ θn−φ1)c∈ˆ BS , and call ˆ Qn:Cν→Cνas the analytical map ˆ Qn((zi)i=1,...,ν) = ˆ Pn(∑i=1,...,νziˆ fi)for all n∈N. To finish, reason as in case (A) but setting φ1,n:=e∑ν i=1ˆ αi,nˆ fiˆ θnand φ2,n:=βφ1,n, where ˆ αn= (ˆ α1,n, . . . , ˆ αν,n)is chosen so that ˆ Qn(ˆ αn) = 0 and {ˆ αn}n∈N→0.  4. MAIN RESULTS The main results of this paper follow as consequence of Lemma 4.1 below. Although the proof of this lemma is inspired by the technique developed in [AFL, Lemma 3.1], it represents a wide generalization of that result. We need the following notations and definitions. Fix a nowhere zero τ0∈Ω0(N)(the existence of such a τ0is well known, anyway see [AFL] for a proof). Then for any compact subset K⊂ N and any θ∈Ω∗ 0(K)we set kθk:=maxK{|θ/τ0|}. This norm induces the topology of the uniform convergence on Ω∗ 0(K). Let K⊂ N be a connected compact region and σ2a Riemannian metric on Kpossibly with singularities. Given P,Q∈Kwe denote by dist(K,σ)(P,Q) = min{lengthσ(α)|αcurve in Kjoining P and Q}. If K1and K2are two compact sets in Kwe set dist(K,σ)(K1,K2) = min{dist(K,σ)(P,Q)|P∈ K1,Q∈K2}. Lemma 4.1. Let M,K be two compact regions in Nwith M ⊂K◦.Assume that M is Runge, K is connected and consider P0∈M◦.Let Ibe a conformal Riemannian metric on K possibly with isolated singularities. Let f= (f1,f2):H1(K,Z)→C2be a group homomorphism, Θ∈℧0(K)and HARMONIC MAPPINGS AND MINIMAL IMMERSIONS IN RN9 Φ= (φ1,φ2)∈Ω0(M)2satisfying φ2 1+φ2 2=Θ|M,f(γ) = Zγ Φ,∀γ∈ H1(M,Z), and either of the following conditions: (A) φ1and φ2are linearly independent in Ω0(M). (B) Θ=0, φ16≡ 0and there is β∈ {ı,−ı}such that f2=βf1and φ2=βφ1. Then, for any ǫ>0there exists Ψ= (ψ1,ψ2)∈Ω0(K)2so that (L1) kΨ−Φk<ǫon M, (L2) ψ2 1+ψ2 2=Θ, (L3) f(γ) = RγΨ,∀γ∈ H1(K,Z), (L4) dist(K,σ(Ψ,I))(P0,∂K)>1/ǫ,where σ2 (Ψ,I):=|ψ1|2+|ψ2|2+I,and (L5) the zeros of Ψon K are those of Φon M (in particular, Ψnever vanishes on K −M). Proof. The proof goes by induction on minus the Euler characteristic of W−M◦. Since Mis Runge then no component of K−M◦is a closed disc, and so −χ(K−M◦)≥0. The basis of the induction is proved in the following Claim 4.2. Lemma 4.1 holds if χ(K−M◦) = 0. Proof. In this case K◦−M=∪k j=1Aj, where Ajare pairwise disjoint open annuli, k∈N. On each Ajwe construct a Jorge-Xavier’s type labyrinth of compact sets as follows (see [JX]). Let zj:Aj→Cbe a conformal parametrization, and let Cj⊂Ajbe a compact region such that Cjcontains no singularities of I,zj(Cj)is a compact annulus of radii rjand Rj, where rj<Rj, and zj(Cj)contains the homology of zj(Aj). This choice is possible since the singularities of Iare isolated. Since I|Cjhas no singularities, we can find a positive constant µwith (4.1) I>µ2|dzj|2on Cj,j=1, . . ., k. Consider a large m∈N(to be specified later) such that 2/m<min{Rj−rj|j=1, . . . , k}. For any j∈ {1, . . . , k}label sj,0 :=Rjand for any n∈ {1, . . . , 2m2}set sj,n:=Rj−n/m3and consider the compact set in Cj(see Figure 4.1): Kj,n=P∈Aj sj,n+1 4m3≤ |zj(P)| ≤ sj,n−1−1 4m3,1 m2≤arg((−1)nzj(P)) ≤2π−1 m2. Then, define Kj= 2m2 [ n=1Kj,nand K= k [ j=1Kj. Consider the pair Ξ= (ϕ1,ϕ2)∈Ω0(M∪K)2given by Ξ|M=Φ,Ξ|Kj=(1 2(λdzj+Θ λdzj),ı 2(λdzj−Θ λdzj)if (A) holds λdzj,βλdzjif (B) holds, j=1, . . . , k, where λ>√2µm4is a constant. Notice that ϕ2 1+ϕ2 2=Θ|M∪K. Let W⊂ N be a domain with finite topology containing K. Applying Lemma 3.3 to the data ˆ W=W,ˆ S=M∪K,ˆ Θ=Θ, and ˆ Φ=Ξ,