Unstable minimal surfaces of annulus type in manifolds

DOI 10.1515 / ADVGEOM.2009.021 de Gruyter 2009

Unstable minimal surfaces of annulus type in manifolds

Hwajeong Kim^∗

(Communicated by F. Duzaar)

Abstract. Unstable minimal surfaces are the unstable stationary points of the Dirichlet integral.

In order to obtain unstable solutions, the method of the gradient flow together with the minimax- principle is generally used, an application of which was presented in [19] for minimal surfaces in Euclidean space. We extend this theory to obtain unstable minimal surfaces in Riemannian manifolds. In particular, we consider minimal surfaces of annulus type.

1 Introduction

For given curvesΓl ⊂ N,l = 1, . . . , m andΓ := Γ1 ∪ · · · ∪Γm, where(N, h)is a Riemannian manifold of dimensionn≥2 with metric(hαβ), we denote the generalized Plateau Problem byP(Γ). This deals with minimal surfaces bounded byΓ, in other words parametrizationsXdefined onΣ⊂R²with∂Σ = Γ, satisfying the following constraints:

(1) τh(X) =0,

(2) |Xu|²_h− |Xv|²_h=hXu, Xvih=0, (3) X|∂Σis weakly monotone and ontoΓ,

whereτh := ∆X^α−Γ^α_βγ∇X^βX^γ =0 is the harmonic equation on(N, h)seen as the Euler–Lagrange equation of the energy functional.

A regular minimal surface is called unstable if its surface area is not a minimum among neighbouring surfaces with the same boundary. Extending the Ljusternik–

Schnirelmann theory on convex sets in Banach spaces, a variational approach to unstable minimal surfaces of disc or annulus type inRⁿwas proposed in 1983 ([21], see also [19], [20]). For the minimal surfaces of higher topological structure inRⁿ, see [11].

Recently in [8], the existence of unstable minimal surfaces of higher topological struc- ture with one boundary in a nonpositively curved Riemannian manifold was studied by applying the method in [19]. In particular, the first part of that paper considers the Jacobi field extension operator as the derivative of the harmonic extension.

∗This paper is based on my thesis [12] supervised by Professor Michael Gr¨uter.

In this article, we study unstable minimal surfaces of annulus type in manifolds. The Euclidean case was tackled already in [20], and our aim is to generalize this result to manifolds satisfying appropriate conditions. Namely, we will consider two boundary curvesΓ1,Γ2in a Riemannian manifold(N, h)such that one of the following holds.

(C1) There existsp∈ N withΓ1,Γ2 ⊂ B(p, r), whereB(p, r)lies within the normal range of all its points. We assume thatr < π/(2√

κ), whereκis an upper bound of the sectional curvature of(N, h).

(C2) Nis compact with nonpositive sectional curvature.

These conditions are related to the existence and uniqueness of the harmonic extension for a given boundary parametrization.

First, we construct suitable spaces of functions, the boundary parametrizations, dis- tinguishing the Cases (C1) and (C2). We introduce a convex set which serves as a tangent space for the given boundary parametrization. Then we consider the following functional:

E(x) :=1 2

Z

|dF(x)|²_h,

whereF(x) denotes the harmonic extension of annulus type or of two-disc type with boundary parametrizationx. We next discuss the differentiability ofE, in particular for the case in which the topology of the surfaces changes (from an annulus to two discs).

Defining critical points ofE, will show the equivalence between the harmonic extensions (inN) of critical points ofE and minimal surfaces inN. The H^2,2-regularity of the harmonic extension of a critical point ofE(see the appendix or [13]) plays an important role in the argument.

In Section 4, we prove the Palais–Smale condition forE. In particular, we investigate carefully the behaviour of boundary mappings which are fixed at only one point. In order to deform level sets ofE, we also construct a suitable vector field and its corresponding flow. Roughly speaking, Lemma 4.3 shows that the energy of some annulus-type har- monic extensions is greater than that of two-disc type harmonic extensions by a uniformly positive constant. Although this result refers to Riemannian manifolds, it turns out to be more restrictive than that of Euclidean spaces, which holds uniformly on any bounded set of boundary parametrizations. This somewhat weaker result is anyhow enough for the present purposes.

Following the arguments set out in [21], we can prove the main theorem of this paper.

This states that if there exists a minimal surface (of annulus type) whose energy is a strict relative minimum inS(Γ1,Γ₂)(suitably defined for each Case (C1) and (C2)), the exis- tence of an unstable minimal surface of annulus type is ensured under certain assumptions related to the solutions ofP(Γ_i). We eventually apply this result to the three-dimensional sphereS³ and the three-dimensional hyperbolic spaceH³, whose curvatures are 1 and

−1, respectively.

2 Preliminaries

2.1 Some definitions. Let(N, h)be a connected, oriented, complete Riemannian man- ifold of dimensionn≥2, embedded isometrically and properly into someR^kas a closed

submanifold by means of the mapη([3]). Moreover,dωandd0denote the area elements inΩ⊂R²and in∂Ωrespectively.

Indicating

B:={w∈R²| |w|<1}

we define

H^1,2∩C⁰(B, N) :={f ∈H^1,2∩C⁰(B,R^k)|f(B)⊂N}

with normkfk1,2;0:=kdfk_L2+kfk_C0. Now set

TfH^1,2∩C⁰(B, N)∼={V ∈H^1,2∩C⁰(B,R^k)|V(·)∈T_f(·)N}

=:H^1,2∩C⁰(B, f^∗T N) with norm

kVk:=

Z

B

|∇^fV|²_hdω 1/2

+kVk_C0 ∼= Z

B

|dV|²_Rkdω 1/2

+kVk_C0. (1) LetΓbe a Jordan curve in N diffeomorphic toS¹ := ∂B. ThenN can be equipped with another metric ˜h such that Γ is a geodesic in (N,˜h). We observe that H^1,2 ∩ C⁰ (B, ∂B),(N,Γ)h˜

andH^1,2∩C⁰ (B, ∂B),(N,Γ)_h

coincide as sets.

Using the exponential map in(N,˜h), we let

H^12,2∩C⁰(∂B; Γ) :={u∈H^12,2∩C⁰(∂B,R^k)|u(∂B) = Γ}, where the norm is given bykuk1

2,2;0 :=kdH(u)k_L2+kuk_C0, andH(u)is the harmonic extension inR^kwithH(u)|∂B(·) =u(·). In addition

TuH^12,2∩C⁰(∂B; Γ)

:={ξ∈H^12,2∩C⁰(∂B, u^∗T N)|ξ(z)∈T_u(z)Γ,for allz∈∂B}

=H^12,2∩C⁰(∂B, u^∗TΓ).

Finally, the energy off ∈H^1,2(Ω, N)is denoted by E(f) := 1

2 Z

Ω

|df|²_hdw.

2.2 The setting. Let Γ₁,Γ₂ be two Jordan curves of classC³ inN with diffeomor- phismsγⁱ:∂B→Γi, i=1,2, anddist(Γ1,Γ2)>0. Forρ∈(0,1)let

A_ρ:={w∈B|ρ <|w|<1}

have boundaryC1:=∂BandCρ:=∂Bρ=:C2(ρfixed), and indicate

X_monⁱ :={xⁱ∈H^12,2∩C⁰(∂B; Γi)|xⁱis weakly monotone and ontoΓiwith degree 1}.

I. We first consider the following condition for(N, h)(⊃Γ1,Γ2).

(C1) There existsp∈ N withΓ1,Γ2 ⊂ B(p, r), whereB(p, r)lies within the normal range of all its points. We assumer < π/(2√

κ), whereκis an upper bound of the sectional curvature of(N, h).

Throughout the paper,B(p, r)denotes a geodesic ball with centerp∈Nas in (C1). We can easily observe the following property (see [13]).

Remark 2.1. IfΓ₁,Γ₂ ⊂ N satisfy (C1), then for each xⁱ ∈ H^12,2∩C⁰(∂B; Γi)and ρ∈(0,1)there existgρ∈H^1,2∩C⁰(Aρ, B(p, r))andgⁱ∈H^1,2∩C⁰(B, B(p, r))with gρ|C₁ =x¹, gρ|C_ρ(·) =x²(_ρ^·)andgⁱ|∂B=xⁱ,i=1,2.

From the results in [7], [9] and the above remark, we have a unique harmonic map of annulus and disc type in B(p, r) ⊂ N for a given boundary mapping in the class H^12,2∩C⁰. Now we define

Mⁱ:={xⁱ∈ X_monⁱ |xⁱpreserves the orientation}.

ThenMⁱis complete, since theC⁰-norm preserves monotonicity. Moreover, let S(Γ1,Γ2) ={X ∈H^1,2∩C⁰(Aρ, B(p, r))|0< ρ <1, X|C_iis weakly monotone},

S(Γi) ={X ∈H^1,2∩C⁰(B, B(p, r))|X|∂Bis weakly monotone}.

II. We now investigate another significant condition for(N, h).

(C2) Nis compact with nonpositive sectional curvature.

A compact Riemannian manifold is homogeneously regular and the condition of nonpos- itive sectional curvature impliesπ2(N) =0. In order to defineMⁱ, we first consider for ρ∈(0,1)the following

G˜ρ:={f ∈H^1,2∩C⁰(Aρ, N)|f|C_iis continuous, weakly monotone and ontoΓi}.

We may take a continuous homotopy class, denoted byF˜_ρ⊂G˜_ρ, so that every two el- ementsf, ginF˜ρare continuously homotopicf ∼g(not necessarily fixing the boundary parametrization). We further demand some relationF˜_ρ∼F˜_σto hold for anyρ, σ∈(0,1).

Precisely, for somef˜∈F˜σ,f ∈F˜ρand some diffeomorphismτ_σ^ρ : [σ,1] →[ρ,1], we requiref˜(r, θ) =f(τ_σ^ρ(r), θ). LetF˜ρbe fixed. Then for anyσ∈(0,1)we can findF˜σ

withF˜_ρ∼F˜_σ.

We now consider all possibleH^1,2∩C⁰-extensions of disc type inN: S(Γi) :={X ∈H^1,2∩C⁰(B, N)|X|∂Bis weakly monotone ontoΓi}, assuming that this set is not empty, for eachi=1,2.

Lemma 2.1. (i) ForX¹∈ S(Γ1)andX²∈ S(Γ2)there existsfρ∈H^1,2∩C⁰(Aρ, N) such thatfρ|C₁(·) =X¹|∂B(·)andfρ|C_ρ(·) =X²|∂B(_ρ^·), forρ∈(0,1).

(ii) Moreover, there existsρ0 ∈(0,1)and a uniform positive constantCsuch that for somefρ∈H^1,2∩C⁰(Aρ, N), withfρ|C_ρ(·) =X²|∂B(_ρ^·)

E(f_ρ)≤C, for allρ≤ρ₀. (2)

Proof. (i) For a given ε > 0, take σ_i > 0 with osc_BσiXⁱ < ε. Choose ρ > 0 with _σ^ρ

2 < σ1, and let H : Bσ₁\B^ρ

σ2

→ R^k be harmonic with X¹|∂B_σ1 −X¹(0) on ∂Bσ₁ andX²|∂B_σ2 −X²(0) on ∂B^ρ

σ2

. This implieskHk_C0 < ε. Now letg ∈ H^1,2∩C⁰(Bσ₁\B^ρ

σ2

, N)withX¹(0)on∂Bσ₁andX²(0)on∂B^ρ

σ2

.

Considering coordinate neighbourhoods for the submanifoldN,→^η R^k, we may take a finite covering offρ((Aρ)), and by projection we obtain a smooth mapr:Nδ(fρ(Aρ))→ N withr|_N

δ(f_ρ(A_ρ))∩N = Idfor someδ > 0, whereNδ(·)isδ-neighbourhood inR^k. SettingT(s, θ) := (¹_sρ, θ)in polar coordinates, we can definefρwith the desired proper- ties:

f_ρ:=











X¹|_B\Bσ

1 onB\Bσ₁,

r◦(g+H) onBσ₁\B^ρ

σ2

, X²(T⁻¹(·)) onB^ρ

σ2

\Bρ.

(3)

(ii) The claim follows from the above construction, since _σ^ρ

2 < σ1,ρ≤ρ0for some

ρ₀ >0. 2

Under the assumption thatS(Γi) 6=∅, for givenΓi ∈ N we have an annulus-type- extension like that of (3), and we take homotopy classes which contain such an extension.

From now on twiddles will be dropped.

Define

S(Γ1,Γ2) :={f ∈Fρ|0< ρ <1}, (4) as well as the two function spaces

M¹:={x¹(·) =f|C1(·), f ∈ S(Γ1,Γ₂)|x¹is orientation preserving with degree 1}, M²:={x²(·) =f|C_ρ(·ρ), f∈ S(Γ₁,Γ₂)|x²is orientation preserving with degree 1}.

Forxⁱ ∈ X_monⁱ ,H_ρ(x¹, x²)denotes the uniqueR^k-harmonic extension onA_ρwithx¹(·) onC₁ andx²(_ρ^·) onCρ, while H(x)is the R^k-harmonic extension of disc type with boundaryx∈ X_monⁱ .

Lemma 2.2. (i) For eachxⁱ₀∈Mⁱ,i=1,2, there existsε(xⁱ₀)>0such that ifxⁱ∈ X_monⁱ withkxⁱ−xⁱ₀k1

2,2;0< ε, thenxⁱ∈Mⁱ. (ii) Mⁱis complete with respect tok · k1

2,2;0.

Proof. (i) Letfρ∈F˜ρwithfρ|C₁=x¹₀andfρ|C_ρ(·) =y²(_ρ^·)for somey²∈M². We consider the smooth retractionr:Nδ(fρ(Aρ))→Nas in the proof of Lemma 2.1.

Letkxⁱ−xⁱ₀k1

2,2;0< ε < δ. Then by Lemma 4.2 from [20], Z

Aρ

|d(r(fρ+Hρ(x¹−x¹₀,0)))|²dω

≤C(kfρk_C0, ε, N) Z

Aρ

|dfρ|²dω+ Z

B

|dH(x¹−x¹₀)|²dω

≤C(kfρk1,2;0, ε, N).

Now, let H(t,·) := (1 −t)Hρ(x¹ −x¹₀,0) : [0,1]×Aρ → R^k with kHk_C0 < ε andG : [0,1]×Aρ → N withG(t,·) = fρ(·)for allt ∈ [0,1]. Sincer(G+H) : [0,1]×Aρ → N is a homotopy betweenfρ andr(fρ +Hρ(x¹ −x¹₀,0)), it follows r(f_ρ +H_ρ(x¹ −x¹₀,0))(∼ f_ρ) ∈ F˜_ρ, andx¹ ∈ M¹. Similarly, we can prove that x²∈M²ifkx²−x²₀k1

2,2;0< ε⁰for some smallε⁰ >0.

(ii) A Cauchy sequence{xⁱ_n} ⊂Mⁱconverges toxⁱ ∈H^12,2∩C⁰(∂B; Γi), and for somen,kxⁱ_n−xⁱk_C0 < ε. ConsideringHρ(x¹−x¹_n,0)andgρ ∈ Fρwithx¹_n onC1

and 0 onCρ, we can find a homotopy inN betweengρandr(gρ+Hρ(x¹−x¹₀,0))as in (i). We may also apply this argument tox². Note thatxⁱis weakly monotone, and hence

xⁱ∈Mⁱ, concluding the proof. 2

From the proof we easily conclude that the set ofxⁱ’s which possess annulus-type- extensions with uniform energy with respect toρ ≤ρ₀is an open and closed subset of X_monⁱ . Thus, it is a non-empty connected component ofX_monⁱ and must coincide withMⁱ, sinceMⁱis a connected subset ofX_monⁱ . Hence we obtain the following property.

Remark 2.2. For eachxⁱ ∈ Mⁱ, i =1,2, there existfρ ∈ S(Γ1,Γ2)andC > 0 with E(fρ) ≤ C for all ρ ≤ ρ0 and someρ0 ∈ (0,1). Clearly, this result also holds for xⁱ∈Mⁱif we assume (C1).

For disc-type extensions ofxⁱ∈Mⁱthe following lemmata will be useful.

Lemma 2.3. Let(N, h)be a homogeneously regular manifold anduan absolutely con- tinuous map of ∂B_r(x₀)into N 3 x₀ withR2π

0 |u⁰(θ)|²_hdθ ≤ ^C_π⁰. Then there exists f ∈ H^1,2(B_r(x₀), N) ∩ C⁰(B_r(x₀), N) with f|_∂Br(x0) = u and E_Br(x0)(f) ≤

C⁰⁰ C⁰

R2π

0 |u⁰(θ)|²_hdθ, whereC⁰⁰, C⁰are the constants defined by homogeneous regularity.

Proof. See [17, Lemma 9.4.8 b)]. 2

Lemma 2.4. Let f_ρ ∈ H^1,2(A_ρ, N), 0 < ρ < 1. For each δ ∈ (ρ,1)there exists τ∈(δ,√

δ)withR2π 0

∂fρ(τ,θ)

∂θ

2

hdθ≤^4E(fρ)

ln¹_δ .

Proof. Similar to the proof of the Courant–Lebesgue lemma. 2

Forxⁱ ∈ Mⁱ, and given the choice ofS(Γ1,Γ2), Remark 2.2 tells that we can find fρ ∈ H^1,2(Aρ, N)with boundaryxⁱ such thatE(fρ) ≤C for allρ ≤ρ0. Then from Lemma 2.4 and Lemma 2.3, we havegτ ∈H^1,2(Bτ, N)with boundaryfρ|∂B_τ for some ρ. Together withgτandfρ|_B\Bτ, we obtain a mapX ∈H^1,2(B, N)with boundaryx¹. Similarly, we haveX˜ ∈H^1,2(B, N)with boundaryx².

Moreover, the harmonic extension of disc type for eachxⁱ ∈ Mⁱ inN is unique, independently of the choice of homotopy classS(Γ1,Γ2), because of the following well- known fact.

Lemma 2.5. π2(N) =0⇔Anyh0, h1∈C⁰(B, N)withh0|∂B=h1|∂Bare homotopic.

On the other hand, using the Construction (3) and the previous lemma we can easily check that the traces of the elements inS(Γi)belong toMⁱ. From [1], [15], [5], we then have the following.

Remark 2.3. (i) Forxⁱ ∈ Mⁱ, there exists a unique harmonic extension of disc type onBand of annulus type onA_ρ,ρ∈(0,1).

(ii) The elements ofMⁱare the traces of the elements ofS(Γi).

III. Now let(N, h)andΓi, i = 1,2 satisfy (C1) or (C2). Observing∂B ∼= R/2π, for a given orientedyⁱ∈ X_monⁱ there exists a weakly monotone mapwⁱ ∈C⁰(R,R)with wⁱ(θ+2π) =wⁱ(θ) +2πsuch thatyⁱ(θ) =γⁱ(cos(wⁱ(θ)),sin(wⁱ(θ))) =:γⁱ◦wⁱ(θ).

In additionwⁱ= ˜wⁱ+ Idfor somew˜ⁱ∈C⁰(∂B,R).

Denoting the Dirichlet integral byDand theR^k-harmonic extension byH, let Wⁱ

R^k :={wⁱ∈C⁰(R,R)|wⁱis weakly monotone, wⁱ(θ+2π) =wⁱ(θ) +2π;

D(H(γⁱ◦wⁱ))<∞}.

Clearly,Wⁱ

R^kis convex (for further details, refer to [21]).

Now takexⁱ∈Mⁱ. Consideringw−wⁱas a tangent vector alongw˜ⁱ, let T_xi =

dγⁱ((w−wⁱ)d

dθ ◦w˜ⁱ)|w∈Wⁱ

R^kandγⁱ◦wⁱ=xⁱ

.

Note thatT_xiis convex inT_xiH^12,2∩C⁰(∂B; Γ_i), sinceWⁱ

R^kis convex. Forξ=dγⁱ((w−

wⁱ)_dθ^d ◦w˜ⁱ)∈ T_xiwe have thatexpg_xiξ=γⁱ(w),expgdenoting the exponential map with respect to the metric˜h.

If (C1) holds, then clearlyexpg_xiξ∈Mⁱforξ∈ T_xi. For the Case (C2), let us recall the proof of Lemma 2.2. SinceNis compact, there existsli >0, depending onγⁱ, such that for anyxⁱ∈Mⁱ,expg_xiξ∈Mⁱ, provided thatkξk_T

xi < li. The following set-up holds true in both Cases (C1) and (C2).

Definition. (i) Let M := M¹×M²×(0,1)with the product topology and x :=

(x¹, x², ρ)∈ M. Then the setTxM:=T_x1× T_x2×Ris convex.

LetF(x) =F(x¹, x², ρ) =Fρ(x¹, x²) :Aρ →N be the unique harmonic exten- sion withx¹onC¹andx²(_ρ^·)onC², and define

E :M −→R: x7−→E(F(x)) := 1 2 Z

A_ρ

|dFρ(x¹, x²)|²_hdω.

(ii) Define∂M:=M¹×M²× {0},Tx∂M:=T_x1× T_x2andM:=M ∪∂M.

LetFⁱ(xⁱ) : Aρ → N be the unique harmonic extension with boundary xⁱ, for x= (x¹, x²,0)∈∂M, and define

E(x) :=E(F¹(x¹)) +E(F²(x²)).

2.3 Harmonic extension operators. LetΩ = A_ρorΩ = B. A weak Jacobi fieldJ with boundaryξalong a harmonic functionf is a weak solution of

Z

Ω

h∇J,∇Xi+htrR(J, df)df, Xidω=0,

for all X∈ H^1,2(Ω, f^∗T N)withX|∂Ω = ξ. Weak Jacobi fields are natural candidate derivatives of the harmonic operatorsFρandFⁱ.

We have the following property of weak Jacobi fields, from [8].

Lemma 2.6. The weak Jacobi fieldJwith boundaryη∈TxⁱH^12,2∩C⁰along a harmonic Fwith boundaryxⁱis well defined in the classH^1,2and continuous up to the boundary.

It satisfies

kJ_Fk_C0 ≤ kJ_F|_∂Ωk_C0, kJ_Fk_1,2;0≤C(N,kfk_1,2:0)kJ_F|_∂Ωk1 2,2;0. Now we can discuss the differentiability of harmonic extension operators.

Lemma 2.7. The operatorsFρ,Fⁱare partially differentiable inx¹(respectivelyx²)for variations inT_x1H^12,2∩C⁰(respectivelyT_x2H^12,2∩C⁰). Their derivatives are continuous Jacobi field operators with respect tox¹, x².

Proof. The proof reproduces an argument we shall explain in full detail in Lemma 3.1, Cases (B), (C), and as such will not be anticipated here. Alternatively, one can follow the

aforementioned [8]. 2

3 The variational problem

3.1 Differentiability ofEonM.

Lemma 3.1. The following hold:

(A) E is continuously partially differentiable inx¹, x²with respect to variations inT_x1, T_x2and the derivatives are continuous onM¹×M².

(B) Eis continuous with respect toρ∈[0,1), even uniformly onNε(xⁱ₀)for someε >0 independent ofxⁱ₀∈Mⁱ,i=1,2.

(C) The partial derivatives inx¹, x²are continuous with respect toρ∈[0,1), uniformly continuous onNε(xⁱ₀)for someε >0independent ofxⁱ₀∈Mⁱ,i=1,2.

(D) Eis differentiable with respect toρ∈(0,1).

Proof. From now on, continuity will be understood in the sense of convergence of subse- quences.

(A) The Dirichlet integral functional is inC^∞, so Lemma 2.7 guarantees thatE is continuously partially differentiable with continuous partial derivatives onM¹×M².

Computation of the derivatives:Letx= (x¹, x², ρ)∈ M,ξ¹ ∈ Tx¹. By Lemma 2.2 there is a smallt₀>0 such thatexpg_x1(tξ¹)∈M¹, 0≤t≤t₀. Thus,

hδ_x1E, ξ¹i:= d dt t=0

E(expg_x1(tξ¹), x², ρ)

= Z

Aρ

hdFρ(x¹, x²),∇D_x1Fρ(x¹, x²)(ξ¹)ihdω

= Z

Aρ

hdFρ(x¹, x²),∇J_Fρ(ξ¹,0)ihdω (by Lemma 2.7), (5)

since by computation we obtain, withFρ(t) :=Fρ(gexp_x1(tξ¹), x²),

∇d dt

F_ρ,i^α (t)dxⁱ⊗ ∂

∂y^α◦ Fρ(t)

=∇d

dtFρ expg_x1(tξ¹), x²

=∇ D_x1F_ρ(x¹, x²)(ξ¹)

, t=0 .

Forξ² ∈ T_x2 Lemma 2.7 yieldshδ_x2E, ξ²i = R

A_ρhdFρ(x¹, x²),∇JF_ρ(0, ξ²(_ρ^·))ihdω.

Similarly, forx= (x¹, x²,0)∈∂M,hδ_xiE, ξⁱi=R

BhdFⁱ(xⁱ),∇J_Fi(ξⁱ)ihdω, i=1,2.

For (B) we shall split the proof into three sub-steps B-I), B-II), B-III). Similarly for (C) we shall have C-I), C-II), C-III).

B-I)The set-up. The claim is thatEis continuous whenρ→ρ0. Fixingρ0 =0 is no great restriction, since the proof forρ0∈(0,1)carries over in an analogous, even easier, fashion. Takingρ0 =0 translates our claim into

Z

A_ρ

dFρ(x¹, x²)

2 hdω−→

Z

B

dF¹(x¹)

2 hdω+

Z

B

dF²(x²)

2

hdω (6)

uniformly onNε(xⁱ₀)for someε >0 independent ofxⁱ₀ ∈Mⁱ, wheneverρ→0.

LetFρ :=Fρ(x¹, x²)andFⁱ := Fⁱ(xⁱ), i= 1,2. By Lemma 2.4, for eachδwith 0< ρ < δ <1 there existsν ∈(δ,√

δ)such that Z 2π

0

∂Fρ(ν, θ)

∂θ _h

dθ≤√ 2π

Z 2π 0

∂Fρ(ν, θ)

∂θ

2

h

dθ ¹₂

≤ C

p|lnδ|. (7) Due to Remark 2.2,Cis independent ofρ≤ρ0, for someρ0∈(0,1).

By means ofFρwe now construct two maps by setting

fν:Aν−→N with fν(re^iθ) :=Fρ(re^iθ), re^iθ∈Aν,

gν⁰ :Aν⁰−→N withgν⁰(re^iθ) :=Fρ(T(re^iθ)), re^iθ∈Aν⁰. (8) The constantsν⁰ := ^ρ_ν,ν ∈ (δ,√

δ)andδ ∈ (ρ,1)satisfy the property (7) in the limit ν⁰, ν → 0 for ρ → 0. (One can take for instance δ = √

ρ). The map T(re^iθ) =

ρ

re^iθ goes fromA_ν⁰ toB_ν\B_ρ surjectively. Then,f_ν andg_ν⁰ are harmonic maps into N withf_ν|_∂B = x¹,g_ν⁰|_∂B = x² andosc_∂Bνf_ν → 0, osc_∂Bν0g_ν⁰ → 0 asρ → 0.

Moreover, sinceTis conformal,E(Fρ) =E(Fρ|A_ν) +E(Fρ|_Bν\Bρ) =E(fν) +E(gν⁰) by conformal invariance of the Dirichlet integral.

B-II)The convergence of{fν},{gν⁰}toFⁱ. We first investigate the modulus of con- tinuity of harmonic maps{hν}:Aν →N which converge uniformly (C⁰-norm) on∂B withE(hν)≤Lfor someL >0, independent ofν ≤ν0for someν0 ∈(0,1). We shall only deal with the assumption (C2), because the argument can clearly be applied to the Case (C1) as well.

LetGR := BR(z) ⊂ Aν for ν ≤ ν˜0. If z ∈ ∂B, considerGR := BR(z)∩Aν. Givenε > 0, by the Courant–Lebesgue lemma there existsδ > 0, independent ofν ≤ ν₀, such that the length of h_ν|∂Gδ does not exceed min{^ε₄,^i(N)₄ }, i(N) > 0. Then h_ν|_∂Gδ ⊂B(q, s)for someq∈N, s≤min{^ε₂,^i(N₂⁾}. Observe thath_νis continuous on

∂Gδ, and there exists anH^1,2-extensionX of disc type, whose image is inB(q, s)with X|∂B_δ = hν|∂B_δ, by the same argument of Remark 2.1. Thus there exists a harmonic extensionh⁰withh⁰(Gδ)⊂B(q, s)⊂B(q,^ε₂), by [7]. From Lemma 2.5,h⁰is homotopic tohonGδ, and from the energy minimizing property of harmonic maps,hν|G_δ = h⁰. Hence, the functionshνwithν ≤ν0have the same modulus of continuity. Furthermore, if these mappings have the same boundary image, they areC⁰-uniformly bounded on each relatively compact domain.

Now apply the above result to{Fρ, ρ ≤ ρ0}inR^k. For someρ0 ∈ (0,1)then, the functionsfνrespectivelygν⁰ have the same modulus of continuity for allρ∈(0, ρ0), and some subsequences, denoted again byfν respectivelygν⁰ are locally uniformly conver- gent. Recall that our maps are continuous, so by localizing in both domain and image, harmonic functions, seen as solutions of Dirichlet problems, may be also regarded as weak solutionsfof the following elliptic systems in local coordinate charts ofN:

d_id_if^α=−Γ^α_βγd_if^βd_if^γ=:G^α(·, f(·), df(·)). (9) We can take the same coordinate charts for the image of{f_ν}_ν≤ν0and{g_ν⁰}_ν0≤ν₀⁰, where ν₀ := ν(ρ₀),ν₀⁰ := ν⁰(ρ₀), to the effect that we have the same weak solution system for (9). Moreover, sincehαβ andΓ^α_βγ are smooth, the structure constants of the weak systems (see [10, Section 8.5]) are independent ofρ≤ρ₀.

Now considerK_σ^σ ={σ ≤ |z| ≤ 1−σ},σ∈(0,1). From the regularity theory of [14] and [10, Section 8.5] and by the covering argument, there existsC ∈ Rsuch that kfν|K_σ^σk_H4,2 ≤Cfor allν ∈ (0, ν0). Hence the Sobolev’s embedding theorem implies that for some sequence{ρi} ⊂ (0,1), limρ_i→0f_ν(ρi)|K_σ^σ = f⁰ inC²(K_σ^σ,Rⁿ), with τh(f⁰) =0 inK_σ^σ.

Forσ:= _n¹, we choose a sequence{f_ν(ρn,i)}as above such that{ρn+1,i}is a subse- quence of{ρn,i}. By diagonalizing we obtain a subsequence{f_ν(ρn,n)}, n≥n0 which converges locally tof⁰in theC²-norm, sof⁰is harmonic onB\(∂B∪ {0}).

On the other handf_ν|∂B =x¹for allν, and thef_ν’s converge uniformly tof⁰in a compact neighbourhood of∂B. Thus,f⁰is continuous onB\{0}withf⁰|_∂B=x¹. Also observe thatosc_∂Brf⁰→0 asr→0, by construction.

For each compactK ⊂ B\{0},R

K|df⁰|²dω = lim_ρi→0R

K|df_ν(ρi)|² ≤ L, withL independent ofK. Thus, f⁰ ∈ H^1,2(B\{0}, N), andf⁰ can be extended to a weakly harmonic map onB ([10, Lemma 8.4.5], see also [18], [4]). Thus,f⁰can be considered weakly harmonic andf⁰∈C⁰(B, N)∩C²(B, N)withf⁰|∂B =x¹, so uniqueness forces f⁰=F¹(x¹).

Similar results hold forgν⁰.

B-III) The convergence of the energy. We consider η◦ f, and denote it again by f := (f^a)a=1,...,k ∈ H^1,2(Ω,R^k)for obvious reasons. Sinceη is isometric, forf :=

(f^α)α=1,...,n ∈ H^1,2(Ω, N)we haveR

Ω|d(f^α)|²_hdω = R

Ω|d(f^a)|²

R^kdω. A harmonic mapf ∈H^1,2(Ω, N)satisfies

Z

Ω

(hdf, dψi − hII◦f(df, df), ψi)dw=0 (10) for anyψ∈H₀^1,2∩C⁰(Ω,R^k), whereIIis the second fundamental form ofη.

SetKσ={σ≤ |z| ≤1}, σ >0 and we considerR^k-harmonic mapsHνandHeνon KσwithHν|∂K_σ =fν|∂K_σandHeν|∂K_σ =F¹|∂K_σ, whereν ∈(0, σ). LetH:B→R^k be harmonic withH|∂B = H_ν|∂B = He_ν|∂B = x¹. Then{Hν},{He_ν} have the same modulus of continuity up to∂B, and we havekH_ν−Hk_C0;K_σ →0, kHe_ν−Hk_C0;K_σ → 0 asν→0. Furthermore, forX_ν:= (f_ν− F¹) + (H_ν−He_ν)∈H₀^1,2∩C⁰(K_σ,R^k), we obtain

kXνk_(C0;K_σ)≤ kfν− F¹k_C0;K_σ+kHν−Hk_C0;K_σ+kH−Heνk_C0;K_σ →0 asν→0.

Now consider Z

K_σ

hd(fν− F¹), d(fν− F¹)idω

= Z

Kσ

hd(fν− F¹), dXνidω

| {z }

=:I

− Z

Kσ

hd(fν− F¹), d(Hν−Heν)idω

| {z }

=:II

.

Whenν →0

|I| ≤ Z

Kσ

hII◦fν(dfν, dfν), Xνidω

+ Z

Kσ

hII◦(dF¹, dF¹), Xνidω

=C(kf_νk_1,2;0),kF¹k_1,2;0)kX_νk_(C0;K_σ)→0 (11) from (10). Moreover, sinceHν−Heνis harmonic inR^k,

|II| ≤ Z

∂Bσ

∂r(Hν−Heν)

dωkfν− F¹k_C0;K_σ →0 asν →0. (12)

ThusR

K_σ|d(fν− F¹)|²dω→0, andR

K_σ|dfν|²dω→R

K_σ|dF¹|²dω, for anyKσ. Since R

B_σ|dF¹|²dω → 0 asσ → 0, we obtain R

A_ν|dfν|²dω → R

B|dF¹|²dω as ν → 0.

SimilarlyR

A_ν0|dg_ν⁰|²dω→R

B|dF²|²dω asν⁰→0.

Now to the uniform convergence onNε(xⁱ₀). Replacef(Aρ)byB(p, r)(for (C1)) orN (for (C2)) in the proof of Lemma 2.2. Then,kFρ(x¹, x²)k_H1,2 ≤ Cuniformly on Nε(xⁱ₀), where the constantCdepends onxⁱ₀, whileεdoes not. The convergence in (11), (12) is uniform onNε(xⁱ₀). The proof of (B) is eventually completed.

C-I)The set-up. We must show that forxⁱ∈Mⁱandξⁱ∈ Txⁱ,

hδ_xiEρ, ξⁱi −→ hδ_xiE, ξⁱi uniformly onNε(xⁱ₀)⊂Mⁱ, i=1,2 asρ→0.

It suffices to show the assertion fori=1. We know that hδx¹Eρ, ξ¹i

= Z

A_ν(ρ)

hdFρ(x¹, x²),∇J_Fρ(ξ¹,0)ihdω+ Z

B_ν(ρ)\Bρ

hdFρ(x¹, x²),∇J_Fρ(ξ¹,0)ihdω

= Z

A_ν(ρ)

hdFρ(x¹, x²),∇J_Fρ(ξ¹,0)ihdω+ Z

A_ν0

hdgν⁰,∇Jg_ν0(0, ζν⁰)idω,

wheregν⁰(·) =Fρ◦T(·)andζν⁰(ν⁰e^iθ) = J_Fρ(ξ¹,0)(νe^iθ)withν⁰ := _ν(ρ)^ρ . Observe J_Fρ(ξ¹,0)◦T is a Jacobi field alonggν⁰, by the conformal property ofT.

C-II)The convergence of Jacobi fields. First, letV_ν :=J_Fρ(ξ¹,0)|_Aν =v^α_ν∂y^∂_α ◦f_ν, for which we will show the existence of aν₀∈(0,1)giving

kDVνk²₂:=

Z

Aν

hαβ◦fνv^α_ν,iv_ν,i^β dω≤Cfor allν∈(0, ν₀). (13) By direct computation kDVνk²₂ ≤ CE(Vν) +C(N,kVνk_C0,kfνk_C0, E(fν)). Since Lemma 2.6 yieldskVνk_C0 ≤ kξ_ν¹k_C0, we only need to show that

E(Vν) :=

Z

A_ν

|∇^fνVν|²dω≤C, ν∈(0, ν0). (14)

LetX_ν := x^α_ν∂y^∂_α ◦f_ν ∈ H^1,2(A_ν, f_ν^∗T N), wherex^α_ν(z) := v^α_2ν

0(τ_ν^2ν0

0 (z)),ν₀ ≤

|z| ≤1 (see Section 2.2 for the definition ofτ_ν^2ν₀⁰) andx^α_ν(z) :=0,ν≤ |z| ≤ν0. Clearly, kDXνk²₂≤C(ν0, N)kDV2ν₀k²₂for allν≤ν0.

By the minimality property of Jacobi fields and Young’s inequality, Z

A_ν

(|∇^fν(Vν)|²− htrR(dfν, Vν)dfν, Vνi)dω

≤ Z

Aν

(|∇^fν(Xν)|²− htrR(dfν, Xν)dfν, Xνi)dω

≤ Z

A_ν

h_αβ◦f_νx^α_ν,ix^β_ν,idω+ε Z

A_ν

|x^α_ν,i ∂

∂y^α◦f_ν|²_hdω +ε⁻¹

Z

Aν

|x^γ_νf_,i^δΓ^β_γδ◦fν

∂

∂y^β ◦fν|²_hdω +

Z

A_ν

h_αβ◦f_νx^γ_νx^λ_νf_,i^δf_,i^µΓ^α_γδ◦f_νΓ^β_λµ◦f_νdω− Z

A_ν

htrR(df_ν, X_ν)df_ν, X_νidω

≤C(N, ε,kfνk_C0, E(fν),kV2ν₀k_C0,kDV2ν₀k²₂).

ButE(Vν)≤C,ν∈(0, ν0), since Z

Aν

htrR(df_ν, V_ν)df_ν, V_νidω≤C(N,kf_νk_C0, E(f_ν),kξ¹k_C0).

Therefore we have (13), and this means that {(v_ν^α) | ν ≤ ν0}α=1,...,n has the same modulus of continuity, see the argument in B-III) and Lemma 2.6.

With the same charts as in (B),(v^α_ν(ρ)) ∈ Rⁿ, ν ≤ ν0 are weak solutions of the Jacobi fields system with uniformly bounded energy and same modulus of continuity on Kσ = {σ ≤ |z| ≤ 1}, withσ > 0 for small ρ, again by Lemma 2.6. Just as in (B), {Vν} converges to the Jacobi field alongF¹|_B\{0} with boundary ξ¹, and for J_F1(ξ¹) =:w^{α ∂}_∂yβ ◦ F¹, we have

k(v^α_ν(z))−(w^α(z))k_C0;K_σ →0,

k(v^α_ν(z))−(w^α(z))k_C2;K →0, asν(orρ)→0, on any compactK⊂B\{0}.

C-III)The convergence of derivatives. TakingK_σ as above, we denotef_ν|_Kσ and F¹|K_σ byfν andF¹, respectively. Note that exp_F1 : U(0) → H^1,2∩C⁰(Kσ, N)is a diffeomorphism on some neighbourhoodU(0)∈ H^1,2∩C⁰(Kσ,(F¹)^∗T N), because d(exp_F1)0 = Id. Moreover, kfν − F¹|K_σk_H1,2∩C⁰ → 0 as ν → 0, so there exists ξν ∈H^1,2∩C⁰(Kσ,(F¹)^∗T N)for smallν >0 withexp_F1ξν =fν.

The mappingξ7→ dexp_F1,ξ depends smoothly onξν ∈T_F1H^1,2∩C⁰(Kσ, N), so dexp_F1,ξ_ν →IdinH^1,2∩C⁰(Kσ), sinceξν→0 inH^1,2∩C⁰(Kσ,(F¹)^∗T N)asν →0.

ForW_ν := w^α_ν∂y^∂_α ◦ F¹ :=dexp⁻¹_F1,ξ

ν(V_ν)we havekw_ν^α(z)−w^α(z)k_C0;K_σ → 0 by C-II). Moreover,dF¹→dfνinL², thusR

K_σ|dexp_F1,ξ_ν(dF¹)−dfν|²dω→0.

We next observe, for∇^F1Wν= (w^α_ν,i+w_ν^γ(F¹)^β_,iΓ^α_βγ(F¹))dzⁱ⊗_∂y^∂α◦ F¹, that Z

Kσ

|dexp_F1,ξν(∇^F1Wν)− ∇^fνVν|²dω→0 asν →0, (15) sincekF¹−fνk1,2;0→0,dexp_F1,ξν →IdinC⁰,∂i(dexp_F1,ξν)→∂i(Id) =0 inL².

Thus, for Xν, Yν ∈ H^1,2 ∩ C⁰(Kσ, T^∗M ⊗ f_ν^∗T N) with R

K_σ|Xν|²dω → 0, R

K_σ|Yν|²dω→0,

dexp_F1,ξ_ν(dF¹) =dfν+Xν, dexp_F1,ξ_ν(∇^F1Wν) =∇^fνVν+Yν.