On the boundary values of Riemann's mapping function
Author:
R. J. V. Jackson
Journal:
Trans. Amer. Math. Soc. 259 (1980), 281297
MSC:
Primary 30C20; Secondary 30C40
MathSciNet review:
561837
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Classically, the calculus of variations is required to prove the existence of a biholomorphism from the unit disk to a given simplyconnected, smooth domain in the complex plane. Here, the problem is reduced to the solution of an ordinary differential equation along the boundary of the domain. The sole coefficient in this equation is identified with the bounded term in the asymptotic expansion of the Bergman kernel function. It is shown that this coefficient can not depend upon any differential expression involving only the curvature function of the boundary.
 [1]
Lars
V. Ahlfors, Conformal invariants: topics in geometric function
theory, McGrawHill Book Co., New YorkDüsseldorfJohannesburg,
1973. McGrawHill Series in Higher Mathematics. MR 0357743
(50 #10211)
 [2]
Stefan
Bergman, The Kernel Function and Conformal Mapping,
Mathematical Surveys, No. 5, American Mathematical Society, New York, N.
Y., 1950. MR
0038439 (12,402a)
 [3]
S.
S. Chern and J.
K. Moser, Real hypersurfaces in complex manifolds, Acta Math.
133 (1974), 219–271. MR 0425155
(54 #13112)
 [4]
R.
Courant, Dirichlet’s Principle, Conformal Mapping, and
Minimal Surfaces, Interscience Publishers, Inc., New York, N.Y., 1950.
Appendix by M. Schiffer. MR 0036317
(12,90a)
 [5]
R. Courant and D. Hilbert, Methods of mathematical physics. II, Interscience, New York, 1962.
 [6]
Jesse
Douglas, Solution of the problem of
Plateau, Trans. Amer. Math. Soc.
33 (1931), no. 1,
263–321. MR
1501590, http://dx.doi.org/10.1090/S00029947193115015909
 [7]
Norberto
Kerzman, The Bergman kernel function. Differentiability at the
boundary, Math. Ann. 195 (1972), 149–158. MR 0294694
(45 #3762)
 [8]
N.
Kerzman and E.
M. Stein, The Cauchy kernel, the Szegö kernel, and the Riemann
mapping function, Math. Ann. 236 (1978), no. 1,
85–93. MR
0486468 (58 #6199)
 [9]
Zeev
Nehari, Conformal mapping, McGrawHill Book Co., Inc., New
York, Toronto, London, 1952. MR 0045823
(13,640h)
 [10]
N.
I. Muskhelishvili, Singular integral equations,
WoltersNoordhoff Publishing, Groningen, 1972. Boundary problems of
functions theory and their applications to mathematical physics; Revised
translation from the Russian, edited by J. R. M. Radok; Reprinted. MR 0355494
(50 #7968)
 [11]
Shlomo
Sternberg, Lectures on differential geometry, PrenticeHall,
Inc., Englewood Cliffs, N.J., 1964. MR 0193578
(33 #1797)
 [12]
A.
Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge
University Press, New York, 1959. MR 0107776
(21 #6498)
 [1]
 L. V. Ahlfors, Conformal invariants, McGrawHill, New York, 1973. MR 0357743 (50:10211)
 [2]
 S. Bergman, The kernel function and conformal mapping, Math. Surveys, no. 5, Amer. Math. Soc., Providence, R. I., 1950. MR 0038439 (12:402a)
 [3]
 S. S. Chern and J. K. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219271. MR 0425155 (54:13112)
 [4]
 R. Courant, Dirichlet's principle, Interscience, New York, 1950. MR 0036317 (12:90a)
 [5]
 R. Courant and D. Hilbert, Methods of mathematical physics. II, Interscience, New York, 1962.
 [6]
 J. Douglas, Solution of the problem of Plateau, Trans. Amer. Math. Soc. 33 (1931), 263321. MR 1501590
 [7]
 N. Kerzman, The Bergman kernel function. Differentiability at the boundary, Math. Ann. 195 (1972), 149158. MR 0294694 (45:3762)
 [8]
 N. Kerzman and E. M. Stein, The Cauchy kernel, the Szegö kernel, and the Riemann mapping function, Math. Ann. 236 (1978), 8593. MR 0486468 (58:6199)
 [9]
 Z. Nehari, Conformal mapping, McGrawHill, New York, 1952. MR 0045823 (13:640h)
 [10]
 N. I. Muskhelishvili, Singular integral equations, Noordhoff, Groningen, 1953. MR 0355494 (50:7968)
 [11]
 S. Sternberg, Lectures on differential geometry, PrenticeHall, Englewood Cliffs, N. J., 1964. MR 0193578 (33:1797)
 [12]
 A. Zygmund, Trigonometric series, 2nd ed., Cambridge Univ. Press, London and New York, 1959. MR 0107776 (21:6498)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
30C20,
30C40
Retrieve articles in all journals
with MSC:
30C20,
30C40
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198005618373
PII:
S 00029947(1980)05618373
Keywords:
Riemann's mapping function,
Plateau's problem,
ordinary differential equations,
potential theory,
Hilbert's transform,
asymptotics of Bergman's kernel function
Article copyright:
© Copyright 1980
American Mathematical Society
