complex analytic methods in minimal surface theoryforstneric/datoteke/beirut...lecture 2: complex...

Complex analytic methods in minimal surfacetheory

Franc Forstneric

University of Ljubljanahttp://www.fmf.uni-lj.si/∼forstneric/

American University in Beirut

24–27 November 2014

Lecture 2: Complex analytic methods

The aim of Lecture 2 is to introduce the complex analytic methods whichwe use in minimal surface theory, and to outline the proof of some of theresults announced in Lecture 1. The emphasis will be on

the theory of Oka manifolds

construction of dominating and period dominating sprays

• Antonio Alarcon and Franc Forstneric, Null curves and directedimmersions of open Riemann surfaces. Invent. Math. 196 (2014), no.3, 733–771

• Franc Forstneric: Stein Manifolds and Holomorphic Mappings (TheHomotopy Principle in Complex Analysis). Ergebnisse der Mathematikund ihrer Grenzgebiete, 3. Folge, 56. Springer-Verlag,Berlin-Heidelberg (2011)

• Franc Forstneric: Oka manifolds: from Oka to Stein and back. Withan appendix by F. Larusson. Ann. Fac. Sci. Toulouse Math. (6) 22(2013) no. 4., 747–809.

Oka theory. . .

. . . is about a tight relationship between homotopy theory and complexgeometry involving maps from Stein manifolds to Oka manifolds.

The Oka Principle: There are only topological obstructions to solvingcertain complex-analytic problems on Stein manifolds and Stein spaces.

1939 Oka 2nd Cousin problem on domains of holomorphy has onlytopological obstructions.

1958 Grauert The Oka-Grauert principle for vector bundles andprincipal fiber bundles over Stein manifolds.

1989 Gromov The Oka principle for maps to elliptic complex manifolds.

2003-5 Larusson Abstract homotopy theoretic framework for Oka theory.

2005-9 Forstneric Convex approximation property implies the Oka property.Introduction of the class of Oka manifolds and Oka maps.

Stein manifolds (Karl Stein, 1951)

A complex manifold S is said to be a Stein manifold if

holomorphic functions separate points:

x , x ′ ∈ S , x 6= x ′ =⇒ f (x) 6= f (x ′) for some f ∈ O(S), and

S is holomorphically convex: For every compact set K ⊂ S , itsO(S)-convex hull K is also compact:

KO(S) = x ∈ S : |f (x)| ≤ supK|f |, ∀f ∈ O(S)

Equivalently, for every discrete sequence aj ∈ S there exists aholomorphic function f on S such that |f (aj )| → +∞ as j → ∞.

A Stein space is a complex space satisfying these axioms.

Embedding Stein manifolds in Euclidean spaces

1949 Behnke-Stein An open Riemann surface is a Stein manifold.

1956-61 Remmert, Bishop, Narasimhan A complex manifold S ofdimension n is Stein if and only if it is embeddable as a closedcomplex submanifold of some CN ; one can take N = 2n + 1.

Stein manifolds are relatives of affine algebraic manifolds.

1984 Stout Every relatively compact domain in a Stein manifold isbiholomorphic to a domain in an affine algebraic manifold.

1992 Eliashberg and Gromov; Schurmann (1997) A Stein manifold of

dimension n > 1 is embeddable in CN with N =[3n2

]+ 1.

1971 Forster This N is optimal for every n > 1.

Problem Is every open Riemann surface biholomorphic to some closednonsingular embedded complex curve in C2?

What should be the dual notion to a Stein manifold?

By definition, a Stein manifold S admits many holomorphic mapsf : S → C. Replace C by a complex manifold X and ask the following

Main question: For which complex manifolds X do there exist manyholomorphic maps S → X from any Stein manifold S?

What is a good way to interpret ‘many maps’?

Start with two classical 19th century theorems:

Weierstrass Theorem. On a discrete subset of a domain Ω ⊂ C we canprescribe the values of a holomorphic function f : Ω→ C.

Runge Theorem. If K ⊂ C is a compact set with no holes then everyholomorphic function K → C can be approximated uniformly on K byentire functions C→ C.

Higher-dimensional analogues, 1950’s

Cartan Extension Theorem. If T is a closed complex subvariety of aStein manifold S , then every holomorphic function T → C extends to aholomorphic function S → C.

Oka-Weil Approximation Theorem. Let K = KO(S) be a compactholomorphically convex subset of a Stein manifold S . Here

KO(S) = p ∈ S : |f (p)| ≤ supK|f | ∀f ∈ O(S).

Then every holomorphic function K → C can be approximated uniformlyon K by holomorphic functions S → C.

These are fundamental properties of Stein manifolds.

We can also view them as properties of the target manifold, thecomplex number field C.

Formulate them as properties of an arbitrary target X , with sourceany Stein manifold (or Stein space).

Oka properties of a complex manifold

The following are Oka properties (anti-hyperbolicity properties) which agiven complex manifold X may or may not have.

Convex Approximation Property (CAP): every holomorphic mapK → X from (a neighborhood of) a compact convex set K ⊂ Cn

can be approximated uniformly on K by holomorphic maps Cn → X .

Basic Oka Property with Approximation (BOPA): Everycontinuous map f0 : S → X from a Stein S which is holomorphic ona compact set K = KO(S) ⊂ S can be deformed to a holomorphicmap f : S → X by a deformation that is arbitrarily small on K .

Basic Oka Property with Interpolation (BOPI): Let T → S be aStein inclusion. Every continuous map S → X which is holomorphicon T is homotopic (rel. T ) to a holomorphic maps S → X .

BOPAI Combine BOPA and BOPI.

POPAI The parametric version of BOPAI.

Main results of Oka theory

2009 Forstneric All Oka properties introduced above are pairwiseequivalent. The class of Oka manifolds.

Example A Riemann surface is Oka iff it is not hyperbolic.

1958-61 Grauert Complex Lie groups and homogeneous manifolds are Oka.

1989 Gromov Elliptic complex manifolds are Oka.

A complex manifold X is elliptic if it admits a dominating spray;i.e., a family of holomorphic maps fx : Cn → X , dependingholomorphically on x ∈ X , such that fx (0) = x anddfx (0) : Cn → TxX is surjective for every x ∈ X .

Example If X admits C-complete holomorphic vector fields V1, . . . , Vn whichspan TxX at every point x ∈ X , then the composition of their flowsis a dominating spray on X :

fx (t1, . . . , tn) = φ1t1 φ2

t2 · · · φntn (x), t1, . . . , tn ∈ C.

The punctured null quadric is an Oka manifold

Recall that A ⊂ Cn is the null quadric

A = z = (z1, . . . , zn) ∈ Cn :n

∑j=1

z2j = 0.

Theorem

A∗ = A \ 0 is an elliptic manifold, and hence an Oka manifold.

Proof. The holomorphic vector fields on Cn,

Vj ,k (z) = zj∂

∂zk− zk

∂

∂zj, 1 ≤ j , k ≤ n

are linear and hence C-complete, their flows preserve A∗, and they spanthe tangent space of A∗ at every point. Thus A∗ is elliptic, and hence anOka manifold by Gromov’s theorem.

Classical approach: the Weierstrass Representation

Let x = (xj )j=1,2,3 : M → R3 be a conformal minimal immersion.

Set

φ = ∂x = (φ1, φ2, φ3),3

∑j=1

φ2j = 0.

Note that φ is a holomorphic 1-form on M with values in A∗ andx = <

∫ zφ.

The stereographic projection of the Gauss map of x is

g =φ3

φ1 − iφ2(meromorphic function on M).

φ = (φj )j=1,2,3 =(12

(1g − g

), i2

(1g + g

), 1)

φ3.

(g , φ3) ≡ Weierstrass Representation of x.

Further instances of the Oka principle

Let π : Z → X be a holomorphic fiber bundle with Oka fiber. Then:

X is Oka if and only if Z is Oka.

If in addition X is Stein, then sections f : X → Z satisfy all forms ofthe Oka principle.

Example: Any cone A ⊂ Cn is a C∗-bundle over a projectivemanifold X ⊂ CPn−1. Hence A∗ is Oka iff X is Oka.For the null cone in C3 we have X ∼= CP1 which is Oka.

If π : Z → X is a holomorphic fiber bundle with an Oka fiber, thenfor any Stein manifold M we have the Oka principle for liftings ofholomorphic maps f : M → X to maps F : M → Z (any continuouslifting can be deformed to a holomorphic lifting).

Example: If A ⊂ C3 is the null quadric then the coordinateprojection π : A∗ → C∗ has Oka fiber (C∗), and hence liftings enjoythe Oka principle.

Dominating holomorphic sprays of maps

Let M be a compact bordered Riemann surface (or a compact complexmanifold with boundary), and let X be a complex manifold.

A r (M, X ): the space of C r maps M → X which are holomorphic in M.

Definition

A holomorphic spray of maps of class A r (M, X ) is a family of mapsft ∈ A r (M, X ) depending holomorphically on a parameter t in aball 0 ∈ B ⊂ CN . The map f0 is the core of the spray.

A spray ft is dominating if ∂t |t=0ft(x) : CN → Tf0(x)X is surjectivefor every x ∈ M.

Lemma (Existence of dominating sprays)

If M is a bordered Riemann surface (or a strongly pseudoconvex domainin a Stein manifold) and X is any complex manifold, then every mapf0 ∈ A r (M, X ) (r ∈ Z+) is the core map of a dominating holomorphicspray ft ∈ A r (M, X ) (t ∈ B ⊂ CN , N large).

The period map

Let M be a compact bordered Riemann surface with bM 6= ∅. Fix anowhere vanishing holomorphic 1-form θ on M.Pick a basis γjlj=1 of the 1st homology group H1(M; Z). Let

P = (P1, . . . ,Pl ) : A 0(M, Cn)→ (Cn)l

be the period map. The j-th component, applied to f ∈ A 0(M, Cn), is

Pj (f ) =∫

γj

f θ =∫ 1

0f (γj (s)) θ(γj (s), γj (s)) ds ∈ Cn.

By Stokes’ theorem, the period map does not change under homotopicdeformations of the loops γj : [0, 1]→ M.

The 1-form f θ is exact if and only if its periods vanish:

Pj (f ) =∫

γj

f θ = 0, j = 1, . . . , l .

Period dominating holomorphic sprays

Let A denote the null quadric in Cn (n ≥ 3) and A∗ = A \ 0.

Lemma (Existence of period dominating sprays)

Let M, θ and γ1, . . . , γl be as above, and let r ∈ Z+.

Assume that f0 : M → A∗ is a map of class A r (M,A∗) which is nonflat,in the sense that its range does not belong to a ray of the null quadric A∗.

Then there exists a dominating and period dominating spray ft : M → A∗of class A r (M) (t ∈ B ⊂ CN , N large):

∂

∂t

∣∣∣∣t=0

P(ft) : CN → (Cn)l is surjective.

Furthermore, every flat map f0 ∈ A r (M,A∗) can be approximatedarbitrarily closely by nonflat maps in A r (M,A∗).

Construction of period dominating sprays

We may assume that M b M is a smoothly bounded domain in an openRiemann surface M. Let Cj = γj ([0, 1]) ⊂ M (j = 1, . . . , l) be a basis ofH1(M, Z) such that C = ∪jCj is O(M)-convex.

The graph G = (x , f0(x)) : x ∈ M has an open Stein

neighborhood Ω ⊂ M ×A∗.

Choose holomorphic vector fields V1, . . . , Vm on Ω which aretangential to A∗ and span the tangent space at each point.

Consider the spray ft given by

ft(x) = φ1h1(x)t1

· · · φNhN (x)tN

(x), t = (t1, . . . , tN ),

where each φj is the flow of one of the vector fields Vk (possiblywith repetitions) and hj are holomorphic functions on M.

We have ∂∂tj

∣∣∣∣t=0

ft(x) = hj (x)Vj (f0(x)). Choose the values of hj in

a suitable way on C = ∪jCj and apply Mergelyan’s approximationtheorem to ensure that ft is dominating and period dominating.

Banach manifold structure

TheoremLet M be a compact bordered Riemann surface and r ∈ Z+. The set

f ∈ A r (M,A∗) : f nonflat, <P(f ) = 0

is a real Banach manifold containing the complex Banach manifold

f ∈ A r (M,A∗) : f nonflat, P(f ) = 0

as a closed Banach submanifold.

Proof.

The set A r (M, X ) is a complex Banach manifold for any r ∈ Z+ andany complex manifold X .The existence of period dominating sprays shows that the equations

<P(f ) = 0, P(f ) = 0 (f ∈ A r (M,A∗))

are of maximal rank at each nonflat point. The claims follow.

Banach manifold structure

By integrating these classes of maps we get the corresponding statementsfor the space of nonflat conformal minimal immersions in Rn:

u : M → Rn, u(x) =∫ x

p0<(f θ) (x ∈ M),

and also for the space of nonflat holomorphic null curves in Cn:

F : M → Cn, F (x) =∫ x

p0f θ (x ∈ M).

TheoremLet M be a compact bordered Riemann surface and n ≥ 3, r ≥ 1.

(a) The space CMIr∗(M, Rn) of nonflat conformal minimal immersionsM → Rn of class C r is a real analytic Banach manifold.

(b) The space NCr∗(M, Cn) of nonflat holomorphic null curves M → Cn

of class C r is a complex Banach manifold.

(c) The space <F : F ∈ NCr∗(M, Cn) is a Banach submanifold of

CMIr∗(M, Rn).

Oka principle

Theorem (Oka principle; Lecture 1, Theorem 1)

Every continuous map f0 : M → A∗ of an open Riemann surface M tothe null quadric A∗ = A \ 0 is homotopic to a holomorphic mapf : M → A∗ such that f θ has vanishing periods, and hence

F (x) = F (p) +∫ x

pf θ, x ∈ M

is a null holomorphic immersion M → Cn. F can be chosen proper.

Proof. Choose an exhaustion M1 ⊂ M2 ⊂ · · · ⊂ ∪∞j=1Mj = M such that

every Mj is a smooth Runge domain in M and Mj+1 is obtained from Mjin one of the following ways:

(a) Noncritical case: Mj+1 can be deformed onto Mj .

(b) Critical case, index 0: A new connected component.

(b) Critical case, index 1: Mj+1 = Mj ∪ (1-handle).

The noncritical case

We inductively construct a sequence of holomorphic maps fj : Mj → A∗such that fj θ has vanishing periods on Mj (finitely many integralconditions) and fj+1|Mj

approximates fj as close as desired.The limit map f = limj→∞ fj : M → A∗ will solve the problem.

The noncritical case: Mj ⊂ Mj+1 are smoothly bounded compact setsin M such that Mj+1 retracts onto Mj (no change of topology).

We embed the map fj : Mj → A∗ into a period dominating

holomorphic spray fj ,t : Mj → A∗ (t ∈ B ⊂ CN).

Since A∗ is an Oka manifold and M ×B is a Stein manifold, we canapproximate the spray fj ,t by a holomorphic sprayfj+1,t : Mj+1 → A∗ for t in a slightly smaller ball 0 ∈ B ′ ⊂ B.

If the approximation is close enough then, because of perioddomination condition, there exists t0 ∈ B′ close to 0 such that themap fj+1,t0 : Mj+1 → A∗ has zero periods.

This completes the noncritical case, and also the critical case of index 0(when a new connected component appears).

The critical case

The critical case: Now Mj+1 retracts onto the union of Mj and asmooth arc Ej

∼= [0, 1] ⊂ Mj+1 whose endpoints aj , bj belong to Mj andwhich is otherwise disjoint from Mj , and Mj ∪ Ej is Runge in Mj+1.

There are two possibilities:

(a) The points aj , bj belong to different connected components of Mj .

(b) The points aj , bj belong to the same connected component of Mj .

In case (a) no new nontrivial loop appears in the 1st homology group andwe proceed as follows:

Extend fj to a smooth map fj : Mj ∪ Ej → A∗, holomorphic on Mj .

Embed fj into a period dominating spray of maps fj ,t (t ∈ B ⊂ CN )of the same class. (See the noncritical case.)

By Mergelyan’s theorem and the Oka property of A∗ we canapproximate fj ,t by a holomorphic spray fj+1,t over Mj+1.

If the approximation is close enough then a member fj+1,t0 of thisspray has vanishing periods over all closed curves in Mj+1.

The critical case–continued

In case (b) the arc Ej closes to a loop Γ ⊂ Mj ∪ Ej ⊂ Mj+1.

We extend fj to a smooth map fj : Mj ∪ Ej → A∗ such that∫Γ

fj θ = 0.

This is possible since the convex hull of the algebraic variety A equals Cn.

The basic idea: Assume that Ej is parametrized by γ : [0, 1]→ Ej withθ(γ(t)) = b > 0 independent of t. Fix a point v ∈ Cn. Choose pointsv1, . . . , vm ∈ A∗ and numbers λ1, . . . , λm > 0 such that

∑j

λj = 1 and ∑j

λjvj = b−1v.

Choose fj : Ej → A∗ so that it spends approximately the time λj at thepoint vj and goes very quickly from one point to another. Then∫

Ej

fj θ =∫ 1

0fj (γ(t))θ(γ(t)) dt ≈ b ∑

j

λjvj = v.

A suitable choice of v gives∫

Γ fj θ = 0. Now proceed as in case (a).

General position theorems

Theorem (Lecture 1, Theorem 3)

Let M be an open Riemann surface.

(a) Every immersed holomorphic null curve F0 : M → Cn (n ≥ 3) canbe approximated uniformly on compacts in M by embeddedholomorphic null curves F : M → Cn.

(b) Every conformal minimal immersion u0 : M → Rn (n ≥ 5) can beapproximated uniformly on compacts in M by conformal minimalembeddings u : M → Rn.

Sketch of proof: It suffices to consider the case when M is a compactbordered Riemann surface. Consider the difference map

δF0 : M ×M → Cn, δF0(x , y) = F0(x)− F0(y).

Clearly F is an embedding if and only if

(δF )−1(0) = DM = (x , x) : x ∈ M ⊂ M ×M.

General position theorems–proof

Since F0 is an immersion, it is locally injective, and there is an openneighborhood U ⊂ M ×M of the diagonal DM such that δF0 does notassume the value 0 ∈ Cn in U \DM .

To prove the theorem, it suffices to find arbitrarily close to F0 anotherholomorphic null curve F : M → Cn whose difference map δF , restrictedto M ×M \ U, is transverse to the origin 0 ∈ Cn.

Since dimC M ×M = 2 < n, this will imply that δF does not assume thevalue zero in (M ×M) \ U, so F (x) 6= F (y) if (x , y) ∈ M ×M \ U. Italso does not assume the value 0 on U \DM if F is close enough to F0.

Hence F is an embedding.

Such a map F is obtained by a transversality argument.

General position theorems–proof

We begin by finding a holomorphic spray of maps H : B×M → Cn,where 0 ∈ B ⊂ CN , such that H(0, · ) = F0 and the difference mapδH : B×M ×M → Cn, defined by

δH(ζ, x , y) = H(ζ, x)−H(ζ, y), ζ ∈ B, x , y ∈ M,

is a submersive family, meaning that its partial differential

dζ |ζ=0 δH(ζ, x , y) : CN → Cn

is surjective for every (x , y) ∈ M ×M \ U.

It follows that dζ(δH) is surjective for all ζ in a neighborhood B′ ⊂ B of

the origin in CN . Hence the map δH : B′ × (M ×M \ U)→ Cn istransverse to any submanifold of Cn, in particular, to the origin 0 ∈ Cn.

For a generic member H(ζ, · ) : M → Cn of this family, the differencemap δH(ζ, · ) is also transverse to 0 ∈ Cn on M ×M \U. By choosing ζclose to 0 we obtain an embedded null curve F = H(ζ, · ) close to F0.

Proper null curves and conformal minimal immersions

Theorem (Lecture 1, Theorem 4)

Every open Riemann surface admits

(a) a proper conformal minimal embedding into R5,

(b) a proper conformal minimal immersion into R3, and

(c) a proper null holomorphic embedding into C3.

We shall present the proof of part (c); the other two are similar.

The main part of the proof is contained in the following lemma whichexplaines how to approximate a given holomorphic null curve on acompact smooth domain U ⊂ M by a holomorphic null curve on a biggerdomain V ⊂ M provided that U is a strong deformation retract of V(i.e., there is no change of topology).

Later on we shall also explain how to handle the critical case where thetopology of the domain changes.

The main lemma

Lemma

Let U ⊂ V ⊂ V be smoothly bounded compact Runge domains in Msuch that U is a deformation retract of V .

Let F = (F1, F2, F3) : U → C3 be a null immersion such that

max|F1(x)|, |F2(x)| > ρ for some ρ > 0 and all x ∈ bU.

Then there is a null immersion F = (F1, F2, F3) : V → C3 such that

(i) F is as close as desired to F in the C1(U) topology,

(ii) max|F1(x)|, |F2(x)| > ρ for all x ∈ V \ U, and

(iii) max|F1(x)|, |F2(x)| > ρ + 1 for all x ∈ bV .

Construction of proper null curves

Proof of the Lemma. Note that V \ U = ∪ii=1Ai where the Ai ’s arepairwise disjoint annuli,

bAi = αi ∪ βi where αi ⊂ bU, βi ⊂ bV .

There exist j ∈N, subsets I1 and I2 of I := 1, . . . , i ×Zj and compactconnected subarcs αi ,j : (i , j) ∈ I satisfying the following conditions:

(a1) ∪jj=1αi ,j = αi for i = 1, . . . , i.

(a2) αi ,j and αi ,j+1 only meet in a common endpoint pi ,j .

(a3) I1 ∪ I2 = I and I1 ∩ I2 = ∅.

(a4) |Fk (x)| > ρ for all x ∈ αi ,j and all (i , j) ∈ Ik , k = 1, 2.

From (a4) one also has that

(a5) if (i , j) ∈ Ih and (i , j + 1) ∈ Il , h 6= l , then |Fk (pi ,j )| > ρ fork ∈ 1, 2.


For every (i , j) ∈ I we choose a smooth embedded arc γi ,j ⊂ Ai withthe following properties:

γi ,j is attached to U at the endpoint pi ,j ,

it intersects the arc αi transversely at that point,

γi ,j ∩ αi = pi ,j,the other endpoint qi ,j of the arc γi ,j lies in βi , γi ,j intersects βitransversely at that point, and γi ,j ∩ βi = qi ,j, and

The arcs γi ,j , j ∈ Zj, are pairwise disjoint.

The situation is shown on the following figure.

The annuli in V \ U

Figure : The annulus Ai .


Let z = (z1, z2, z3) be the coordinates on C3 and πk : C3 → C thecoordinate projections π(z1, z2, z3) = zk for k = 1, 2, 3.

Choose compact smooth embedded arcs λi ,j ⊂ C3, (i , j) ∈ I , satisfying:

(b1) λi ,j agrees with F (γi ,j ) near the endpoint F (pi ,j ).

(b2) |πk (z)| > ρ for all z ∈ λi ,j−1 ∪ λi ,j , (i , j) ∈ Ik , k = 1, 2.

(b3) |πk (z)| > ρ + 1 for all z ∈ vi ,j−1, vi ,j, (i , j) ∈ Ik , k = 1, 2, where

vi ,l ∈ C3 denotes the other endpoint of the arc λi ,l .

(b4) The unit tangent vector field to the arc λi ,j assumes values in A∗.

The existence of such arcs λi ,j is clear. Since pi ,j−1 and pi ,j are theendpoints of αi ,j , properties (b1) and (b2) are compatible thanks to (a4).On the other hand, (b3) is always possible.

Taking into account (b1), we can find a smooth map G : V → Cn thatagrees with F in an open neighborhood of U and maps the arc γi ,jdiffeomorphically onto λi ,j for every (i , j) ∈ I .


The setS := U ∪

(∪(i ,j)∈I γi ,j

)⊂ M

is admissible. Condition (b4) shows that the map G |S : S → C3 is ageneralized null curve. Hence the Mergelyan theorem furnishes a nullimmersion G = (G1, G2, G3) : V → C3 such that

(c1) G is as close as desired to F in the C1(U) topology,

(c2) |Gk (x)| > ρ for all x ∈ γi ,j−1 ∪ αi ,j ∪ γi ,j , (i , j) ∈ Ik , k = 1, 2, and

(c3) |Gk (x)| > ρ + 1 for all x ∈ qi ,j−1, qi ,j, (i , j) ∈ Ik , and k = 1, 2.

To obtain (c2) we take into account (a4) and (b2), whereas (c3) followsfrom (b3). Furthermore, (c2) and (c3) give

(c4) if (i , j) ∈ Ih and (i , j + 1) ∈ Il , h 6= l , then for k ∈ 1, 2 we have

|Gk (x)| > ρ for all x ∈ γi ,j , and |Gk (qi ,j )| > ρ + 1.

Therefore G satisfies the Lemma on the admissible set S .


Let βi ,j be the Jordan arc in βi which connects the points qi ,j−1 and qi ,jand does not intersect the set qi ,h : h ∈ Zj \ j − 1, j.For every (i , j) ∈ I we denote by Di ,j the closed disc in Ai bounded by

the arcs αi ,j , γi ,j−1, γi ,j , and βi ,j . Then Ai = ∪jj=1Di ,j for every i .

Since G is continuous, properties (c2) and (c3) extend to small openneighborhoods of the compact sets γi ,j−1 ∪ αi ,j ∪ γi ,j and qi ,j−1, qi ,j,respectively. Therefore, we can choose for every k ∈ 1, 2 and every(i , j) ∈ Ik a closed disc Ki ,j ⊂ Di ,j \ (γi ,j−1 ∪ αi ,j ∪ γi ,j ) such that

(d1) Ki ,j ∩ βi ,j is a compact connected Jordan arc,

(d2) |Gk (x)| > ρ for all x ∈ Di ,j \Ki ,j , and

(d3) |Gk (x)| > ρ + 1 for all x ∈ βi ,j \Ki ,j .

Obviously we have

V \ U = ∪(i ,j)∈I(Ki ,j ∪Di ,j \Ki ,j

), (1)

bV = ∪(i ,j)∈I((βi ,j ∩Ki ,j ) ∪ βi ,j \Ki ,j

). (2)


Assume that I1 6= ∅; otherwise I2 = I 6= ∅ and reason symmetrically.

We now deform G into another null immersion H : V → C3 satisfying theLemma on the set U ∪

(∪(i ,j)∈I1 Di ,j

). We will apply to G a perturbation

that is large in Ki ,j for all (i , j) ∈ I1, small on U, and controlledelsewhere. Such control will be insured by demanding that H1 = G1.

The compact set

S1 :=(

U ∪(∪(i ,j)∈I2 Di ,j

))∪(∪(i ,j)∈I1Ki ,j

)⊂ M

is admissible. Note that the compact sets U ∪(∪(i ,j)∈I2 Di ,j

)and

∪(i ,j)∈I1Ki ,j are disjoint.

Choose a point ξ = (0, ξ2, ξ3) ∈ C3 ∩ z1 = 0 such that

|ξ2 + G2(x)| > ρ + 1 for all x ∈ ∪(i ,j)∈I1Ki ,j . (3)

In fact, any ξ with ξ2 large enough satisfies this condition.


The map H = (H1, H2, H3) : S1 → C3, given by

(e1) H(x) = G (x) for all x ∈ U ∪(∪(i ,j)∈I2 Di ,j

), and

(e2) H(x) = (G1(x), G2(x) + ξ2, G3(x) + ξ3) for all x ∈ ∪(i ,j)∈I1Ki ,j ,

is a null immersion.

Mergelyan’s Theorem provides a null holomorphic immersionH = (H1, H2, H3) : V → C3 such that

H1 = G1 (4)

and the following properties hold:

(f1) H is as close to H as desired in the C 1(S1) topology.

(f2) |Hk (x)| > ρ for all x ∈ Di ,j \Ki ,j , (i , j) ∈ Ik , k = 1, 2.

(f3) |Hk (x)| > ρ + 1 for all x ∈ βi ,j \Ki ,j , (i , j) ∈ Ik , k = 1, 2.

(f4) |H2(x)| > ρ + 1 for all x ∈ Ki ,j and all (i , j) ∈ I1 = I \ I2.


If I2 = ∅ then the Lemma already holds with F = H.

Indeed, in such case I = I1; hence, taking into account (1), properties(f2) and (f4) imply item (ii) in the lemma. Likewise, (iii) follows from(f3), (f4), and (2). Finally, item (i) is insured by (f1) and (c1).

Assume now that I2 6= ∅. In the next step we deform H to obtain a nullimmersion F : V → C3 satisfying the Lemma. The deformation is big on∪(i ,j)∈I2Di ,j and small elsewhere. The deformation procedure issymmetric with respect to the one in the previous step of the proof.

The properness theorem for null curves M → C3 is obtained by anobvious inductive application of the Lemma.

complex analytic methods in minimal surface theoryforstneric/datoteke/beirut...lecture 2: complex...

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