Glueing theorems
Theorems that establish the existence of analytic functions subject to prescribed relations on the boundary of a domain.
Lavrent'ev's glueing theorem [1]: Given any analytic function on
with
and
, then one can construct two analytic functions
and
, where
and
, mapping the rectangles
,
and
,
univalently and conformally onto disjoint domains
and
, respectively, in such a way that
. This theorem was used (see [6]) to prove the existence of a function
,
,
, realizing a quasi-conformal mapping of the disc
onto the disc
and possessing almost-everywhere a given characteristic
, where
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and is a measurable function defined for almost-all
,
. A modified form of Lavrent'ev's theorem was also used to solve the problem of mapping a simply-connected Riemann surface conformally onto the disc [5].
Other glueing theorems (with weaker restrictions on the functions of type , see [2]) have played a major role in the theory of Riemann surfaces. Another example is as follows (see [3], [5]): Suppose one is given an arc
on the circle
with end points
and
,
, and a function
on
with the properties: 1) at all the interior points of
,
is regular and
; 2) the function
establishes a one-to-one mapping of
onto the complementary arc
on
leaving
and
invariant. Then there is a function
![]() |
regular for except at
, such that
at the interior points of
.
It has also been proved that there is a univalent function with these properties (see [4], Chapt. 2).
References
[1] | M.A. Lavrent'ev, "Sur une classe de répresentations continues" Mat. Sb. , 42 : 4 (1935) pp. 407–424 |
[2] | L.I. Volkovyskii, "On the problem of the connectedness type of Riemann surfaces" Mat. Sb. , 18 : 2 (1946) pp. 185–212 (In Russian) |
[3] | A.C. Schaeffer, D.C. Spencer, "Variational methods in conformal mapping" Duke Math. J. , 14 : 4 (1947) pp. 949–966 |
[4] | A.C. Schaeffer, D.C. Spencer, "Coefficient regions for schlicht functions" , Amer. Math. Soc. Coll. Publ. , 35 , Amer. Math. Soc. (1950) |
[5] | G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian) |
[6] | P.P. Belinskii, "General properties of quasi-conformal mappings" , Novosibirsk (1974) pp. Chapt. 2, Par. 1 (In Russian) |
Glueing theorems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Glueing_theorems&oldid=47102