Parametrization of extremals of Grötzsch’s problem
We give parametric expressions for extremals of Grötzsch’s problem.
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irk-123456789-1244032017-09-25T03:03:19Z Parametrization of extremals of Grötzsch’s problem Tamrazov, P.M. We give parametric expressions for extremals of Grötzsch’s problem. 2010 Article Parametrization of extremals of Grötzsch’s problem / P.M. Tamrazov // Український математичний вісник. — 2010. — Т. 7, № 4. — С. 570-583. — Бібліогр.: 9 назв. — англ. 1810-3200 2010 MSC. 30C, 30D, 30E, 31A, 31B, 31C, 33E, 39B. http://dspace.nbuv.gov.ua/handle/123456789/124403 en Український математичний вісник Інститут прикладної математики і механіки НАН України |
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We give parametric expressions for extremals of Grötzsch’s problem. |
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Tamrazov, P.M. Parametrization of extremals of Grötzsch’s problem Український математичний вісник |
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Tamrazov, P.M. |
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Tamrazov, P.M. |
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Parametrization of extremals of Grötzsch’s problem |
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Parametrization of extremals of Grötzsch’s problem |
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Parametrization of extremals of Grötzsch’s problem |
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Parametrization of extremals of Grötzsch’s problem |
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Parametrization of extremals of Grötzsch’s problem |
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parametrization of extremals of grötzsch’s problem |
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Інститут прикладної математики і механіки НАН України |
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Parametrization of extremals of Grötzsch’s problem / P.M. Tamrazov // Український математичний вісник. — 2010. — Т. 7, № 4. — С. 570-583. — Бібліогр.: 9 назв. — англ. |
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AT tamrazovpm parametrizationofextremalsofgrotzschsproblem |
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Український математичний вiсник
Том 7 (2010), № 4, 570 – 583
Parametrization of extremals of
Grötzsch’s problem
Promarz M. Tamrazov
(Presented by V. Ya Gutlyanskĭı)
Abstract. We give parametric expressions for extremals of Grötzsch’s
problem.
2010 MSC. 30C, 30D, 30E, 31A, 31B, 31C, 33E, 39B.
Key words and phrases. Grötzsch’s problem, conformal mappings,
hyperbolic capacity, quadratic differentials, graphs.
Introduction
Let C be the complex plane, C be the Riemann sphere, K := {z ∈
C : |z| < 1} be the unit disk, and T := ∂K.
The formulation of Grötzsch’s problem is the following:
Among all continua containing a fixed finite point collection (i.e. col-
lection of points) belonging to K, to find such one which has the minimal
hyperbolic capacity.
In [2] H. Grötzsch (in analogy with his results concerning Tchebo-
taröv’s problem) stated the existence and uniqueness of the solution of
his problem as well as his constructive property for it.
G. V. Kuz’mina gave more detailed treatment of this problem (see
her book [3, Chapter 3]).
In the present work we establish some (1.1)-parametrization of ex-
tremals of Grötzsch’s problem via proper class of simple rectilinear
graphs, isomorphic and in a sense isometric to extremal continua (namely
via the same class of graphs which participated in parametrization of ex-
tremals of Tchebotaröv’s problem, see [7, 9]).
We shall consider Grötzsch’s problem in the following completely
equivalent form (see [3, Chapter 5]). Let m ≥ 1 be an integer, and
Received 21.04.2009
ISSN 1810 – 3200. c© Iнститут математики НАН України
P. M. Tamrazov 571
{aj} := {aj}
m+1
j=1 be a fixed unordered collection of distinct points
a1, . . . , am+1 ∈ C \ K.
Grötzsch’s problem. Among all doubly connected domains D ⊂ C\K
embracing K and separating K from the whole point collection {aj}, to
find such one which has the maximal modulus (i.e. the maximal modulus
of the family of closed curves contained in D, embracing K and separating
the boundary components of D).
For convenience sake we shall consider the following completely equiv-
alent form of this problem:
Problem H. To consider this problem under additional requirement that
am+1 = ∞.
It is known (see for instance [3, Chapter 5]) that the mentioned maxi-
mum modulus in the problem is attained, and we denote it by Mod({aj}).
Moreover the doubly connected domain D =: Dmax({aj}) under question
giving the maximal value for the mentioned modulus is uniquely defined
by the point collection {aj}, and its inner boundary component coincides
with T.
The parametrization of Grötzsch’s problem was considered in [8]. But
the statement of that work about the extremality of its function (3) in
Grötzsch’s problem is wrong (cf. with [9]). Therefore theorems of the
work [8] must be corrected in terms not related to its function (3) (cf.
with [9]).
The author tried to consider one more representation of extremals of
Grötzsch’s (and Tchebotaröv’s) problems, but Prof. G. V. Kuz’mina and
Prof. E. G. Emelyanov have shown to me a counter example to it. I am
greatly thankful to them.
1. Parametric formulation
The particular case m = 1 of Grötzsch’s problem under consideration
is a well-known Grötzsch’s distortion theorem. Therefore for simplicity
of formulations in the sequel we shall exclude this case and assume that
m ≥ 2.
As usual (cf. with [4, 6, 7]), we may omit all unessential points of
the collection {aj} not influencing upon the extremal of the problem,
and hence we shall consider only those collections {aj}
m+1
j=1 for which all
points aj influence upon the extremal of the problem (that is equivalent
to the requirement that all aj are end points of the set B := C \ (K ∪
Dmax({aj})), see [4] and cf. with [7], and therefore are simple poles of
572 Parametrization of extremals...
the quadratic differential Q(w)dw2 of the problem (see [3, Chapter 5])).
The class of all such collections will be denoted by M.
For any r ∈ (1,∞), let us denote Sr := {z ∈ C : 1 < |z| < r}.
For the above given collection {aj}, let r({aj}) denote that r > 1 for
which the ring Sr is conformally equivalent to the domain Dmax({aj}).
Denote by f{aj} the univalent conformal mapping of the ring Sr({aj})
onto the domain Dmax({aj}) for which f{aj}(T ) = T and f{aj}(r({aj})) =
∞.
Such a mapping exists, is unique and has the continuous (with re-
spect to topology of C in the image) extension f{aj} onto Sr({aj}) and
f{aj}(Sr({aj})) = C \K (for instance see [3, Chapter 5] and cf. with [8]).
The function f{aj} will be called the extremal of the Problem H (cor-
responding to the element {aj} ∈ M).
The quadratic differential Q(w)dw2 of Problem H can be defined via
its extremal function f{aj}(z) as the left part of the functional-differential
equation
−
( df{aj}(z)
zf ′
{aj}
(z)
)2
= −
(dz
z
)2
.
Because of our assumption, all points aj belonging to the fixed point
collection {aj} ∈ M are endpoints of the set B. So every other point
of B is either a zero bn of the order νn ≥ 1 of Q(w)dw2 (and then it
is a branch point of B), or this point is a regular, non-critical point of
Q(w)dw2 (and then it belongs to a critical trajectory of this quadratic
differential). Thus exactly as in [6,7], we may consider B as an undirected,
connected, simple, acyclic, plane graph on C \ K consisting of nodes of
order one at all points aj and only at them, nodes of orders νn + 2 at all
zeros bn of degrees νn ≥ 1, and only at them, and of all critical analytic
trajectories of Q(w)dw2 (contained in B and ending at zeros or simple
poles of Q(w)dw2) as edges of the graph. This curvilinear geometric
graph is connected and has no cycles. So it is a tree. And we shall
denote it by L({aj}), assuming that am+1 = ∞.
The total multiplicity of all zeros of Q(w)dw2 on C \K equals m− 1.
Let now k be the number of different zeros of Q(w)dw2 on C \ K.
Then the number of edges of the graph under consideration equals m+k
(on the basis of the Euler’s theorem).
Any elements {aj} and {a′j} of M are called similar if there is t ∈ T
such that {a′j} = t{aj}.
Let M̃ be the factor set of M with respect to similarity.
One can show that for every {aj} ∈ M and any t ∈ T there holds
r(t{aj}) = r({aj}).
P. M. Tamrazov 573
Our main results in this problem can be expressed in the following
way. We have defined a parametric set G̃ consisting of simple, transparent
geometric objects Γ̃ (both of these notations are defined in Section 3
of [7], see also below Sections 3 and 4). Then for every such Γ̃ and every
t ∈ T we have defined a point collection {aj(Γ̃, t)} ∈ M and a univalent
conformal mapping f
Γ̃,t
: S
r({aj(Γ̃,t)})
→ C \ {aj(Γ̃, t)} with f
Γ̃,t
(T ) =
T, f
Γ̃,t
(r({aj(Γ̃, t)})) = ∞, f
Γ̃,t
(1) = t, such that this f
Γ̃,t
is extremal in
Grötzsch’s problem for the mentioned point collection {aj(Γ̃, t)}. We have
also
f
Γ̃,t
= tf
Γ̃,1
.
The point collection {aj(Γ̃, t)} is a function of data given by Γ̃ and t.
And conversely, for every collection {aj} ∈ M there are (the unique)
t ∈ T and Γ̃ ∈ G̃ such that the function f
Γ̃,t
is extremal in Grötzsch’s
problem for the mentioned point collection {aj} and f
Γ̃,t
(1) = t.
Thus we have established the parametric (1,1)-correspondence be-
tween all couples (Γ̃, t) and extremals f{aj} of the problem H for all col-
lections {aj} ∈ M.
Properties of the mentioned objects are described in Theorems 3.1–
3.3.
2. Construction of extremals
Below we shall construct extremals of Problem H (normalized by the
condition that ∞ is an end point of the boundary of the extremal image,
or equivalently that ∞ is a simple pole of the quadratic differential of
this problem). This class will be parametrized by means of geometric
rectilinear graphs Γ ∈ G defined in [6] and in Section 3 of the work
[7]. For convenience sake in this Section we repeat some definitions and
notations from [6, 7] being used below in construction of the extremals
for Problem H.
Let G be the class of all finite, undirected, connected plane graphs Γ
each of which satisfies the following conditions:
1) each edge γ of Γ is a rectilinear open interval in C of the length |γ| >
0, and these intervals mutually do not intersect each other, while
nodes of the graph coincide with the endpoints of these intervals;
2) Γ does not contain nodes of order 2 and cycles;
3) the sum of lengths |γ| of all intervals γ of the graph Γ equals π;
574 Parametrization of extremals...
4) the point ζ = 0 is a node of Γ of order 1, and the edge of Γ incident
to this point is contained in the real half-axis Re ζ > 0.
Let Supp Γ denote the closure in C of the geometric union of all edges
of the graph Γ ∈ G. Starting at the node 0, let us run along Γ in the
direction in which the complementary to Γ domain C \ (Supp Γ) remains
on the left. Such a pass of Γ will be called natural. For every point ζ
on an edge γ ∈ Γ, let r1(Γ, ζ) and r2(Γ, ζ) denote the length of the pass
respectively to the first and to the second reaching the point ζ, while
r1(Γ, 0) = 0, r2(Γ, 0) = 2π. Under a single such pass along an edge γ
the growth of each of functions r1 and r2 equals |γ|. For every node v of
the order τ(v), let r1(Γ, v), . . . , rτ(v)(Γ, v) denote the length till the first,
. . . , τ(v)th pass of v.
For every ζ ∈ Supp Γ and all j = 1, . . . , τ(ζ) let us denote
εΓ,j(ζ) := exp (irj(Γ, ζ)).
Let Γ′ be another graph from G, and for every ζ ′ ∈ Supp Γ′ the
objects τ ′(ζ ′) and εΓ′,j′(ζ
′) be defined exactly as analogous objects were
defined for Γ and ζ ∈ Supp Γ.
Then the graphs Γ and Γ′ will be called equivalent, if there exists the
isomorphism η : Γ → Γ′ such that η(0) = 0 and for every node v of Γ we
have
rj(Γ
′, η(v)) = rj(Γ, v) ∀ j = 1, . . . , τ(v).
If graphs Γ, Γ′ ∈ G are equivalent, then for every ζ ∈ Supp Γ there
corresponds a uniquely defined ζ ′ ∈ Supp Γ′ for which
εΓ,j(ζ) = εΓ′,j(ζ
′) ∀ j = 1, . . . , τ(ζ).
For a graph Γ ∈ G, let V (Γ) be the set of all its nodes of order 1, and
W (Γ) be the set of all other its nodes (of orders ≥ 3). Let V be the set
of all points εΓ,1(p) (∈ T ), when p runs through the set V (Γ). Denote by
Wv the set of all points εΓ,j(v) (∈ T ), when v ∈ W (Γ) is fixed and j runs
through the set of values 1, . . . , τ(v). Denote also
W :=
⋃
v∈W (Γ)
Wv.
Clearly the point z = 1 is contained in V .
The cardinal numbers of the sets V (Γ) and W (Γ) are m + 1 and k,
respectively, and Γ contains exactly m + k edges.
For u and ζ in C \ {0}, let us introduce the function
ρ(u, ζ) :=
1
|u|
(u − ζ)
(
u −
1
ζ
)
.
P. M. Tamrazov 575
Fix any r > 1.
Introduce the function s : z 7→ rz.
Consider the ring Dr := {z ∈ C : 1/r < |z| < r}.
3. Main results
For a fixed graph Γ ∈ G, r ∈ (1,∞), t ∈ T under the above notations
and assumptions, we get the following result.
Theorem 3.1. For every Γ, r, t under consideration, there is a function φ
with the following properties. It is holomorhic and univalent in Dr, sym-
metric with respect to T, continuous on Sr\{r}, continuous in generalized
sense (with respect to topology of C in the image) on Sr and posessing
the following properties. For every point ζ0 ∈ Supp Γ the function φ glues
rational-analytically all points rεΓ,j(ζ0) (j = 1, . . . , τ(ζ0)) into one point
denoted by Y (ζ0), and φ is continuously and meromorphically extendable
into a neighborhood of every point z ∈ Sr \ s(W ) (holomorphically for
every such z 6= r). Moreover φ(Sr) = C \ K, φ(r) = ∞, while at the
inner boundary of Sr there holds φ(T ) = T, φ(1) = t, and the restriction
φr of the function φ to Sr is extremal in Problem H for the collection
of all points A(p) := y(p) where p runs over the whole set V (Γ). The
quadratic differential of this problem is given by the formula:
Q(w)dw2 = −
dw2
w2
∏
v∈W (Γ)
ρ(Y (v), w)τ(v)−2
∏
α∈V \{1}
ρ(φ(rα), w)
.
On the set C \K, the collection of all (simple) poles of Q(w)dw2 belongs
to M and coincides with the set of all m+1 points A(p), while the set of
all zeros of Q(w)dw2 coincides with the set of all points B(v) := Y (v),
where v runs over all k points of the set W (Γ), and the order of each zero
B(v) of Q(w)dw2 equals τ(v)−2. Each point A(p) is an endpoint of some
single critical trajectory of Q(w)dw2. The boundary of the domain φ(Sr)
with respect to C\K is the union of m+k critical trajectories of Q(w)dw2,
their m+1 endpoints A(p) (∀ p ∈ V (Γ)) and k points B(v) (∀ v ∈ W (Γ)).
The functions φ and φr with the above properties are uniquely defined by
the fixed Γ, r, t.
Let Γ ∈ G be the fixed graph from Theorem 3.1 with all related to
it objects and notations (in particular, A(p) for all p ∈ V (Γ)). Using
the notation L({aj}) from Section 1 defined for every point collection
{aj} ∈ M, let’s define a particular collection {aj} as the set {A(p)}p∈V
from Theorem 3.1 and denote L({A(p)}p∈V ) =: Γ∗.
Then we get the following result.
576 Parametrization of extremals...
Theorem 3.2. The graph Γ∗ is isomorphic to Γ, with the correspondence
of the node ζ = 0 of Γ to the node w = A(0) = ∞ of Γ∗, and the pass
of Γ∗ in the direction in which the domain C \ (Γ∗ ∪ K) remains on the
left, corresponds to the pass of Γ in the natural direction (in which the
complementary to Γ domain C \ (Supp Γ) remains on the left). Then the
length of every pass along Γ∗ in the metric |Q1/2dw| equals to the length
of its pre-image on Γ with respect to the natural length measuring on Γ
(see Section 2), and
2π∫
0
|Q1/2(eiθ)| dθ = 2π.
Thus the graphs Γ and Γ∗ are isomorphic, equally oriented relative
to their complementary (with respect to C \ K) domains and isometric
in the sense of Theorem 3.2 (this isometry being consistent with the
isomorphism and the direction of pass).
From the definitions we see that for every equivalent graphs Γ′, Γ′′ ∈ G
and related to them objects corresponding to each other in this equiva-
lence (including objects of the form p, v, εΓ,1(p), εΓ,j(v), V (Γ), W (Γ)
for these graphs), the objects V, W, τ(v), Wv, A(p), B(v), φ of similar
form coincide.
Let G̃ denote the factor-set of G with respect to the equivalence, and
Γ̃ denote its general element.
For a graph Γ ∈ G, let {Γ} denote the class of all graphs from G
equivalent to Γ.
Let F : G̃ × (1, +∞) × T → M be the mapping defined for each
Γ̃ ∈ G̃, r ∈ (1, +∞), t ∈ T as the collection {φ(rεΓ,1(p)}p∈V (Γ), where φ
is the function mentioned in Theorem 1 (with the requirement φ(1) = t
for arbitrary Γ ∈ Γ̃ and the corresponding V (Γ)).
Theorem 3.3. The class of all extremals of Problem H is parametrized
by elements of the set G̃× (1, +∞)×T, and this parametrization is one-
to-one correspondence:
1) to every element Γ̃ ∈ G̃, each finite r > 1 and any t ∈ T there
corresponds one (and only one) point collection {Aj(Γ̃, r, t)} ∈ M
for which the function φr mentioned in Theorem 3.1 (with any graph
Γ ∈ Γ̃) is extremal in Problem H; this {Aj(Γ̃, r, t)} is the collection
of all points A(p) := φ(rεΓ,1(p)) where p runs over the whole set
V (Γ), and in notation of Section 1 we have r({Aj(Γ̃, r, t)}) = r
(and φ(1) = t, φ(r) = ∞).
2) and conversely, for every collection of points {aj} ∈ M there exists
one and only one class Γ̃ ∈ G̃, the single finite r > 1 and the single
P. M. Tamrazov 577
t ∈ T such that the function φr with φ mentioned in Theorem 3.1
(with arbitrary Γ ∈ Γ̃) is extremal in Problem H for {aj}, and
hence {aj} = F (Γ̃, r, t) = {Aj(Γ̃, r, t)}, r({aj}) = r, φ(1) = t (and
also φ(r) = ∞).
The proofs use arguments and facts established in Sections 5–10 of
the work [7] with some essential modifications.
4. Couples of arcs and stars
Let ∆ denote the class of all (unordered) couples {δ+, δ−} consisting
of non-intersecting open arcs δ+, δ− on the unit circle T ⊂ C. For a couple
{δ+, δ−} =: δ ∈ ∆ let us denote 〈δ〉 := δ+∪δ−. This 〈δ〉 will be called the
support of δ. Let also |δ+| and |δ−| be lengths of δ+ and δ−, respectively.
Let ∆0 be the set of all δ := {δ+, δ−} for which the closure Clos 〈δ〉
of the set 〈δ〉 is connected and does not coincide with T, and in this case
let P (δ) denote the common endpoint of δ+ and δ−.
For any n ∈ N, an unordered collection {δ1, . . . , δn} =: A will be
called one-sheeted, if 〈δ1〉, . . . , 〈δn〉 are mutually non-intersecting. We say
that δ1, . . . , δn are members of A. The set
n⋃
k=1
〈δk〉 =: suppA
will be called the support of A.
A one-sheeted unordered collection A = {δ1, . . . , δn} will be called
a star, if every member δj of A has a connected component S(δj , A) of
the set T \ 〈δj〉 which contains supports of all other members of A. In
such a case the other connected component of T \ 〈δj〉 will be denoted by
S0(δj , A) and called the shadow of δj with respect to A. The set
n⋃
k=1
S0(δj , A) =: S0(A)
will be called the shadow of A.
A star A will be called connectable, if the set T \ (S0(A) ∪ suppA)
consists of a finite number of points.
5. Constellations
A finite, non-empty collection of connectable stars will be called a
constellation.
A constellation C will be called irreducible, if every its star contains
at least three members.
578 Parametrization of extremals...
Let A1 and A2 be different stars from C, and δ be a member of both
of these stars. Then A1 and A2 will be called neighbors, and δ will be
called a link in C (between these neighbors).
Let k ≥ 1 be an integer. A sequence δ1, . . . , δk of links in C is called
a chain (of the length k) in C, if there exists a sequence A1, . . . , Ak+1 of
stars in C with the following properties:
1) Aj 6= Aj+1 and δj is the link between Aj and Aj+1 for each j =
1, . . . , k;
2) if k ≥ 2, then also δj 6= δj+1 for all j = 1, . . . , k − 1.
The stars A1, . . . , Ak+1 are called vertices of the chain δ1, . . . , δk.
A constellation C will be called acyclic, if for every chain δ1, . . . , δk in
it with the vertices A1, . . . , Ak+1 there holds A1 6= Ak+1, and if moreover
each δ ∈ ∆ may be a member of at most two stars from C and 〈δ〉 does
not intersect supports of other stars from C.
Let C be an acyclic, irreducible constellation.
Every two neighbors from C have only one link between them in C.
It follows from our definitions and assumptions that for every chain
of the length k ≥ 2 all links of the chain are mutually different, and all
vertices A1, . . . , Ak+1 of the chain are mutually different as well.
A chain δ1, . . . , δk in C will be called maximal, if it is not contained
in a longer chain in C (this means that there is no link δ in C such that
some of the sequences δ, δ1, . . . , δk or δ1, . . . , δk, δ is a chain in C).
Every chain in C is contained in a maximal chain in C.
If a sequence δ1, . . . , δk in C is a maximal chain in C and A1, . . . , Ak+1
are the vertices of this chain, then all members δ0 of A1 different from
δ1, and all members δk+1 of Ak+1 different from δk satisfy the following
condition: their shadows S0(δ0, A1) and S0(δk+1, A
k+1) do not intersect
supports of other stars from C.
Every A ∈ C containing at most one link will be called a margin star
of C.
It is easily verified that C contains at least one margin star.
We see also that a margin star has at most one neighbor star.
A constellation C will be called connected if for each two different (if
any) stars A′, A′′ from C there exists a chain δ1, . . . , δk in C such that δ1
is a member of A′ and δk is a member of A′′.
P. M. Tamrazov 579
6. Proof of the main statement
Let Γ ∈ G.
Using the functions εΓ,j(ζ) (see Section 2), we shall define the follow-
ing objects.
For every edge γ of Γ let δ+
γ and δ−γ be the images of γ in the maps
ζ 7→ εΓ,1(ζ) and ζ 7→ εΓ,2(ζ), respectively, and in notation of Section 8
of [7] we have (δ+
γ , δ−γ ) =: δγ ∈ ∆1.
For every v ∈ W (Γ), let Av be the collection of all couples δγ corre-
sponding to all edges γ incident to v. Then Av is a connectable star (see
Section 4). Moreover, the collection
{Av}v∈W (Γ) =: C(Γ) =: C
is a constellation which is acyclic, irreducible and connected (see Sec-
tion 5).
Consider the corresponding Riemann surface R(C) built in Section 8
of [7] for the constellation C := C(Γ). It is an oriented, compact, sim-
ply connected, schlichtartig Riemann surface conformally equivalent to a
Riemann sphere (see Sections 8, 9 of [7]).
Let us denote
K1/r := {z ∈ C : |z| < 1/r},
T1/r := {z ∈ C : |z| = 1/r},
and
R
+ := R(C(Γ)) \ K1/r.
The Riemann surface R
+ is a simply connected hyperbolic closed
domain with the border T1/r.
Let now R denote the duplicate of R
+, and
R
− := R \ (R+ \ T1/r).
The surface R is a Riemann surface topologically equivalent to C and
symmetric with respect to T1/r. So it is conformally equivalent to C.
Let λΓ,t denote the conformal homeomorphism of the Riemann sur-
face R onto C normalized by the conditions λΓ,t(T1/r) = T, λΓ,t(1/r) =
t, λΓ,t(1) = ∞.
Denote by ̺ and σ the inversions of R and C with respect to T1/r and
T, respectively.
We have
s−1(T ) = T1/r,
̺(T1/r) = Tr.
580 Parametrization of extremals...
Therefore all just the mentioned mappings admit reflections within re-
spective surfaces with respect to mentioned circles, and for the functions
extended by reflections we preserve the same notations.
Then because of the symmetries we have
λΓ,t ◦ ̺ = σ ◦ λΓ,t.
Let us consider the mapping ηC(Γ) := ηC defined in Section 8 of [7] for
the constellation C = C(Γ), the corresponding superposition φ : S̄r →
C \ K defined by the formula
φ := φΓ,r,t := λΓ,t ◦ ηC(Γ) ◦ s−1,
and its inversion φ−1.
Obviously
ηC(Γ)(T1/r) = T1/r.
Let
at,j := φ(rεΓ,1(pj)),
where pj runs over the whole set V corresponding to Γ (j = 1, . . . , m+1).
Let us show that the restriction of the function φ to Sr is the extremal
of Grötzsch’s problem for the point collection {at,j} introduced in this
Section.
Let h : Sr → C \ (K ∪ {at,j}) be any meromorphic univalent func-
tion with h(T ) = T. Then for every edge γ of Γ, the corresponding
δγ := (δ+
γ , δ−γ ) and any couple (z+, z−) of δγ-corresponding points (see
Section 8 of [7]), the closure of the union of φ-images of radii of Sr pass-
ing through the points z+ and z− contains two non-intersecting arcs
contained in h(Sr) and connecting the boundary components of the set
h(Sr). Therefore on the basis of Theorems 5 and 6 from our work [5] one
can show (cf. with [7]) that the restriction to Sr of the function φ is the
extremal of Grötzsch’s problem for {at,j}.
Therefore the function
(φ−1 · (φ′ ◦ φ−1)
φ ◦ φ−1
)2
is a rational function on C, and φ satisfies the equation (cf. with [3,
Chapter 5]): (zφ′(z)
φ(z)
)2
=
p(φ(z))
q(φ(z))
,
where
p(w) :=
m∏
j=1
ρ(at,j , w)
P. M. Tamrazov 581
and
q(w) :=
∏
v∈W (Γ)
(ρ(b(v), w)τ(v)−2
with b(v) := φ(rβ) ∀β ∈ Wv.
For every γ ∈ Γ and each ζ ∈ 〈δγ〉, let ζ̂ denote the point on 〈δγ〉
which is δγ−corresponding to ζ (see Section 8 of [7]).
Let a collection {aj} ∈ M of points be given. Let f := f{aj} be
the extremal of Grötzsch’s problem for this collection. Lavrentiev [4] has
established the result on the isometry (with respect to harmonic measure)
of sewing of opposite sides of analytic arcs constituting the boundary of
the extremal image in Tchebotaröv’s problem (this result is generalized
by Goluzin [1, p. 152–157]).
The following is a simple explanation of the isometry of the extremal
f of Grötzsch’s problem along critical trajectories. We have
Q(f)df2 = −dz2/z2.
Hence
f ′(z)2 = −1/(Q(f(z))z2),
and since Q is a rational function, therefore |f ′(z)| is the same for every
couple of boundary points being sewed by f . So f is isometric along
critical trajectories in the metric |f ′(z)dz| which coincides with isometry
with respect to harmonic measure.
Let L({aj}) be the curvilinear graph (see above Section 1) gener-
ated by f := f{aj}. From here there follows the existence of a graph
Γ := Γ{aj} ∈ G which is isomorphic to L({aj}) (with the correspondence
of the node am+1 ∈ L({aj}) to the node pm+1 ∈ V (Γ), see Theorem 3.1),
isometric to it (with respect to natural measure in Γ ∈ G and harmonic
measure in L({aj})), and equally oriented relative to the complements.
The function f := f{aj} extended to the circle Tr gives one more real-
ization of the Riemann surface R+ (see above this Section) with similar
normalizations at the circles T (for f) and T1/r (for φ) and at the pre-
image of ∞. Therefore the function f/f(1) coincides with the function
φΓ{aj}
φΓ{aj}
(1)
,
where φ =: φΓ{aj}
is the function constructed for this Γ{aj}. From here
there follows that
aj/f(1) =
φΓ{aj}
(αj)
φΓ{aj}
(1)
∀αj ∈ V (Γ),
582 Parametrization of extremals...
and the function
f(1)
φΓ{aj}
φΓ{aj}
(1)
is extremal in Grötzsch’s problem for the given {aj}.
Note that for every fixed Γ̃ ∈ G̃ the collection {φ(aj)} related to Γ ∈ Γ̃
does not depend of the specific choice of the mentioned Γ from the fixed
Γ̃, and the corresponding function φ is extremal in Grötzsch’s problem
for the collection {φ(aj)}.
Conversely, for any collection {aj} ∈ M there exist (and are unique)
Γ̃ from G̃ and c ∈ C \ {0} such that for every Γ ∈ Γ̃ the function cφ is
extremal for the collection {aj}.
By the way, we can freely choose a Γ from G, and by this we can pre-
determine the topological structure of the curvilinear graph L({aj}) in
the w-plane, and even its metric properties in the sense of Theorem 3.2.
From the above facts it is easy to derive all other statements of The-
orems 3.1–3.3.
Remark 6.1. After receiving the referee report (November, 2010), I
have to note that formerly this work is a part of another manuscript
containing one more analytical statement and submitted for publication
in 2008. But when G. V. Kuz’mina and E. G. Emelyanov shown me
that the mentioned statement is wrong, I cancelled the submission. Re-
cently the several new papers have been published: E. G. Emelyanov,
A sewing theorem for quadratic differentials // Zapiski Nauchnuh Semi-
narov POMI, 371 (2009), 69–77; A. Yu. Solynin, Quadratic differentials
and weighted graphs on compact surfaces, Analysis and Mathematical
Physics. Trends in Mathematics, Birkhauser, Springer 473–505, 2009; P.
Tamrazov, Parametrization of extremals for some generalization of Cheb-
otarev’s problem // Georgian Math. J., 17 (2010), 597–619. In Solynin’s
paper a general problem of constructing quadratic differentials starting
from graphs is posed. The paper by Emel’yznov contains a different ap-
proach for the proof of the sewing theorems. Finally, in my paper it is
established the parametrization of extremals for the problem which is one
of the known generalizations of Chebotarev’s problem. The parametriza-
tion of this problem essentially differs from that of Chebotarev’s problem
and Grötzsch’s problem.
References
[1] G. M. Goluzin, Geometric Theory of Functions of a complex Variable, AMS,
Rhode Island, 1969 (Transl. from Russian).
[2] H. Grötzsch, Über ein Variationsproblem der konformen Abbildung // Ber. Ferh.
Sächs. Akad. Wiss. Leipzig, 82 (1930), 251–263.
P. M. Tamrazov 583
[3] G. V.Kuz‘mina, Modules of curves families and quadratic differentials // Trudy
Matematich. Inst. im. Steklova, CXXXXIX, Leningrad, “Nauka”, Leningradskoe
Otdelenie, 1980 (in Russian).
[4] M. A. Lavrentyev, On the theory of conformal mappings // Trudy Fiz.-Mat. Inst.
Akad. Nauk SSSR, 5 (1934), 159–246 (in Russian).
[5] P. M. Tamrazov, Theorems on the covering of lines under a conformal mapping //
Mat. Sb., 66 (108) (1965), 502–524. (in Russian).
[6] P. M. Tamrazov, Tchebotaröv’s problem // Comptes Rendus Acad. Sci. Paris,
Ser. I, 341 (2005), 405–408.
[7] P. M. Tamrazov, Tchebotaröv’s extremal problem // Centr. Europ. J. Math., 3(4)
(2005), 591–605.
[8] P. M. Tamrazov, Grötzsch’s problem // Zbirnyk prac’ Instytutu Matematyky
Nacional’noi Akademii Nauk Ukrainy, Kiev, 2 (2005), N 3, 252–261.
[9] P. M. Tamrazov, Erratum to “Tchebotaröv’s extremal problem” // Centr. Europ.
J. Math., 7(3) (2009), 568–570.
Contact information
Promarz M.
Tamrazov
Institute of Mathematics,
National Academy of Sciences of Ukraine
3, Tereschenkivska st.
01601, Kiev-4
Ukraine
E-Mail: proma@imath.kiev.ua
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