On the Problem of Plateau / Subharmonic Functions

177 Pages·1971·13.692 MB·English

by  Tibor Radó (auth.)

Tibor Rad6
On the Problem
of Plateau
ubharmonic Functions
Reprint
Springer-Verlag Berlin Heidelberg New York 1971
AMS Subject Classifications (1970): 49FlO, 3lC05
ISBN-13:978-3-S40-05479-5 e-ISBN-13:978-3-642-65236-3
DOl: 10.1007/978-3-642-65236-3
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is
concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction l),y
photocopying machine or similar means, and storage in data banks
Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable
to the publisher, the amount of the fee to be determined by agreement with the publisher
Library of Congress Catalog Card Number 71-160175.
Herstellung: Strauss & Cramer, Leutershausen

Tibor Rad6
On the Problem
of lateau
Subharmonic Functions
Reprint
Springer-Verlag New York Heidelberg Berlin 1971
AMS Subject Classifications (1970): 49FlO, 31C05
ISBN 3-540-05479-0' Springer-Verlag Berlin Heidelberg New York
ISBN 0-387'-05479-0 Springer-Verlag New York Heidelberg Berlin
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is
concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by
photocopying machine or similar means, and storage in data banks
Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable
to the publisher, the amount of the fee to be determined by agreement with the publisher
Library of Congress Catalog Card Number 71-160175. Printed in Germany
Herstellung: Strauss & Cramer, Leutershausen
ERGEBNISSE OER MATHEMATIK
UND IHRER GRENZGEBIETE
HERAUSGEGEBEN VON DER SCHRIFTLEITUNG
DES
"ZENTRALBLATT FOR MATHEMATIK"
ZWEITER BAND
---------------2---------------
ON THE PROBLEM
OF PLATEAU
BY
TIBOR RAD6
WITH 1 FIGURE
BERLIN
VERLAG VON JULIUS SPRINGER
1933
Contents.
Page
Introduction. . . . . . 1
1. Curves and surfaces 2
II. Minimal surfaces in the small . 19
III. Minimal surfaces in the large. 31
IV. The non~parametric problem . 49
V. The problem of Plateau in the parametric form. 68
VI. The simultaneous problem in the parametric form. Generaliza-
tions ......................... 90
ERGEBNISSE DER MATHEMATIK
UND IHRER GRENZGEBIETE
HERAUSGEGEBEN VON DER SCHRIFTLEITUNG
DES
"ZENTRALBLATT FOR MATHEMATIK"
FONFTER BAND
SUBHARMONIC
FUNCTIONS
BY
TIBOR RADO
BERLIN
VERLAG VON JULIUS SPRINGER
1937
Preface.
A convex function f may be called sublinear in the following sense;
if a linear function l is ::=: j at the boundary points of an interval,
then l:> j in the interior of that interval also. If we replace the
terms interval and linear junction by the terms domain and harmonic
function, we obtain a statement which expresses the characteristic
property of subharmonic functions of two or more variables. This ge
neralization, formulated and developed by F. RIEsz, immediately at
tracted the attention of many mathematicians, both on account of its
intrinsic interest and on account of the wide range of its applications.
+
If f (z) is an analytic function of the complex variable z = x i y.
If I
then (z) is subharmonic. The potential of a negative mass-distribu
tion is subharmonic. In differential geometry, surfaces of negative
curvature and minimal surfaces can be characterized in terms of sub
harmonic functions. The idea of a subharmonic function leads to
significant applications and interpretations in the fields just referred
to, and· conversely, every one of these fields is an apparently in
exhaustible source of new theorems on subharmonic functions, either
by analogy or by direct implication. The purpose of this report is
first to give a detailed account of those facts which seem to consti
tute the general theory of subharmonic functions, and second to pre
sent a selected group of facts which seem to be well adapted to illu
strate the relationships between subharmonic functions and other
theories. Roughly, Chapters I, II, III, V, VI are devoted to the first
purpose, while Chapters IV and VII are devoted to the second one.
The presentation is formulated for the case of two independent vari
ables, but both the methods and the results remain valid in the general
case, except for obvious modifications, unless the contrary is explicitly
stated.
Subharmonic functions have a long and interesting history. F. RIESZ
points out that various methods, due to POINCARE, PERRON, REMAK
in potential theory and to HARTOGS and R. NEVANLINNA in the theory
of functions of a complex variable, are based essentially on the idea
of a subharmonic function. The reader should consult RIESZ [4], [5].
for detailed historical references. Readers interested in the possibilities
of applying subharmonic functions may read, for general information,
RIESZ [4J, [5J, BECKENBACH-RADO [1], [2], EVANs [4], FROSTMAN [1].