Laminated currents

source: https://doi.org/10.7892/boris.37830 | downloaded: 1.2.2022

doi:10.1017/S0143385707000880 Printed in the United Kingdom

Laminated currents

JOHN ERIK FORNÆSS†, YINXIA WANG‡ and ERLEND FORNÆSS WOLD§

† Mathematics Department, The University of Michigan, East Hall, Ann Arbor, MI 48109, USA

(e-mail: fornaess@umich.edu)

‡ Department of Mathematics, Henan Polytechnic University, Jiaozuo, 454000, China (e-mail: yinxiawang@gmail.com)

§ Mathematisches Institut, Universit¨at Bern, Sidlerstr. 5, CH-3012 Bern, Switzerland (e-mail: erlendfw@math.uio.no)

(Received9March2007and accepted in revised form26October2007)

Abstract. In this paper we prove the equivalence of two definitions of laminated currents.

1. Introduction

Let K be a relatively-closed subset of the bidisc1²(z, w)= {(z, w); |z|,|w|<1}. We suppose that K is a disjoint union of holomorphic graphs, w= f_α(z), where f_α is a holomorphic function on the unit disc with f_α(0)=αand|f_α(z)|<1. We letLdenote the lamination ofK.

There are two notions of laminated currents that we will discuss. LetT be a positive closed (1,1)-current supported on K. We assume that T is the restriction of a positive closed current defined on a neighborhood of 1². We denote by [V_α] the current of integration along the graph of f_α.Letλdenote a continuous(1,0)-form which at(z, f_α(z)) equals a non-zero multiple ofdw− f_α⁰(z)d z.

Definition 1. We say that T is a laminated current directed byL if λ∧T =0 for any suchλ.

These are the same as Sullivan’sstructure currents[10]. The present terminology was introduced by Berndtsson and Sibony in [1], and such currents were treated further in [4].

In accordance with Dujardin [3] we also define the following.

Definition 2. We say thatT is alaminated current subordinate toLif there is a positive measureµsuch thatT=R

α[V_α]dµ(α).

Our main result is the following.

MAIN THEOREM. The current T is subordinate toLif and only if it is directed byL.

We note that this is a result by Sullivan in the case of the lamination being smooth, i.e.

the graphs vary smoothly withα[10]. In the continuous setting Dujardin has shown that if a currentT is dominated by a current subordinate toLthenT is subordinate toL.

The part of Sullivan’s proof that does not go through automatically in the non-smooth case is a certain approximation step, and so in the present article we are concerned with approximation of partially-smooth functions. In [5] the authors proved such an approximation theorem in the case of laminations in R² and inR³.In the last section we show that the main theorem breaks down for Riemann-surface laminations in higher dimension.

For related material on laminated currents the reader may consult the paper of Bedford et al[2].

2. Holomorphic motions and preliminary estimates for slopes of holomorphic graphs We need to know how the laminationLdefined above varies with the parameterλ, and we use the fact that it defines a holomorphic motion. Let1:= {z∈C: |z|<1}denote the unit disc inC.A holomorphic motion is a subset E of the complex planeC(or the Riemann spherebC) and a map f :1×E→C(orbC) such that f(0,·)=id, f(λ,·)is injective for eachλ, and f(·,z)is holomorphic for eachz. The laminationL defines a holomorphic motion.

Let us briefly recall some facts. It is known [9] that any holomorphic motion has an extension to a holomorphic motion f :1×C→C. This means that we may regard K as a subset of a lamination of 1×C. From [8] we have that f is automatically jointly continuous in(λ,z); in fact the map(λ,z)7→(λ, f_λ(z))is a homeomorphism onto 1×C. Moreover, f(λ,·)is quasi-conformal for eachλ, and f(λ,·)distorts cross-ratios by a bounded amount depending on|λ|. In particular we have the following. If C is compact inC^∗andx,y,z are three distinct points inCwithc0=(x−y)/(z−y)∈C, then(f_λ(x)− f_λ(y))/(f_λ(z)− f_λ(y)) is close toc₀ depending only on|λ| (for a fixed C). To see this one can consider the mapλ7→(f_λ(x)− f_λ(y))/(f_λ(z)− f_λ(y)), a map from the unit disk toC\ {0,1}, and use the fact that it has to be distance-decreasing in the Poincar´e metric. Finally we recall that f(λ,·)is H¨older continuous with exponent 1+(|λ|).

Next we need a basic estimate on slopes of the graphs. For the benefit of the reader we include the details of this well-known fact. We denote byO()the space of holomorphic functions on.Letk · k_∞denote the sup norm. Set

H^∞=H^∞(1)= {f ∈O(1): kfk_∞<∞}. Also, if 0<C<∞we set

H_C^∞=H_C^∞(1)= {f ∈O(1): kfk_∞<C}. LEMMA1. If f ∈H₁^∞(1)and f(z)6=0for all z∈1,then

|f⁰(0)| ≤2|f(0)|log(1/|f(0)|).

Proof. Pick a holomorphic function f(z)on the unit disc such that 06= |f(z)|<1 for all z∈1.We can replace f(z)by e^iθf(z)for any realθ.This does not change|f(0)|or

|f⁰(0)|.Hence we can assume that f(0) >0.

We set h(z):=log f(z).Then h(z)is a holomorphic function on the unit disc and Re(h(z)) <0.We can also choose a branch of the logarithm so that log(f(0))= −a<0.

If k(z)=h(z)/a, thenk(z) is a holomorphic function on the unit disc andk(0)= −1, Re(k(z)) <0. We define L(w)=(w+1)/(w−1).Then L(−1)=0 and if Re(w) <0 then|L(w)|<1.Then0(z):=L(k(z)) is a holomorphic function from the unit disc to the unit disc. Moreover 0(0)=L(k(0))=L(−1)=0. Since 0(0)=0 and |0(z)|<1 we can apply the Schwarz lemma. So we can conclude that |0⁰(0)| ≤1. By the chain rule, 0⁰(0)=L⁰(k(0))k⁰(0)=L⁰(−1)k⁰(0).SinceL⁰(w)= −2/(w−1)²we get0⁰(0)=

−2/(−1−1)²k⁰(0) and therefore k⁰(0)= −20⁰(0). Hence we get |k⁰(0)| ≤2. Since k(z)=h(z)/a,we can conclude next that|k⁰(0)| = |h⁰(0)|/a.Hence|h⁰(0)| =a|k⁰(0)| ≤ a·2, so|h⁰(0)| ≤2a. Next recall thath(z)=log f(z), so f(z)=e^h(z).Hence f⁰(z)= e^h(z)h⁰(z).Therefore f⁰(0)=e^h(0)h⁰(0)= f(0)h⁰(0).Hence|f⁰(0)| ≤ |f(0)||h⁰(0)|.This implies that |f⁰(0)| ≤2a|f(0)|. Now recall that log f(0)= −a. But we have set this up so that log f(0)=log|f(0)| +iarg f(0) is real-valued. So log|f(0)| = −a, i.e.

log(1/|f(0)|)=a.Therefore|f⁰(0)| ≤2a|f(0)| =2|f(0)|log(1/|f(0)|).This concludes

the proof of the lemma. 2

COROLLARY1. Suppose that we have two functions f and g holomorphic on the unit disc with f −g∈H₁^∞(1). Suppose that f(z)6=g(z)for each z∈1.We then have the estimate|f⁰(z)−g⁰(z)| ≤4|f(z)−g(z)|log(1/|f(z)−g(z)|)for all z∈1,|z|<1/2. Proof. Pick z,|z|<1/2. We define G(w)= f(z+w/2)−g(z+w/2). Then G(w) satisfies the conditions of Lemma 1. Hence|G⁰(0)| ≤2|G(0)|log(1/|G(0)|). Therefore,

1

2|f⁰(z)−g⁰(z)| ≤2|f(z)−g(z)|log 1

|f(z)−g(z)|. 2 3. Approximation for complex curves inC²

We assume that for everyc=(a,b)=(a+i b)∈Cwe have a holomorphic graph0cgiven byw=y1+i y2= f_c(z),z=x1+i x2∈1.We assume that all surfaces are disjoint and that there is a surface through every point in1×C.We assume that f_c(0)=c.

Letπ:1×C→Cbe defined byπ(z, fc(z))=c. By the discussion in the previous section the functionπis continuous.

Fix a positive constantR. By Corollary 1 there exists a positive real numberδ0>0 such that ifz∈(1/2)1and ifc,c⁰∈R1with|c−c⁰|< δ0then

∂

∂z f_c⁰(z)− ∂

∂z f_c(z)

≤4· |f_c⁰(z)− f_c(z)|log 1

|f_c⁰(z)− fc(z)|. (1) We define a class of partially-smooth functions:

A:=

φ∈C(1×C):φ(z, fc(z))∈C¹(0c), 8(x₁,x₂, w):= ∂

∂x1φ(x₁,x₂, f_c(x₁,x₂)), w= f_c(x₁,x₂)∈C(1×C), 9(x1,x2, w):= ∂

∂x₂φ(x1,x2, fc(x1,x2)), w= fc(x1,x2)∈C(1×C)

.

THEOREM1. Letφ∈A, let R be a positive real number and let >0. Then there exists a functionψ∈C¹(1×R1)such that for every point(x₁,x₂, w)=(x₁,x₂, f_c(x₁,x₂))∈ 1×R1:

|ψ(x1,x2, w)−φ(x1,x2, w)|< ,

∂

∂x1

[ψ(x₁,x₂, f_c(x₁,x₂))] − ∂

∂x1

[φ(x₁,x₂, f_c(x₁,x₂))] < ,

∂

∂x2

[ψ(x₁,x₂, f_c(x₁,x₂))] − ∂

∂x2

[φ(x₁,x₂, f_c(x₁,x₂))] < . We will prove the theorem using the following result.

PROPOSITION1. Let g∈A,g(x₁,x₂, f_{a+i b}(x₁,x₂))=a, and let R be a positive real number. There exists a positive real number t0 such that the following holds. For all >0 there exists a function h∈C¹(t₀1×R1)such that for every point(x₁,x₂, w)= (x1,x2, fc(x1,x2))∈t01×R1:

|h(x₁,x₂, w)−g(x₁,x₂, w)|< ,

∂

∂x1

[h(x₁,x₂, f_c(x₁,x₂))] < ,

∂

∂x₂[h(x1,x2, f_c(x1,x2))] < . The same result holds if we replaceabybin the definition ofg.

Proof of Theorem 1 from Proposition 1.

LEMMA2. Let p∈1 be a point, and let R,t0 be positive real numbers such that 1t0(p)⊂⊂1. Consider the lamination restricted to 1t0(p)×C. If the conclusion of Proposition 1 holds on1t0(p)×R1(with respect to projection onto{p} ×C), then the conclusion of Theorem 1 holds on1t₀(p)×R1.

Proof. Letπ=(π1, π2)denote the projection onto{p} ×C. For each j,k∈Zandδ >0 we letc^δ(j,k)denote the point (p, jδ+kδi). Let3^δ_j denote theC¹-smooth function defined by 3^δ_j(t)=cos²[π/2δ(t−jδ)] when (j−1)δ≤t≤(j+1)δ and 0 otherwise.

For eachc^δ(j,k)we first define a function

ψ^δj k(z):=φ(z, f_cδ(j,k)(z)), and then we define a preliminary approximation

ψ^δ(z, w)=X

j,k

ψ^δ_{j k}(z)3j(π1(z, w))3k(π2(z, w)).

Let (z₀, w0)∈1t0(p)×R1. Then π(z₀, w0) is contained in a square with corners c^δ(j,k),c^δ(j+1,k),c^δ(j,k+1)andc^δ(j+1,k+1), and

ψ^δ(z₀, w0)= X

m=j,j+1,n=k,k+1

ψ_mn^δ (z₀)3m(π1(z₀, w0))3n(π2(z₀, w0)).

We have

|ψ^δ(z0, w0)−φ(z0, w0)| =

X

m=j,j+1,n=k,k+1

[ψmn^δ (z0)−φ(z0, w0)]

×3^δ_m(π1(z0, w0))·3^δ_n(π2(z0, w0))

≤max_{m=j,j+1,n=k,k+1}|ψ_mn^δ (z0)−φ(z0, w0)|. Since the map from1t₀(p)×Cdefined by(z, α)7→(z, f_α(z)) is a homeomorphism it follows thatψ^δ→φuniformly asδ→0.

Next we approximate derivatives along leaves. Let α be such that (z₀, w0)= (z₀, f_α(z₀)). Since the functions3^δ_j ◦πi are constant along leaves,

∂

∂x_i[ψ^δ(z₀, f_α(z₀))−φ(z₀, f_α(z₀))]

=

X

m=j,j+1,n=k,k+1

∂

∂x_i[ψ_mn^δ (z0)−φ(z0, f_α(z0))]

×3^δ_m(π1(z0, f_α(z0)))·3^δ_n(π2(z0, f_α(z0)))

≤maxm=j,j+1,n=k,k+1

∂

∂x_i[ψmn^δ (z0)−φ(z0, f_α(z0))] . It follows thatψ^δ→φalso inC¹-norm on leaves.

Now the conclusion of Lemma 2 follows because the functionsπj can be approximated

uniformly and inC¹-norm on leaves. 2

For each point p∈1there exists by Proposition 1 a positive real numbert_psuch that constant approximation is possible on1t_p(p)×R1. Hence by Lemma 2 approximation of functions inAis possible.

We may then choose a locally-finite cover {U_α}_α∈N of 1 by disks such that approximation by functions in Ais possible on eachU_α×R4. Let{ϕ_α}be a partition of unity subordinate to{U_α}. For eachαletC_α= k∇ϕ_αk.

For a given_αletg_αbe an_α-approximating function ofφonU_α×R4. We will show that there is a sequence{α}such that the function

ψ=X

α

ϕ_α· g_α satisfies the claims of the theorem.

Letz₀∈U_α, and let{α1, . . . , αm}be the finite set ofα’s, such that the support ofφ_α intersectsU_α. Then

ψ(z, f_c(z))=

m

X

i=1

ϕ_αi(z)· g_αi(z, f_c(z)),

for allznearz₀. Then

|ψ(z0, f_c(z0))−φ(z0, f_c(z0))|

=

m

X

i=1

ϕ_αi(z₀)· g_αi(z₀, f_c(z₀))

−φ(z₀, f_c(z₀))

≤

m

X

i=1

ϕ_αi(z₀)· |g_αi(z₀, f_c(z₀))−φ(z₀, f_c(z₀))|

≤max{_αi}. Further

∂

∂x₁[ψ(z0, fc(z0))−φ(z0, fc(z0))]

=

∂

∂x1

" m

X

i=1

ϕ_αi(z)· g_αi(z₀, f_c(z₀))

−φ(z₀, f_c(z₀))

#

=

m

X

i=1

∂

∂x1

[ϕ_αi(z₀)·(g_αi(z₀, f_c(z₀))−φ(z₀, f(z₀)))]

=

m

X

i=1

∂

∂x1

[ϕ_αi(z₀)] ·(g_αi(z₀, f_c(z₀)))−(φ(z₀, f(z₀))) +

m

X

i=1

ϕ_αi(z₀)· ∂

∂x₁[g_αi(z₀, f_c(z₀))−φ(z₀, f(z₀))]

≤m·max{C_αi} ·max{_αi} +max{_αi}. Similarly we get that

∂

∂x2

[ψ(z₀, f_c(z₀))−φ(z, f_c(z₀))]

≤m·max{C_αi} ·max{_αi} +max{_αi}. It is clear that we may choose_αi fori=1, . . . ,mto get the desired estimate for all pointsz0∈U_α for this particularα. Running through allαwe find that any particularαi

will only come under consideration a finite number of times. Hence we may choose the

sequence{α}. 2

We proceed to prove the proposition.

Fix δ0 to get the estimate (1) (in the beginning of §3) for all |c−c⁰|< δ0 with

|c|,|c⁰| ≤2R. For anyδwith 0< δ < δ0we letc^δ(j,k)=(j+k·i)·δfor j,k∈Z.Let χ: [0,1] →Rbe a smooth function such thatχ(t)=0 for 0≤t≤1/4 andχ(t)=1 for 3/4≤t≤1.LetCbe a constant such that|χ⁰(t)| ≤Cfor allt∈ [0,1].

We first define a functionh_δon the surfaces0_cδ(j,k)simply byh_δ|₀

cδ(j,k) ≡jδ. We want to interpolate this function between the surfaces.

For a fixedzconsider the sets of points

Q_cδ(j,k)(z):= {f_cδ(j,k)(z), f_cδ(j+1,k)(z), f_cδ(j,k+1)(z), f_cδ(j+1,k+1)(z)}.

We first show that these sets move nicely withzfor small enough|z|and independent of δ. In particular we want to know that we may define quadrilateral regionsR_δ,j,k(z), with straight edges and cornersQ_cδ(j,k)(z), and that these sets have disjoint interior.

We make the change of coordinates in thewvariable, by setting w(˜ z, w)= ˜wj k(z, w)= w− f_cδ(j,k)(z)

f_cδ(j+1,k)− f_cδ(j,k)(z). We get

w(˜ z, f_cδ(j,k)(z))≡0, w(˜ z, f_cδ(j+1,k)(z))≡1.

From the discussion on holomorphic motions in §2 we get the following.

LEMMA3. Fix N. Then there exists a real number t0>0independent ofδ such that if

|l|,|m|<N then| ˜wj k(z, f_cδ(j+l,k+m)(z))− ˜wj k(z, f_cδ(j+l,k+m)(0))|<1/10for all|z|<

t0and any j,k.

From now on we assume that|z| ≤t₀.

LEMMA4. The quadrilaterals have disjoint interiors.

Proof. Pick(j,k). We use the linear change of coordinates in thewdirection for fixedz:

w˜j k(z, w)= w− f_cδ(j,k)(z) f_cδ(j+1,k)(z)− f_cδ(j,k)(z).

This sends f_cδ(j+l,k+m)(z)close to(j+l,k+m)on a small disc in thez direction for uniformly bounded(l,m). Hence it is clear that the quadrilaterals are disjoint. 2 Next we define preliminary functions h^δ_{j k} on the respective quadrilaterals. First we define a functiont_z(y₁,y₂)to be constant equal to 0 on the line between f_cδ(j,k)(z)and f_cδ(j,k+1)(z), and constant equal to 1 on the line between f_cδ(j+1,k)(z)and f_cδ(j+1,k+1)(z).

We extendt_z continuously to be affine on the two other edges, and then we extendt_z to be constant equal tovon the line between f_cδ(j,k)(z)+v·(f_cδ(j+1,k)(z)− f_cδ(j,k)(z))and

f_cδ(j,k+1)(z)+v·(f_cδ(j+1,k+1)(z)− f_cδ(j,k+1)(z)). Finally we defineh^δ_{j k}by h^δ_{j k}(z,y₁,y₂)= jδ+δ·(χ◦t_z) (y₁,y₂).

Theh^δ_{j k}patch up smoothly along the vertical sides of the quadrilaterals where the functions are constant. To be able to patch them together in the ‘horizontal’ directions we first extend eachh^δ_{j k}across the ‘horizontal’ edges.

To do this we use the coordinates defined byw˜. Consider the normalization w˜j k(z, w)= w− f_cδ(j,k)(z)

f_cδ(j+1,k)(z)− f_cδ(j,k)(z).

Let h˜^δ_{j k} be defined by h˜^δ_{j k}◦ ˜w=h^δ_{j k}. We want to glue together the two functions on the quadrilaterals sharing (in the new coordinates) the line segmentγ between(0,0)and (1,0), i.e. the functionh˜^δ_{j k}defined aboveγ and the functionh˜^δ_j(k−1)belowγ.

We start by extending the function h˜^δ_{j k}. Note first that by Lemma 3 the quadrilaterals R_δ,j,k andR_δ,j,k−1in the new coordinates – henceforth denotedR˜_δ,j,k and

R˜_δ,j,k−1– have corners within(1/10)-distance from the points(l,m)forl,m∈ {0,1,−1}. Note also that if we define a function t˜_z(y˜₁,y˜₂) (w˜ = ˜y₁+iy˜₂) along lines in the quadrilateralR˜_δ,j,k(z)in the new coordinates as we did when we definedt_z(y1,y2)above, thenh^δ_{j k}=(jδ+δ(χ◦ ˜t))◦ ˜w. Because of the placing of the corners we see that there exists a constantK independent ofδ, j,ksuch thatk∇_w˜(jδ+δ(χ◦ ˜t))k ≤Kδ.

Continue the lines inR˜_δ,j,kthat pass through the interval[(1/8),1−(1/8)]and extend h˜^δ_{j k} to be constant on these lines. By the placing of the corners there is a constant µ – independent ofδ and j,k – such that these lines can be extended to the line between (0,−µ)and(1,−µ). LetP˜_δ,j,kdenote the extended setR˜_δ,j,k∪(R˜_δ,j,k−1∩ {y₂≥ −µ}); we see thath˜^δ_{j k} extends to be constant on the part ofP˜_δ,j,kwhere it is not already defined.

Extendh˜^δ_j(k−1)similarly in the other direction.

To glue the functions together we choose a smooth functionϕ(z,y˜₁,y˜₂)=ϕ(y˜₂)such thatϕ(y˜₂)=1 ify₂≥µand such thatϕ(y˜₂)=0 ify₂≤ −µ. We define our final function

h_δ(z, w):=(ϕ◦ ˜wj k) (z, w)·h^δ_{j k}(z, w)+(1−ϕ◦ ˜wj k) (z, w)·h^δ_j(k−1)(z, w). (2) Fix a constantM such thatk∂ϕ/∂y˜2k =M.

LEMMA5. There are constants N1and N2such that for each j,k, δwe have h^δ_{j k}(z, w)= jδ if |w− f_cδ(j,k)(z)| ≤N₁|f_cδ(j+1,k)(z)− f_cδ(j,k)(z)|. Moreover there is a smooth functiong˜^δ_{j k}(z,y˜₁,y˜₂)such that h^δ_{j k}= ˜g^δ_{j k}◦ ˜wandk∇_w˜g˜^δ_{j k}k ≤N₂δ.

Proof. The existence of the constantN1can be seen by our description of the function in local coordinates where we used Lemma 3. To see the rest let us give the function g˜^δ_{j k} explicitly.

Fix z. Let (a₁,a₂) denote the corner of R˜^δ_j,k that is close to (0,1), and define a map Az(y˜1,y˜2):=(y˜1− ˜y2(a1/a2),y˜2(1/a2)). Then Az changes smoothly with z and kA_zk<2 for all the possibilities of(a₁,a₂)we are considering.

Next we define a functionbt on the quadrilateral A_z(R˜^δ_j,k)along lines as above. Let (b1,b2)denote the corner close to(1,1)and fixby=(by1,by2). We have that the two vertical sides ofA_z(R˜^δ_j,k)meet at the point(0,−L)whereL=b₂/(b₁−1). Calculating the slope of the line from the pointbyto the point(bt(by),0), we get thatby1/(L+

by2)= ˜t(y)/L, which gives us

bt(by)= by₁·L L+

by2

= by₁·b₂ b2+

by2(b1−1).

We have thatbt varies smoothly with(b₁,b₂)and we see thatbt has bounded derivatives for the cases of(b1,b2)we are considering. Defineg˜^δ_{j k}by

g˜^δ_{j k}= jδ+δ(χ◦bt◦A_z),

and the functionh^δ_{j k}is given byh^δ_{j k}= ˜g^δ_{j k}◦ ˜w. 2 LEMMA6. h_δ→g in sup norm on1t0×R1.

Proof. It is clear thath_δ(0,·)→g(0,·)uniformly. The claim then follows from Lemma 8

below. 2

LEMMA7. If t₀andδ are small enough, then |f_cδ(j,k)(z)− f_cδ(j+1,k)(z)| ≥δ²for all z with|z| ≤t₀and all j,k such that|c^δ(j,k)| ≤2R.

Proof. This follows from the H¨older continuity of the holomorphic motion. 2

LEMMA8. Let c∈R1. The function h_δ(z, f_c(z))is small inC¹-norm along the graph0c. Proof. We need to estimate the derivatives of the functionh_δ(z, fc(z))at an arbitrary point (z₀, f_c(z₀)), and this point is contained in some extended quadrilateralP_δ,j,k. We estimate

∂/∂x=∂/∂x1– the case of∂/∂x2 is similar. Since we are working on lines we use the notation(x,y₁,y₂)for coordinates.

If the point is close to the vertical edges, then the functionh_δis locally constant, so we are done. We can assume that also(z₀, f_c(z₀))∈P_δ,j,k\P_δ,j,k+1. We divide the proof into two cases. Assume first that(z0, fc(z0))is not in P_δ,j,k−1.Then the functionh_δis simply equal to the functionh^δ_{j k} (see (2)).

We have that

∂

∂x(h^δ_{j k}(x, f(x)))= ∂h^δ_{j k}

∂x ,∂h^δ_{j k}

∂y₁ ,∂h^δ_{j k}

∂y₂

(x, f(x))·

1,∂f₁

∂x,∂f₂

∂x

(x)

= ∂h^δ_{j k}

∂x (x, f(x))+ ∂h^δ_{j k}

∂y1 ,∂h^δ_{j k}

∂y2

(x, f(x))· ∂f1

∂x,∂f2

∂x

(x).

(3) For fixeds, vwe may define a curve(x,g(x)):

g(x)=(1−s)[(1−v)f_cδ(j,k)(x)+vf_cδ(j+1,k)(x)] +s[(1−v)f_cδ(j,k+1)(x)+vf_cδ(j+1,k+1)(x)].

Thenh^δ_{j k}(x,g(x))≡jδ+χ(v)δ. Choosesandv so that(x₀,g(x₀))=(x₀, f_c(x₀)). We get that

0= ∂

∂x(h^δ_{j k}(x,g(x)))

= ∂h^δ_{j k}

∂x (x,g(x))+ ∂h^δ_{j k}

∂y1 ,∂h^δ_{j k}

∂y2

(x,g(x))· ∂g₁

∂x,∂g₂

∂x

(x), (4) and so substracting (4) from (3) we get

∂

∂x(h^δ_{j k}(x0, f(x0)))= ∂h^δ_{j k}

∂y₁ ,∂h^δ_{j k}

∂y₂

(x0,g(x0))· ∂f₁

∂x −∂g₁

∂x ,∂f₂

∂x −∂g₂

∂x

(x0).

Using Lemma 3 we see that kf_c(x₀)− f_cδ(j+l,k+m)(x₀)k ≤2kf_cδ(j+1,k)(x₀)− f_cδ(j,k)(x0)kforl,m∈ {0,1}, and so

∂

∂x(fc− f_cδ(j+l,k+m)) (x0)

≤4k(fc− f_cδ(j+l,(k+m)) (x0))klog 1

k(f_c− f_cδ(j+l,k+m)) (x₀)k

≤8k(f_cδ(j+1,k)− f_cδ(j,k)) (x0)klog 1

2k(f_cδ(j+1,k)− f_cδ(j,k)) (x₀)k. It follows that

∂

∂x(h_δ(x, f(x)))

≤8·

∂h_δ

∂y1,∂h_δ

∂y2

· k(f_cδ(j+1,k)− f_cδ(j,k)) (x0)k

× log 1

2k(f_cδ(j+1,k)− f_cδ(j,k)) (x₀)k.

We proceed to estimatek(∂h_δ/∂y₁, ∂h_δ/∂y₂)k. We change coordinates according to Lemma 5 and writeh_δas a compositiong˜_δ◦ ˜w(y). We getkD_ww˜k =1/(kf_cδ(j+1,k)(x0)−

f_cδ(j,k)(x₀)k), and we have thatk∇_w˜g˜_δk ≤N₂δ. This shows that

∂h_δ

∂y₁,∂h_δ

∂y₂

≤N2δ 1

kf_cδ(j+1,k)(x₀)− f_cδ(j,k)(x₀)k. This gives

∂

∂x(h_δ(x, f(x)))

≤8N2δlog 1

kf_cδ(j+1,k)(x₀)− f_cδ(j,k)(x₀)k. We have by Lemma 7 thatkf_cδ(j+1,k)(x₀)− f_cδ(j,k)(x₀)k ≥δ², and so

∂

∂x(h_δ(x0, f(x0)))

≤8N₂δlog 1

2δ² →0 asδ→0.

The other case we have to consider is when (z0, fc(z0)) is contained in an overlap where we glued our functions together. In that case we may assume that(z₀, f_c(z₀))is also contained inP^δ_j(k−1)(see (2)).

LetvEdenote the vectorvE=∂/∂x(x0, f_c(x0)). We have that

∇h_δ(x0, fc(x0))· Ev= ∇[ϕ◦ ˜w·h^δ_{j k}](x0, fc(x0))· Ev

+ ∇[(1−ϕ)◦ ˜w·h^δ_j(k−1)](x0, fc(x0))· Ev

=h^δ_{j k}(x₀, f_c(x₀))· ∇[ϕ◦ ˜w](x, f_c(x₀))· Ev +(ϕ◦ ˜w) (x₀, f_c(x₀))· ∇[h^δ_{j k}](x₀, f_c(x₀))· Ev +h^δ_j(k−1)(x₀, f_c(x₀))· ∇[(1−ϕ)◦ ˜w](x, f_c(x₀))· Ev +((1−ϕ)◦ ˜w) (x0, fc(x0))· ∇[h^δ_j(k−1)](x0, fc(x0))· Ev.

By the above calculations we need not worry about the second and fourth term in this sum so we have to check that

(h^δ_{j k}(x₀, f_c(x₀))−h^δ_j(k−1)(x₀, f_c(x₀)))· ∇[ϕ◦ ˜w](x₀, f_c(x₀))· Ev→0 asδ→0.

First of all we have that |h^δ_{j k}(x₀, f_c(x₀))−h^δ_j(k−1)(x₀, f_c(x₀))| ≤2δ. Further, |∇[ϕ

◦ ˜w](x₀, f_c(x₀))· Ev| ≤M· kD[ ˜w](x₀, f_c(x₀)) (v)Ek.

Now

D[ ˜w](x0, f_c(x0)) (v)E = ∂

∂x

"

x, fc(x)− f_cδ(j,k)(x) f_cδ(j+1,k)(x)− f_cδ(j,k)(x)

# (x0).

Ignoring the constant term (it gets killed byδ), we get that kD[ ˜w](x0, fc(x0)) (v)E k ≤

|f_c⁰(x0)− f⁰

c^δ(j,k)(x0)|

|f_cδ(j+1,k)(x₀)− f_cδ(j,k)(x₀)| +

|f_c(x0)− f_cδ(j,k)(x0)| · |f⁰

c^δ(j+1,k)(x0)− f⁰

c^δ(j,k)(x0)|

|f_cδ(j+1,k)(x0)− f_cδ(j,k)(x0)|²

≤

|fc(x0)− f_cδ(j,k)(x0)|

|f_cδ(j+1,k)(x0)− f_cδ(j,k)(x0)|log 1

|fc(x0)− f_cδ(j,k)(x0)| +

|fc(x0)− f_cδ(j,k)(x0)| · |f_cδ(j+1,k)(x0)− f_cδ(j,k)(x0)|

|f_cδ(j+1,k)(x0)− f_cδ(j,k)(x0)|²

×log 1

|f_cδ(j+1,k)(x0)− f_cδ(j,k)(x0)|.

By Lemma 3,|fc(x0)− f_cδ(j,k)(x0)|/|f_cδ(j+1,k)(x0)− f_cδ(j,k)(x0)| ≤2, and so kD[ ˜w](x₀, f_c(x₀)) (v)E k ≤2·log 1

|f_cδ(j+1,k)(x0)− f_cδ(j,k)(x0)|

+2 log 1

|f_c(x₀)− f_cδ(j,k)(x₀)|.

By Lemma 5, our function is constant unless|f_c(x0)− f_cδ(j,k)(x0)| ≥N1|f_cδ(j+1,k)(x0)

− f_cδ(j,k)(x₀)| ≥N₁δ²(by Lemma 7), and so we may assume that kD[ ˜w](x0, fc(x0)) (v)Ek ≤2 log 1

δ²+2 log 1 N₁δ². All in all:

|(h^δ_{j k}(x₀, f_c(x₀))−h^δ_j(k−1)(x₀, f_c(x₀)))· ∇[ϕ◦ ˜w](x₀, f_c(x₀))· Ev|

≤4Mδ log 1

δ² +log 1 N₁δ²

→0 asδ→0. 2

4. Proof of the main theorem

We are ready to prove the main theorem. As pointed out in §2, by the theorem of Slodkowski [9, 11], we can assume that Lis a lamination of 1×Cas in the previous section.

Proof of the main theorem. Suppose thatT is a positive closed(1,1)-current on1²(0,1), supported on the laminated set K described in the introduction. We assume that T is subordinate to the lamination L of K. Hence there is a positive measure µ such that

T =R

[V_α]dµ(α).Suppose thatλ=dw− f_α⁰(z)d z. We want to show thatλ∧T =0. Letφbe any smooth(1,0)test form. We need to show thathλ∧T, φi =0.This follows since

hλ∧T, φi = Z

(λ∧T)∧φ

= Z

T ∧(λ∧φ)

= Z

α

Z

V_α

λ∧φ

dµ(α)

= Z

α0=0.

Assume next thatT is directed byL. SinceLis a lamination of1×Cwe may invoke the approximation result from the previous section. With the approximation result at hand the implication follows from Sullivan’s proof of the smooth case [10]. We include the proof for the benefit of the reader.

Step 1 is to show that there exists a family of probability measuresσ_α such thatσ_α is supported on0α, and a measureµ⁰on theα-plane such that for all test formsω,

T(ω)= Z Z

0α

ωdσ_α dµ⁰.

Let ω be a (1,1) test form and let λ(z, w)=dw− f_α⁰(z)d z for w= f_α(z). Let vE1(z, w)=(1, f_α⁰(z)) and let vE2(z, w)=(i,i· f_α⁰(z)) for w= f_α(z), and define the 2-tangent fieldv(z, w)=(vE1(z, w),vE2(z, w)).

Switching basis,

ω=ψ1d z∧d z+ψ2d z∧λ+ψ3d z∧λ+ψ4λ∧λ,

for some functionsψi, and by assumption, T(ω)=T(ψ1d z∧d z). The function ψ1 is given byψ1=(1/2i)ω(v), and so

T(ω)=T 1

2iω(v)d z∧d z

.

On the other hand we may use T to define a linear functional L on C0(1×C) by L(ψ)=T(ψd z∧d z), and so by the Riesz representation theorem there is a measure ν such that

L(ψ)=

Z ψdν.

This means that

T(ω)= Z 1

2iω(v)dν.

Now the measureν disintegrates [6]: there exists a family of probability measures σ_α such thatσ_α is supported on0_α, and a measure µ⁰ on theα-plane such that for all ψ∈C₀(1×C),

Z

ψdν= Z Z

0α

ψdσ_α

dµ⁰.

We define currentsT_αbyT_α(ω)=R

0α(1/2i)ω(v)dσ_α, and we get that T(ω)=

Z

T_α(ω)dµ⁰.

The next step is to show that T_α is closed for µ⁰-almost all α. Let {ωj}be a dense set ofC¹-smooth(0,1)test forms and fix a j∈N. Letgbe a continuous function in the α-variable and extendgconstantly along leaves. We want to show that

Z

g·T_α(∂ω)dµ⁰=0,

because this would imply that∂T_α=0 forµ⁰-almost allα(sincegis arbitrary).

By Theorem 1 there exists a sequence g_i of smooth functions such that g_i→g uniformly and inC¹-norm on leaves. SinceT is closed,

0= Z

T_α(∂(gωj))dµ⁰= Z

T_α(∂g_i ∧ωj)dµ⁰+ Z

g_i·T_α(∂ωj)dµ⁰. SinceT_α(∂gi∧ω)→0 we get that

Z

g·T_α(∂ωj)dµ⁰= lim

i→∞

Z

g_i·T_α(∂ωj)dµ⁰=0.

Running through allωj we see thatT_αis closed forµ⁰-almost allα. The only possibility then is that the measures σ_α are constant multiples of d z∧d z, i.e. σ_α=ϕ(α)d z∧d z whereϕis a measurable function [7]. Defineµ:=ϕ·µ⁰.

5. Two counterexamples

In [5] the authors proved versions of the main theorem for real laminations inR²andR³. In those results we added an extra slope condition on the laminations which is analogous to the estimate in Corollary 1. We give here a simple example of a lamination of curves inR²where the slope condition is not satisfied. Also, the conclusion of the main theorem fails. The analogue of Theorem 1, i.e. approximation of partially-smooth functions, fails as well.

For eacht∈R, we let γt be the curve y= f_t(x)=(x−t)³inR².Clearly this gives a continuous lamination ofR²by curves. The curves are all tangent to thex-axis. This implies that the current of integration of thex-axis is annihilated by the 1-formλdefined byd y− f_t⁰(x)d x onγt.However, this current is not an integral of currents[γt].We also observe that the functiona(x,y)defined bya(x, f_t(x))=t cannot be approximated by C¹functions, because any such approximation will have to have a small derivative along thex-axis.

We can also modify this example so that we have a Riemann surface lamination in C³. For t∈C, let γt be the complex curve γt(s)=(z, w, τ)=(s, (s−t)², (s−t)³).

These curves laminate C³, and γt is tangent to the z-axis at (t,0,0).Hence the z-axis is annihilated by any continuous 1-forms defining the lamination. Hence the current of integration of the z-axis is directed. But clearly it is not subordinate to the lamination.

Again the functiona(z, w, τ)defined bya|_γ

t =tcannot be approximated byC¹functions.

Acknowledgements. The first author is supported by an NSF grant. The third author is supported by Schweizerische Nationalfonds grant 200021-116165/1.

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