We Prove the Existence of a Smooth Minimizer of the Willmore

93 (2013) 471-530

MINIMIZERS OF THE WILLMORE FUNCTIONAL UNDER FIXED CONFORMAL CLASS

Ernst Kuwert & Reiner Schatzle¨

Abstract

We prove the existence of a smooth minimizer of the Willmore energy in the class of conformal immersions of a given closed Rie- mann surface into IRn,n =3, 4, if there is one conformal immersion with Willmore energy smaller than a certain bound n,p depending on codimension and genus p of the Riemann surface.W For tori in codimension 1, we know 3,1 =8π . W

1. Introduction For an immersed closed surface f :Σ IRn the Willmore functional is deﬁned by → 1 2 (f)= H dg, W 4 | | Σ ∗ where H denotes the mean curvature vector of f, g = f geuc the pull- back metric and g the induced area measure on Σ . For orientable Σ of genus p , the Gauß equations and the Gauß-Bonnet theorem give rise to equivalent expressions

1 2 1 0 2 (1.1) (f)= A dg + 2π(1 p)= A dg + 4π(1 p) W 4 | | − 2 | | − Σ Σ 0 1 where A denotes the second fundamental form, A = A 2 g H its tracefree part. − ⊗ We always have (f) 4π with equality only for round spheres, see [Wil82] in codimensionW ≥ one that is n = 3 . On the other hand, if (f) < 8π then f is an embedding by an inequality of Li and Yau inW [LY82].

E. Kuwert and R. Schätzle were supported by the DFG Forschergruppe 469, and R. Schätzle was supported by the Hausdorff Research Institute for Mathematics in Bonn. Main parts of this article were discussed during a stay of both authors at the Centro Ennio De Giorgi in Pisa. Received 06/01/2011.

471 472 E. KUWERT & R. SCHATZLE¨

Critical points of are called Willmore surfaces or more precisely Willmore immersions.W They satisfy the Euler-Lagrange equation which is the fourth order, quasilinear geometric equation

0 ∆gH + Q(A )H = 0 where the Laplacian of the normal bundle along f is used and Q(A0) acts linearly on normal vectors along f by Q(A0)φ := gikgjlA0 A0 , φ . ij kl The Willmore functional is scale invariant and moreover invariant under the full Möbius group of IRn . As the Möbius group is non-compact, this invariance causes difficulties in the construction of minimizers by the direct method. In [KuSch11], we investigated the relation of the pull-back metric g to constant curvature metrics on Σ after dividing out the Möbius group. More precisely, we proved in [KuSch11] Theorem 4.1 that for immersions f :Σ IRn,n = 3, 4, Σ closed, orientable and of genus p = p(Σ) 1 satisfying→ (f) δ for some δ > 0 , where ≥ W ≤Wn,p −

= min 8π, 4π + (β3 4π) p = p, 0 p

u ∞ , u 2 C(p, δ). L (Σ) ∇ L (Σ,g˜)≤ In this paper, we consider conformal immersions f :Σ IRn of a ﬁxed closed Riemann surface Σ and prove existence of smooth→ minimizers in this conformal class under the above energy assumptions.

Theorem 7.3 Let Σ be a closed Riemann surface of genus p 1 ≥ with smooth conformal metric g0 with (Σ, g ,n) := W 0 inf (f) f :Σ IRn smooth immersion conformal to g < , {W | → 0 } Wn,p where is deﬁned in (1.2) and n = 3, 4 . Wn,p MINIMIZERS OF THE WILLMORE FUNCTIONAL 473

Then there exists a smooth conformal immersion f :Σ IRn which minimizes the Willmore energy in the set of all smooth conformal→ immersions. Moreover f satisfies the Euler-Lagrange equation 0 ik jl 0 (1.4) ∆gH + Q(A )H = g g Aijqkl on Σ, where q is a smooth transverse traceless symmetric 2-covariant tensor with respect to the Riemann surface Σ , that is with respect to g = ∗ f geuc . For the proof, we select by the compactness theorem [KuSch11] Theorem 4.1 a minimizing sequence of smooth immersions fm :Σ n → IR conformal to g0 , that is (f ) (Σ, g ,n), W m →W 0 converging f f weakly in W 2,2(Σ, IRn), m → where the limit f defines a uniformly positive definite pull-back metric ∗ 2u g := f geuc = e g0 with u L∞(Σ) . We call such f a W 2,2 immersion uniformly con- ∈ − formal to g0 . For such f we can define the Willmore functional and obtain by lower semicontinuity

(f) lim (fm)= (Σ, g0,n). W ≤ m→∞ W W Now there are two questions on f : Is f a minimizer in the larger class of uniformly conformal • W 2,2 immersions? Is f −smooth? • For the ﬁrst question we can approximate uniformly conformal f by 2,2 [SU83] strongly in W by smooth immersions f˜m , but these f˜m are in general not conformal to g0 anymore. To this end, we develop a correction in conformal class and get smooth conformal immersions f¯m converging strongly in W 2,2 to f . We obtain that the inﬁmum over smooth conformal immersions and over uniformly conformal immersions coincide.

Theorem 5.1 For a closed Riemann surface Σ of genus p 1 ≥ with smooth conformal metric g0 (Σ, g ,n)= W 0 = inf (f) f :Σ IRn smooth immersion conformal to g {W | → 0 } = inf (f) f :Σ IRn is a W 2,2 immersion uniformly {W | → − conformal to g . 0 } 474 E. KUWERT & R. SCHATZLE¨

Smoothness of f is obtained by the usual whole filling procedure as in the proof of the existence of minimizers for fixed genus in [Sim93]. Here again we need the correction in conformal class in an essential way. Our method applies to prove smoothness of any uniformly conformal W 2,2 minimizer. − Theorem 7.4 Let Σ be a closed Riemann surface of genus p 1 n ≥ with smooth conformal metric g0 and f :Σ IR be a uniformly 2,2 ∗→ 2u conformal W immersion, that is g = f geuc = e g0 with u L∞(Σ) , which minimizes− the Willmore energy in the set of all smooth∈ conformal immersions (f)= (Σ, g ,n), W W 0 then f is smooth and satisfies the Euler-Lagrange equation 0 ik jl 0 ∆gH + Q(A )H = g g Aijqkl on Σ, where q is a smooth transverse traceless symmetric 2-covariant tensor with respect to g .

Finally to explain the correction in conformal class, we recall by Poincaré’s theorem, see [Tr], that any smooth metric g on Σ is conformal to a unique smooth unit volume constant curvature metric gpoin = e−2ug . Denoting by et the set of all smooth metrics on Σ , by M etpoin the set of all smooth unit volume constant curvature metrics M ≈ on Σ , and by := Σ Σ the set of all smooth diffeomorphisms D { −→ } of Σ , we see that etpoin/ is the moduli space of all conformal structures on Σ . ActuallyM toD deal with a space of better analytical properties, we consider := φ φ id and the Teichmüller D0 { ∈ D | ≃ Σ } space := etpoin/ 0 . This is a smooth open manifold of dimension 2 for pT= 1M and of dimensionD 6p 6 for p 2 and the projection π : et is smooth, see [FiTr84− ] and [Tr≥ ]. We define in 3 the firstM variation→ T in Teichmüller space of an immersion f :Σ IRn§ and a variation V C∞(Σ, IRn) by → ∈ d 0= π((f + tV )∗g ) = δπ .V dt euc |t=0 f and consider the subspace ∞ n := δπ .C (Σ, IR ) T ∗ . Vf f ⊆ π(f geuc)T In case of f = Tπ , which we call f of full rank in Teichmüller space, the correctionV T in Teichmüller space is easily achieved by implicit function theorem. A severe problem arises in the degenerate case, when f is not of full rank in Teichmüller space. In this case, we prove in Proposition 4.1 that dim = dim 1, Vf T − MINIMIZERS OF THE WILLMORE FUNCTIONAL 475

ik jl 0 g g Aijqkl = 0 for some non-zero smooth transverse traceless symmetric 2-covariant tensor q with respect to g , and f is isothermic locally around all but finitely many points of Σ , that is around all but finitely many points of Σ there exist local conformal principal curvature coordinates. Then by implicit function theorem, we can do the correction in . In the Vf one dimensional orthogonal complement spanned by some e f , we do the correction with the second variation by monotonicity.⊥V To deal with the positive and the negative part along span e , we need two { } variations V± satisfying δ2π (V ), e > 0, δπˆ .V = 0, ± f ± f ± whoose existence is established in a long computation in Proposition 4.2. Recently, Kuwert and Li and independently Riviere have extended the existence of non-smooth conformally constrained minimizers in any codimension, where branch points may occur if (Σ, g0,n) 8π , see [KuLi10] and [Ri10]. In [Ri10] the obtained conformallyW constrained≥ minimizers are smooth and satisfy the Euler-Lagrange equation (1.4) in the full rank case, whereas in the degenerate case these are proved to be isothermic, but the smoothness and the verification of (1.4) as in Theorem 7.4 are left open. Everything that is done in this article for (Σ, g0,n) < n,p for n = 3, 4, can be done with [Sch12] Theorem 4.1W for (Σ, gW,n) < 8π for any n . W 0 Acknowledgement. The main ideas of this paper came out during a stay of both authors at the Centro Ennio De Giorgi in Pisa. Both authors thank very much the Centro Ennio De Giorgi in Pisa for the hospitality and for providing a fruitful scientific atmosphere.

2. Direct method Let Σ be a closed, orientable surface of genus p 1 with smooth metric g satisfying ≥ (Σ,g,n) := W inf (f) f :Σ IRn smooth immersion conformal to g < , {W | → } Wn,p as in the situation of Theorem 7.3 where n,p is defined in (1.2) above and n = 3, 4 . To get a minimizer, we considerW a minimizing sequence of immersions f :Σ IRn conformal to g in the sense m → (f ) (Σ,g,n). W m →W After applying suitable Möbius transformations according to [KuSch11] 2,2 1,∞ Theorem 4.1 we will be able to estimate fm in W (Σ) and W (Σ) , see below, and after passing to a subsequence, we get a limit f W 2,2(Σ) W 1,∞(Σ) which is an immersion in a weak sense, see (2.6)∈ ∩ 476 E. KUWERT & R. SCHATZLE¨ below. To prove that f is smooth, which implies that it is a minimizer, and that f satisfies the Euler-Lagrange equation in Theorem 7.3 we will consider variations, say of the form f +V . In general, these are not conformal to g anymore, and we want to correct it by f + V + λrVr for suitable selected variations Vr . Now even these are not conformal to g since the set of conformal metrics is quite small in the set of all metrics. To increase the set of admissible pull-back metrics, we observe that it suffices for (f + V + λrVr) φ being conformal to g for some diffeomorphism φ of Σ . In other◦ words, the pullback metric ∗ (f + V + λrVr) geuc need not be conformal to g , but has to coincide only in the modul space. Actually, we will consider the Teichmüller space, which is coarser than the modul space, but is instead a smooth open manifold, and the bundle projection π : et of the sets of metrics et into the Teichmüller space Mis smooth,→T see [FiTr84], [Tr]. ClearlyM (Σ,g,n) depends only onT the conformal structure defined by g , inW particular it descends to Teichmüller space and leads to the following definition.

Deﬁnition 2.1. We deﬁne n,p : [0, ] for p 1,n 3, by selecting a closed, orientable surfaceM ΣT of → genus∞p and ≥ ≥

(τ) := inf (f) f :Σ IRn smooth immersion,π(f ∗g )= τ . Mn,p {W | → euc } We see (π(g)) = (Σ,g,n). Mn,p W n Next, infτ n,p(τ)= βp for the infimum under fixed genus defined in M n (1.3), and, as the minimum is attained and 4π < βp < 8π , see [Sim93] and [BaKu03], n 4π < min n,p(τ)= βp < 8π. τ∈T M In the following proposition, we consider a slightly more general situation than above.

n Proposition 2.2. Let fm :Σ IR ,n = 3, 4, be smooth immersions of a closed, orientable surface→Σ of genus p 1 satisfying ≥ (2.1) (f ) δ W m ≤Wn,p − and

(2.2) π(f ∗ g ) τ in . m euc → 0 T Then replacing f by Φ f φ for suitable M¨obius transformations m m ◦ m ◦ m Φm and diﬀeomorphisms φm of Σ homotopic to the identity, we get

(2.3) lim sup fm W 2,2(Σ)< , m→∞ ∞ MINIMIZERS OF THE WILLMORE FUNCTIONAL 477

∗ 2um and fmgeuc = e gpoin,m for some unit volume constant curvature metrics gpoin,m with

um L∞(Σ), um L2(Σ,g )< , (2.4) ∇ poin,m ∞ g g smoothly poin,m → poin with π(gpoin)= τ0 . After passing to a subsequence

2,2 ∗ 1,∞ fm f weakly in W (Σ), weakly in W (Σ), (2.5) → u u weakly in W 1,2(Σ), weakly∗ in L∞(Σ), m → and ∗ 2u (2.6) f geuc = e gpoin where (f ∗g )(X,Y ) := ∂ f, ∂ f for X,Y T Σ . euc X Y ∈ Proof. Clearly, replacing fm by Φm fm φm as above does neither change the Willmore energy nor the projection◦ ◦ into the Teichm¨uller space. By [KuSch11] Theorem 4.1 after applying suitable M¨obius transfor- ∗ mations, the pull-back metric gm := fmgeuc is conformal to a unique −2um constant curvature metric e gm =: gpoin,m of unit volume with

osc u , u 2 C(p, δ). Σ m ∇ m L (Σ,gm)≤ Combining the M¨obius transformations with suitable homotheties, we may further assume that fm has unit volume. This yields

2um 1= dg = e dg , m poin,m Σ Σ and, as gpoin,m has unit volume as well, we conclude that um has a zero on Σ , hence

u ∞ , u 2 C(p, δ) m L (Σ) ∇ m L (Σ,gpoin,m)≤ ∗ 2um and, as fmgeuc = gm = e gpoin,m , f ∞ C(p, δ). ∇ m L (Σ,gpoin,m)≤ Next 2um 2um ∆gpoin,m fm = e ∆gm fm = e Hfm and

2 2 max um 2 2um ∆g fm 2 e Hf e dg poin,m L (Σ,gpoin,m)≤ | m | poin,m Σ

(2.7) = 4e2 max um (f ) C(p, δ). W m ≤ To get further estimates, we employ the convergence in Teichmüller space (2.2). We consider a slice of unit volume constant curvature S 478 E. KUWERT & R. SCHATZLE¨ metrics for τ0 , see [FiTr84], [Tr]. There exist uniqueg ˜poin,m with π(˜g ∈)= T π(g ) τ for m large enough, hence ∈ S poin,m m → 0 ∗ φmg˜poin,m = gpoin,m for suitable diffeomorphisms φm of Σ homotopic to the identity. Re- placing f by f φ , we get g =g ˜ and m m ◦ m poin,m poin,m ∈ S g g smoothly poin,m → poin with g ,π(g )= τ . Then translating f suitably, we obtain poin ∈ S poin 0 m f ∞ C(p, δ). m L (Σ)≤ Moreover standard elliptic theory, see [GT] Theorem 8.8, implies by (2.7), f 2,2 C(p, δ, g ) m W (Σ)≤ poin for m large enough. 2,2 Selecting a subsequence, we get fm f weakly in W (Σ), f 1,∞ → ∈ W (Σ),Dfm Df pointwise almost everywhere, um u weakly in W 1,2(Σ), pointwise→ almost everywhere, and u L∞(Σ)→ . Putting g := f ∗g , that is g(X,Y ) := ∂ f, ∂ f , we see∈ euc X Y g(X,Y ) ∂ f , ∂ f = g (X,Y ) ← X m Y m m = e2um g (X,Y ) e2ug (X,Y ) poin,m → poin for X,Y T Σ and pointwise almost everywhere on Σ , hence ∈ ∗ 2u f geuc = g = e gpoin. q.e.d. 1,∞ n ∗ 2u We call a mapping f W (Σ, IR ) with g := f geuc = e gpoin, u ∞ ∈ ∈ L (Σ), gpoin a smooth unit volume constant curvature metric on Σ as in (2.6) a lipschitz immersion uniformly conformal to gpoin or in short a uniformly conformal lipschitz immersion. If further f W 2,2(Σ, IRn) , we call f a W 2,2 immersion uniformly conformal∈ to g or a − poin uniformly conformal W 2,2 immersion. In this case, we see g, u W 1,2(Σ) , hence we can define− weak Christoffel symbols Γ L2 in∈ local charts, a weak second fundamental form A and a weak∈ Riemann tensor R via the equations of Weingarten and Gauß

k ∂ijf =Γij∂kf + Aij, R = A , A A , A . ijkl ik jl − ij kl Of course this deﬁnes H , A0, K for f as well. Moreover we deﬁne the tangential and normal projections for V W 1,2(Σ, IRn) L∞(Σ, IRn) ∈ ∩ ij cπf .V := g ∂if, V ∂j f, (2.8) π⊥.V := V π .V W 1,2(Σ, IRn) L∞(Σ, IRn). f − Σ ∈ ∩ MINIMIZERS OF THE WILLMORE FUNCTIONAL 479

n Molliﬁng f as in [SU83] 4 Proposition, we get smooth fm :Σ IR with § → (2.9) f f strongly in W 2,2(Σ), weakly∗ in W 1,∞(Σ) m → and, as Df W 1,2 , that locally uniformly ∈ sup d(Dfm(x),Df(y)) 0. |x−y|≤C/m → This implies for that the pull-backs are uniformly bounded from below and above (2.10) c g f ∗ g Cg 0 ≤ m euc ≤ for some 0 < c0 C < and m large, in particular fm are smooth immersions. ≤ ∞

3. The full rank case For a smooth immersion f :Σ IRn of a closed, orientable surface Σ of genus p 1 and V C∞→(Σ, IRn) , we see that the variations ≥ ∈ ∗ f +tV are still immersions for small t . We put gt := (f +tV ) geuc, g = ∗ g0 = f geuc and deﬁne

(3.1) δπf .V := dπg.∂tgt|t=0. The elements of := δπ .C∞(Σ, IRn) T Vf f ⊆ πgT can be considered as the infinitesimal variations of g in Teichmüller space obtained by ambient variations of f . We call f of full rank in Teichmüller space, if dim f = dim . In this case, the necessary corrections in Teichmüller spaceV mentionedT in 2 can easily be achieved by the inverse function theorem, as we will see§ in this section. 2u Writing g = e gpoin for some unit volume constant curvature metric gpoin by Poincaré’s theorem, see [Tr] Theorem 1.3.7, we see π(gt) = −2u r π(e gt) , hence for an orthonormal basis q (gpoin), r = 1,..., dim , T T T of transvere traceless tensors in S2 (gpoin) with respect to gpoin −2u (3.2) δπf .V = dπg.∂tgt|t=0 = dπgpoin .e ∂tgt|t=0

dim T = e−2u∂ g .qr(g ) dπ .qr(g ). t t|t=0 poin gpoin gpoin poin r=1 Calculating in local charts (3.3) g = g + t ∂ f, ∂ V + t ∂ f, ∂ V + t2 ∂ V, ∂ V , t,ij ij i j j i i j 480 E. KUWERT & R. SCHATZLE¨ we get e−2u∂ g ,qr(g ) t t|t=0 poin gpoin ik jl −2u r = g g e ∂tgt,ij q (gpoin) dg poin poin |t=0 kl poin Σ ik jl r = 2 g g ∂if, ∂jV q (gpoin) dg kl Σ ik jl g g r = 2 g g f, V q (gpoin) dg − ∇j ∇i kl Σ ik jl g r 2 g g ∂if, V q (gpoin) dg − ∇j kl Σ ik jl 0 r = 2 g g A ,V q (gpoin) dg, − ij kl Σ

T T T T as q S2 (gpoin) = S2 (g) is divergence- and tracefree with respect to g .∈ Therefore dim T ik jl r r δπf .V = 2 g g ∂if, ∂jV q (gpoin) dg dπg .q (gpoin) kl poin r=1 Σ

dim T ik jl 0 r r (3.4) = 2 g g A ,V q (gpoin) dg dπg .q (gpoin), − ij kl poin r=1 Σ and δπ and are well deﬁned for uniformly conformal lipschitz im- f Vf mersions f and V W 1,2(Σ) . First, we select variations∈ whoose image via δπ form a basis of . f Vf Proposition 3.1. For a uniformly conformal lipschitz immersion f and ﬁnitely many points x ,...,x Σ , there exist V ,...,V 0 N ∈ 1 dim Vf ∈ C∞(Σ x ,...,x , IRn) such that 0 − { 0 N } (3.5) = span δπ .V s = 1,..., dim . Vf { f s | Vf }

∞ n Proof. Clearly δπf : C (Σ, IR ) Tπgpoin is linear. For suitable W C∞(Σ, IRn) , we obtain a direct→ sum decompositionT ⊆ C∞(Σ, IRn)= ker δπ W f ⊕ and see that δπ W im δπ is bijective, hence dim W = dim f | → f Vf and a basis V1,...,Vdim Vf of W satisﬁes (3.5). Next we choose cutoﬀ functions ϕ C∞( N B (x )) such that 0 ̺ ∈ 0 ∪k=0 ̺ k ≤ ϕ 1, ϕ =1 on N B (x ) and ϕ C̺−1 , hence 1 ϕ ̺ ≤ ̺ ∪k=0 ̺/2 k |∇ ̺|g ≤ − ̺ ∈ MINIMIZERS OF THE WILLMORE FUNCTIONAL 481

∞ C0 (Σ x0,...,xN ), 1 ϕ̺ 1 on Σ x0,...,xN , Σ ϕ̺ g dg −{ } − → −{∞ } |∇ | n ≤ C̺ 0 for ̺ 0 . Clearly ϕmVs C0 (Σ x0,...,x N , IR ) and by (3.4)→ → ∈ − { } δπ .(ϕ V ) δπ .V , f m s → f s hence ϕmV1,...,ϕmVdim Vf satisﬁes for large m all conclusions of the proposition. q.e.d. We continue with a convergence criterion for the ﬁrst variation. Proposition 3.2. Let f :Σ IRn be a uniformly conformal lips- → chitz immersion approximated by smooth immersions fm with pull-back ∗ 2u ∗ 2um metrics g = f geuc = e gpoin, gm = fmgeuc = e gpoinm for some smooth unit volume constant curvature metrics gpoin, gpoin,m and satisfying

f f weakly in W 2,2(Σ), weakly∗ in W 1,∞(Σ), m → (3.6) Λ−1g g Λg , poin ≤ m ≤ poin u ∞ Λ m L (Σ)≤ for some Λ < . Then for any W L2(Σ, IRn) ∞ ∈ π .W π .W fm → f as m . →∞ Proof. By (3.6)

g = f ∗ g f ∗g = g weakly in W 1,2(Σ), m m euc → euc weakly∗ in L∞(Σ), (3.7) Γk Γk weakly in L2, gm,ij → g,ij A , A0 A , A0 weakly in L2, fm,ij fm,ij → f,ij f,ij K d 1 A 2 d C, | gm | gm ≤ 2 | fm | gm ≤ Σ Σ hence by [FiTr84], [Tr],

1,2 r r weakly in W (Σ), q (gpoin,m) q (gpoin)  (3.8) →  weakly∗ in L∞(Σ), dπ .qr(g ) dπ .qr(g ). gpoin,m poin,m → gpoin poin Then by (3.4)

δπfm .W dim T ik jl 0 r r = 2 g g A ,W q (gpoin,m) dg dπg .q (gpoin,m) − m m fm,ij kl m poin,m r=1 Σ 482 E. KUWERT & R. SCHATZLE¨

δπ .W. → f q.e.d. In the full rank case, the necessary corrections in Teichm¨uller space mentioned in 2 are achieved in the following lemma by the inverse function theorem.§ Lemma 3.3. Let f :Σ IRn be a uniformly conformal lipschitz → immersion approximated by smooth immersions fm satisfying (2.2)- (2.6). If f is of full rank in Teichm¨uller space, then for arbitrary x 0 ∈ Σ, neighbourhood U∗(x0) Σ of x0 and Λ < , there exists a neigh- ⊆ ∞ ∞ bourhood U(x0) U∗(x0) of x0 , variations V1,...,Vdim T C0 (Σ n ⊆ ∈ − U(x0), IR ) , satisfying (3.5), and δ > 0,C < ,m0 IN such that ∞ n ∞ ∈ for any V C0 (U(x0), IR ) with fm + V a smooth immersion for some m ∈m , and V = 0 or ≥ 0 Λ−1g (f + V )∗g Λg , poin ≤ m euc ≤ poin V 2,2 Λ, (3.9) W (Σ)≤ A 2 d ε (n), | fm+V | fm+V ≤ 0 U∗(x0) where ε (n) is as in Lemma A.1, and any τ with 0 ∈T (3.10) d (τ,τ ) δ, T 0 ≤ there exists λ IRdim T satisfying ∈ ∗ π((fm + V + λrVr) geuc)= τ and ∗ λ CdT π((fm + V ) geuc),τ . | |≤ Proof. By (2.3), (2.4) and Λ large enough, we may assume

um,Dfm W 1,2(Σ)∩L∞(Σ) Λ, (3.11) ≤ Λ−1g f ∗ g Λg , poin ≤ m euc ≤ poin in particular

2 (3.12) Af df C(Σ, gpoin, Λ). | m | m ≤ Σ 2 2 Putting νm := gpoin fm gpoin gpoin , we see νm(Σ) C(Λ, gpoin) and |∇ | ∗ 0 ∗ ≤ for a subsequence νm ν weakly in C0 (Σ) . Clearly ν(Σ) < , and there are at most ﬁnitely→ many y ,...,y Σ with ν( y ) ∞ε , 1 N ∈ { i} ≥ 1 where we choose ε1 > 0 below. ∞ As f is of full rank, we can select V1,...,Vdim T C0 (Σ x ,y ,...,y , IRn) with span δπ .V = T by Proposition∈ 3.1.− { 0 1 N } { f r} gpoin T MINIMIZERS OF THE WILLMORE FUNCTIONAL 483

We choose a neighbourhood U (x ) U (x ) of x with a chart 0 0 ⊆ ∗ 0 0 ϕ : U (x ) ≈ B (0), ϕ (x )=0, 0 0 0 −→ 2 0 0 supp V U (x )= for r = 1,..., dim , r ∩ 0 0 ∅ T put x U (x )= ϕ−1(B (0)) for 0 < ̺ 2 and choose x U(x ) 0 ∈ ̺ 0 0 ̺ ≤ 0 ∈ 0 ⊆ U1/2(x0) small enough, as we will see below. dim T Next for any x r=1 supp Vr , there exists a neighbourhood ∈ ∪ ≈ U (x) of x with a chart ϕ : U (x) B (0), ϕ (x) = 0, U (x) 0 x 0 −→ 2 x 0 ∩ U (x )= ,ν(U (x)) < ε and in the coordinates of the chart ϕ 0 0 ∅ 0 1 x ik jl r s (3.13) g g gpoin,rsΓ Γ dg ε1. poin poin gpoin,ij gpoin,kl poin ≤ U0(x) Putting x U (x)= ϕ−1(B (0)) U (x) for 0 < ̺ 2 , we see that ∈ ̺ x ̺ ⊂⊂ 0 ≤ there are ﬁnitely many x ,...,x dim T supp V such that 1 M ∈∪r=1 r dim T supp V M U (x ). ∪r=1 r ⊆∪k=1 1/2 k Then there exists m IN such that for m m 0 ∈ ≥ 0 2 2 fm dg < ε1 for k = 1,...,M. |∇gpoin |gpoin poin U1(xk) For V and m m as above, we put f˜ := f + V + λ V . Clearly ≥ 0 m,λ m r r supp (f f˜ ) M U (x ). m − m,λ ⊆∪k=0 1/2 k By (3.9), (3.11), (3.13), and λ < λ 1/4 small enough independent | | 0 ≤ of m and V , f˜m,λ is a smooth immersion with (3.14) (2Λ)−1g g˜ := f˜∗ g 2Λg , poin ≤ m,λ m,λ euc ≤ poin and if V = 0 by (3.9) and the choice of U (x) that 0 2 A ˜ dg˜ ε0(n) for k = 0,...,M | fm,λ | m,λ ≤ U1(xk) for C(Λ, g )(ε +λ ) ε (n). If V = 0 , we see supp (f f˜ ) poin 1 0 ≤ 0 m − m,λ ⊆ M U (x ) . Further by (3.12) ∪k=1 1/2 k 1 2 Kg˜ dg˜ A ˜ dg˜ | m,λ | m,λ ≤ 2 | fm,λ | m,λ Σ Σ

M 1 2 1 2 Af df + A ˜ dg˜ ≤ 2 | m | m 2 | fm,λ | m,λ k=0 Σ U1/2(xk) C(Σ, g , Λ) + (M + 1)ε (n). ≤ poin 0 484 E. KUWERT & R. SCHATZLE¨

This verifies (A.3) and (A.4) for f = fm, f˜ = f˜m,λ, g0 = gpoin and different, but appropriate Λ . (A.2) follows from (2.4) and (3.12). Then −2˜u for the unit volume constant curvature metricg ˜poin,m,λ = e m,λ g˜m,λ conformal tog ˜m,λ by Poincaré’s theorem, see [Tr] Theorem 1.3.7, we get from Lemma A.1 that

(3.15) u˜ ∞ , u˜ 2 C m,λ L (Σ) ∇ m,λ L (Σ,gpoin)≤ with C < independent of m and V . ∞ 2,2 1,∞ From (3.9), we have a W W bound on f˜0 , hence for f˜m,λ . ∩ − 2,2 On Σ U(x0) , we get f˜m,λ = fm + λrVr f weakly in W (Σ − ∗ 1,∞ → − U(x0)) and weakly in W (Σ U(x0)) for m0 , λ0 0 by − → ∞ → 2,2 (2.5). If V = 0 , then f˜m,λ = fm + λrVr f weakly in W (Σ) ∗ 1,∞ → and weakly in W (Σ) for m0 , λ0 0 by (2.5). Hence letting m , λ 0,U(x ) x ,→∞ we conclude→ 0 →∞ 0 → 0 → { 0} (3.16) f˜ f weakly in W 2,2(Σ), weakly∗ in W 1,∞(Σ), m,λ → in particular

1,2 ∗ 2u weakly in W (Σ), g˜m,λ f geuc = e gpoin =: g  →  weakly∗ in L∞(Σ).

Together with (3.14), this implies by [FiTr84 ], [Tr], (3.17) π(˜g ) τ . m,λ → 0 We select a chart ψ : U(π(g )) IRdim T and putπ ˆ := poin ⊆T → ψ π, δπˆ = dψ dπ, ˆ := dψ . . By (3.17) for m large enough, ◦ ◦ Vf πgpoin Vf 0 λ0 and U(x0) small enough independent of V , we get π(˜gm,λ) U(τ0) and deﬁne ∈

Φm(λ) :=π ˆ(˜gm,λ). This yields by (3.4), (3.14), (3.15), (3.16) and Proposition 3.2

DΦm(λ) = (δπˆ ˜ .Vr)r=1,...,dim T (δπˆf .Vr)r=1,...,dim T fm,λ → =: A IRdim T×dim T . ∈ As span δπ .V = T = IRdim T , the matrix A is invertible. { f r} gpoin T ∼ Choosing m0 large enough, λ0 and U(x0) small enough, we obtain DΦ (λ) A 1/(2 A−1 ), m − ≤ hence by standard inverse function theorem for any ξ IRdim T with ξ Φ (0) < λ /(2 A−1 ) there exists λ B (0)∈ with | − m | 0 ∈ λ0

Φm(λ)= ξ, λ 2 A−1 ξ Φ (0) . | |≤ | − m | MINIMIZERS OF THE WILLMORE FUNCTIONAL 485

As ∗ dT π((fm + V ) geuc),τ dT (π(˜g0),τ0)+ dT (τ0,τ) < dT (π(˜g0),τ0)+ δ ≤ by (3.10), we see for δ small enough, m0 large enough and U(x0) small enough by (3.17) that there exists λ IRdim T satisfying ∈ ∗ π((fm + V + λrVr) geuc)= π(˜gm,λ)= τ, ∗ λ CdT π((fm + V ) geuc),τ , | |≤ and the lemma is proved. q.e.d.

4. The degenerate case In this section, we consider the degenerate case when the immersion is not of full rank in Teichmüller space. First we see that the image in Teichmüller space looses at most one dimension. Proposition 4.1. For a uniformly conformal immersion f W 2,2(Σ, IRn) we always have ∈ (4.1) dim dim 1. Vf ≥ T − f is not of full rank in Teichmüller space if and only if ik jl 0 (4.2) g g Aijqkl = 0 for some non-zero smooth transverse traceless symmetric 2-covariant ∗ tensor q with respect to g = f geuc . In this case, f is isothermic locally around all but finitely many points of Σ , that is around all but finitely many points of Σ there exist local conformal principal curvature coordinates. −2u Proof. Let gpoin = e g be the unit volume constant curvature ∗ T T metric conformal to g = f geuc . For q S2 (gpoin), weput Λq : C∞(Σ, IRn) IR ∈ → ik jl 0 Λq.V := 2 g g A ,V qkl dg − ij Σ and define the annihilator Ann := q ST T (g ) Λ = 0 . { ∈ 2 poin | q } T T As dπgpoin S2 (gpoin) Tgpoin is bijective, we see by (3.4) and elementary| linear algebra→ T (4.3) dim = dim + dim Ann. T Vf Clearly, (4.4) q Ann gikgjlA0 q = 0, ∈ ⇐⇒ ij kl 486 E. KUWERT & R. SCHATZLE¨ which already yields (4.2) Choosing a conformal chart with respect to 2v 1,2 ∞ the smooth metric gpoin , we see gij = e δij for some v W L 0 0 0 0 ∈ 0 ∩ and A11 = A22, A12 = A21,q11 = q22,q12 = q21 , as both A and q T T − − 2u ∈ S2 (gpoin) are symmetric and tracefree with respect to g = e gpoin . This rewrites (4.4) into (4.5) q Ann A0 q + A0 q = 0. ∈ ⇐⇒ 11 11 12 12 T T The correspondence between S2 (gpoin) and the holomorphic quadratic differentials is exactly that in conformal coordinates (4.6) h := q iq is holomorphic. 11 − 12 Now if (4.1) were not true, there would be two linearly independent q1,q2 Ann by (4.3). Likewise the holomorphic functions hk := k ∈k q11 iq12 are linearly independent over IR , in particular neither of them− vanishes identically, hence these vanish at most at finitely many points, as Σ is closed. Then h1/h2 is meromorphic and moreover not a real constant. This implies that Im(h1/h2) does not vanish identically, hence vanishes at most at finitely many points. Outside these finitely many points, we calculate

1 1 q11 q12 Im(h1/h2)= h2 −2Im(h1h2)= h2 −2 det   , | | | | 2 2 q11 q12   hence 1 1 q11 q12 det   vanishes at most at finitely many points 2 2 q11 q12   and by (4.5) (4.7) A0 = 0. Approximating f by smooth immersions as in (2.10), we get H = 2gij A0 = 0 weakly, ∇k ∇i jk where = D⊥ denotes the normal connection in the normal bundle along f∇. Therefore H is constant and | | ∂ H = gij ∂ H , ∂ f ∂ f = gij H , A ∂ f weakly. k k j i − jk i 2u 2u 1,2 ∞ Using ∆gpoin f = e ∆gf = e H and u W L , we conclude successively that f C∞ . Then (4.7) implies∈ that∩ f parametrizes a round sphere or a plane,∈ contradicting p 1 , and (4.1) is proved. Next if f is not of full rank in Teichmüller≥ space, there exits q Ann 0 = by (4.3), and the holomorphic function h in (4.6)∈ vanishes− { } at most∅ at finitely many points. In a neighbourhood of a point where h does not vanish, there is a holomorphic function w with (w′)2 = ih . Then w has a local inverse z and using w as MINIMIZERS OF THE WILLMORE FUNCTIONAL 487 new local conformal coordinates, h transforms into (h z)(z′)2 = i , hence q = 0,q =1 in w coordinates. By (4.5) ◦ − 11 12 − A12 = 0, and w are local conformal principal curvature coordinates. q.e.d. Since we loose at most one dimension in the degenerate case, we will do the necessary corrections in Teichmüller space mentioned in 2 by investigating the second variation in Teichmüller space. To be more§ precise, for a smooth immersion f :Σ IRn with pull-back metric g = ∗ → f geuc conformal to a unit volume constant curvature metric gpoin = −2u ∗ e f geuc by Poincaré’s theorem, see [Tr] Theorem 1.3.7, we select a chart ψ : U(π(g )) IRdim T , putπ ˆ := ψ π, ˆ := poin ⊆T → ◦ Vf dψπgpoin . f , and define the second variation in Teichmüller space of f with respectV to the chart ψ by 2 2 d ∗ (4.8) δ πˆf (V ) := πˆ((f + tV ) geuc)|t=0. dt If f is not of full rank in Teichmüller space, we know by the previous Proposition 4.1 that dim = dim 1 , and we do the corrections Vf T − in the remaining direction e ˆ , e = 1 , which is unique up to the ⊥ Vf | | sign, by variations V± satisfying δ2πˆ (V ), e > 0. ± f ± We construct V± locally around an appropriate x0 Σ in a chart ≈ ∈ ϕ : U(x0) B1(0) of conformal principal curvature coordinates that is −→ 2v (4.9) g = e geuc, A12 = 0, which exists by Proposition 4.1, and moreover assuming that x0 is not umbilical that is (4.10) A0(x ) = 0, 0 as Σ is not a sphere. We choose some unit normal 0 at f in f(0) , some η C∞(B (0)) and put η (y) := η(̺−1y) , N ∈ 0 1 ̺ V := η . ̺N0 For ̺ small, the support of V is close to 0 and V is nearly normal. ∗ −2u Putting ft := f + tV,gt := ft geuc, g = g0 , we seeπ ˆ(gt) =π ˆ(e gt) and calculate 2 2 d −2u δ πˆf (V )= πˆ(e gt)|t=0 dt −2u 2 −2u −2u = dπˆgpoin .(e (∂ttgt)|t=0)+ d πˆgpoin (e (∂tgt)|t=0, e (∂tgt)|t=0). We will see that the second term, which is quadratic in g , is much smaller than the first one for ̺ small. To calculate the effect of the first 488 E. KUWERT & R. SCHATZLE¨

T T term on e , we consider the unique qe S2 (g) with dπˆgpoin .qe(gpoin)= e and see by (3.3) that ∈

dπˆ .(e−2u(∂ g ) ), e gpoin tt t |t=0 ik jl −2u = g g e (∂ttgt,ij) qe,kl dg poin poin |t=0 poin Σ ik jl = 2 g g ∂iV, ∂jV qe,kl dg. Σ

Now qe satisﬁes (4.2), hence by (4.9) and (4.10) that qe,11 = qe,22 0,q = q q IR 0 in B (0) and − ≡ e,12 e,21 ≡ ∈ − { } 1

2 −2v(0) 2 δ πˆf (V ), e 4qe ∂1η∂2η d for ̺ 0. → L → B1(0)

∞ Now it just requires to find appropriate η C0 (B1(0)) to achieve a positive and a negative sign. Our actual choice∈ is given after (4.26) below. In the following long computation, we make the above considerations work. Firstly we show that the second variation in Teichmüller space is well defined for uniformly conformal W 2,2 immersions f and V W 1,2(Σ) . Then we estimate rigorously the lower− order approximations∈ which were neglected above. We proceed for smooth f and V by decomposing

(4.11) e−2u(∂ g ) = σg + g + q, t t |t=0 poin LX poin

∞ T T with σ C (Σ), X (Σ),q S2 (gpoin) and continue by recalling σg +∈ g =∈Xker dπˆ ∈ with { poin LX poin} gpoin

2 −2u δ πˆf (V ) = dπˆgpoin .(e (∂ttgt)|t=0) +d2πˆ (σg + g + q,σg + g + q) gpoin poin LX poin poin LX poin −2u 2 = dπˆgpoin .(e (∂ttgt)|t=0)+ d πˆgpoin(q,q) +d2πˆ (σg + g , σg + g + 2q) gpoin poin LX poin poin LX poin −2u = dπˆgpoin . e (∂ttgt)|t=0 X X gpoin −L L 2 2σ X gpoin 2σq 2 X q + d πˆgpoin (q,q). − L − − L MINIMIZERS OF THE WILLMORE FUNCTIONAL 489

r For an orthonormal basis q (gpoin), r = 1,..., dim , of transverse T T T traceless tensors in S2 (gpoin) with respect to gpoin , we obtain dim T 2 r δ πˆf (V )= αr dπˆgpoin .q (gpoin) r=1 (4.12) dim T 2 r s + βrβs d πˆgpoin (q (gpoin),q (gpoin)), r,s=1 where

ik jl −2u αr := gpoingpoin e (∂ttgt)|t=0 X X gpoin −L L Σ r 2σ X gpoin 2σq 2 X q qkl(gpoin) dgpoin , − L − − L ij

ik jl r (4.13) βr := g g qijq (gpoin) dg . poin poin kl poin Σ Since g LX LX poin,ij mo = gpoin Xm o X gpoin,ij + iXm X gpoin,jo + jXm X gpoin,oi , ∇ L ∇ L ∇ L we get integrating by parts

ik jl −2u αr = gpoingpoin e (∂ttgt)|t=0 2σ X gpoin − L Σ r 2σq 2 X q qkl(gpoin) dgpoin − − L ij ik jl mo (4.14) + gpoingpoingpoin oXm X gpoin,ij iXm X gpoin,jo ∇ L −∇ L Σ r jXm X gpoin,oi qkl(gpoin) dgpoin −∇ L ik jl mo r + g g g Xm X gpoin,ij oq (gpoin) dg . poin poin poin L ∇ kl poin Σ For a uniformly conformal immersion f W 2,2(Σ, IRn) and V W 1,2(Σ, IRn) , we see g W 1,2(Σ), u W 1,2∈(Σ) L∞(Σ), (∂ g ) ∈ ∈ ∈ ∩ t t |t=0 ∈ L2(Σ), and (∂ g ) L1(Σ) by (3.3). Then we get a decomposition tt t |t=0 ∈ as in (4.11) with σ L2(Σ), X 1(Σ),q ST T (Σ) S (Σ) . Observ- ∈ ∈X ∈ 2 ⊆ 2 ing that qr(g ) ST T (Σ) S (Σ) , we conclude that δ2πˆ is well poin ∈ 2 ⊆ 2 f deﬁned for uniformly conformal immersions f W 2,2(Σ, IRn) and V W 1,2(Σ, IRn) by (4.12), (4.13) and (4.14). ∈ ∈ 490 E. KUWERT & R. SCHATZLE¨

Proposition 4.2. For a uniformly conformal immersion f W 2,2(Σ, IRn) , which is not of full rank in Teichm¨uller space, and ﬁnitely∈ ∞ many points x1,...,xN Σ , there exist V1,...,Vdim T−1,V± C0 (Σ x ,...,x , IRn) such∈ that ∈ − { 1 N } (4.15) ˆ = span δπˆ .V s = 1,..., dim 1 Vf { f s | T − } and for some e ˆ , e = 1, ⊥ Vf | | (4.16) δ2πˆ (V ), e > 0, δπˆ .V = 0. ± f ± f ± Proof. By Proposition 4.1, there exists x Σ x ,...,x such 0 ∈ − { 1 N } that f is isothermic around x0 . Moreover, since Σ is not a sphere, 0 hence A does not vanish almost everywhere with respect to g , as we have seen in the argument after (4.7) in Proposition 4.1, we may assume that

(4.17) x supp A0 2 . 0 ∈ | | g ∞ By Proposition 3.1, there exist V1,...,Vdim T−1 C0 (Σ x0,...,xN , n ∈ −{ ≈ } IR ) which satisfy (4.15). Next we select a chart ϕ : U(x0) B1(0) of conformal principal curvature coordinates that is −→

2v (4.18) g = e geuc, A12 = 0, in local coordinates of ϕ and where v W 1,2(B (0)) L∞(B (0)) . ∈ 1 ∩ 1 Moreover, we choose U(x ) so small that U(x ) supp V = for s = 0 0 ∩ s ∅ 1,..., dim 1, and U(x0) x1,...,xN = . For any V0 1,2 nT − ∞ ∩ { } ∅ ∈ W (Σ, IR ) L (Σ) with supp V0 U(x0) , there exists a unique dim T−1∩ ⊆ 1,2 n γ IR such that for V := V0 γsVs W (Σ, IR ), supp V Σ ∈ x ,...,x − ∈ ⊆ − { 1 N }

(4.19) δπˆf .V = 0.

By (3.4)

0 (4.20) γ C δπˆ .V C A 2 V ∞ , | |≤ | f 0|≤ L (suppV0,g) 0 L (U(x0)) where C does not depend on V0 . By (3.2), we get for the decomposi- r tion in (4.11) that q = 0 . We select the orthonormal basis q (gpoin), r = 1,..., dim , of ST T (g ) with respect to g , in such a way that T 2 poin poin dπˆ .qr(g ) ˆ for r = 2,..., dim , and dπˆ .q1(g ), e > gpoin poin ∈ Vf T gpoin poin MINIMIZERS OF THE WILLMORE FUNCTIONAL 491

0. Putting

I(V0) :=

ik jl −2u 1 gpoingpoin e (∂ttgt)|t=0 2σ X gpoin qkl(gpoin) dgpoin − L ij Σ ik jl mo + gpoingpoingpoin oXm X gpoin,ij iXm X gpoin,jo (4.21) ∇ L −∇ L Σ 1 jXm X gpoin,oi qkl(gpoin) dgpoin −∇ L ik jl mo 1 + g g g Xm X gpoin,ij oq (gpoin) dg . poin poin poin L ∇ kl poin Σ we obtain from (4.12), (4.13) and (4.14) δ2πˆ (V ), e = dπˆ .q1(g ), e I(V ). f gpoin poin 0 As I(V ) I(V ) for V V in W 1,2(Σ, IRn) , it suﬃces to ﬁnd 0,m → 0 0,m → 0 V respectively V W 1,2(Σ) such that 0 ∈ (4.22) I(V ) > 0. ± 0 ∞ n Recalling δπˆf .W ˆf e for any W C (Σ, IR ) , we see in the same way by (3.4)∈ thatV ⊥ ∈

ik jl 0 1 1 0= δπˆf .W, e = 2 g g A ,W q (gpoin) dg dπg .q (gpoin), e , − ij kl poin Σ 1 ∞ n hence, as dπˆgpoin .q (gpoin), e > 0 and W C (Σ, IR ) is arbi- ik jl 0 1 ∈ 0 1 trary, g g Aijqkl(gpoin) = 0 . We get by (4.18) that A11q11(gpoin) = 1 1 0 in B1(0) ∼= U(x0) and by (4.17), as q11(gpoin) iq12(gpoin) is holomorphic, −

q1 (g )= q1 (g ) = 0, 11 poin − 22 poin (4.23)  in B1(0) ∼= U(x0) q1 : q1 IR 0  12 ≡ ∈ − { } in local coordinates of ϕ .  Next we consider V0 to be normal at f and calculate by (3.3) that (∂ g ) = ∂ f, ∂ V + ∂ f, ∂ V t t,ij |t=0 i j j i = 2 A ,V γ ∂ f, ∂ V γ ∂ f, ∂ V , − ij 0− s i j s− s j i s hence by (4.20)

(∂ g ) 2 2 A 2 V ∞ +C γ t t |t=0 L (Σ)≤ L (suppV0,g) 0 L (Σ) | | C A 2 V ∞ . ≤ L (suppV0,g) 0 L (Σ) As above we can select σ L2(Σ), X 1(Σ) in (4.11) such that ∈ ∈X σ 2 , X 1,2 C A 2 V ∞ , L (Σ) W (Σ)≤ L (suppV0,g) 0 L (Σ) 492 E. KUWERT & R. SCHATZLE¨ where C does not depend on V0 . We get from (4.21)

ik jl −2u 1 I(V0) gpoingpoine (∂ttgt,ij)|t=0 qkl(gpoin) dgpoin − (4.24) Σ 2 2 C A 2 V ∞ , ≤ L (suppV0,g) 0 L (Σ) where C does not depend on V0 . We continue with (3.3) and get, as supp V0 supp Vs = for s = 1,..., dim 1 , ∩ ∅ T − (∂ g ) = 2 ∂ V, ∂ V = 2 ∂ V , ∂ V + 2γ γ ∂ V , ∂ V . tt t,ij |t=0 i j i 0 j 0 r s i r j s As ik jl −2u 1 gpoingpoine 2γrγs ∂iVr, ∂jVs qkl(gpoin) dgpoin Σ 2 2 2 C γ C A 2 V ∞ ≤ | | ≤ L (suppV0,g) 0 L (Σ) and by (4.18)

ik jl −2u 1 g g e 2 ∂iV0, ∂jV0 q (gpoin) dg poin poin kl poin Σ

1 −2v 2 = 4q e ∂1V0, ∂2V0 d , L B1(0) where we identify U(x0) ∼= B1(0) , we get from (4.24) 1 −2v 2 I(V0) 4q e ∂1V0, ∂2V0 d − L (4.25) B1(0) 2 2 C A 2 V ∞ , ≤ L (suppV0,g) 0 L (Σ) where C does not depend on V0 . Perturbing x0 ∼= 0 in U(x0) ∼= B1(0) slightly, we may assume that 0 is a Lebesgue point for f and v . We select a unit vec- n ∇ tor 0 IR normal at f in f(0) and deﬁne via normal projection N ⊥∈ 1,2 n := πf 0 W (B1(0), IR ) , see (2.8). Clearly, (0) = 0 , N N ∈ ∞ N N and 0 is a Lebesgue point of . For η C0 (B1(0)) , we put η (y) := η(̺−1y) and V := η andN calculate ∈ ̺ 0 ̺N −2v 2 ∂1V0, ∂2V0 e d L B1(0)

2 −2v 2 −2v 2 = ∂1η̺∂2η̺ e d + (∂1η̺)η̺ , ∂2 e d |N | L N N L B1(0) B1(0)

−2v 2 2 −2v 2 + η̺∂2η̺ ∂1 , e d + η ∂1 , ∂2 e d . N N L ̺ N N L B1(0) B1(0) MINIMIZERS OF THE WILLMORE FUNCTIONAL 493

As η 2 = η 2 , we see ∇ ̺ L (B̺(0)) ∇ L (B1(0)) −2v 2 lim ∂1V0, ∂2V0 e d ̺→0 L B1(0)

2 −2v(̺y) = lim ∂1η(y)∂2η(y) (̺y) e dy ̺→0 |N | B1(0)

−2v(0) 2 = e ∂1η∂2η d . L B1(0) Since V ∞ η ∞ = η ∞ 0 L (Σ)≤ ̺ L (B̺(0)) L (B1(0)) and

A 2 0 for ̺ 0, L (suppV0,g)→ → we get

1 −2v(0) 2 (4.26) lim I(η̺ ) = 4q e ∂1η∂2η d . ̺→0 N L B1(0)

Introducing new coordinatesy ˜ := (y + y )/√2, y˜ := ( y + y )/√2 , 1 1 2 2 − 1 2 that is we rotate the coordinates by 45 degrees, and putting η(˜y1, y˜2)= ξ(˜y )τ(˜y ) with ξ,τ C∞(] 1/2, 1/2[) , we see 1 2 ∈ 0 − 2 1 ′ 2 2 1 2 ′ 2 ∂y1 η∂y2 η d = ξ τ ξ τ . L 2 | | | | − 2 | | | | B1(0)

∞ Choosing τ C0 (] 1/2, 1/2[),τ 0 and ξ(t) := τ(2t) , we get ξ′ 2 = 2 τ∈′ 2, 2 ξ−2 = τ 2 and≡ | | | | | | | | 2 3 2 ′ 2 ∂y1 η∂y2 η d = τ τ > 0. L 4 | | | | B1(0) Exchanging ξ and τ , we produce a negative sign. Choosing ̺ small enough and approximating V0 = η̺ smoothly, we obtain the desired V C∞(Σ x ,...,x , IRn) .N q.e.d. ± ∈ 0 − { 1 N } Next we extend the convergence criterion in Proposition 3.2 to the second variation. Proposition 4.3. Let f :Σ IRn be a uniformly conformal 2,2 → W immersion approximated by smooth immersions fm with pull- − ∗ 2u ∗ 2um back metrics g = f geuc = e gpoin, gm = fmgeuc = e gpoinm for 494 E. KUWERT & R. SCHATZLE¨ some smooth unit volume constant curvature metrics gpoin, gpoin,m and satisfying

f f weakly in W 2,2(Σ), weakly∗ in W 1,∞(Σ), m → (4.27) Λ−1g g Λg , poin ≤ m ≤ poin u ∞ Λ m L (Σ)≤ for some Λ < . Then for any chart ψ : U(π(g )) IRdim T , ∞ poin ⊆T → πˆ := ψ π, δπˆ = dψ δπ,δ2πˆ deﬁned in (4.8) and any W W 1,2(Σ, IRn) ◦ ◦ ∈ δπˆ .W δπˆ .W, δ2πˆ (W ) δ2πˆ (W ) fm → f fm → f as m . →∞

Proof. By Proposition 3.2, we know already δπˆfm .W δπˆf .W and get further (3.7) and (3.8). → Next we select a slice (g ) of unit volume constant curvature S poin metrics for π(g ) =: τ around g with π : (g ) = U(τ ) , poin 0 ∈T poin S poin ∼ 0 and qr(˜g ) ST T (˜g ) for π(˜g ) U(τ ) , see [FiTr84], [Tr]. poin ∈ 2 poin poin ∈ 0 As π(g ) = π(g ) π(g ) U(τ ) = (g ) , there exist poin,m m → poin ∈ 0 ∼ S poin for m large enough smooth diﬀeomorphisms φm of Σ homotopic to ∗ the identity with φmgpoin,m =:g ˜poin,m (gpoin). As π(˜gpoin,m) = π(g )= π(g ) π(g ) andg ˜ ∈ S (g ) , we get poin,m m → poin poin,m ∈ S poin (4.28)g ˜ g smoothly. poin,m → poin The Theorem of Ebin and Palais, see [FiTr84], [Tr], imply by (3.7), (4.27) and (4.28) that after appropriately modifying φm (4.29) φ , φ−1 id weakly in W 2,2(Σ), weakly∗ in W 1,∞(Σ), m m → Σ in particular, −1 (4.30) Dφ 1,2 ∞ , D(φ ) 1,2 ∞ C m W (Σ)∩L (Σ) m W (Σ)∩L (Σ)≤ with C independent of m and V . For the second variations, we see by (3.3) and (4.27) ∂ ((f + tW )∗g ) = ∂ f , ∂ W + ∂ f , ∂ W t m euc ij |t=0 i m j j m i ∂ f, ∂ W + ∂ f, ∂ W = ∂ ((f + tW )∗g ) → i j j i t euc ij |t=0 weakly in W 1,2(Σ). Following (4.11), we decompose

φ∗ ∂ ((f˜ + tW )∗g ) = σ g˜ + g˜ + q m t m euc |t=0 m poin,m LXm poin,m m with q ST T (˜g ) and moreover recalling (4.30), we can achieve m ∈ 2 poin,m σ 1,2 , X 2,2 , q 2 C. m W (Σ) m W (Σ) m C (Σ)≤ MINIMIZERS OF THE WILLMORE FUNCTIONAL 495

1,2 For a subsequence, we get σm σ weakly in W (Σ), Xm X weakly 2,2 → 1 T T → in W (Σ),qm q strongly in C (Σ) with q S2 (gpoin) and by (4.29) → ∈ (4.31) ∂ ((f + tW )∗g ) = σg + g + q. t euc |t=0 poin LX poin Observing from (3.3) that (∂ g ) = 2 ∂ W, ∂ W , tt t,ij |t=0 i j we get from (4.12), (4.13) and the equivariance ofπ ˆ

dim T 2 r δ πˆfm (W )= αm,r dπˆg˜poin,m .q (˜gpoin,m) r=1

dim T 2 r s + βm,rβm,s d πˆg˜poin,m (q (˜gpoin,m),q (˜gpoin,m)), r,s=1 where

ik jl r αm,r := g g 2 ∂iW, ∂jW q (gpoin,m) dg m m kl m Σ ik jl g˜poin,mg˜poin,m Xm Xm g˜poin,m − L L Σ r + 2σm Xm g˜poin,m qkl(˜gpoin,m) dg˜poin,m L ij ik jl r g˜poin,mg˜poin,m 2σmqm + 2 Xm qm qkl(˜gpoin,m) dg˜poin,m , − L ij Σ

ik jl r βm,r := g˜ g˜ qm,ijq (˜gpoin,m) dg˜ . poin,m poin,m kl poin,m Σ By the above convergences in particular by (3.7), (3.8) and (4.28), we obtain

ik jl r αm,r αr := g g 2 ∂iW, ∂jW q (gpoin) dg → kl Σ ik jl r gpoingpoin X X gpoin + 2σ X gpoin qkl(gpoin) dgpoin − L L L ij Σ ik jl r gpoingpoin 2σq + 2 X q qkl(gpoin) dgpoin , − L ij Σ

ik jl r βm,r βr := g g qijq (gpoin) dg . → poin poin kl poin Σ 496 E. KUWERT & R. SCHATZLE¨

We see from (4.28) d2πˆ (qr(˜g ),qs(˜g )) d2πˆ (qr(g ),qs(g )). g˜poin,m poin,m poin,m → gpoin poin poin Observing (4.31), we get from (4.12), (4.13) 2 2 1,2 n (4.32) δ πˆ ˜ (W ) δ πˆf (W ) for any W W (Σ, IR ), fm → ∈ and the proposition is proved. q.e.d.

Remark. The bound on the conformal factor um in the above proposition is implied by Proposition A.2, when we replace the weak 2,2 convergence of fm f in W (Σ) by strong convergence. Now we can extend→ the correction lemma 3.3 to the degenerate case. Lemma 4.4. Let f :Σ IRn be a uniformly conformal W 2,2 im- → − mersion approximated by smooth immersions fm satisfying (2.2)- (2.6). If f is not of full rank in Teichm¨uller space, then for arbitrary x0 Σ, neighbourhood U (x ) Σ of x , and Λ < , there exists a neigh-∈ ∗ 0 ⊆ 0 ∞ bourhood U(x0) U∗(x0) of x0 , variations V1,...,Vdim T−1,V± ∞ ⊆n ∈ C0 (Σ U(x0), IR ) , satisfying (4.15) and (4.16), and δ > 0,C < − ∞ n ,m0 IN such that for any V C0 (U(x0), IR ) with fm + V a smooth∞ ∈ immersion for some m m∈ , and V = 0 or ≥ 0 Λ−1g (f + V )∗g Λg , poin ≤ m euc ≤ poin V 2,2 Λ, (4.33) W (Σ)≤ A 2 d ε (n), | fm+V | fm+V ≤ 0 U∗(x0) where ε (n) is as in Lemma A.1, and any τ with 0 ∈T (4.34) d (τ,τ ) δ, T 0 ≤ there exists λ IRdim T−1, IR , satisfying = 0 , ∈ ± ∈ + − ∗ π((fm + V + λrVr + ±V±) geuc)= τ,

1/2 ∗ λ , ± CdT π((fm + V ) geuc),τ . | | | |≤ Further for any λ0 > 0 , one can choose m0, δ in such a way that for m m , ≥ 0 V 2,2 δ, W (Σ)≤ there exists λ˜ IRdim T−1, ˜ IR , satisfying ˜ ˜ = 0 , ∈ ± ∈ + − ∗ π((fm + V + λ˜rVr + ˜±V±) geuc)= τ, λ˜ , ˜ λ , | | | ±|≤ 0 MINIMIZERS OF THE WILLMORE FUNCTIONAL 497 and ˜ 0, ˜ = 0, if = 0, + + ≤ − + ˜ 0, ˜ = 0, if = 0. − − ≤ + − Proof. By (2.3), (2.4) and Λ large enough, we may assume

um,Dfm 1,2 ∞ Λ, (4.35) W (Σ)∩L (Σ)≤ Λ−1g f ∗ g Λg , poin ≤ m euc ≤ poin in particular

2 (4.36) Af df C(Σ, gpoin, Λ). | m | m ≤ Σ 2 2 Putting νm := gpoin fm gpoin gpoin , we see νm(Σ) C(Λ, gpoin) and |∇ | ∗ 0 ∗ ≤ for a subsequence νm ν weakly in C0 (Σ) . Clearly ν(Σ) < , and there are at most ﬁnitely→ many y ,...,y Σ with ν( y ) ∞ε , 1 N ∈ { i} ≥ 1 where we choose ε1 > 0 below. By Proposition 4.2, we can select V1,...,Vdim T−1,V+ = Vdim T ,V− = V C∞(Σ x ,y ,...,y , IRn) satisfying (4.15) and (4.16). dim T +1 ∈ 0 − { 0 1 N } We choose a neighbourhood U0(x0) U∗(x0) of x0 with a chart ≈ ⊆ ϕ : U (x ) B (0), ϕ (x )=0, 0 0 0 −→ 2 0 0 supp V U (x )= for r = 1,..., dim + 1, r ∩ 0 0 ∅ T put x U (x )= ϕ−1(B (0)) for 0 < ̺ 2 and choose x U(x ) 0 ∈ ̺ 0 0 ̺ ≤ 0 ∈ 0 ⊆ U1/2(x0) small enough, as we will see below. Next for any x dim T +1supp V , there exists a neighbourhood ∈ ∪r=1 r U (x) of x with a chart ϕ : U (x) ≈ B (0), ϕ (x) = 0, U (x) 0 x 0 −→ 2 x 0 ∩ U (x )= ,ν(U (x)) < ε and in the coordinates of the chart ϕ 0 0 ∅ 0 1 x ik jl r s (4.37) g g gpoin,rsΓ Γ dg ε1. poin poin gpoin,ij gpoin,kl poin ≤ U0(x) Putting x U (x)= ϕ−1(B (0)) U (x) for 0 < ̺ 2 , we see that ∈ ̺ x ̺ ⊂⊂ 0 ≤ there are ﬁnitely many x ,...,x dim T +1supp V such that 1 M ∈∪r=1 r dim T +1supp V M U (x ). ∪r=1 r ⊆∪k=1 1/2 k Then there exists m IN such that for m m 0 ∈ ≥ 0 2 2 fm dg < ε1 for k = 1,...,M. |∇gpoin |gpoin poin U1(xk)

For V and m m0 as above, we put f˜m,λ, := fm +V +λrVr +±V± . Clearly ≥ supp (f f˜ ) M U (x ). m − m,λ, ⊆∪k=0 1/2 k 498 E. KUWERT & R. SCHATZLE¨

By (4.33), (4.35), (4.37), and λ , < λ 1/4 small enough inde- | | | | 0 ≤ pendent of m and V , f˜m,λ, is a smooth immersion with (4.38) (2Λ)−1g g˜ := f˜∗ g 2Λg , poin ≤ m,λ, m,λ, euc ≤ poin and if V = 0 by (4.33) and the choice of U (x) that 0 2 A ˜ dg˜ ε0(n) for k = 0,...,M | fm,λ, | m,λ, ≤ U1(xk) for C(Λ, g )(ε +λ ) ε (n). If V = 0 , we see supp (f f˜ ) poin 1 0 ≤ 0 m− m,λ, ⊆ M U (x ) . Further by (4.36) ∪k=1 1/2 k 1 2 Kg˜ dg˜ A ˜ dg˜ | m,λ, | m,λ, ≤ 2 | fm,λ, | m,λ, Σ Σ

M 1 2 1 2 Af df + A ˜ dg˜ ≤ 2 | m | m 2 | fm,λ, | m,λ, k=0 Σ U1/2(xk)

C(Σ, g , Λ) + (M + 1)ε (n). ≤ poin 0 This veriﬁes (A.3) and (A.4) for f = fm, f˜ = f˜m,λ,, g0 = gpoin and different, but appropriate Λ . (A.2) follows from (2.4) and (4.36). Then for −2˜u the unit volume constant curvature metricg ˜poin,m,λ, = e m,λ, g˜m,λ, conformal tog ˜m,λ, by Poincar´e’s theorem, see [Tr] Theorem 1.3.7, we get from Lemma A.1 that

(4.39) u˜ ∞ , u˜ 2 C m,λ, L (Σ) ∇ m,λ, L (Σ,gpoin)≤ with C < independent of m and V . ∞ From (4.33), we have a W 2,2 W 1,∞ bound on f˜ , hence ∩ − m,0,0 for f˜ . On Σ U(x ) , we get f˜ = f + λ V + V m,λ, − 0 m,λ, m r r ± ± → f weakly in W 2,2(Σ U(x )) and weakly∗ in W 1,∞(Σ U(x )) for − 0 − 0 m , λ 0 by (2.5). If V = 0 , then f˜ = f + λ V + 0 → ∞ 0 → m,λ, m r r V f weakly in W 2,2(Σ) and weakly∗ in W 1,∞(Σ) for m , ± ± → 0 → ∞ λ0 0 by (2.5). Hence letting m0 , λ0 0,U(x0) x0 , we conclude→ → ∞ → → { } (4.40) f˜ f weakly in W 2,2(Σ), weakly∗ in W 1,∞(Σ), m,λ, → Then by (4.38), (4.39), (4.40) and Proposition 4.3 for any W W 1,2(Σ, IRn) ∈

2 2 (4.41) δπˆ ˜ .W δπˆf .W, δ πˆ ˜ (W ) δ πˆf (W ) fm,λ, → fm,λ, → for m , λ 0,U(x ) x . 0 →∞ 0 → 0 → { 0} MINIMIZERS OF THE WILLMORE FUNCTIONAL 499

Further from (4.40)

1,2 ∗ 2u weakly in W (Σ), g˜m,λ, f geuc = e gpoin =: g  →  weakly∗ in L∞(Σ),

1,2 ∗ and by (4.39), we seeu ˜m,λ, u˜ weakly in W (Σ) and weakly in L∞(Σ) , hence →

weakly in W 1,2(Σ), −2˜um,λ, 2(u−u˜) g˜poin,m,λ, = e g˜m,λ, e gpoin  →  weakly∗ in L∞(Σ), which together with (4.38) implies  (4.42) π(˜g ) τ . m,λ, → 0 We select a chart ψ : U(π(g )) IRdim T and putπ ˆ := poin ⊆T → ψ π, δπˆ = dψ dπ, ˆ := dψ . , δ2πˆ deﬁned in (4.8). By (4.42) ◦ ◦ Vf πgpoin Vf for m0 large enough, λ0 and U(x0) small enough independent of V , we get π(˜g ) U(τ ) and deﬁne m,λ, ∈ 0 Φm(λ, ) :=π ˆ(˜gm,λ,). We see by (4.41)

DΦm(λ, ) = (δπˆ ˜ .Vr)r=1,...,dim T +1 (δπˆf .Vr)r=1,...,dim T +1. fm,λ, → After a change of coordinates, we may assume ˆ = IRdim T−1 0 Vf × { } and e := e ˆ . Writing Φ (λ, ) = (Φ (λ, ), ϕ (λ, )) dim T ⊥ Vf m m,0 m ∈ IRdim T−1 IR IR , we get × × (dim T−1)×(dim T−1) ∂λΦm,0(λ, ) (π ˆ δπˆf .Vr)r=1,...,dim T−1 =: A IR . → Vf ∈ From (4.15), we see that A is invertible, hence after a further change of coordinates we may assume that A = I(dim T−1) and (4.43) ∂ Φ (λ, ) I 1/2 λ m,0 − (dim T−1) ≤ for m0 large enough, λ0 and U(x0) small enough independent of V . Next by (4.16), we obtain

∂± Φm(λ, ) δπˆf .V± = 0, (4.44) → ϕ (λ, ) (δπˆ .V ) , e = 0, ∇ m → f r r=1,...,dim T +1 as δπˆ .V ˆ e , hence f r ∈ Vf ⊥ (4.45) ∂ Φ (λ, ) , ϕ (λ, ) ε | ± m | |∇ m |≤ 2 for any ε2 > 0 chosen below, if m0 large enough, λ0 and U(x0) small enough independent of V . 500 E. KUWERT & R. SCHATZLE¨

The second derivatives 2 ∂ssΦm(λ, )= δ πˆ ˜ (Vs), (4.46) fm,λ, 2 2 4∂srΦm(λ, )= δ πˆ ˜ (Vs + Vr) δ πˆ ˜ (Vs Vr) fm,λ, − fm,λ, − are given by the second variation in Teichm¨uller space. From (4.41) for W = V ,V V C∞(Σ, IRn) , we conclude for m large enough, s s ± r ∈ 0 λ0 and U(x0) small enough independent of V that (4.47) D2Φ (λ, ) Λ | m |≤ 1 for some 1 Λ < and ≤ 1 ∞ 2 ∂++ Φm(λ, ) δ πˆf (V+), (4.48) → ∂ Φ (λ, ) δ2πˆ (V ), −− m → f − hence by (4.16) 2 lim ∂±± ϕm(λ, )= δ πˆf (V±), e > 0 ± m,λ, ± and (4.49) ∂ ϕ (λ, ) γ ± ±± m ≥ for some 0 < γ 1/4 and m0 large enough, λ0 and U(x0) small enough independent≤ of V . Now, we choose ε > 0 to satisfy CΛ ε 2 1 2 ≤ γ/4 . Choosing m0 even larger and U(x0) even smaller we get by (4.42) d (π(˜g ),τ ) δ, T 0 0 ≤ where we choose δ now. As ∗ dT π((fm + V ) geuc),τ dT (π(˜g0),τ0)+ dT (τ0,τ) < 2δ ≤ 2 2 by (4.34), we choose Cψδ Λ1λ0, λ0/8, γλ0/32 and conclude from ≤ dim T−1 Proposition B.1 that there exists λ IR ,± IR with +− = 0 and satisfying ∈ ∈

∗ π((fm + V + λrVr + ±V±) geuc)= π(˜gm,λ,)= τ, 1/2 ∗ λ , ± CdT π((fm + V ) geuc),τ . | | | |≤ To obtain the second conclusion, we consider λ0 > 0 such small that CΛ1ε2 + CΛ1λ0 γ/2 and ﬁx this λ0 . We assume V W 2,2(Σ) δ ≤ 2,2≤ and see as in (4.40) that f˜m,0,0 = fm + V f weakly in W (Σ) and weakly∗ in W 1,∞(Σ) for m , δ 0→ by (2.5). Again we get 0 → ∞ → (4.41) and (4.44) for λ, = 0 , hence for m0 large enough, δ small enough, but ﬁxed U(x0) , ϕ (0) σ |∇ m |≤ MINIMIZERS OF THE WILLMORE FUNCTIONAL 501 with CΛ ε + Cσλ−1 + CΛ λ γ . Then by Proposition B.1, there 1 2 0 1 0 ≤ exist further λ˜ IRdim T−1, ˜ IR with ˜ ˜ = 0 and satisfying ∈ ± ∈ + − ˜ ∗ π((fm + V + λrVr + ˜±V±) geuc)= π(˜gλ,˜ ˜)= τ, λ˜ , ˜ λ , | | | ±|≤ 0 ˜ 0, ˜ = 0, if = 0, + + ≤ − − ˜ 0, ˜ = 0, if = 0, − − ≤ + + and the lemma is proved. q.e.d.

5. Elementary properties of minimization As a first application of our correction Lemmas 3.3 and 4.4 we estab- lish continuity properties of the minimal Willmore energy under fixed Teichmüller class . Mn,p n Proposition 5.1. n,p : [βp , ] is upper semicontinuous. Secondly, for a sequenceMτ Tτ →in ,n∞= 3, 4 , m → T lim inf n,p(τm) < n,p = n,p(τ) lim inf n,p(τm). m→∞ M W ⇒ M ≤ m→∞ M In particular is continuous at τ with Mn,p ∈T (τ) . Mn,p ≤Wn,p Proof. For the upper semicontinuity, we have to prove

(5.1) lim sup n,p(τ) n,p(τ0) for τ0 . τ→τ0 M ≤ M ∈T

It suffices to consider n,p(τ0) < . In this case, there exists for any ε > 0 a smoothM immersion ∞f :Σ IRn with π(f ∗g ) = → euc τ0 and (f) < n,p(τ0)+ ε , where Σ is a closed, orientable surface ofW genus p M1 . ≥ By Lemmas 3.3 and 4.4 applied to the constant sequence fm = f and V = 0 , there exist for any τ close enough to τ0 in Teichmüller space λ IRdim T +1 with τ ∈ ∗ π((f + λτ,rVr) geuc)= τ, λ 0 for τ τ . τ → → 0 Clearly f + λ V f smoothly, hence τ,r r → lim sup n,p(τ) lim (f + λτ,rVr)= (f) n,p(τ0)+ ε τ→τ0 M ≤ τ→τ0 W W ≤ M and (5.1) follows. To prove the second statement, we may assume after passing to a subsequence that τm τ and n,p(τm) < n,p 2δ for some δ > 0 . We select smooth immersions→ fM:Σ IRnWwith−π(f ∗ g )= τ and m → m euc m (5.2) (f ) (τ ) + 1/m, W m ≤ Mn,p m 502 E. KUWERT & R. SCHATZLE¨ hence for m large enough (f ) δ . Replacing f by Φ W m ≤Wn,p − m m ◦ fm φm as in Proposition 2.2 does neither change the Willmore energy◦ nor its projections in Teichmüller space, and we may assume that fm,f satisfy (2.2) - (2.6). By Lemmas 3.3 and 4.4 applied to fm,V = 0 and τ , there exist λ IRdim T +1 for m large enough with m ∈ ∗ π((fm + λm,rVr) geuc)= τ, λ 0 for m . m → →∞ This yields

n,p(τ) lim inf (fm + λm,rVr) M ≤ m→∞ W lim inf( (fm)+ C λm ) = lim inf (fm). ≤ m→∞ W | | m→∞ W which is the second statement. Finally, if were not continuous at τ with (τ) Mn,p ∈ T Mn,p ≤ n,p , by upper semicontinuity of n,p proved above, there exists a sequenceW τ τ in with M m → T lim inf n,p(τm) < n,p(τ) n,p. m→∞ M M ≤W Then by above n,p(τ) lim inf n,p(τm), M ≤ m→∞ M which is a contradiction, and the proposition is proved. q.e.d. Secondly, we prove that the inﬁmum taken in Deﬁnition 2.1 over smooth immersions is not improved by uniformly conformal W 2,2 immersions. − Proposition 5.2. Let f :Σ IRn be a uniformly conformal W 2,2 immersion with pull-back metric→ f ∗g = e2ug conformal − euc poin to a smooth unit volume constant curvature metric gpoin . Then there n exists a sequence of smooth immersions fm :Σ IR satisfying (2.2) - (2.6), →

∗ ∗ fmgeuc and f geuc are conformal, f f strongly in W 2,2(Σ), m → 1 2 n,p(π(gpoin)) lim (fm)= (f)= Af df + 2π(1 p). M ≤ m→∞ W W 4 | | − Σ

Proof. We approximate f by smooth immersions fm as in (2.9) ∗ ∗ and (2.10). Putting g := f geuc, gm := fmgeuc and writing g = 2u 2um e gpoin, gm = e gpoin,m for some unit volume constant curvature metrics gpoin, gpoin,m by Poincar´e’s theorem, see [Tr] Theorem 1.3.7, we get by Proposition A.2

(5.3) u ∞ , u 2 C m L (Σ) ∇ m L (Σ)≤ MINIMIZERS OF THE WILLMORE FUNCTIONAL 503 for some C < independent of m . In local charts, we see ∞ g g strongly in W 1,2, weakly∗ in L∞, m → Γk Γk strongly in L2, gm,ij → g,ij and A = gm gm f ∂ f Γk ∂ f = g gf = A fm,ij ∇i ∇j m → ij − g,ij k ∇i ∇j f,ij strongly in L2. Therefore by (1.1)

1 2 (5.4) (fm) (f)= Af dg + 2π(1 p) W →W 4 | |g − Σ and (5.5) π(g ) π(g ). m → poin Then there exist diﬀeomorphisms φ :Σ ≈ Σ, φ id , such that m −→ m ≃ Σ g˜ := φ∗ g g smoothly. poin,m m poin,m → poin Next by (5.3) −1 (5.6) Dφ ∞ , Dφ ∞ C m L (Σ) m L (Σ) ≤ and

−2um g 1,2 e 1,2 ∞ g 1,2 ∞ C poin,m W (Σ) ≤ W (Σ)∩L (Σ) m W (Σ)∩L (Σ) ≤ and for a subsequence g g,˜ u u˜ weakly in W 1,2(Σ) , in poin,m → m → particularg ˜ g = e−2um g e−2˜ug , and ← poin,m m → −2˜u 2(u−u˜) (5.7)g ˜ = e g = e gpoin is conformal to the smooth metric gpoin . The Theorem of Ebin and Palais, see [FiTr84], [Tr], imply after appropriately modifying φm (5.8) φ id weakly in W 2,2(Σ), weakly∗ in W 1,∞(Σ) m → Σ ∗ 2(u−u˜) and gpoin = idΣg˜ = e gpoin by (5.7), hence (5.9) u u˜ = u weakly in W 1,2(Σ), weakly∗ in L∞(Σ). m → Putting f˜ := f φ , u˜ := u φ , we see m m ◦ m m m ◦ m ˜∗ ∗ 2um 2˜um (5.10) fmgeuc = φm(e gpoin,m)= e g˜poin,m and by (5.6), (5.8) and (5.9)

f˜ f weakly in W 2,2(Σ), weakly∗ in W 1,∞(Σ), m → u˜ u weakly in W 1,2(Σ), weakly∗ in L∞(Σ). m → 504 E. KUWERT & R. SCHATZLE¨

Therefore f˜m,f satisfy (2.2) - (2.6). By Lemmas 3.3 and 4.4 applied to V = 0 , there exist λ IRdim T +1 for m large enough with m ∈ ∗ π((f˜m + λm,rVr) geuc)= π(gpoin), λ 0 for m , m → →∞ ˜∗ ¯ as π(fmgeuc) = π(gm) π(gpoin) by (5.5). Then putting fm := f + λ (V φ−1) , we→ see m m,r r ◦ m ¯∗ π(fmgeuc)= π(gpoin), f¯ f strongly in W 2,2(Σ), weakly∗ in W 1,∞(Σ), m → c f¯∗ g C 0 ≤ m euc ≤ −1 for some 0 < c0 C < and m large, when observing that φm ≤2,2 ∞ 1,∞ 2 −1 is bounded in W (Σ) W (Σ) by (5.6), (5.8) and D (φm ) C D2(φ ) φ−1 . ∩ | | ≤ | m ◦ m | This means f¯m approximate f as in (2.9), (2.10), and additionally induce the same Teichm¨uller class as f . Therefore we can replace fm by f¯m at the beginning of this proof or likewise, we can additionally ∗ assume that π(fmgeuc)= π(gpoin) which improves (5.5) to

(5.11) π(gm)= π(gpoin,m)= π(gpoin). ∗ Then φm can be chosen to getg ˜poin,m = φmgpoin,m = gpoin , and ˜∗ 2˜um ∗ by (5.10) that fmgeuc = e gpoin is conformal to gpoin and f geuc = 2u e gpoin . We know already that f˜m,f satisfy (2.2) - (2.6). Moreover by (5.4) (f˜ )= (f ) (f), W m W m →W hence

2˜um 2u 2 ∆g f˜m = e H ˜ e H f =∆g f strongly in L poin fm → poin and f˜ f strongly in W 2,2(Σ). m → Finally n,p(π(gpoin)) lim (fm)= (f), M ≤ m→∞ W W and the proposition follows. q.e.d. As a particular consequence, we get for Riemann surfaces the following theorem. Theorem 5.1. For a closed Riemann surface Σ of genus p 1 ≥ with smooth conformal metric g0 (Σ, g ,n) W 0 = inf (f) f :Σ IRn smooth immersion conformal to g {W | → 0 } MINIMIZERS OF THE WILLMORE FUNCTIONAL 505

= inf (f) f :Σ IRn is a {W | → W 2,2 immersion uniformly conformal to g . − 0 } Proposition 5.2 implies strong convergence of minimizing sequences. Proposition 5.3. Let f :Σ IRn be a uniformly conformal 2,2 → W immersion approximated by smooth immersions fm satisfying (2.2)− - (2.6) and (5.12) (f ) (π(f ∗ g )) + ε W m ≤ Mn,p m euc m with ε 0 . Then m → (5.13) f f strongly in W 2,2(Σ) m → and (5.14) (f)= (τ ). W Mn,p 0 Proof. By (2.3), (2.4), (2.5) and Λ0 large enough, we may assume after relabeling the sequence fm

Df 1,2 ∞ Λ , m W (Σ)∩L (Σ)≤ 0 1 2 gpoin gpoin,m 2gpoin, (5.15) ≤ ≤ Λ−1g g = f ∗ g Λ g , 0 poin ≤ m m euc ≤ 0 poin A 2 d Λ . | fm | fm ≤ 0 Σ In local charts, we see

g g weakly in W 1,2, weakly∗ in L∞, m → Γk Γk weakly in L2, gm,ij → g,ij and A = gm gm f ∂ f Γk ∂ f fm,ij ∇i ∇j m → ij − g,ij k = g gf = A weakly in L2. ∇i ∇j f,ij ∗ We conclude by Propositions 5.1, 5.2 and (5.12), as π(fmgeuc) τ0 by (2.2), → ∗ (f) lim inf (fm) lim inf n,p(f geuc) n,p(τ0) (f), W ≤ m→∞ W ≤ m→∞ M m ≤ M ≤W hence (5.14) and (5.16) (f ) (f), W m →W in particular H H strongly in L2 . This yields using (2.4) fm → f ∆ f = e2um H e2uH =∆ f strongly in L2 gpoin,m m fm → f gpoin and 506 E. KUWERT & R. SCHATZLE¨

ij ij ij ∂i gpoin√gpoin∂j(fm f) = ∂i (gpoin√gpoin gpoin,m√gpoin,m)∂jfm − − +√g ∆ f √g ∆ f 0 strongly in L2 poin,m gpoin,m m − poin gpoin → 1,2 recalling that ∂jfm is bounded in W and gpoin,m gpoin smoothly by (2.4), hence → f f strongly in W 2,2(Σ), m → and the proposition is proved. q.e.d. Remark. From the proof above, we see that strong convergence in (5.13) is obtained for any closed surface Σ , when (2.2) and (5.12) are replaced by (5.16).

6. Decay of the second derivative

In this section, we add to our assumptions on f,fm , as considered in 2 - 4, that fm is approximately minimizing in its Teichm¨uller class, §see (6.1).§ The aim is to prove in the following proposition a decay for the second derivatives which implies that the limits in C1,α . Proposition 6.1. Let f :Σ IRn be a uniformly conformal 2,2 → W immersion approximated by smooth immersions fm satisfying (2.2)− - (2.6) and (6.1) (f ) (π(f ∗ g )) + ε W m ≤ Mn,p m euc m with ε 0 . Then there exists α> 0,C < such that m → ∞ 2 2 2α (6.2) f dg C̺ for any x Σ,̺> 0, |∇gpoin |gpoin poin ≤ ∈ gpoin B̺ (x) in particular f C1,α(Σ) . ∈ Proof. By (2.3), (2.4) and Λ0 large enough, we may assume after relabeling the sequence fm

u ,Df 1,2 ∞ Λ , m m W (Σ)∩L (Σ)≤ 0 1 g g 2g , 2 poin ≤ poin,m ≤ poin (6.3) Λ−1g g = f ∗ g Λ g , 0 poin ≤ m m euc ≤ 0 poin 2 Af df Λ0. | m | m ≤ Σ 2 2 Putting νm := gpoin fm gpoin gpoin , we see νm(Σ) C(Λ0, gpoin) and |∇ | ∗ 0 ∗ ≤ for a subsequence νm ν weakly in C0 (Σ) with ν(Σ) < . We consider x Σ→ with a neighbourhood U (x ) satisfying∞ 0 ∈ 0 0 (6.4) ν(U (x ) x ) < ε , 0 0 − { 0} 1 MINIMIZERS OF THE WILLMORE FUNCTIONAL 507 where we choose ε1 = ε1(Λ0,n) > 0 below, together with a chart ≈ −1 ϕ : U0(x0) B2̺0 (0), ϕ(x0) = 0,U̺(x0) := ϕ (B̺(0)) for 0 < ̺ 2̺ 2, such−→ that ≤ 0 ≤ 1 (6.5) g (ϕ−1)∗g 2g 2 euc ≤ poin ≤ euc and in the coordinates of the chart ϕ

ik jl r s (6.6) g g gpoin,rsΓ Γ dg < ε1. poin poin gpoin,ij gpoin,kl poin B̺0 (0)

Moreover we select

(6.7) U (x ) U(x ) U (x ), ̺1 0 ⊆ 0 ⊆ ̺0 0 variations V ,...,V ,V C∞(Σ U(x ), IRn) and δ > 0,C = 1 dim T−1 ± ∈ 0 − 0 Cx0,ϕ < ,m0 IN , as in Lemmas 3.3 and 4.4 for x0,U̺0 (x0) and Λ:= ∞ ∈ ∗ C(Λ0) deﬁned below. As π(fm) π(gpoin) by (2.2), we get for m0 large enough →

(6.8) d (π(f ∗ g ),π(g )) < ε for m m . T m euc poin ≥ 0

Clearly for each x0 Σ , there exist U0(x0), ̺0 as above, since gpoin ∈ ν(B̺ (x0) x0 ) ν( )=0 for ̺ 0 . For x ,U−(x { )} as→ above∅ and 0 <→ ̺ ̺ , there exists m 0 0 0 ≤ 0 1 ≥ m0 such that νm(U̺0 (x0) U̺/2(x0)) < ε1 for m m1 , hence in the coordinates of the chart ϕ− ≥

2 2 2 D fm d | | L B̺0 (0)−B̺/2(0)

ik jl C g g ∂ijfm, ∂klfm dg ≤ poin poin poin B̺0 (0)−B̺/2(0)

2 2 g fm dg ≤ |∇ poin |gpoin poin (6.9) U̺0 (0)−U̺/2(x0)

ik jl r s + 2 g g Γ Γ ∂rfm, ∂sfm dg poin poin gpoin,ij gpoin,kl poin B̺0 (x)

ik jl r s 2ε1 + C(Λ0) g g gpoin,rsΓ Γ dg ≤ poin poin gpoin,ij gpoin,kl poin B̺0 (0) C(Λ )ε . ≤ 0 1 508 E. KUWERT & R. SCHATZLE¨

There exists σ ]3/̺/4, 7̺/8[ satisfying by Co-Area formula, see [Sim] 12, ∈ § 7̺/8 2 2 1 −1 2 2 1 D fm d 8̺ D fm d dr | | H ≤ | | H ∂Bσ(0) 3̺/4 ∂Br(0)

−1 2 2 2 −1 (6.10) = 8̺ D fm d C(Λ0)ε1̺ . | | L ≤ B7̺/8(0)−B3̺/4(0) First we conclude that

2 1 1/2 osc Dfm C D fm d C(Λ0)ε , ∂Bσ(0)| |≤ | | H ≤ 1 ∂Bσ (0) hence for any x ∂Bσ(0) and the aﬃne function l(y) := fm(x)+ Df (x)(y x) ∈ m − −1 1/2 σ f l ∞ + D(f l) ∞ ) C(Λ )ε . m − L (∂Bσ(0)) m − L (∂Bσ (0)) ≤ 0 1 Moreover by (6.3) and (6.5) c (Λ )(δ ) ( ∂ l, ∂ l ) = g (x) C(Λ )(δ ) 0 0 ij ij ≤ i j ij m ≤ 0 ij ij hence Dl IR2×2 is invertible and ∈ Dl , (Dl)−1 C(Λ ). ≤ 0 2 Next by standard trace extension lemma, there exists f˜m C (Bσ(0)) such that ∈

f˜m = fm, Df˜m = Dfm on ∂Bσ(0), σ−1 f˜ l + D(f˜ l) | m − | | m − | −1 C(σ f l ∞ + D(f l) ∞ ) (6.11) ≤ m − L (∂Bσ (0)) m − L (∂Bσ(0)) C(Λ )ε1/2, ≤ 0 1 2 2 2 2 2 1 D f˜m d Cσ D fm d C(Λ0)ε1. | | L ≤ | | H ≤ Bσ(0) ∂Bσ(0) We see Df˜ Dl C(Λ )ε1/2 (Dl)−1 −1 /2 m − ≤ 0 1 ≤ for ε1 = ε1(Λ0) 1 small enough, and Df˜m is of full rank everywhere with ≤ Df˜ , (Df˜ )−1 C(Λ ). m m ≤ 0 n Extending f˜m = fm on Σ Uσ(x0) , we see that f˜m :Σ IR is a C1,1 immersion with pullback− metric satisfying by (6.5) → (6.12)− c (Λ )g c (Λ )g g˜ := f˜∗ g C(Λ )g C(Λ )g . 0 0 poin ≤ 0 0 euc ≤ m m euc ≤ 0 euc ≤ 0 poin MINIMIZERS OF THE WILLMORE FUNCTIONAL 509 and by (6.6), (6.9) and (6.11)

2 2 2 2 (6.13) D f˜m d , g f˜m dg C(Λ0)ε1, | | L |∇ poin |gpoin poin ≤ B̺0 (0) B̺0 (0)

in particular, as A ˜ ∂ijf˜m in local coordinates, | fm,ij| ≤ | |

2 (6.14) A ˜ dg˜ C(Λ0)ε1 ε0(n) | fm | m ≤ ≤ U̺0 (x0)

for ε1 = ε1(Λ0,n) small enough. Putting V := f˜m fm , we see supp V U (x ) U(x ) for 0 < ̺ ̺ from (6.7) and− ⊂⊂ ̺ 0 ⊆ 0 ≤ 1

V 2,2 C V 2,2 C(Λ ). W (Σ)≤ W (Bσ (0))≤ 0

Together with (6.12) and (6.14), this veriﬁes (3.9) and (4.33)for Λ= C(Λ0) in Lemmas 3.3 and 4.4, respectively. After slightly smoothing V , there exists λ IRdim T +1 by Lemmas 3.3 and 4.4 and (6.8) with m ∈

˜ ∗ ∗ π((fm + λm,rVr) geuc)= π(fmgeuc), (6.15) 1/2 ˜∗ ∗ λm Cx0,ϕdT π(fmgeuc),π(fmgeuc) . | |≤

By the minimizing property (6.1), and the Gauß-Bonnet theorem in (1.1), we get

1 2 ∗ Af df εm n,p(π(f geuc)) + 2π(p 1) 4 | m | m − ≤ M m − Σ 1 2 A ˜ d ˜ , ≤ 4 | fm+λm,rVr | fm+λm,rVr Σ

hence, as f˜m = fm in Σ Uσ(x0) and supp Vr Uσ(x0)= , (6.16) − ∩ ∅ 2 2 Af df A ˜ d ˜ + Cx0,ϕ(Λ0, gpoin) λm + 4εm. | m | m ≤ | fm | fm | | Uσ(x0) Uσ(x0) 510 E. KUWERT & R. SCHATZLE¨

We continue, using A ˜ ∂ijf˜m in local coordinates, (6.10) and | fm,ij| ≤ | | (6.11),

2 A ˜ d ˜ | fm | fm Uσ(x0)

2 2 2 C(Λ0) D f˜m d ≤ | | L Bσ(0)

2 2 2 C(Λ0) D fm d (6.17) ≤ | | L B7̺/8(0)−B3̺/4(0)

2 2 C(Λ0) fm dg ≤ |∇gpoin | poin U7̺/8(x0)−U3̺/4(x0) 2 2 + C(Λ0) Γg ∞ ̺ poin L (B̺0 (0)) 2 C(Λ0)νm U7̺/8(x0) U3̺/4(x0) + Cx0,ϕ(Λ0, gpoin)̺ . ≤ −

We calculate in local coordinates in Bσ(0) that

2um 2um ∆gpoin,m fm = e ∆gm fm = e Hfm and

∆ (f f˜ ) C(Λ )( H + D2f˜ + Γ Df˜ ). | gpoin,m m − m |≤ 0 | fm | | m| | gpoin,m | | m| By standard elliptic theory, see [GT] Theorem 8.8, from (2.4) for m ≥ m1 large enough, as fm = f˜m on ∂Bσ(0) , we get

2 2 2 D fm d | | L Bσ(0) 2 Cx0,ϕ(Λ0, gpoin) Hf df ≤ | m | m Uσ(x0)

2 + νm U7̺/8(x0) U3̺/4(x0) + ̺ , − MINIMIZERS OF THE WILLMORE FUNCTIONAL 511 where we have used (6.17). Putting (6.16), (6.17) and (6.18) together yields

νm(U̺/2(x0))

2 2 = fm dg (6.19) |∇gpoin | poin U̺/2(x0)

2 Cx0,ϕ(Λ0, gpoin) νm U7̺/8(x0) U3̺/4(x0) + ̺ + λm + εm . ≤ − | | ˜∗ ∗ To estimate λm , we continue observing that fmgeuc =g ˜m and fmgeuc = g coincide on Σ U (x ) , m − ̺ 0 ˜∗ ∗ 2 dT (π(fmgeuc),π(fmgeuc)) 2d (π(f˜∗ g ),π(g ))2 + 2d (π(f ∗ g ),π(g ))2 ≤ T m euc poin T m euc poin 1 ij Cτ0 gpoin,ijg˜m g˜m √gpoin dx ≤ 2 − Σ 1 ij +Cτ0 gpoin,ijgm√gm √gpoin dx 2 − Σ

1 ij = 2Cτ0 gpoin,ijgm√gm √gpoin dx 2 − Σ−U̺(x0)

1 ij +C 0 g g˜ g˜ √g dx τ 2 poin,ij m m − poin U̺(x0)

1 ij +Cτ0 gpoin,ijgm√gm √gpoin dx 2 − U̺(x0)

1 ij 2 2Cτ0 gpoin,ijgm√gm √gpoin dx + Cτ0 (Λ0)̺ . ≤ 2 − Σ−U̺(x0) 2u where we have used (6.3) and (6.12). As gm g = e gpoin pointwise and bounded on Σ , we get from Lebesgue’s convergence→ theorem ˜∗ ∗ lim sup dT (π(fmgeuc),π(fmgeuc)) Cτ0 (Λ0)̺ m→∞ ≤ and from (6.15) 1/4 lim sup λm Cx0,̺0gpoin (Λ0)̺ . m→∞ | |≤ Plugging into (6.19) and passing to the limit m , we obtain →∞ 1/4 ν(U̺/2(x0)) Cx0,ϕ(Λ0, gpoin)ν U̺(x0) U̺/2(x0) +Cx0,ϕ(Λ0, gpoin)̺ ≤ − 512 E. KUWERT & R. SCHATZLE¨ and by hole-ﬂlling

ν(U (x )) γν(U (x )) + C (Λ , g )̺1/4 ̺/2 0 ≤ ̺ 0 x0,ϕ 0 poin with γ = C/(C + 1) < 1 . Iterating with [GT] Lemma 8.23, we arrive at

gpoin 2α −2α (6.20) ν(B̺ (x )) C (Λ , g )̺ ̺ for all ̺> 0 0 ≤ x0,ϕ 0 poin 1 gpoin and some 0 < α = αx0,ϕ(Λ0, gpoin) < 1 . Since ν(B̺ (x0)) ν( x ) for ̺ 0 , we ﬁrst conclude → { 0} → (6.21) ν( x ) = 0. { 0} Then we can improve the choice of U0(x0) in (6.4) to

ν(U0(x0)) < ε1, and we can repeat the above iteration for any x U (x ), 0 < ̺ ∈ ̺1/2 0 ≤ ̺1/2 to obtain

gpoin 2α −2α ν(B̺ (x)) C (Λ , g )̺ ̺ for all x U (x )̺> 0. ≤ x0,ϕ 0 poin 1 ∈ ̺1/2 0 By a ﬁnite covering, this yields (6.2). Since in the coordinates of the chart ϕ

2 2 2 D fm d | | L B̺(x)

ik jl C g g ∂ijfm, ∂klfm dg ≤ poin poin poin B̺(x)

2 2 g fm dg ≤ |∇ poin |gpoin poin B̺(x)

ik jl r s +2 g g Γ Γ ∂rfm, ∂sfm dg poin poin gpoin,ij gpoin,kl poin B̺(x) C (Λ , g )̺2α̺−2α + C(Λ , g )̺2, ≤ x0,ϕ 0 poin 1 0 poin we conclude by Morrey’s lemma, see [GT] Theorem 7.19, that f C1,α(Σ) , and the proposition is proved. q.e.d.∈

7. The Euler-Lagrange equation The aim of this section is to prove the Euler-Lagrange equation for the limit of immersions approximately minimizing under ﬁxed Teichm¨uller class. From this we will conclude full regularity of the limit. MINIMIZERS OF THE WILLMORE FUNCTIONAL 513

Theorem 7.1. Let f :Σ IRn be a uniformly conformal W 2,2 im- → − mersion approximated by smooth immersions fm satisfying (2.2) - (2.6) and (7.1) (f ) (π(f ∗ g )) + ε W m ≤ Mn,p m euc m with εm 0 . Then f is a smooth minimizer of the Willmore energy under fixed→ Teichmüller class (7.2) (f)= (τ ) W Mn,p 0 and satisfies the Euler-Lagrange equation 0 ik jl 0 (7.3) ∆gH + Q(A )H = g g Aijqkl on Σ, where q is a smooth transverse traceless symmetric 2-covariant tensor ∗ with respect to g = f geuc .

Proof. By Propositions 3.1 and 4.2, we select variations V1,...,Vdim T ∞ n C0 (Σ, IR ) , satisfying (3.5) or variations V1,...,Vdim T−1,V± ∈ ∞ n ∈ C0 (Σ, IR ) , satisfying (4.15) and (4.16), depending on whether f has full rank in Teichm¨uller space or not. For V C∞(Σ, IRn) and putting f := f +tV +λ V + V , ∈ m,t,λ, m r r ± ± we see for t t0 for some t0 = t0(V, Λ0,n) > 0 small enough and λ , λ |for| ≤ some λ = λ (V ,V , Λ ) > 0 small enough that | | | |≤ 0 0 0 r ± 0

Df 1,2 ∞ 2Λ , m,t,λ, W (Σ)∩L (Σ)≤ 0 (2Λ )−1g f ∗ g 2Λ g 0 poin ≤ m,t,λ, euc ≤ 0 poin and all m IN . As f f strongly in W 2,2(Σ) and weakly∗ in ∈ m,t,λ, → W 1,∞(Σ) for m , t, λ, 0 by Proposition 5.3, we get from → ∞ → Proposition 4.3 and the remark following for any chart ψ : U(π(gpoin)) dim T ˆ 2 ⊆ IR , andputπ ˆ := ψ π, δπˆ = dψ δπ, f := dψπgpoin . f , δ πˆ definedT → in (4.8), and any W ◦C∞(Σ, IRn) ◦ V V ∈ ∗ π(f geuc) τ0, (7.4) m,t,λ, → δπˆ .W δπˆ .W, δ2πˆ (W ) δ2πˆ (W ) fm,λ, → f fm,t,λ, → f as m , t, λ, 0. If f is of full rank in Teichmüller space, then →∞dim T → ˆf = IR , and we put d = dim . If f is not of full rank in VTeichmüller space, we may assume afterT a change of coordinates, ˆ = IRdim T−1 0 Vf × { } and put d := dim 1 and e := edim T ˆf . By (3.5) or (4.15), we T − ⊥dimVT see for the orthogonal projection π ˆ : IR ˆf that Vf → V d×d (7.5) (π ˆ δπˆf .Vr)r=1,...,d =: A IR Vf ∈ 514 E. KUWERT & R. SCHATZLE¨ is invertible, hence after a further change of variable, we may assume that A = Id . In the degenerate case, we further know (7.6) δπˆ .V , e = 0. f r By (4.16) (7.7) δ2πˆ (V ), e 2γ, δπˆ .V = 0, ± f ± ≥ f ± for some γ > 0 . Next we put for m large enough and t0, λ0 small enough ∗ Φm(t, λ, ) :=π ˆ(fm,t,λ,geuc).

Clearly, Φm is smooth. We get from (7.4), (7.5) (7.6) and (7.7) for some Λ < and any 0 < ε 1 that 1 ∞ ≤ D2Φ (t, λ) Λ , m ≤ 1 2 osc D Φm ε, (7.8) ≤ ∂ Φ (t, λ) I ε 1/2, λ m − d ≤ ≤ ∂ Φ (t, λ) (δπˆ .V ) , λ m → f r r=1,...,d in the full rank case, and writing Φm(t, λ, )=(Φm,0(t, λ, ), ϕm(t, λ, )) in the degenerate case that

D2Φ (t, λ, ) Λ , m ≤ 1 osc D2Φ ε, m ≤ ∂ Φ (t, λ, ) I ε 1/2, λ m,0 − d ≤ ≤ ∂±± ϕm(t, λ, ) γ, (7.9) ± ≥ ∂ Φ (t, λ, ) , Dϕ (t, λ, ) ε, | m | | m |≤ ∂ Φ (t, λ, ) (δπˆ .V ) , λ m → f r r=1,...,d ∂ ϕ (t, λ, ) δ2πˆ (V ), e , ±± m → f ± ∂ Φ (t, λ, ), Dϕ (t, λ, ) 0, m m → all for m m0 large enough and t t0, λ , λ0 small enough or respectively≥ t, λ, 0 . We choose| | ≤ε, λ | smaller| | | ≤ to satisfy CΛ ε+ → 0 1 CΛ1λ0 γ/2 . Moreover choosing m0 large enough and t0 small enough,≤ we can further achieve Dϕ (t, 0, 0) σ | m |≤ with CΛ ε + Cσλ−1 + CΛ λ γ , and 1 0 1 0 ≤ Φ (t, 0, 0) Φ (0, 0, 0) Ct Λ λ2, λ /8, γλ2/32. | m − m |≤ 0 ≤ 1 0 0 0 MINIMIZERS OF THE WILLMORE FUNCTIONAL 515

By Proposition B.1 there exist λ (t) , (t) , λ˜ (t) , ˜ (t) | m,r | | m,± | | m,r | | m,± |≤ λ0 with m,+(t)m,−(t) = 0, ˜m,+(t)˜m,−(t)=0 and

(7.10) Φm(t, λm(t),m(t)) =Φm(0, 0, 0) =Φm(t, λ˜m(t), ˜m(t)), which means ∗ ∗ π (fm + tV + λm,r(t)Vr + m,±(t)V±) geuc = π(fmgeuc) (7.11) ∗ = π (fm + tV + λ˜m,r(t)Vr + ˜m,±(t)V±) geuc , and

m,+(t)˜m,+(t) 0, ˜m,−(t) = 0, if m,+(t) = 0, (7.12) ≤ (t)˜ (t) 0, ˜ (t) = 0, if (t) = 0. m,− m,− ≤ m,+ m,− By (7.8), (7.9) and (7.10), we get from a Taylor expansion of the smooth function Φm at 0 that

t∂tΦm(0) + λm(t)∂λΦm(0) + m(t)∂Φm(0)+ 1 T 2 + (t, λm(t),m(t)) D Φm(0)(t, λm(t),m(t)) 2 2 2 2 Cε( t + λm(t) + m(t) ), ≤ | | | | | | hence, as m,+(t)m,−(t)=0,

1 2 t∂tΦm(0)+λm(t)∂λΦm(0)+m,±(t)∂± Φm(0)+ m,±(t) ∂±± Φm(0) 2 CΛ ε−1( t 2 + λ (t) 2)+ Cε (t) 2). ≤ 1 | | | m | | m,± | Passing to the limit m , we get for subsequences λm(t) λ(t), (t) (t) with →λ(t ∞) , (t) λ and by (7.8) and (7.9) → m,± → | | | |≤ 0 1 2 2 tδπˆf .V + λr(t)δπˆf .Vr + ±(t) δ πˆf (V±) (7.13) 2 −1 2 2 2 CΛ1ε ( t + λ(t) )+ Cε ±(t) . ≤ | | | | | | In the degenerate case, we recall δπˆ .V ˆ e , hence by (7.7) and f (r) ∈ Vf ⊥ (7.13) 1 γ (t)2 (t)2 δ2πˆ (V ), e ± ≤ 2 ± f ± 1 2 2 tδπˆf .V + λr(t)δπˆf .Vr + m,±(t) δ πˆf (V±) ≤ 2 −1 2 2 2 CΛ1ε ( t + λ(t) )+ Cε ±(t) ≤ | | | | | | and for Cε γ/2 small enough ≤ (7.14) (t) C( t + λ(t) ). | ± |≤ | | | | Then again by (7.13) (7.15) tδπˆ .V + λ (t)δπˆ .V C( t 2 + λ(t) 2). | f r f r|≤ | | | | 516 E. KUWERT & R. SCHATZLE¨

In the full rank case, we have ± = 0 , and (7.15) directly follows from (7.13). As (δπˆf .Vr)r=1,...,d are linearly independent by (7.5), we continue λ(t) C( t + λ(t) 2) C t + Cλ λ(t) , | |≤ | | | | ≤ | | 0| | hence for λ small enough λ(t) C t . We get from (7.14) 0 | |≤ | | (7.16) (t) C t | ± |≤ | | and from (7.15) tδπˆ .V + λ (t)δπˆ .V C t 2. | f r f r|≤ | | Therefore λ is diﬀerentiable at t = 0 with (7.17) λ′ (0)δπˆ .V = δπˆ .V. r f r − f −1 Writing for the inverse A = (brs)r,s=1,...,d in (7.5), we continue with (3.4) λ′ (0) = b δπˆ .V, e r − rs f s dim T ik jl 0 σ σ = 2 g g A ,V q (gpoin) dg dπˆg .q (gpoin), brses , ij kl poin σ=1 Σ hence putting dim T qr := 2qσ (g ) dπˆ .qσ(g ), b e ST T (g ), kl kl poin gpoin poin rs s∈ 2 poin σ=1 we get

′ ik jl 0 r (7.18) λ (0) = g g A ,V q dg. r ij kl Σ Moreover f + tV + λ (t)V + (t)V f + tV + λ (t)V + (t)V m m,r r m,± ± → r r ± ± strongly in W 2,2(Σ) and weakly∗ in W 1,∞(Σ) , hence recalling (7.11) (f + tV + λ (t)V + (t)V ) (f + tV + λ (t)V + (t)V ) W r r ± ± ←W m m,r r m,± ± (π(f ∗ g )) (f ) ε (f). ≥ Mn,p m euc ≥W m − m →W Since (t, λ, ) (f + tV + λrVr + ±V±) is smooth, we get again by a Taylor expansion,→ W (7.16) and (7.17) 0 (f + tV + λ (t)V + (t)V ) (f) ≤W r r ± ± −W = tδ .V + λ (t)δ .V + (t)δ .V + O( t 2). Wf r Wf r ± Wf ± | | As +(t)−(t) = 0 and by (7.12), we can adjust the sign of ±(t) according to the sign of δ .V and improve to Wf ± 0 tδ .V + λ (t)δ .V + O( t 2). ≤ Wf r Wf r | | MINIMIZERS OF THE WILLMORE FUNCTIONAL 517

Diﬀerentiating by t at t = 0 , we conclude from (7.17) and (7.18)

′ ik jl 0 r δ f .V = λ (0)δ f .Vr = g g A ,V q δ f .Vr dg, W − r W − ij kl W Σ hence putting q := qr δ .V ST T (g ) , we get kl − kl Wf r ∈ 2 poin ik jl 0 ∞ n (7.19) δ f .V = g g A ,V qkl dg for all V C (Σ, IR ). W ij ∈ Σ As f W 2,2 C1,α by Proposition 6.1, we can write f as a graph, more ∈ ∩ precisely for any x0 Σ there exists a neighbourhood U(x0) of x0 such that after a translation,∈ rotation and a homothetie, which leaves as conformal transformation invariant, there is a (W 2,2 C1,α) W ≈ ∩ − inverse chart φ : B1(0) U(x0), φ(0) = x0 , with f˜(y) := (f φ)(y)= −→2,2 1,α n−2 ◦ (y, u(y)) for some u (W C )(B1(0), IR ) . Moreover, we may assume u , Du 1∈ and from∩ (6.2) that | | | |≤ 2 2 2 2α (7.20) D u d C̺ for any Ball B̺. | | L ≤ B̺ We calculate the square integral of the second fundamental form for a graph as

2 ij kl r s 2 (u) := A ˜ d ˜ = (δrs drs)g g ∂iku ∂jlu √g d , A | f | f − L B1(0) B1(0) ij −1 kl r s where gij := δij + ∂iu∂ju, (g ) = (gij) , drs := g ∂ku ∂lu , see [Sim93] p. 310. By the Gauß-Bonnet theorem in (1.1), we see for any v C∞(B (0), ∈ 0 1 IRn−2) that 4 (graph(u + v)) = (u + v) + 8π(1 p) , hence from (7.19) W A −

ik jl 1 ij kl m δ u.v = g g g g q˜kl ∂iju Γij ∂mu, v dg, A − 2 − B1(0) whereq ˜ = φ∗q, Γm = gmk ∂ u, ∂ u . From this we conclude that ij ij k (7.21) ijkl s imkl s t ijkl s r ∂jl(2ars ∂iku ) ∂j (∂∂j u ats )∂iku ∂mlu = brs (∂u)˜qkl∂iju − weakly for testfunctions v C∞(B (0), IRn−2) , where ∈ 0 1 ijkl ij kl ars (Du) := (δ d )g g √g, rs − rs ijkl ik jl 1 ij kl brs (Du) = (g g g g )(δrs drs)√g. − 2 − 3,2 Then we conclude from [Sim] Lemma 3.2 and (7.20) that u (Wloc 2,α ∈ ∩ Cloc )(B1(0)) . 518 E. KUWERT & R. SCHATZLE¨

Full Regularity is now obtained by [ADN59], [ADN64]. First we conclude by finite differences that u W 4,2(B (0)) and ∈ loc 1 2aijkl(Du)∂ us + ˜b (Du, D2u) (1 + D3u)+ b (Du) q˜(.) D2u = 0 rs ijkl r ∗ r ∗ ∗ ijkl 2 strongly in B1(0), with ars , ˜br, br are smooth in Du and D u , ∗ 1,2 0,α 4,2 3,p whereasq ˜ = φ q (W C )(B1(0)) . As W ֒ W for all 1 ∈ ∩4,p 3,α → ≤ p < , we see u Wloc (B1(0)) ֒ Cloc (B1(0)) and then u 4,α ∞ ∈ → ∈ Cloc (B1(0)) . Now we proceed by induction assuming u, f˜ Ck,α(B (0)) for some ∈ loc 1 k 4 . We seeg ˜ := f˜∗g Ck−1,α , hence we get locally confor- ≥ euc ∈ loc mal Ck,α charts ϕ : U B (0) ≈ Ω IR2 with ϕ−1,∗g˜ = e2vg . − ⊆ 1 −→ ⊆ euc On the other hand, as gpoin is smooth, there exists a smooth confor- ≈ 2 −1,∗ 2u0 mal chart ψ : U(x0) Ω0 IR with ψ gpoin = e geuc , when −→ ⊆ ∗ 2u choosing U(x0) small enough. As g = f geuc = e gpoin , we see that −1 ψ φ ϕ :Ω Ω0 is a regular conformal mapping with respect to standard◦ ◦ euclidian→ metric, in particular holomorphic or anti-holomorphic, k,α ∗ hence smooth. We conclude that φ Cloc (B1(0)) andq ˜ = φ q k−1,α T T ∈ ∈ Cloc (B1(0)) , as q S2 (gpoin) is smooth. Then we conclude k+1,α ∈ ˜ ˜ −1 u Cloc (B1(0)) and by induction u, f, φ and f = f φ are smooth.∈ ◦ In [KuSch02] 2, the first variation of the Willmore functional with a different factor was§ calculated for variations V to be

d 1 0 δ (f).V := (f + tV )= ∆gH + Q(A )H ,V dg, W dtW 2 Σ and we obtain from (7.19) 0 ik jl 0 ∆gH + Q(A )H = 2g g Aijqkl on Σ, which is (7.3) up to a factor for q . (7.2) was already obtained in Proposition 5.3 (5.14). This concludes the proof of the theorem. q.e.d. As a corollary we get minimizers under fixed Teichmüller or conformal class, when the infimum is smaller than the bound in (1.2). Wn,p Theorem 7.2. Let Σ be a closed, orientable surface of genus p 1 and τ satisfying ≥ 0 ∈T (τ ) < Mn,p 0 Wn,p where n,p is defined in (1.2) and n = 3, 4 . ThenW there exists a smooth immersion f :Σ IRn which minimizes → ∗ the Willmore energy in the fixed Teichmüller class τ0 = π(f geuc) (f)= (τ ). W Mn,p 0 MINIMIZERS OF THE WILLMORE FUNCTIONAL 519

Moreover f satisﬁes the Euler-Lagrange equation 0 ik jl 0 ∆gH + Q(A )H = g g Aijqkl on Σ, where q is a smooth transverse traceless symmetric 2-covariant tensor ∗ with respect to g = f geuc .

Proof. We select a minimizing sequence of smooth immersions fm : Σ IRn with π(f ∗ g )= τ → m euc 0 (7.22) (f ) (τ ). W m → Mn,p 0 We may assume that (fm) n,p δ for some δ > 0 . Replac- ing f by Φ f φW for suitable≤ W − Möbius transformations Φ m m ◦ m ◦ m m and diffeomorphisms φm of Σ homotopic to the identity, which does neither change the Willmore energy nor the projection into the Te- ichmüller space, we may further assume by Proposition 2.2 that fm f weakly in W 2,2(Σ) and satisfies (2.2) - (2.6). Since (7.1) is implied by→ (7.22), Theorem 7.1) yields that f is a smooth immersion which mini- ∗ mizes the Willmore energy in the fixed Teichmüller class τ0 = π(f geuc) and satisfies the above Euler-Lagrange equation. q.e.d. Theorem 7.3. Let Σ be a closed Riemann surface of genus p 1 ≥ with smooth conformal metric g0 with (Σ, g ,n) := W 0 inf (f) f :Σ IRn smooth immersion conformal to g < , {W | → 0 } Wn,p where n,p is defined in (1.2) and n = 3, 4 . ThenW there exists a smooth conformal immersion f :Σ IRn which minimizes the Willmore energy in the set of all smooth conformal→ immersions. Moreover f satisfies the Euler-Lagrange equation 0 ik jl 0 (7.23) ∆gH + Q(A )H = g g Aijqkl on Σ, where q is a smooth transverse traceless symmetric 2-covariant tensor with respect to the Riemann surface Σ , that is with respect to g = ∗ f geuc .

Proof. We put τ0 := π(g0) and see by invariance (τ )= Mn,p 0 inf (f) f :Σ IRn smooth immersion conformal to g < . {W | → 0 } Wn,p Therefore by Theorem 7.2, there exists a smooth immersion f :Σ IRn which minimizes the Willmore energy in the fixed Teichmüller class→ ∗ τ0 = π(f geuc) and satisfies the above Euler-Lagrange equation. More- over there exists a diffeomorphism φ of Σ homotopic to the identity ∗ such that (f φ) geuc is conformal to g0 . Then f˜ := f φ is a smooth conformal immersion◦ of the Riemann surface Σ which◦ minimizes the Willmore energy in the set of all smooth conformal immersions and moreover satisfies the Euler-Lagrange equation. q.e.d. 520 E. KUWERT & R. SCHATZLE¨

In any case, we get that minimizers under fixed Teichmüller class or fixed conformal class are smooth and satisfy the Euler-Lagrange equation.

Theorem 7.4. Let Σ be a closed Riemann surface of genus p 1 n ≥ with smooth conformal metric g0 and f :Σ IR be a uniformly 2,2 ∗→ 2u conformal W immersion, that is g = f geuc = e g0 with u L∞(Σ) , which minimizes− the Willmore energy in the set of all smooth∈ conformal immersions

(f)= (Σ, g ,n), W W 0 then f is smooth and satisﬁes the Euler-Lagrange equation

0 ik jl 0 ∆gH + Q(A )H = g g Aijqkl on Σ, where q is a smooth transverse traceless symmetric 2-covariant tensor with respect to g .

Proof. By Proposition 5.2 there exists a sequence of smooth immersions f :Σ IRn satisfying (2.2) - (2.6) and m →

∗ ∗ fmgeuc and f geuc are conformal, f f strongly in W 2,2(Σ), m → ∗ ∗ lim (fm)= (f)= n,p(π(f geuc)) = n,p(π(f geuc)). m→∞ W W M M m This implies (7.1), and the conclusion follows directly from Theorem 7.1. q.e.d.

Appendix A. Conformal factor Lemma A.1. Let Σ be a closed, orientable surface of genus p 1 , ≥ g0 a given smooth metric on Σ , x1,...,xM Σ with charts ϕk : ≈ ∈ U(x ) B (0), ϕ (x ) = 0,U (x ) := ϕ−1(B (0)) for 0 < ̺ 1, k −→ 1 k k ̺ k k ̺ ≤ (A.1) Λ−1g (ϕ−1)∗g Λg for k = 1,...,M. euc ≤ k 0 ≤ euc Let f :Σ IRn be a smooth immersion with g := f ∗g = e2ug → euc poin for some unit volume constant curvature metric gpoin and

Λ−1g g Λg , 0 ≤ ≤ 0 (A.2) u ∞ , Kg dg Λ, L (Σ) | | ≤ Σ MINIMIZERS OF THE WILLMORE FUNCTIONAL 521 and f˜ :Σ IRn a smooth immersion with g˜ := f˜∗g = e2˜ug˜ → euc poin for some unit volume constant curvature metric g˜poin ,

Λ−1g g˜ Λg , 0 ≤ ≤ 0 (A.3) Kg˜ dg˜ Λ, | | ≤ Σ supp (f f˜) M U (x ) and − ⊆∪k=1 1/2 k 2 (A.4) A˜ dg˜ ε0(n) for k = 1,...,M | | ≤ U1(xk) for some universal 0 < ε0(n) < 1 . Then

(A.5) u˜ ∞ , u˜ 2 C(Σ, g , Λ,p). L (Σ) ∇ L (Σ,g˜)≤ 0 Proof. We know (A.6) ∆ u + K e−2u = K , ∆ u˜ + K e−2˜u = K on Σ. − g p g − g˜ p g˜ Observing K e−2˜u d = K = 4π(p 1) and multiplying Σ |− p | g˜ − p − (A.6) byu ˜ λ˜ for any λ˜ IR , we get recalling (A.3) − ∈ 2 2 c0(Λ) u˜ dg0 u˜ dg˜ |∇ |g0 ≤ |∇ |g˜ Σ Σ

−2˜u = ( Kpe + Kg˜)(˜u λ˜) dg˜ C(Λ,p) u˜ λ˜ ∞ , − − ≤ − L (Σ) Σ ˜ hence for λ = u˜ dg0 by Poincar´einequality −Σ (A.7)

˜ 2 2 u˜ λ L (Σ,g0) C(Σ, g0) u˜ L (Σ,g0) C(Σ, g0, Λ,p) oscΣu.˜ − ≤ ∇ ≤ Next by the uniformization theorem, see [FaKr] Theorem IV.4.1, we can parametrize f˜ ϕ−1 : B (0) IRn conformally with respect to ◦ k 1 → the euclidean metric on B1(0) , possibly after replacing B1(0) by a slightly smaller ball. Then by [MuSv95] Theorem 4.2.1 for ε0(n) small enough, there exist v C∞(U (x )) with k ∈ 1 k ∆ v = K on U (x ), − g˜ k g˜ 1 k (A.8) 2 v ∞ C A˜ d C ε (n) 1. k L (U1(xk))≤ n | | g˜ ≤ n 0 ≤ U1(xk) We get (A.9) −2˜u ∆g˜(˜u vk)= Kpe 0, − − − ≥  on U1(xk) ∆ (˜u v )+ K (e−2˜u e−2vk )= K e−2vk 0,  − g˜ − k p − − p ≥  522 E. KUWERT & R. SCHATZLE¨ for k = 1,...,M , and, as g =g ˜ on Σ M U (x ) , −∪k=1 1/2 k (A.10) −2˜u −2u ∆g(˜u u)+ Kp(e e ) = 0, M − − −  on Σ k=1U1/2(xk). ∆ (˜u u) 0,  −∪ − g − + ≤ Thereforeu ˜ u cannot have positive interior maxima nor negative − interior minima in Σ M U (x ) as K 0 by standard maximum −∪k=1 1/2 k p ≤ principle, hence putting Γ := M ∂U (x ) we get ∪k=1 3/4 k sup (˜u u)± = max(˜u u)±. M − Γ − Σ−∪k=1U3/4(xk) From above, we see from (A.1), (A.3), (A.8), (A.9),

0 ∂ g˜ij g˜ ∂ (˜u v ) + K g˜(e−2˜u e−2vk ) ≤− i j − k p − (A.11) = K e−2vk g˜ C(Λ,p), − p ≤ hence using [GT] Theorem 8.16

sup (˜u vk)± max(˜u vk)± + C(Λ,p). Γ U3/4(xk) − ≤ − Together, we get from (A.2) and (A.8)

(A.12) max u˜± max u˜± + C(Λ,p). Σ ≤ Γ From (A.1), (A.2), (A.3), and gpoin having unit volume, we get (A.13) c (Λ) (Σ), (Σ), (Σ) C(Λ). 0 ≤ g g0 g˜ ≤ Now if minΣ u˜ C(Λ,p) , there exists x Γ withu ˜(x) minΣ u˜ + C(Λ,p) 0 .≤− Asu ˜ v min u˜ 1∈ =: λ , we get≤ identifying ≤ − k ≥ Σ − ϕk : U1(xk) ∼= B1(0) by the weak Harnack inequality, see [GT] Theorem 8.18, from (A.11)

2 u˜ vk λ L (B1/8(x)) C(Λ) inf (˜u vk λ) − − ≤ B1/8(x) − − C(Λ)(˜u(x) min u˜ + 2) C(Λ,p), ≤ − Σ ≤ and u˜ min u˜ L2(B (x)) C(Λ,p). − Σ 1/8 ≤ We see from (A.1) and (A.7) ˜ ˜ c0 λ min u˜ λ min u˜ L2(B (x)) | − Σ | ≤ − Σ 1/8

˜ 2 2 C(Λ) u˜ λ L (Σ,g0) + u˜ min u˜ L (B (x)) ≤ − − Σ 1/8 C(Σ, g , Λ,p)(1 + osc u˜). ≤ 0 Σ Using (A.13), we have proved min u˜ C(Λ,p)= Σ ≤− ⇒ MINIMIZERS OF THE WILLMORE FUNCTIONAL 523

2 (A.14) u˜ min u˜ L (Σ,g0) C(Σ, g0, Λ,p)(1 + oscΣu˜). − Σ ≤ If further min u˜ C(Σ, g , Λ,p)(1 + √osc u˜) , we put A := Σ ≪ − 0 Σ [minΣ u˜ u˜ minΣ u/˜ 2] and seeu ˜ minΣ u˜ minΣ u/˜ 2 on Σ A and using≤ (A.3)≤ − ≥ | | − 1 min u˜ g˜(Σ A) u˜ min u˜ dg˜ 2| Σ | − ≤ | − Σ | Σ−A

Λ u˜ min u˜ dg0 C(Σ, g0, Λ,p)(1 + oscΣu˜), ≤ | − Σ | ≤ Σ hence C(Σ, g , Λ,p)(1 + √osc u˜) (Σ A) 0 Σ c (Λ)/2, g˜ − ≤ min u˜ ≤ 0 | Σ | if min u˜ C(Σ, g , Λ,p)(1 + √osc u˜) is large enough. This yields | Σ |≫ 0 Σ using (A.13) and g˜poin (Σ) = 1

2˜u c0(Λ)/2 g˜(A)= e dg˜ g˜ (Σ) exp(min u˜) = exp(min u˜), ≤ poin ≤ poin Σ Σ A and we conclude

(A.15) min u˜ C(Σ, g0, Λ,p)(1 + oscΣu˜). Σ ≥− In the same way as above, if maxΣ u˜ C(Λ,p) , we get from (A.12) that there exists x Γ withu ˜(x) max≥−u˜ C(Λ,p) 0 . Asu ˜ v ∈ ≥ Σ − ≥ − k ≤ maxΣ u˜ +1 =: λ , we get by the weak Harnack inequality, see [GT] Theorem 8.18, from (A.11)

2 u˜ vk λ L (B1/8(x)) C(Λ) inf (λ u˜ + vk) − − ≤ B1/8(x) − C(Λ)(max u˜ u˜(x) + 2) C(Λ,p), ≤ Σ − ≤ and

max u˜ u˜ L2(B (x)) C(Λ,p). Σ − 1/8 ≤ We see from (A.1) and (A.7) ˜ ˜ c0 max u˜ λ max u˜ λ L2(B (x)) | Σ − | ≤ Σ − 1/8

˜ 2 2 C(Λ) u˜ λ L (Σ,g0) + max u˜ u˜ L (B (x)) ≤ − Σ − 1/8

C(Σ, g0, Λ,p)(1 + oscΣu˜). ≤ Using (A.13), we have proved max u˜ C(Λ,p)= Σ ≥ ⇒ 524 E. KUWERT & R. SCHATZLE¨

2 (A.16) max u˜ u˜ L (Σ,g0) C(Σ, g0, Λ,p)(1 + oscΣu˜). Σ − ≤ Now if maxΣ u˜ C(Σ, g0, Λ,p)(1 + √oscΣu˜) , we need a lower bound onu ˜ . If min ≫u˜ C(Λ,p) , we see from (A.14) and (A.16) that Σ ≤− min u˜ max u˜ C(Σ, g0, Λ,p)(1 + oscΣu˜) 0, Σ ≥ Σ − ≥ therefore minΣ u˜ C(Λ,p). Weput A := [maxΣ u/˜ 2 u˜ maxΣ u˜] and see max u˜ ≥−u˜ max u/˜ 2 on Σ A , hence using≤ (A.7)≤ Σ − ≥ Σ − 1 −2˜u max u˜ g˜ (Σ A) max u˜ u˜ e dg˜ 2 Σ poin − ≤ | Σ − | Σ−A

C(Λ,p) max u˜ u˜ dg0 C(Σ, g0, Λ,p)(1 + oscΣu˜) ≤ | Σ − | ≤ Σ and C(Σ, g , Λ,p)(1 + √osc u˜) 1 (Σ A) 0 Σ , g˜poin − ≤ max u˜ ≤ 2 Σ if maxΣ u˜ C(Σ, g0, Λ,p)(1 + √oscΣu˜) is large enough. This yields (A) ≫1/2 and by (A.13) g˜poin ≥ 2˜u C(Λ) g˜(Σ) e dg˜ g˜ (A) exp(max u˜) exp(max u˜)/2, ≥ ≥ poin ≥ poin Σ ≥ Σ A and we conclude

(A.17) max u˜ C(Σ, g0, Λ,p)(1 + oscΣu˜). Σ ≤ (A.15) and (A.17) yield

oscΣu˜ = max u˜ min u˜ Σ − Σ 1 C(Σ, g , Λ,p)(1 + osc u˜) C(Σ, g , Λ,p)+ osc u,˜ 0 Σ 0 2 Σ ≤ ≤ hence oscΣu˜ C(Σ, g0, Λ,p) . Then (A.5) follows from (A.7), (A.15) and (A.17), and≤ the lemma is proved. q.e.d. Here, we use this lemma to get a bound on the conformal factor for sequences strongly converging in W 2,2 . Proposition A.2. Let f :Σ IRn be a uniformly conformal 2,2 → W immersion approximated by smooth immersions fm with pull- − ∗ 2u ∗ 2um back metrics g = f geuc = e gpoin, gm = fmgeuc = e gpoinm for some smooth unit volume constant curvature metrics gpoin, gpoin,m and satisfying f f strongly in W 2,2(Σ), weakly∗ in W 1,∞(Σ), m → Λ−1g g Λg . poin ≤ m ≤ poin MINIMIZERS OF THE WILLMORE FUNCTIONAL 525

Then

∞ 2 sup um L (Σ), um L (Σ,gm) < . m∈IN ∇ ∞

Proof. We want to apply Lemma A.1 to f = f1, f˜ = fm, g0 = gpoin . (A.2) and (A.3) are immediate by the above assumptions for appropriate possibly larger Λ < . In local charts, we∞ have

g g strongly in W 1,2, weakly∗ in L∞(Σ), m → Γk Γk strongly in L2, gm,ij → g,ij and A = gm gm f ∂ f Γk ∂ f fm,ij ∇i ∇j m → ij − g,ij k = g gf = A strongly in L2. ∇i ∇j f,ij 2 2 1 Therefore A √gm A √g strongly in L , hence for each x | fm |gm → | f |g ∈ Σ there exists a neighbourhood U(x) of x with

2 Af dg ε0(n) for all m IN, | m |gm m ≤ ∈ U(x) where ε0(n) is as in Lemma A.1. Choosing U(x) even smaller, we ≈ may assume that there are charts ϕx : U(x) B1(0) with ϕx(x) = 0 and c g (ϕ−1)∗g C g . Selecting−→ a ﬁnite cover Σ = 0,x euc ≤ poin ≤ x euc M ϕ−1(B (0)) , we obtain (A.1) and (A.4) for appropriate Λ < . ∪k=1 xk 1/2 ∞ As clearly supp (f1 fm) Σ , the assertion follows from Lemma A.1. − ⊆ q.e.d.

Appendix B. Analysis Proposition B.1. Let Φ = (Φ , ϕ) : BM+2(0) IRM+1 = IRM 0 λ0 → × IR,M IN, be twice diﬀerentiable satisfying for ξ = (λ, , ν) ∈ ∈ BM+2(0) IRM IR IR λ0 ⊆ × × ∂ Φ I 1/2, λ 0 − M ≤ ∂ Φ , ∂ Φ , ∂ ϕ ε, | 0| | ν 0| | λ |≤ D2Φ Λ, ≤ ∂ ϕ, ∂ ϕ γ − νν ≥ with 0 < ε, γ, λ 1/4, 1 Λ < , 0 ≤ ≤ ∞ (B.1) CΛε γ. ≤ 526 E. KUWERT & R. SCHATZLE¨

Then for η = (η , η¯) IRM IR with 0 ∈ × Φ (0) η min(Λλ2, λ /8), | 0 − 0|≤ 0 0 ϕ(0) η¯ γλ2/32, | − |≤ 0 there exists ξ BM+2(0) with ν = 0 and satisfying ∈ λ0 Φ(ξ)= η with (B.2) ξ Cγ−1/2 Φ(0) η 1/2. | |≤ | − | If further Dϕ(0) σ, (B.3) | |≤ CΛε + Cσλ−1 + CΛλ γ, 0 0 ≤ there exists a solution ξ˜ BM+2(0) of Φ(ξ˜)= η, ˜ν˜ = 0 with ∈ λ0 ˜ 0, ν˜ = 0, if ν = 0, ≤ νν˜ 0, ˜ = 0, if = 0. ≤ Proof. After the choice in (B.10) below, we will need only one variable or ν . Therefore to simplify the notation, we put ν = 0 and omit ν . First there exists a twice diﬀerentiable function λ :] λ0/2, λ0/2[ M − → Bλ0/2(0)

(B.4) Φ0(λ(),)= η0. Indeed putting T (λ) := λ Φ (λ, )+ η for λ λ /2, − 0 0 | |≤ 0 we see T ′ I ∂ Φ 1/2, ≤ M − λ 0 ≤ hence 1 T (λ ) T (λ ) λ λ . | 1 − 2 |≤ 2| 1 − 2| As T (0) = Φ (0,) η Φ (0,) Φ (0, 0) + Φ (0) η | | | 0 − 0| ≤ | 0 − 0 | | 0 − 0| ∂ Φ + λ /8 < (ε/2 + 1/8)λ λ /4, ≤ 0 | | 0 0 ≤ 0 we get T (λ) λ /2+ T (0) < λ /2 | | ≤ | | | | 0 and by Banach’s ﬁxed point theorem, (B.4) has a unique solution λ = λ() BM (0) . In particular ∈ λ0/2 (B.5) λ(0) 2 T (0) 2 Φ (0) η 2Λλ2. | |≤ | 0 |≤ | 0 − 0|≤ 0 MINIMIZERS OF THE WILLMORE FUNCTIONAL 527

By implict function theorem, λ is twice diﬀerentiable and

∂λΦ0∂λ + ∂Φ0 = 0, T ∂λ ∂λλΦ0∂λ + 2∂λΦ0∂λ + ∂λΦ0∂λ + ∂Φ0 = 0. This yields

∂λ 2 ∂Φ0 2ε 1/2, (B.6) | |≤ | |≤ ≤ ∂ λ 2 ∂ λT ∂ Φ ∂ λ + 2∂ Φ ∂ λ + ∂ Φ CΛ. | |≤ | λλ 0 λ 0 0|≤ Next we put ψ() := ϕ(λ(),) η¯ for < λ /2 − | | 0 and see by (B.5)

ψ(0) ϕ(λ(0), 0) ϕ(0) + ϕ(0) η¯ (B.7) | | ≤ | − | | − | ∂ ϕ λ(0) + γλ2/32 CΛελ2 + γλ2/32 γλ2/16. ≤ λ | | 0 ≤ 0 0 ≤ 0 Clearly ψ is twice diﬀerentiable and

∂ψ = ∂λϕ∂λ + ∂ϕ, (B.8) T ∂ψ = ∂λ ∂λλϕ∂λ + 2∂λϕ∂λ + ∂λϕ∂λ + ∂ϕ. This yields using (B.6)

T ∂ψ ∂ϕ ∂λ ∂λλϕ∂λ + 2∂λϕ∂λ + ∂λϕ∂λ (B.9) | − | ≤ | | C D2ϕ ∂ λ + ∂ ϕ ∂ λ CΛε. ≤ | | λ | |≤ Since the assumptions and conclusions are the equivalent for Φ and η , we assume after possibly exchanging by ν that − − (B.10) ψ(0) 0. ≤ Replacing by , we may further assume ∂ψ(0) 0 . On the other hand by (B.7)− and (B.9) ≥ 2 lim inf ψ() ψ(0) + ∂ψ(0)λ0/2 + (inf ∂ψ/2)(λ0/2) րλ0/2 ≥ γλ2/16 + (γ CΛε)λ2/8 > 0, ≥− 0 − 0 when using (B.1). Therefore there exists 0 < λ /2 with ψ()=0, ≤ 0 hence putting ξ := (λ(), , 0) BM+2(0) ∈ λ0

Φ(ξ)= Φ0(λ(),), ϕ(λ(),) = η0, ψ() +η ¯ = η. Choosing λ0 small enough, we further obtain (B.2). If further (B.3) is satisﬁed for the original λ0 , we see from (B.5), (B.6) and (B.8) that ∂ ψ(0) C Dϕ(λ(0), 0) C Dϕ(0) +C D2ϕ λ(0) Cσ+CΛλ2. | |≤ | |≤ | | | |≤ 0 528 E. KUWERT & R. SCHATZLE¨

Proceeding as above with (B.7) and (B.9), we calculate 2 lim inf ψ() ψ(0) + ∂ψ(0)λ0/2 + (inf ∂ψ/2)(λ0/2) ց−λ0/2 ≥ γλ2/16 (Cσ + CΛλ2)λ + (γ CΛε)λ2/8 ≥− 0 − 0 0 − 0 (γ/16 CΛε Cσλ−1 CΛλ )λ2 > 0, ≥ − − 0 − 0 0 when using (B.3). Therefore there exists λ0/2 < ˜ 0 with ψ(˜)= ˜ −M+2 ≤ ˜ 0 , hence putting ξ := (λ(˜), ,˜ 0) Bλ0 (0) , we get Φ(ξ) = η and ˜ 0 . ∈ q.e.d. ≤ References

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Mathematisches Institut der Albert-Ludwigs-Universitat¨ Freiburg Eckerstraße 1, D-79104 Freiburg, Germany E-mail address: [email protected] 530 E. KUWERT & R. SCHATZLE¨

Mathematisches Institut der Eberhard-Karls-Universitat¨ Tubingen¨ Auf der Morgenstelle 10 D-72076 Tubingen,¨ Germany E-mail address: [email protected]