GEORGIAN MATHEMATICAL JOURNAL: Vol. 4, No. 3, 1997, 201-218

PARAMETRIZATION OF A FAMILY OF MINIMAL SURFACES BOUNDED BY THE BROKEN LINES IN ℝ^{3}

R. ABDULAEV

Abstract. Consideration is given to a family of minimal surfaces bounded by the broken lines in ℝ^{3} which are locally injectively projected onto the coordinate plane. The considered family is bijectively mapped by means of the Enepper–Weierstrass representation onto a set of circular polygons of a certain type. The parametrization of this set is constructed, and the dimension of the parameter domain is established.

The Dirichlet problem for an equation of minimal surfaces in the nonconvex domain does not always have a solution even under infinitely smooth boundary conditions. Geomerically, this means that a minimal surface bounded by a given curve injectively projected onto the coordinate plane is not always injectively projected onto the same plane. For references on this topic see [1] and [2]. One way to investigate the problem of projecting a minimal surface onto the plane is as follows: instead of studying the solvability of an individual Dirichlet boundary value problem one should consider a sufficiently well surveyable set of spatial curves and find out which part of this set is filled up by the curves for which the above problem is solvable.

The first step in this direction was made in [3] where it was shown that there exist no minimal surfaces which are bounded by four ribs of a tetrahedron of variable height and injectively projected onto the tetrahedron base.

In this paper, for some family of spatial broken lines we investigate a subfamily of such broken lines, which bound the minimal surfaces locally injectively projected onto the coordinate plane. The parametrization of this subfamily is constructed and the dimension of the parameter domain is established.

201

1991 Mathematics Subject Classification. 49Q05, 53A10.

Key words and phrases. Minimal surface, locally injective projection, closed broken line, Enneper–Weierstrass formula, parametrization, quadratic differential.

1072-947X/97/0500-0201$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1997 Plenum Publishing Corporation

§ 1. INTRODUCTION

Let S be a simply connected minimal surface parametrized by an infinitely smooth mapping x : E = {|z|^{2} = ξ^{2} + η^{2} < 1} → ℝ^{3}. We denote by π the orthogonal projection π : ℝ^{3} → Π = {x ∈ ℝ^{3}, x^{3} = 0} and call S the d-surface if π ∘ x is a homeomorphism, and the c-surface if π ∘ x is a local homeomorphism. Since z → (z, x(z)) is a homeomorphism, S is the d-surface iff π|_{x(E)} is injective. Assume that S is the c-surface, x ∈ C(\overline{E}), and the set of values of the restriction π|_{∂S} consists of boundary points of the bounded simply connected domain Q. We shall show that under these assumptions S is the d-surface. Indeed, it is easy to show that since π ∘ x : E → Π is a local homeomorphism, we have (π ∘ x)(E) ∩ (Π\\overline{Q}) = ∅ because otherwise x would be unbounded in E. Therefore π ∘ x is an unlimited non-ramified covering of the domain Q by a circle. The mapping π ∘ x is injective by virtue of the theorem on monodromy [4].

In this paper the mapping x will be represented by the following Enneper– Weierstrass formulas [2]:

\begin{aligned}2x^{1}(z) & = Re ∫_{0}^{z} F′(t)(1 − ω^{2}(t))dt + c_{1}, \\ 2x^{2}(z) & = Re i ∫_{0}^{z} F′(t)(1 + ω^{2}(t))dt + c_{2}, \\ x^{3}(z) & = Re ∫_{0}^{z} F′(t)ω(t)dt + c_{3}, \end{aligned}  (1)

where F (z) and ω(z) are the holomorphic functions in E, c_{j}, j = \overline{1, 3}, are the real constants.

If a minimal surface is the c-surface, then for any point M ∈ S there exists a neighborhood V_{s}(M) such that π|_{V_{s}(M)} is injective, and therefore the surface V_{s}(M) can be represented as x^{3} = u(x^{1}, x^{2}), (x^{1}, x^{2}) = π(V_{s}(M)) where u ∈ C^{2}(π(V_{s}(M))) and

(1 + u_{x^{2}}^{2})u_{x^{1}x^{1}} − 2u_{x^{1}x^{2}}u_{x^{1}}u_{x^{2}} + (1 + u_{x^{1}}^{2})u_{x^{2}x^{2}} = 0.  (2)

Introducing the notation p = u_{x^{1}}, q = u_{x^{2}}, W = \sqrt{1 + p^{2} + q^{2}} for an appropriate orientation of the surface, we obtain [2] the following expression of the unit normal vector ν⃗(M) = (ν_{1}(M), ν_{1}(M), ν_{3}(M)):

ν⃗(M) = (\frac{2 Re ω(z)}{|ω(z)|^{2} + 1}, \frac{2 Im ω(z)}{|ω(z)|^{2} + 1}, \frac{|ω(z)|^{2} − 1}{|ω(z)|^{2} + 1}) =
= [W (π(M))]^{−1} · (p(π(M)), q(π(M)), −1).  (3)

Let us show that for S to be the c-surface it is necessary and sufficient that F′(z) ≠ 0, |ω(z)| ≠ 1, z ∈ E. We write π ∘ x in the form

ζ(z) = x^{1}(z) + ix^{2}(z) = \frac{1}{2}[ ∫_{0}^{z} \overline{F′dt} − ∫_{0}^{z} F′ω^{2}dt + c_{1} + ic_{2}].

Hence \overline{ζ}_{z} = \frac{1}{2}F′(z), \overline{ζ}_{z} = −\frac{1}{2}F′(z)ω^{2}(z) and d\overline{ζ} = F′(z)dz − F′(z)ω^{2}(z)dz,

|\overline{ζ}_{z}|^{2} − |\overline{ζ}\overline{z}|^{2} = \frac{1}{4}|F′(z)|^{2}(1 − |ω(z)|^{2}). The latter equality provides the sufficiency of the conditions. Let S be the c-surface and |ω(z_{0})| = 1. This equality can be fulfilled only when W (π ∘ x(z_{0})) = ∞, which contradicts the definition of the c-surface. If |ω(z_{0})| < 1 but F′(z_{0}) = 0, then on a sufficiently small circumference z = z_{0} + εe^{iθ} we shall have

\overline{ζ}′_{θ} = F′_{θ}(1 − \overline{F} ′_{θ}(z_{0} + εe^{iθ})(F′_{θ}(z_{0} + εe^{iθ}))^{−1}\overline{ω^{2}}(z_{0} + εe^{iθ})).

Hence, taking into account |\overline{F} ′_{θ}ω^{2} ·(F′_{θ})^{−1}| < 1, we obtain \frac{1}{2π} ∫_{0}^{2π} d Arg \overline{ζ}_{θ} = \frac{1}{2π} ∫_{0}^{2π} d Arg F′_{θ} ≥ 2, which also contradicts the fact that the mapping π ∘ x is a local homeomorphism.

In what follows, by a curve we shall mean both a class of the equivalence of continuous mappings of a segment and an individual representative of this class. The curve will be called an arc if it is an injective mapping. For l : [a, b] → ℝ^{m} we shall denote by l^{0} the restriction of l on (a, b). The set of values of the curve l will be denoted by [l] (if a = b, then [l^{0}] = ∅). By l_{1} · l_{2} we shall mean the product of curves in the usual sense when the end point of the curve l_{1} coincides with the initial point of the curve l_{2}. The notation πβ(l_{1}, l_{2}), |β(l_{1}, l_{2})| ≤ 1 will denote the angle between the positive tangent at the end point of l_{1} and the positive tangent of l_{2} at the initial point and counted from l_{1}. Finally, |l| will denote the length of l.

Let Γ̃ be a closed broken line in ℝ^{3} not lying in one plane and satisfying the following condition: if π(x_{1}) = π(x_{2}), x_{1} ∈ Γ̃, x_{2} ∈ Γ̃, then x_{1} and x_{2} belong to the same segment of Γ̃. This condition immediately implies that π[Γ̃] cuts the plane into two components. We orient Γ̃ so that the orientation induced on π∘Γ̃ would be positive with respect to the unbounded component. Let us number the vertices M_{i} of Γ̃ according to the chosen orientation and denote by Γ̃_{i} the oriented segment of Γ̃ whose initial point is M_{i} and whose end point is M_{i+1} (M_{n+1} = M_{1}). By T_{i} = (X_{i}, Y_{i}Z_{i}) we denote the unit vector co-directed with Γ̃_{i}. It is assumed that M_{i}, M_{i+1}, M_{i+2} are not collinear and therefore T_{i} ≠ T_{i+1}. Moreover, since Γ̃ does not lie in the plane, among the vectors T_{i} there are three noncomplanar vectors with successive indices, and without loss of generality we can consider T_{n−2}, T_{n−1}, T_{n} as such.

Denote by G a set of broken lines possessing the following property: if the initial point and orientation of Γ are appropriately chosen, then Γ_{i} is codirected with Γ̃_{i}. We denote by Λ a subspace of ℝ^{n} defined by the equation ∑_{i=1}^{n} λ_{i}T_{i} = 0. Let λ : G → Λ_{+}Λ ∩ ℝ_{+}^{n}, λ(Γ) = (|Γ_{1}|, |Γ_{2}|, …, |Γ_{n}|). Since T_{n−2}, T_{n−1}, T_{n} are noncoplanar, the mapping τ : Λ_{+} → ℝ_{+}^{n−3} τ(λ_{1}, λ_{2}, …, λ_{n}) = (λ_{1}, λ_{2}, …, λ_{n−3}) is injective. The definition of Λ_{+} readily implies that Λ′_{+} = (τ ∘ λ)(G) is an open connected convex subset in ℝ_{+}^{n−3}.

Since for y ∈ Λ′_{+} and t > 0 we have ty ∈ Λ′_{+}, after introducing the notation Λ̃ = Λ′_{+} ∩ S^{n−4}, where S^{n−4} is the unit sphere in ℝ^{n−3}, we shall have the injective mapping 𝔭 : G → Λ̃

𝔭(Γ) = ( ∑_{i=1}^{n−3} |Γ_{i}|^{2})^{− \frac{1}{2}} (|Γ_{1}|, |Γ_{2}|, …, |Γ_{n−3}|).

We denote by G_{c} the subset of G consisting of broken lines bounding the minimal c-surfaces, and by 𝔖 the set of minimal c-surfaces bounded by the broken lines of the family G_{c}. Let P (S) = (𝔭 ∘ τ ∘ λ)∂S, P = P (𝔖).

§ 2. ω-Images of Surfaces of the Family 𝔖

If a minimal surface is parametrized by formulas (1), g is the Gaussian mapping of the surface, and σ is the stereographic projection, then ω = σ ∘ g ∘ x [2].

Let t_{j} = x^{−1}(M_{j}) = e^{iθ_{j}}, 0 < θ_{1} < θ_{2} < ⋯ < 2π, l_{j} = {t = e^{iθ}, θ_{j} ≤ θ ≤ θ_{j+1}\}. The function ω(z) is analytically continuable through l_{j}^{0}, j = \overline{1, n}, and, by virtue of (3), satisfies on l_{j}^{0} the equation

2X_{j} Re ω(t) + 2Y_{j} Im ω(t) + Z_{j}(|ω(t)|^{2} − 1) = 0.  (4)

There exists the following representation [5]:

2x^{1}(z) = Re ∫_{0}^{z} [Φ^{2}(t) − Ψ^{2}(t)]dt + c_{1},

2x^{2}(z) = Re i ∫_{0}^{z} [Φ^{2}(t) + Ψ^{2}(t)]dt + c_{2},

x^{3}(z) = Re ∫_{0}^{z} Φ(t)Ψ(t)dt + c_{3},

(5) where Φ^{2}(z)dz and Ψ^{2}(z)dz are the holomorphic differentials in E. The following equalities are fulfilled in the neighborhood of t_{j}:

\begin{matrix}Φ(z) = C_{1}(z − t_{j})^{\frac{1−γ_{j}}{2}} Φ_{1}(z) + C_{2}(z − t_{j})^{− \frac{1−γ_{j}}{2}} Ψ_{1}(z) \\ Ψ(z) = C_{3}(z − t_{j})^{\frac{1−γ_{j}}{2}} Φ_{2}(z) + C_{4}(z − t_{j})^{− \frac{1−γ_{j}}{2}} Ψ_{2}(z), \end{matrix}  (6)

where C_{j}, j = \overline{1, 4}, are the constants; C_{1}C_{4} − C_{2}C_{3} = 1; Φ_{1}(z), Φ_{2}(z), Ψ_{1}(z) and Ψ_{2}(z) are a holomorphic in the neighborhood V_{j} of the point

t_{j}, 0 < γ_{j} < 1, while (z − t_{j})^{± \frac{1−γ_{j}}{2}} is holomorphic branch in V_{j} ∩ E. The relation of F (z) and ω(z) with Φ(z) and Ψ(z) is given by the equalities F′(z) = Φ^{2}(z) and ω(z) = Φ^{−2}(z) · Ψ^{2}(z). This immediately implies that, firstly, ω′(z) can have only a finite number of zeros on l_{j}^{0}, j = \overline{1, n}, and, secondly, ω(z) has a finite or infinite limit ω(t_{j}) = ω_{j} for z → t_{j}. Hence on account of the chosen orientation of the surface and equation (4) we conclude that if X_{j−1}Y_{j} − X_{j}Y_{j−1} ≠ 0, then lim_{z→t_{j}} ω(z) = (ν_{1}^{(j)} + iν_{2}^{(j)})(1 − ν_{3}^{(j)})^{−1}, where (ν_{1}^{(j)}, ν_{2}^{(j)}, ν_{3}^{(j)}) = − sign(X_{j−1}Y_{j} −X_{j}Y_{j−1})·|T_{j−1} ×T_{j}|^{−1} ·T_{j−1} ×T_{j}. The case X_{j−1}Y_{j} − X_{j}Y_{j−1} = 0 will be considered below.

Let α_{j}, |α_{j}| < 1, j = \overline{1, n}, be the numbers defined by the equalities sin πα_{j} = (T_{j−1}, T_{j}, ν^{(j)}), cos πα_{j} = T_{j−1} · T_{j}.

Lemma 1. Let S ∈ 𝔖 and α_{j} < 0. Then the plane passing through [Γ_{j−1}] and [Γ_{j}] crosses S in any neighborhood of the point M_{j}.

Proof. The formulation of the lemma implies that without loss of generality it can be assumed that Γ_{j−1} and Γ_{j} lie in the plane Π and M_{j} = 0. By the symmetry principle the harmonic function x^{3}(z) = Re ∫_{0}^{z} F′(t)ω(t)dt + c_{3} continues through the arc θ_{j−1} < θ < θ_{j+1}, while the holomorphic function F′(z) in the semi-neighborhood V (t_{j}) ∩ E of the point t_{j} can be written in the form F′(z) = (z − t_{j})^{−α_{j}}F_{0}(z), where F_{0}(z) is holomorphic and non-vanishing [6]. Since in t_{j} the function F′(z)ω(z) cannot have zero of nonintegral order, by virtue of the boundedness of the harmonic function ω(z), x^{3}(z) has zero of at least second order at the point t_{j}. Hence, on account of the familiar result ([7, Theorem 2.1]) we conclude that x^{3}(z) changes its sign in V (t_{j}) ∩ E.

Let now S ∈ 𝔖 and X_{j−1}Y_{j} − X_{j}Y_{j−1} = 0, i.e., the plane Q passing through [Γ_{j−1}] and [Γ_{j}] is orthogonal to Π. Then by the proven lemma, for any δ > 0 in V_{s}(M_{j}, δ) = B(M_{j}, δ)∩S, where B(M, δ) is the ball with center M and radius δ, there are points lying in different half-spaces into which R^{3} is divided by Q. We take two such points and connect them by the curve l, [l] ⊂ V_{s}(M_{j}, δ), intersecting Q in some point p_{0} and let p_{1} ∈ [Γ_{j−1}] ∪ [Γ_{j}] π(p_{1}) = π(p_{0}). Let Π_{1} = {x ∈ ℝ^{3}, ax^{1} + bx^{2} = c} be the plane orthogonal to the plane Π and passing through p_{0}, t′ = e^{iθ′} = x^{−1}(p_{1}), z_{0} = x^{−1}(p_{0}),

l′ [l′] ⊂ E be the curve \frac{a}{2}[ Re ∫_{0}^{z} F′(1 − ω^{2})dt + c_{1}] + \frac{b}{2}[ Re i ∫_{0}^{z} F′(1 +

ω^{2})dt + c_{2}] = c connecting the points z_{0} and t′, and l′′ = x ∘ l′. Assuming l′′ to be parametrized by the natural parameter x = x(s), 0 ≤ s ≤ s_{0}, x(0) = p_{0}, x(s_{0}) = p_{1}, we shall consider the function d(s) = |π(p(s))−π(p_{1})| on 0 ≤ s ≤ s_{0}. Since d(s) ≥ 0, d(s) ∈ C([0, s_{0}]) and d(0) = d(s_{0}) = 0, there exists s^{∗}, 0 < s^{∗} < s_{0}, such that d(s^{∗}) is a maximum value. It can be easily verified that the point x(s^{∗}) cannot have in S a neighborhood in which π is injective. If however d(s) ≡ 0, then l′′ is a segment. Thus if S ∈ 𝔖, then it is necessary that (T_{j−1}, T_{j}, ν_{j}) > 0, which uniquely fixes the position of ω_{j} for |ω_{j}| = 1.

Denote by B_{j} a circumference described by equation (4). Let ω_{j} ≠ ω_{j+1}. Denote by b̃_{j} an arc of B_{j} with the initial point ω_{j}, the end point ω_{j+1} and wholly lying in the closed unit circle. Such arcs are available, since |ω_{j}| ≤ 1 and B_{j} are the stereographic images of circumferences of a sphere. If ω_{j} = ω_{j+1}, then [b̃_{j}] = ω_{j}. Let b̃_{j_{1}}, b̃_{j_{2}}, …, b̃_{j_{m_{0}}} be all the arcs of positive length.

It will always be assumed here that b̃ = ∏_{k=1}^{m_{0}} b̃_{j_{k}} is a Jordan curve. Denote by D_{0} the component ℂ\[b̃] which lies in the unit circle.

Lemma 2. If ω parametrizes S ∈ 𝔖, then ω(z) is univalent in D_{0} and ω(E) ⊆ D_{0}.

Proof. Since [b_{j}] ⊂ [B_{j}], where b_{j} = ω ∘ l_{j}, for ω ∉ ∪^{n}_{j=1}[B_{j}] we have

1 ≤ \frac{1}{2π} ∫_{0}^{2π}d Arg(ω(e^{iθ}) − ω_{0}) = \frac{1}{2π} ∫ ; _{b} d Arg(ω − ω_{0})

which, by virtue of the Jordanian property of b̃, implies

ind_{ω_{0}} b̃ = \frac{1}{2π} ∫ ; _{b} d Arg(ω − ω_{0}) = \begin{cases}1, ω_{0} ∈ D_{0}, \\ 0, ω_{0} ∈ C\overline{D}_{0}. \end{cases} \begin{matrix}(7) \\ (7′) \end{matrix}

The latter equality implies ω′(E)\ ∪^{n}_{j=1} [B_{j}] ⊂ D_{0}. On assuming that there

is a point z′ ∈ E such that ω(z′) ∈ ∪^{n}_{j=1}[B_{j}] ∩ C\overline{D}_{0} and recalling that the

mapping ω is open, there will exist z′′ ∈ E, ω(z′′) ∉ ∪^{n}_{j=1}[B_{j}] such that

ω(z′′) ∈ C\overline{D}_{0}.

If 𝔖 ≠ ∅, then equality (7) holds for any ω_{0} ∈ D_{0} and can be regarded as a necessary condition for the family 𝔖 to be nonempty. In this connection note that for the family of boundaries considered in [3] the topological indices corresponding to these boundaries are negative.

Let us discuss the degree of freedom of function ω(Z). Writing S in terms of isothermic coordinates

x^{j}(z) = Re ∫_{0}^{z} φ_{j}(t)dt + c_{j}, j = \overline{1, 3},

where φ_{j}(z) are holomorphic in E, we obtain [2] ω(z) = φ_{3}(z)·(φ_{1} −iφ_{2})^{−1}. Therefore ω(z) is defined up to a conformal automorphism E and hence for uniqueness of ω(z) we should use some way of normalizing the mapping x. But since normalization of x is equivalent to the normalization of ω = σ ∘ g ∘ x, we can use one of the standard ways of normalizing the conformal homeomorphism ω. Namely, we choose the points ω_{0} ∈ D_{0}\ ∪^{n}_{j=1} [B_{j}] and

ω_{1} ∈ b̃_{n} and normalize ω(z) by the condition

ω(0) = ω_{0}, ω(1) = ω_{1}.  (8)

The function ω corresponding to the surface S and normalized by the condition (8) will be denoted by ω_{s}(z).

§ 3. Admissible Boundaries

Denote by Ω a class of functions ω(z) univalent holomorphic in E and continuous in \overline{E}, which are normalized by (8) and satisfy the following conditions:

(a) |ω(z)| ≤ 1, z ∈ \overline{E};

(b) for each ω(z) ∈ Ω there exist n points (depending on ω) t_{j}(ω) = e^{iθ_{j}(ω)}, j = \overline{1, n}, 0 < θ_{1}(ω) < ⋯ < θ_{n}(ω) < 2π, such that ω(z) is analytically continuable through l_{j}^{0}(ω), where l_{j}(ω) = {t = e^{iθ}, θ_{j}(ω) ≤ θ ≤ θ_{j+1}(ω)};

(c) ω(t) ∈ [B_{j}], t ∈ l_{j}, j = \overline{1, n}, θ_{n+1} = θ_{1}.

Denote by μ_{ω}(z) the order of zero of ω′ at the point z.

Lemma 3. For ω ∈ Ω we have

(1) μ_{ω}(z) ≤ 1, z ∈ l_{j}^{0}, j = \overline{1, n};

(2) ∑_{z∈l_{j}^{0}} μ_{ω}(z) ≤ 2, j = \overline{1, n}.

Proof. (1) Let μ(z_{0}) ≥ 2, z_{0} ∈ l_{j}^{0}. Then in the neighborhood U(z_{0}) of the point z_{0} we have ω(z)−ω(z_{0}) = (z−z_{0})^{μ+1}v(z), where v(z) is a nonvanishing holomorphic function in U(z_{0}). Denote by v_{1}(z) an arbitrary regular branch of \sqrt[μ+1]{v(z)} in U(z_{0}). As can be easily verified, ζ = (z − z_{0})v_{1}(z) · ζ(z) is a univalent function in some neighborhood U_{1}(z_{0}). Denoting by z = h(ζ), ζ ∈ ζ(U_{1}(z)) the inverse function of ζ(z), we obtain ω(h(z_{1}))−ω(z_{0}) = ζ^{μ+1}, which means that ω(h(z)) cannot be a univalent function in any component of ζ(U_{1}(z_{0})\[l_{j}]). Thus, taking into account that ζ(z) is univalent, we obtain a contradiction with the assumption that ω(z) is univalent.

(2) Let t_{j,k} = e^{iθ_{j,k}}, k = \overline{1, m_{j} − 1}, θ_{j} = θ_{j,0} < θ_{j,1} < θ_{j,2} < θ_{j,3}, ⋯ < θ_{j,m_{j}} = θ_{j+1}, be the critical points of ω on l_{j}^{0}. Let l_{j,k} = {t = e^{iθ}, θ_{j,k} < θ < θ_{j,k+1}\}, k = \overline{0, m_{j} − 1}, and b_{j,k} = ω ∘ l_{j,k}. Since ω(e^{iθ}) ∈ [B_{j}], e^{iθ} ∈ l_{j} and, as it was proved, all critical points are simple, we obtain ω(U(e^{iθ_{j,1}}) ∩ E) ⊃ U(ω(e^{iθ_{j,1}}))∩[b_{j,0}], and hence [b_{j,1}^{0}]∩[b_{j,0}^{0}] ≠ ∅. If we assume that ω(e^{iθ_{j,2}}) ∈ [b_{j,0}], then we have ω(U(e^{iθ_{j,2}})) ∩ E) ⊃ U(ω(e^{iθ′})\[b_{j,1}], where θ_{j,0} < θ′ < θ_{j,1}, which contradicts the assumption that ω(z) is univalent. Therefore ω(e^{iθ_{j,2}}) ∉ [b_{j,0}]. Since ω(U(e^{iθ_{j,2}})∩E) ⊃ U(ω(e^{iθ_{j,2}}))\[b_{j,2}^{0}], we have [b_{j,3}^{0}]∩ [b_{j,2}^{0}] ≠ ∅ and therefore either ω(e^{iθ_{j,3}}) ∈ [b_{j,2}^{0}] or ω(e^{iθ_{j,1}}) ∈ [b_{j,3}^{0}]. In both cases, repeating the above reasoning, we come to a contradiction with the property that ω(z) is univalent.

Lemmas 2 and 3 imply that if on l_{j}^{0} there are two critical points, then ω(e^{iθ_{j,2}}) ∉ [b_{j,0}] ∪ [b̃_{j}]. Denote by n_{j} the number of critical points ω(z) on l_{j}^{0}. Let ω ∈ Ω, |b̃_{j}| ≠ 0. If n_{j} = 0, then b_{j} = b̃_{j}. If n_{j} = 1, then either [b_{j,0}] ⊃ [b̃_{j}], or [b_{j,1}] ⊃ [b̃_{j}]. In the former case b_{j} = b̃_{j}b_{j,1}^{−1} · b_{j,1}, in the latter case b_{j} = b_{j,0} · b_{j,0}^{−1} · b̃_{j}. If n_{j} = 2, then b_{j} = b_{j,0} · b_{j,0}^{−1} · b̃_{j} · b_{j,2}^{−1} · b_{j,2}. For |b̃_{j}| = 0, by the definition of the class Ω we have n_{j} ≥ 1, and thus we obtain b_{j} = b_{j,0} · b_{j,0}^{−1} for n_{j} = 1 and b_{j} = b_{j,0} · b_{j,0}^{−1} · b_{j,2}^{−1} · b_{j,2} for n_{2} = 2. Introducing the notation b_{j,0} · b_{j,0}^{−1} = p_{2j−1} and

p_{2j} = \begin{cases}b_{j,1}^{−1} · b_{j,1} for n_{j} = 1, \\ b_{j,2}^{−1} · b_{j,2} for n_{j} = 2 \end{cases}

in all cases we shall have b_{j} = p_{2j−1}b̃_{j}p_{2j}, where each of the factors can have a zero length but |b_{j}| ≠ 0. Note that if |b̃_{j}| ≠ 0, then [p_{2j}] ∩ [p_{2j−1}] = ∅. If however |b̃_{j}| = 0 and |p_{2j−1}| · |p_{2j}| = ∅, then [p_{2j−1}] ∩ [p_{2j}] = ω_{j}.

Let ω_{j−1} ≠ ω_{j} = ω_{j+1} = ⋯ = ω_{j+ν_{j}} ≠ ω_{j+ν_{j}+1}, ν_{j} ≥ 1. We write it in the form j ∈ T and introduce the notation j̃ = {j, j + 1, …, j + ν_{j} − 1}, j^{∗} = {2j − 2, 2j − 1, …, 2j + 2ν_{j} − 1}.

Let B′_{2k−1} : z = φ_{2k−1}(s), 0 ≤ s ≤ δ_{2k−1}, B′_{2k} : z = φ_{2k}(s), 0 ≤ s ≤ δ_{2k}, φ_{2k−1}(0) = ω_{k}, φ_{2k}(0) = ω_{k+1} be the arcs of the circumference B_{k} of positive length, satisfying the following conditions:

(a) If φ_{m}(s′) ∉ D_{0}, then φ_{m}(s) ∉ D_{0}, s′ < s ≤ δ_{m};

(b) [B′_{2k−1}] ∩ [b̃_{k}] = ω_{k}, [B′_{2k}] ∩ [b̃_{k}] = ω_{k+1}, [(B′_{2k−1})^{0}] ∩ [(B′_{2k})^{0}] = ∅;

(c) If k ∈ j̃, then β(b̃_{j−1}, B′_{2k−1}) ≥ 0.

We denote the restriction of B′_{m} on [0, s] by B′_{m}(s). Let σ_{m} = max s : \{s ∈ [0, δ_{m}], φ_{m}(s) ∈ \overline{D}_{0}. Let V denote a subset of the set ℕ_{n} = \{1, 2, …, n\} such that σ_{2m−2} > 0, m ∈ V.

We denote by L the set of curves of the form

L = ∏_{k=1}^{n} b_{k} = ∏_{k=1}^{n} p_{2k−1}b̃_{k}p_{2k},  (9)

where p_{m} = B′_{m}(s)·(B′_{m}(s))^{−1}, 0 ≤ s ≤ s_{m} < σ_{m} and for which there exists ω ∈ Ω such that L = ω ∘ ∂E when the initial point of ∂E coincides with t_{1}(ω). The curves contained in L are called admissible boundaries.

Theorem 1. For the set L to be non-empty it is necessary and sufficient that, together with condition (7), for each j ∈ T there would exist a set P_{j}, P_{j} ⊂ j^{∗}, such that

(1) σ_{m} > 0 for m ∈ P_{j};

(2) P_{j} ∩ \{2j + 2k − 1, 2j + 2k\} ≠ ∅, k = \overline{0, ν_{j−1}};

(3) for k, m ∈ P_{j} and k < m there holds

β(b̃_{j−1}B′_{k}) > β(b̃_{j−1}B′_{m}).  (10)

Proof. The necessity of condition (7) is proved by repeating the arguments used in proving Lemma 2. Let L ≠ ∅, L ∈ L, and ω_{L}(z) ∈ Ω be the function conformally mapping E onto D_{0}\[L]. Let j ∈ T. As P_{j} we shall choose a subset of m ∈ j^{∗} for which |p_{m}| ≠ 0. Conditions (1) and (2) will be fulfilled by virtue of the definition of the class Ω. To prove (3) we choose, on l_{k}^{0}, a point z_{k} and connect it with t_{j−1}(ω) by a simple curve γ in E. Let D_{γ} be the domain lying in E and bounded by the arc γ and the arc of the unit circumferece from t_{j−1} to z_{k} and containing the point t_{k}. The assumption β(b̃_{j−1}, B′_{k}) < β(b̃_{j−1}, B′_{m}) would imply [b_{m}^{0}] ∩ D_{γ} ≠ ∅, which obviously contradicts the fact that the function ω is univalent.

To prove the sufficiency we choose ε > 0 so small as to make the set

D_{0}\ ∪^{2n}_{m=1} [B′_{m}(ε)] connected. Next we consider a curve of form (9), where

0 < |p_{m}| < 2ε, m ∈ ∪_{j∈T}(P_{j}), and |p_{m}| = 0 for m ∉ ∪_{j∈T}P_{j}. Let again ω_{L}(z)

be the function conformally mapping E onto the domain D_{1} = D_{0}\[L] and normalized by condition (8), A_{k} = φ_{k}(ε), k ∈ ∪_{j∈T}P_{j}, and ε_{1} > 0

be so small that U(A_{k}, ε_{1})\[B′_{k}(ε)] ⊂ D_{1}. Let a(ω) be a homographic transformation corresponding to the rotation of the sphere mapping the point (X_{k}, Y_{k}, Z_{k}) into (0, 1, 0). The function f(z) = \sqrt{a(ω(z)) − a(A_{k})} maps ω^{−1}(U(A_{k}, ε_{1}))\[B′_{k}(ε)] onto the semi-circle and therefore f(z) analytically continues through the arc of the unit circumference. But ω(z) = a^{−1}(f^{2}(z) + a(A_{k})) and hence ω(z) is holomorphic in the neighborhood of ω^{−1}(A_{k}). Moreover, since L consists of a finite number of circumference arcs, then each point of L is an accessible point for D_{1} and therefore ω(z) continues up to the homeomorphism ω^{∗} : \overline{E} → D_{1}∪L^{∗} [8], where L^{∗} denotes the set of prime ends of the domain D_{1}. By inequality (10) and the assumption that ω is univalent, the pre-image of the prime end ω_{k} will precede the pre-image of the prime end ω_{m} when L is described in the positive direction with respect to D_{1}. Denoting by l_{k} an arc of the unit circumference having initial point at (ω^{∗})^{−1}(ω_{k}) and end at (ω^{∗})^{−1}(ω_{k+1}) and described counteclockwise, we shall have ω^{∗} ∘ l_{k} = B′_{k}(ε) · (B′_{k}(ε))^{−1}.

Theorem 1 (as follows from its proof) gives a criterion for of a curve of form (9) to belong to the set L : L ∈ L ≠ ∅ iff D_{0}\[L] is connected, |b_{n}| ≠ 0, k = \overline{1, n}, and for any k ∈ j^{∗}, m ∈ j^{∗}, σ_{m} > 0, σ_{k} > 0, k < m and |p_{k}(L)|, |p_{m}(L)| ≠ 0 there holds

β(b̃_{j−1}, p_{k}(L)) > β(b̃_{j−1}, p_{m}(L)).  (11)

Moreover, in view of the equality β(b̃_{j−1}, B′_{2k}) = β(b̃_{j−1}, B′_{2k−1}) − 1, the proven theorem readily implies

Corollary 1. If L ∈ L ≠ ∅ and for k ∈ j̃, |p_{2k−1}(L)| ≠ 0, then |p_{2m}(L)| = 0, m = \overline{j, k − 1}. If |p_{2k}(L)| ≠ 0, then |p_{2m−1}(L)| = 0, m = \overline{k + 1, j + ν_{j}}.

Corollary 2. If L ∈ L ≠ ∅ and β_{k}(L) = β(b_{k−1}(L), b_{k}(L)) < 0, then σ_{2k−2} > 0 and |p_{2k−1}(L)| = |p_{2k−2}(L)| = 0.

§ 4. Parametrization of the Set L

Let I : L → ℕ_{2n} = {1, 2, …, 2n}, where I(L) = {i_{k_{1}}, i_{k_{2}}, …, i_{k_{ρ(L)}}} is the ordered set of indices of curves p_{i_{ks}} (L) of positive length contained in L ∈ L. We introduce the notation F_{k} = {2k−1, 2k} and assume that i_{k_{s}} ∈ F_{k_{s}}. Let further I = I(L), β^{−}(L) be a subset of the set {β_{1}(L), β_{2}(L), …, β_{n}(L)}, consisting of negative numbers and I^{−}(L) = {i ∈ ℕ_{n}, β_{i} < 0}, κ(L) = card β^{−}(L).

Two curves L_{1} ∈ L and L_{2} ∈ L will be called equivalent (L_{1} ∼ L_{2}) if I(L_{1}) = I(L_{2}). We denote by L_{i} the equivalence class I(L) = i if i ∈ I. Since ρ(L_{1}) = ρ(L_{2}) and κ(L_{1}) = κ(L_{2}) obviously hold for L_{1} ∼ L_{2}, the notations ρ(i) and κ(i) are correct.

Let a ⊂ ℕ_{2n}, i ∈ I, and a ∩ i = ∅. We write a ∈ E(i) if i ∪ a ∈ I. Let a ⊂ ℕ_{2n}, i ∈ I and a ⊆ i. We write a ∈ R(i) if i\a ∈ I. i ∈ I is called maximal if E(i) = ∅. The set of maximal i’s is denoted by I′. For R(i) ≠ ∅, i is called reducible.

Let L = ∏_{k=1}^{n} p_{2k−1}b̃_{k}p_{2k} ∈ L, i_{k} ∈ I(L) and t ≥ 0. Denote by r_{i_{k}}^{t}(L) the curve defined by the following conditions:

|p_{m}(r_{i_{k}}^{t}(L))| = |p_{m}(L)|, m ≠ i_{k}
|p_{i_{k}}(r_{i_{k}}^{t}(L))| = t|p_{i_{k}}(L)|.

It is obvious that for any L ∈ L there exists ε > 0 such that r_{i_{k}}^{t}(L) ∈ L, 1 − ε < t < 1 + ε, i_{k} ∈ I(L). From the definition of R(i) it follows that a = {i_{m_{1}}, i_{m_{2}}, …, i_{m_{p}}} ∈ R(i) iff r_{a}(L) = r_{m_{1}}^{0} ∘ r_{m_{2}}^{0} ∘ ⋯ r_{m_{p}}^{0}(L) ∈ L.

Let now β_{k}(L) ∈ β^{−}(L) ≠ ∅, L ∈ L. By virtue of Corollary 2 of Theorem 1 σ_{2k−2} > 0 and therefore there exists ε, 0 < ε < min(σ_{2k−2}, σ_{2k−1}), such that a curve e_{2k}^{ε}(L) of form (9) defined by the conditions |p_{m}(e_{2k−2}^{ε}(L))| = |p_{m}(L)|, m ≠ 2k − 2, |p_{2k−2}(e_{2k−2}^{ε}(L))| = ε, belongs to L. In a similar manner we define e_{2k−1}^{ε}(L):

|p_{m}(e_{2k−1}^{ε}(L))| = |p_{m}(L)|, m ≠ 2k − 1, |p_{2k−1}(e_{2k−1}^{ε}(L))| = ε.

We introduce the notation H_{k} = {2k − 2, 2k − 1}. Let i ∈ I, β^{−}(i) = {β_{k_{1}}, β_{k_{2}}, …, β_{κ(i)}}, {k′_{1}, k′_{2}, …, k′_{m}} ∈ I^{−}(L), L ∈ L_{i}, a = (a′_{1}, a′_{2}, …, a′_{m}) ∈ H_{k′_{1}} ×H_{k′_{1}} ×⋯×H_{k′_{m}}. For L ∈ L_{i} we write the notation e_{a}^{ε} = e_{a′}^{ε′} ∘e_{a′_{2}}^{ε_{2}} ∘⋯∘ e_{a′_{m}}^{ε_{m}}(L). It can be easily verified that r_{a}(i) = I(r_{a}^{0}(L)) and e_{a}(i) = I(e_{a}^{ε}(L)) do not depend on an order of r_{m_{k}}^{0} and e_{a′_{k}}^{ε_{k}} in the definition of r_{a}(L) and e_{a}^{ε}(L).

Lemma 4. If (a) σ_{2k} = 0 or (b) β((B′_{2k−3})^{−1}, B_{2k}) > 0, 2k − 3 ∈ j^{∗}, 2k ∈ j^{∗} holds for any k ∈ j̃, j ∈ T, then i ∩ j^{∗} = i′ ∩ j^{∗} for any i ∈ I, i′ ∈ I.

Proof. In the case (a) we have |p_{2k−1}(L)| ≠ 0, L ∈ L and therefore for j ≤ m ≤ k, by virtue of Corollary 1 of Theorem 1, we obtain |p_{2m}(L)| = 0 for any L ∈ L. If σ_{2m} > 0 holds for some m, k < m ≤ j + ν_{j} − 1, then β(b̃_{j−1}, B′_{2m}) + 1 > β(b̃_{j−1}, B′_{2k−1}) and hence |p_{2m−1}(L)| = 0 for arbitrary m, k < m ≤ j + ν_{j} − 1, and any L ∈ L. In the case (b), by Corollary 2 we have |p_{2k−2}(L)| = |p_{2k−1}(L)| = 0 and thus |p_{2k−3}(L)| · |p_{2k}(L)| ≠ 0 holds also for any L ∈ L.

Lemma 5. i ∈ I′ iff κ(i) = 0, i ∈ I.

Proof. Let κ(i) = 0. As the preceding lemma suggests, it is of interest for us to consider only the case where k ∈ j̃, j ∈ T, σ_{2k−2} > 0, β((B′_{2k−3})^{−1}, B_{2k}) < 0. Then either F_{k−1} ⊂ i and F_{k} ∩i = 2k and therefore 2k − 1 ∉ E(i) or F_{k} ⊂ i and F_{k−1} ∩ i = 2k − 3 and hence 2k − 2 ∉ E(i), i.e., i ∈ I′. Conversely, let β_{k} < 0. Then by Corollary 2 we have 2k − 2 ∈ E(i), 2k − 1 ∈ E(i) for L ∈ L_{i} and

β_{k}(e_{2k−1}^{ε}(L)) > 0, β_{k}(e_{2k−2}^{ε}(L)) > 0.  (12)

From the definition of r_{a}(i) and e_{a}(i) and inequalities (12) it follows that

κ(r_{a}(i)) = κ(i) + card a, ρ(r_{a}(i)) = κ(i) − card a,
κ(e_{a}(i)) = κ(i) − card a, ρ(e_{a}(i)) = κ(i) + card a.

(13)

Lemma 6. If in I there exists a maximal non-reducible multi-index i, then I = i.

Proof. Let i_{k_{s}} ∈ i. Since i is non-reducible, we have k_{s} ∉ V, |b̃_{k_{s}}| = 0 and i_{k_{s}} = F_{k_{s}} ∩ i = 2k_{s} − 1. Therefore i_{k_{s}} ∈ i′ for any i′ ∈ I, i.e., i ⊆ i′ for any i ∈ I. But since i is maximal, we obtain i = i′.

Theorem 2. ρ(i) + κ(i) ≡ ρ = const, i ∈ I.

Proof. First, we shall prove ρ(i) ≡ const, i ∈ I′. As suggested by Lemma 6, we should consider only the case with reducible i ∈ I′. Let i_{k_{s}} ∈ R(i), i ∈ I′. Then by Corollary 2 card(i∩H_{k_{s}}) = 1. Therefore ρ(i) = ρ(i′), i, i′ ∈ I′. If however i ∉ I′, then κ(i) > 0 and therefore there exists β_{k}(i) < 0. But then e_{2k−1}(i) ∈ I, and by (13) we have ρ(e_{2k−1}(i)) + κ(e_{2k−1}(i)) = ρ(i) + κ(i).

Let i ∈ I′, and τ_{i} : I′ → ℕ_{ρ} be the order preserving bijection. We define the mapping χ : L′ = ∪_{i∈I′}L_{i} → ℝ_{ρ} as follows:

χ_{m}(L) = (−1)^{τ_{L}^{−1}(m)} · |p_{τ_{L}^{−1}(m)}(L)|, m ∈ ℕ_{ρ}, where τ_{L} = τ_{I(L)}.

Let us show that χ is injective. The statement is obvious for L_{1} ∼ L_{2}. If I(L_{1}) ≠ I(L_{2}), then since for k ∈ j̃ ⊂ V, I(L_{1}) ∩ H_{k} ≠ I(L_{2}) ∩ H_{k} there exists m ∈ ℕ_{ρ} such that τ_{L_{1}}^{−1}(m) ∈ H_{k}, τ_{L}^{−1}(m) ∈ H_{k}, τ_{L_{1}}^{−1}(m) ≠ τ_{L_{2}}^{−1}(m) and therefore χ_{m}(L_{1}) and χ_{m}(L_{2}) will have different signs. Let now L ∈ L_{i}, i ∉ I′, a ∈ H_{m_{1}} × H_{m_{2}} × ⋯ × H_{m_{κ(L)}}, m_{j} ∈ I^{−}(L), j = \overline{1, κ(L)}. Then I(e_{a}^{ε}(L)) ∈ I′. Define χ(L) by setting

χ_{m}(L) = \begin{cases}χ_{m}(e_{a}^{ε}(L)), & τ_{e_{a}^{ε}(L)}^{−1}(m) ∈ I(L), \\ 0, & τ_{e_{a}^{ε}(L)}^{−1}(m) ∉ I(L). \end{cases}  (14)

It is obvious that χ defined in this manner does not depend on the choice of a. The injectivity of the mapping χ defined by (14) is proved similarly.

Theorem 3. χ(L) is a domain.

Proof. We shall show that χ(L) is open. Let χ(L) = (χ_{1}(L), …, χ_{ρ}(L)), L ∈ L, and χ_{k}(L) ≠ 0. We introduce the notation U_{k} = {χ_{k} = χ_{k}(r^{t}(L)), 1 − ε_{k} < t < 1 + ε_{k}}, where ε_{k} > 0 is so small that D_{0}\[r_{τ_{L}(k)}^{t}(L)] is connected for any t, |t − 1| < ε. If χ_{k}(L) = 0, then, taking ε > 0 so small that D_{0}\[e_{a}^{ε}(L)], a ∈ H_{m_{1}} × H_{m_{2}} × ⋯ × H_{m_{κ}(L)}, is connected, we find that τ_{e_{a}^{ε}(L)}^{−1}(k) is equal either to 2m_{j} − 2 or to 2m_{j} − 1, m_{j} ∈ I^{−}(L). We write

U_{k}^{+} = {χ_{j} = χ_{j}(L), j ≠ k, χ_{k} = χ_{k}(e_{2m_{j}−2}^{t}(L)), 0 ≤ t < ε},
U_{k}^{−} = {χ_{j} = χ_{j}(L), j ≠ k, χ_{k} = χ_{k}(e_{2m_{j}−1}^{t}(L)), 0 < t < ε},
U_{k} = U_{k}^{+} ∪ U_{k}^{−}.

It can be easily verified that U(L) = U_{1} × U_{2} × · × U_{ρ} ⊂ χ(L) is the neighborhood of χ(L).

Now we shall show that χ(L) is connected. Fix ε_{0} > 0 such that

D_{0}\ ∪^{2n}_{m=1} [B′_{m}(ε_{0})] is connected. Let p(L)= min_{i_{s}∈I(L)} |p_{i_{s}}(L)| and assume that

L_{1} ∼L_{2}, I(L_{1})=i ∈ I′. Fix the number ε_{1}, 0<ε_{1} <min (ε_{0}, |p(L_{1})|, |p(L_{2})|). Let ψ_{s}(t) be a monotonically decreasing function on [s−1, s], ψ_{s}(s−1) = 1, ψ_{s}(s) = ε_{1} · |p_{τ_{i}^{−1}(s)}(L_{k})|^{−1}, s = \overline{1, ρ}. Denote by g_{s}(L_{k}) a segment g_{s}(L_{k}) =

{χ_{m} = χ_{m}(L_{k}), m ≠ τ_{i}(i_{s}); χ_{τ_{i}(i_{s})} = χ_{τ_{i}(i_{s})}(r_{i_{s}}^{ψ_{s}(t)}(L_{k})). By construction, [g_{s}(L_{k})] ⊂ χ(L), k = 1, 2, s = \overline{1, ρ}, and therefore the graph of the broken line g(L_{1}, L_{2}) = g(L_{1}) · (g(L_{2}))^{−1}, where g(L_{k}) = ∏_{s=1}^{ρ} g_{s}(L_{k}) and which connects χ(L_{1}) and χ(L_{2}), is contained in χ(L).

Let now I(L_{1}) ∈ I′, I(L_{2}) ∈ I′, I(L_{1}) ≠ I(L_{2}). We write L_{k}^{ε_{1}} = χ^{−1}(g(L_{k})(ρ)), k = 1, 2. As suggested by Lemma 4, it is of interest for us to consider only the case where I(L_{1})∩j^{∗} ≠ I(L_{2})∩j^{∗}, σ_{2k} > 0, k ∈ j̃ ⊂ V, and β((B′_{2k−3})^{−1}, B′_{2k}) < 0 for any k ∈ j̃, 2k − 3 ∈ j^{∗}, 2k ∈ j^{∗}.

Let F_{k_{1}} ⊂ I(L_{1}), F_{k_{2}} ⊂ I(L_{2}), k_{1} ∈ j̃, k_{2} ∈ j̃, k_{1} < k_{2}. Then by Corollary 1 of Theorem 1 we have 2m−1 ∈ I(L_{1}), j ≤ m ≤ k_{1}, 2m ∉ I(L_{1}), j ≤ m ≤ k_{1} − 1; 2m − 1 ∉ I(L_{1}), k_{1} + 1 ≤ m < j + ν_{j} − 1; 2m ∈ I(L_{1}), k_{1} + 1 ≤ m < j + ν_{j} − 1; 2m − 1 ∈ I(L_{2}), j ≤ m ≤ k_{2}; 2m ∈ I(L_{2}), j ≤ m ≤ k_{2} − 1; 2m − 1 ∉ I(L_{2}), k_{2} + 1 ≤ m ≤ j + ν_{j} − 1; 2m ∈ I(L_{2}), k_{2} + 1 ≤ m ≤ j + ν_{j} − 1. Since 2k_{1} ∈ R(I(L_{1})), the segment

h_{2k_{1}}(L_{1}) = {χ_{q} = χ_{q}(L_{1}^{ε}), q ≠ τ_{L_{1}}(2k_{1}); χ_{τ_{L_{1}} (2k_{1})} =
= χ_{τ_{L_{1}} (2k_{1})}(r_{2k_{1}}^{φ_{2k_{1}} (t)}(L_{1}^{ε_{1}}))},

where φ_{2k_{1}}(t) is a decreasing function on [0, 1], φ_{2k_{1}}(0) = 1, φ_{2k_{1}}(1) = 0, is contained in χ(L), including the point h_{2k_{1}}(1), and β_{k_{1}}(r_{2k_{1}}^{φ_{2k_{1}} (1)}(L_{1}^{ε_{1}})) = β_{k_{1}}(r_{2k_{1}}^{0}(L_{1}^{ε_{1}})) < 0. Therefore 2k_{1}+1 ∈ E(r_{2k_{1}}^{0}(L_{1}^{ε_{1}})) and hence the segment

h_{2k_{1}+1}^{∗} = {χ_{q} = χ_{q}(r_{2k_{1}}^{0}(L_{1}^{ε})), q ≠ τ_{L_{1}}(2k_{1}),
χ_{τ_{L_{1}} (2k_{1})} = χ_{τ_{L_{1}} (2k_{1})}(e_{2k_{1}+1}^{φ_{2k_{1}+1}^{∗}(t)}(r_{2k_{1}}^{0}(L_{1}^{ε_{1}}))},

where φ_{2k_{1}+1}^{∗}(t) is an increasing function on [1, 2], φ_{2k_{1}+1}^{∗}(1) = 0, φ_{2k_{1}+1}^{∗}(2) = ε_{1}, is contained in χ(L), including the end point. By con-

struction, we have F_{k_{1}+1} ⊂ I(e_{2k_{1}+1}^{φ_{2k_{1}+1}^{∗}(2)}(r_{2k_{1}}^{0}(L_{1}^{ε_{1}})) ∈ I′. Continuing the process, after a finite number of steps we shall obtain the broken line h = h_{2k_{1}} · h_{2k_{1}+1}^{∗} · h_{2k_{1}+2} · h_{2k_{1}+3}^{∗} ⋯ h_{2k_{2}−2} · h_{2k_{2}−1}^{∗} connecting the points χ(L_{1}) and χ(L_{2}) in χ(L).

Finally, let L ∈ L, I(L) ∉ I′, and I^{−}(L) = {m_{1}, m_{2}, …, m_{κ(L)}}, a ∈ H_{m_{1}} × H_{m_{2}} × … × H_{m_{κ(L)}}. Then the broken line f = f_{1} · f_{2} ⋯ f_{κ(L)}, where f_{s} is a segment in χ(L) : f_{s} = {χ_{q} = χ_{q}(L), q ≠ τ_{e_{a}^{ε_{1}} (L)}(a_{s});

χ_{τ_{e^{ε_{1}} (L)}(a_{s})} = χ_{τ_{e^{ε_{1}} (L)}(a_{s})}(e_{a_{s}}^{σ_{s}(t)}(e_{a_{s−1}}^{ε_{1}} ∘ ⋯ e_{a_{1}}^{ε_{1}}(L))), where σ_{s}(t) is an increasing function on [s − 1, s], σ(s − 1) = 0, σ(s) = ε_{1}, is also contained in χ(L). By construction, I(e_{a_{κ(L)}}^{ε_{1}} ∘ e_{a_{κ(L)−1}}^{ε_{1}} ∘ ⋯ e_{a_{1}}^{ε_{1}}) ∈ I′ and we have thus proved the connectedness of χ(L).

§ 5. Construction of a Minimal Surface by a Given Admissible Boundary

Let S ∈ 𝔖 be represented by formulas (1). By differentiating (1) with respect to θ, e^{iθ} ∈ l_{k}^{0}, k = \overline{1, n}, we obtain

\begin{aligned}(x^{1})_{θ} & = −\frac{1}{2} Im F′(e^{iθ})(1 − ω^{2}(e^{iθ}))e^{iθ}, \\ (x^{2})_{θ} & = −\frac{1}{2} Re F′(e^{iθ})(1 + ω^{2}(e^{iθ}))e^{iθ}, \\ (x^{3})_{θ} & = − Im F′(e^{iθ})ω(e^{iθ})e^{iθ}. \end{aligned}  (15)

Since S ∈ 𝔖 we have ((x^{1})_{θ}, (x^{2})_{θ}, (x^{3})_{θ}) = d_{k}(θ)T, d_{k} > 0 for e^{iθ} ∈ l_{k}^{0}. Consider (15) as a system of equations with respect to Re F′(e^{iθ}) and Im F′(e^{iθ}). The compatibility condition of system (15) is expressed by equation (4) and hence is fulfilled automatically. By solving (15) we obtain

F′(e^{iθ}) = −2i\frac{X_{k} − iY_{k} + Z_{k}\overline{ω}(e^{iθ})}{1 + |ω(e^{iθ})|^{2}}e^{iθ}d_{k}(θ)  (16)

or

Re(Q(t)tF′(t)) = 0,  (17)

where

Q(t) = X_{k} + iY_{k} + Z_{k}ω(t), t ∈ e^{iθ} ∈ l_{k}, k = \overline{1, n}.  (18)

Thus F′(z) is a solution of the Riemann–Hilbert boundary value problem (17) with a piecewise-continuous coefficient Q(t) which is analytic on each arc l_{k}^{0}. Moreover, as follows from [6], F′(z) is bounded in the neighborhood of the points t_{k_{j}}(ω), j = \overline{1, α^{−}}, corresponding to the negative values of α_{j} or, speaking in terms of [9], F′(z) belongs to the class h(t_{k_{1}}, t_{k_{2}}, …, t_{k_{α^{−}}}). The number α^{−} of points in whose neighborhood F′(z) is bounded is defined by the initial broken line Γ̃ and does not depend on L ∈ L.

We rewrite (17) as

Q(t)tF′(t) + \overline{Q}(t)\overline{t}\overline{F}′(t) = 0.

Since F^{−}(z) = \overline{F} (1/\overline{z}) is bounded at infinity, (F^{−}(z))′ has zero of second order at infinity. Moreover, for z = 1/\overline{ζ} we have

(\overline{\frac{dF}{dz}} ) = −ζ^{2} \frac{dF^{−}}{dζ}  (19)

and therefore on the unit circumference the boundary values \frac{dF^{+}}{dt} = \frac{dF}{dt}

and \frac{dF^{−}}{dt} will be connected through the relation

\overline{\frac{dF^{+}}{dt}} = −t^{2} \frac{dF^{−}}{dt}  (20)

which makes it possible to rewrite the boundary conditon (17) as

\frac{dF^{+}}{dt} = e^{2i Arg \overline{Q}(t)} \frac{dF^{−}}{dt} .  (21)

By direct calculations we ascertain that sin Arg(\overline{Q}(t_{k} − 0))/\overline{Q}(t_{k} + 0)) = − sin πα_{k}, cos Arg(\overline{Q}(t_{k} − 0))/\overline{Q}(t_{k} + 0)) = cos πα_{k}, k = \overline{1, n}, t_{n+1} = t_{1}.

On each arc l_{k} we choose a branch Arg \overline{Q}(t) such that

\frac{1}{π} [ Arg \overline{Q}(t_{k} − 0) − Arg(t_{k} + 0)] = −α_{k}.

By simple calculations we obtain \frac{1}{π} [ Arg \overline{Q}(t_{1} − 0) − Arg(t_{1} + 0)] = 2 − α_{1}. Let us rewrite the boundary condition (21) as

\frac{dF^{+}}{dt} = \frac{e^{2i Arg \overline{Q}(t)}}{t^{2}} t^{2} \frac{dF^{−}}{dt}  (22)

Introducing the notations Φ^{+}(z) = \frac{dF^{+}}{dz} and Φ^{−}(z) = z^{2} \frac{dF^{−}}{dz} and recalling that z^{2} \frac{dF^{−}}{dz} is bounded at infinity, we find that the index of the boundary value problem in the class of functions bounded at infinity and belonging to h(t_{k_{1}}, t_{k_{2}}, …, t_{k_{α^{−}}}) is equal to zero. Thus a general solution of problem (22) is given by the formula

Φ(z) = C exp \frac{1}{π} ∫_{0}^{2π} \frac{Arg \overline{Q}(e^{iθ}) − θ}{e^{iθ} − z} de^{iθ},  (23)

where C is an arbitrary complex constant. From (23) we have

\frac{dF^{+}}{dz} = C exp \frac{1}{π} ∫_{0}^{2π} \frac{Arg \overline{Q}(e^{iθ}) − θ}{e^{iθ} − z} de^{iθ}, |z| < 1,

\frac{dF^{−}}{dz} = \frac{1}{z^{2}} C exp \frac{1}{π} ∫_{0}^{2π} \frac{Arg \overline{Q}(e^{iθ}) − θ}{e^{iθ} − z} de^{iθ}, |z| > 1.

(24)

Condition (19) is fulfilled by an appropriate choice of the constant C. Using simple transformations [9], we find that C must satisfy the equality

\overline{C} exp ( − \frac{i}{π} ∫_{0}^{2π}[ Arg \overline{Q}(e^{iθ}) − θ]dθ = −C.  (25)

Assuming C = λ_{0}e^{iθ}, λ_{0} ≥ 0 we obtain

α = −\frac{1}{2π} ∫_{0}^{2π}[ Arg \overline{Q}(e^{iθ}) − θ]dθ ± \frac{π}{2} .  (26)

The substitution of (25) into (24) gives

\frac{dF}{dz} = ±λ_{0} exp ( − \frac{1}{2πi} ∫_{0}^{2π}[ Arg \overline{Q}(e^{iθ}) − θ]\frac{e^{iθ} + z}{e^{iθ} − z} dθ).  (27)

Let now S^{ω} be a minimal surface defined by (1), where F′(z) and Q(t) are given by (27) and (18), respectively. In (27) we take the sign “−” to show that S^{ω} ∈ 𝔖. For t ∈ l_{k}^{0} (27) implies

Arg F′(t) = Arg(X_{k} − iY_{k} + Z\overline{ω}(t)) − \frac{π}{2} − θ.  (28)

On account of (28), (4), and the equality |X_{k} − iY_{k} + Z\overline{ω}| = 1 we obtain by direct calculations

(x^{1})_{θ} = \frac{λ_{0}}{2} X_{k}|F′(e^{iθ})|(1 + |ω(e^{iθ})|^{2}),

(x^{2})_{θ} = \frac{λ_{0}}{2} Y_{k}|F′(e^{iθ})|(1 + |ω(e^{iθ})|^{2}),

(x^{3})_{θ} = \frac{λ_{0}}{2} Z_{k}|F′(e^{iθ})|(1 + |ω(e^{iθ})|^{2}).

Thus S^{ω} ∈ 𝔖 and heence we have the bijection δ : 𝔖 → φ.

Denote by Γ^{ω} the boundary S^{ω}. Passing to the limit in (27) as z → t ≠ t_{k}, we obtain [9]

|F′(e^{iψ})| = λ_{0} exp \frac{1}{2π} ∫_{0}^{2π}[ Arg \overline{Q}(e^{iθ}) − θ]ctg \frac{θ − ψ}{2} dθ,

which implies [2]

|Γ_{k}^{ω}| = \frac{λ_{0}}{2} ∫_{θ_{k}}^{θ_{k+1}}exp \{\frac{1}{2π} ∫_{0}^{2π}[ Arg \overline{Q}(e^{iθ}) − θ]ctg \frac{θ − ψ}{2} dθ\}(1 + |ω|e^{iθ}))^{2})dψ.

Fix λ_{0} so that ∑_{k=1}^{n−3} |Γ_{k}^{ω}| = 1. We obtain the bijective mapping P ∘ δ ∘ χ^{−1}(L) : χ(L) → P and hence the parametrization of the set P by the domain χ(L).

§ 6. Dimension of the Domain χ(L).

Let L ∈ L, ω_{L}(z) and F′_{ω}(z) be the functions constructed in the preceding sections. For each k, k ∈ ℕ_{n} the vector (ω_{L}(e^{iθ}))_{θ} is oriented along the tangent to circumference (4) at the point t = e^{iθ} and hence X_{k} + iY_{k} + Z_{k}ω(t) = if_{k}(t)(ω_{L}(e^{iν}))_{θ}, where f_{k}(t), t ∈ l_{k}, is the real function. This gives us

0 = Re (X_{k} + iY_{k} + Z_{k}ω(t))tF′(t) = f_{k}(t) Im tF′(t)(ω_{L}(e^{iθ}))_{θ}

so that Im F′(t)t(ω_{L}(t))_{θ}dθ^{2} = 0, from which in view of (ω_{L}(t))_{θ} = t\frac{dω_{L}(t)}{dt} we obtain

Im F′(t)ω′(t)t^{2}dθ^{2} = − Im F′(t)ω′(t)dt^{2} = 0.

Thus the quadratic differential η(z) = F′(z)ω′(z)dz^{2} is analytically continuable onto the entire Riemann sphere if for the point z^{∗} symmetric to z with respect to the unit circumference we set [10] η(z^{∗}) = \overline{η(z)}. Taking into account that ω′_{L}(z) ≠ 0, |z| < 1, and in the neighborhood of the point t_{k} we have ω′_{L}(z) = (z − t_{k})^{−β_{k}(L)}ω_{k}(z), where ω_{k}(z) is holomorphic and nonvanishing, in this neighborhood [11], and recalling the behavior of F′(z) in this neighborhood [6], we conclude that zeros and poles of the differential η(z) are located on the unit circumference. Moreover, by the definition of α_{k} and β_{k} and also on account of Lemma 3 the poles and zeros of η(z) can be only simple. Let I(L) ∈ I′. Then by virtue of Lemma 5 κ(L) = 0 and therefore η(z) cannot have zero at any of the points t_{k}(ω), k = \overline{1, n}. The poles of η(z) coincide with t_{k} corresponding to α_{k} > 0. Hence the number P (η) of the poles of η(z) on the unit circumference (and on the entire sphere) is equal to n − α^{−}. Furthermore, the differential η(z) cannot have zero at any of the points t_{k}(ω) and hence the number N(η) of zeros of η(z) coincides with the number of critical points ω_{L}(z) on ∪^{n}_{k=1}l_{k}^{0} or, as follows from Theorem 2, with the number ρ. On the other hand, we have the equality [10] ord η(z) = N(η) − P (η) = −4. Recalling that ρ is constant on I and substituting N(η) = ρ and P (η) = n − α^{−} into the above equality, we obtain

Theorem 4. If L ≠ ∅, then ρ = n − α^{−} − 4.

References

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(Received 22.01.1996)

Author’s address:

A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Aleksidze St., Tbilisi 380093 Georgia