Allen Shields*

Carath6odory and Conformal Mapping

In 1912 and 1913 (that is, 75 years ago), Constantin Carath6odory published three fundamental papers on conformal mapping in the Mathematische Annalen (Car- a th6odory [1912], [19131,2] ). The results in these papers are still of great importance. By a domain we mean a connected open set in the complex plane C; also, D denotes the open unit disk. An Appendix con- tains the statements of two theorems referred to in the text.

In the first paper Carath6odory considers sequences {Gn} of simply connected domains that all contain the origin. For simplicity he first assumes that they all lie inside some fixed disk. Let f, denote the Riemann mapping function that maps D onto G, with f, (0) = 0, and fn' (0) > 0. He considers the following problem: find conditions on {G~} under which {f,(z)} converges for all z in D. He first shows that if ~f,(z)} does con- verge, to a function f(z) (z E D), then either f = 0 or f is one-to-one.

Carath6odory defines the kernel G of the sequence {G,} as follows. If there is no neighborhood of 0 con- tained in all the domains Gn, then G = {0}; otherwise, G is the largest domain with the property that every compact set, contained in G and containing 0 in its in- terior, is contained in all but finitely many of the do- mains G,. (He shows the existence and uniqueness of G by an explicit construction). The sequence {G,} is said to converge to its kernel G if G is also the kernel of each infinite subsequence of {G,}. Finally, he proves that the sequence {fn(z)} converges in D (call the limit function flz)) if and only if the sequence {G,} converges to its kernel G. Further, if G = {0} then f is the zero

function; otherwise, f is the Riemann map of D onto G. In the third part of the paper Carath6odory intro-

duces, with a rather cumbersome definition, a class of domains that have come to be known as Carathdodory domains. Simply stated, these are the bounded, simply connected domains G such that every neighborhood of every bounda ry point contains points that can be joined to oo by a path that avoids the closure of G. These domains have turned out to be useful in polyno- mial approximation (see Farrell [1934], Marku~evi~ [1936], and Rubel, Shields [1964], for example). It turns out that Carath~odory domains are character- ized (among bounded, simply connected domains) by the following property: if f is any analytic function in G, with [fl ~ 1, then there exists a sequence {Pn} of polynomials such that [Pn[ ~ 1 in G for each n, and {p,(z)} converges to f(z) for all z in G.

This paper is noteworthy in another respect. In it for the first time the Schwarz Lemma is pointed out and named, and its importance emphasized. (That Cara- th6odory was the first to name it is suggested in foot- note 2 to Carath6odory [1914].) He uses the Iemma to prove the un iqueness of the conformal map of a simply connected domain onto D, when the point sent to 0 has been specified and the mapping function has a positive derivative at this point. (He states that the uniqueness was first proved in Poincar6 [1884], p. 231.)

Shortly afterward Bieberbach [1913] (he was then writing his dissertation in Basel) and Koebe [1914] pointed out that the Schwarz Lemma could be dis-

* Column editor's address: Department of Mathematics, University of Michigan, A n n Arbor, MI 48109 USA

18 THE MATHEMATICAL INTELL1GENCER VOL. 10, NO. 1 �9 1988 Springer-Verlag New York

pensed with and the theorem generalized. We quote from Bieberbach.

This adherence to the Schwarz Lemma probably hindered Carath6odory from presenting his theorem in the gener- ality in which it can be formulated. It holds not only for mappings to a disk but also to any other domain, simply or multiply connected. The methods that we use . . . are very elementary. Not even for the new uniqueness theorem in w 5 do we require any other tool than Montel's theorem, which dominates my entire work.

Bierberbach states the un iqueness theorem as follows. Let B be a bounded domain containing the origin and let f be an analytic function in B, different from z. Let f(B) C B, riO) = 0, and either f' (0) = 1 or ~'(0)1 # 1. Then f cannot be a one-to-one map of B onto itself.

Bieberbach uses an equivalent definition of kernel of a sequence {B,} of domains all containing the origin but not necessarily simply connected. The kernel B is the largest domain conta ining the origin each of whose points is in all but finitely many of the domains {Bn}. He general izes Cara th6odory ' s theorem as follows. Let {B,} be a sequence of domains, uniformly bounded, each containing the origin and converging to their kernel B. For each n, let f, be one-to-one and analytic in B~ and map it onto a domain A,, so that the domains {An} are uniformly bounded. Assume further that for each n, f,(0) = 0, and either f,'(0) = I or ~,'(0)1

1. Then {f,(z)} is convergent at all points of B (to a limit function f(z)) if and only if {A,} converges to its kernel A. In this case f maps B onto A.

In the second paper [19131] Carath6odory proves that if G is a Jordan domain (that is, G is bounded and the boundary of G is homeomorphic to a circle), then the Riemann mapping function from D to G extends to be a homeomorphism between the closed domains. This result was conjectured by Osgood [1901], p. 56. The conjecture was proved independently and more or less simultaneously by Osgood and Taylor [1913]. (Carath6odory had presented his solution to a meeting in Karlsruhe in 1911. At the end of the printed paper he gives Easter, 1912 as the date of completion of the manuscript. Osgood and Taylor date their manuscript September 1912.)

Incidentally, the first complete proof that every simply connected plane domain with more than one boundary point can be mapped homeomorphically by an analytic function onto D (i.e., the Riemann map- ping theorem for plane domains) was given by Os- good in 1900. He states that his method is similar to that used by Poincar6 [1883]. (Osgood concludes his paper with the words , " O n the Atlantic, June 11, 1900.") Earlier work by Schwarz and Painlev6 and others had considered mainly Jordan domains with piecewise smooth boundaries, and had shown that the analytic homeomorphism of the domain onto D ex- tended to a homeomorphism between the closures.

Constantin Carath4odory. (Photo from p. 237 of Hilbert, by Constance Reid, Springer-Verlag New York 1983.)

Carath6odory's proof of the result for general Jordan domains is based on a theorem of Fatou (1906): If f is bounded and analytic in D, then the limit, as r in- creases to 1, of f(re it) exists, except perhaps for a set of t of linear Lebesgue measure zero (0 ~ t < 2=). He gave a new proof of Fatou's theorem, based on Lebesgue's theorem on differentiating an integral. Carath6odory only needed the special case that the limit exists for a dense set of t, but even this case seemed to require Lebesgue's theory. He writes (p. 307): "The proof of Osgood's conjecture was probably not possible at the time it was made (1900). Since then, however, analysis has been enriched by one of the most far-reaching dis- coveries that this discipline has k n o w n - - I mean the Lebesgue integral and the theorems that Herr Le- besgue has proved with its aid." Koebe responded to this, and in his review (JFM, 44, p. 760) of his own paper [1913] he reports that he is able to prove the theorem using only elementary complex analysis, "so that the use of the Lebesgue theory appears indeed as a significant complication."

LindelOf [1914] gave an extremely simple proof of Osgood's conjecture (without using Lebesgue theory). Courant [1914] proved a theorem giving conditions on a sequence {G,} of Jordan domains so that the domains "converge" to a limit Jordan domain G, with the nor-

THE MATHEMATICAL INTELLIGENCER VOL. 10, NO. 1, 1988 19

m a l i z e d m a p p i n g f u n c t i o n s f rom D to the Gn converging uniformly on the closure of D to the map- ping function for G.

Courant 's theorem was applied to polynomial ap- proximation in Walsh [1926]. Earlier Walsh [1924] had given a rather awkward proof that if f is continuous on the closed domain b o u n d e d by an analytic Jordan curve, and holomorphic in the interior, then f can be uniformly approximated by polynomials on the closed domain. The 1926 paper proves this for arbitrary Jordan domains. In the introduction Walsh states that Carath6odory had suggested that Courant's theorem should be applied to the problem. With this tool the proof is rather easy. Indeed, consider first the special case G = D. We replace f by fr (here fr (z) = f(rz), 0 < r < 1). Then fr is holomorphic in a neighborhood of the closed unit disk and so, by Runge's theorem, it may be approximated uniformly on the closed disk by rational functions and, in fact, by polynomials. Finally, fr is uniformly close to f on the closed disk, and so f can be approximated uniformly by polynomials. Now, for the case of a general Jordan domain G, Courant's theorem allows one to argue in a similar manner (replacing the larger disks {Izl < 1/r} by larger Jordan domains shrinking down to G). It seems likely that Carath6o- dory realized all this, but instead of publishing the re- sult himself, he gave it to the younger mathematician (Walsh was 31 in 1926, and Carath6odory was 53).

In the third paper (Carath6odory [1913]), a general theory is developed for the boundary behavior of the Riemann map from a general simply connected do- main to the unit disk D. This can be viewed as a vast extension of the result for Jordan domains. It involves a subtle compactification of the domain and the exten- sion of the mapping to the resulting "boundary." The theory is closely connected to the theory of cluster sets

If there had been Fields Medals at that time, Carath~odory might have been a candidate on the basis of this work.

of analytic functions. For more recent treatments of the theory see Collingwood, Lohwater [1966], Chap. 9, and Pommerenke [1975], Chap. 9. Quite recently the theory has found new applications at the hands of Sullivan [1985]. For another side of the theory, see Pir- anian [1958].

If there had been Fields Medals at that time, Carath- 6odory might have been a candidate on the basis of this work. He could not have received a medal how- ever, even if they had been instituted. He was 40 years old in 1913, and the fifth International Congress of Mathematicians had just taken place in 1912 in Cam- bridge. Because of the war and its aftermath the next congress was not held until 1924, but even if there had

been a congress in 1916 he would have been too old for a medal. When the first medals were given, at the Oslo Congress in 1936, Carath6odory was on the jury and gave the speech on the work of the two recipients: L. Ahlfors (born 1907) and J. Douglas (born 1897).

Biography

What follows is based on and excerpted from an un- finished autobiographical article Carath6odory pre- pared at the request of the Austrian Academy [1954- 57], Vol. V, pages 389-408. The article is most inter- esting and should be read in its entirety.

Constantin Carath6odory was born September 13, 1873, in Berlin, to Greek parents. At that time there were many ethnic Greeks in the Turkish foreign min- istry, including the ambassadors to Berlin, London, Paris, Rome, Stockholm, and Washington. Carath6o- dory's grandfather was the personal physician to the Sultan, and his father was the ambassador to Belgium. His father had a wide circle of friends from many lands; guests were always at their house, people of considerable culture who spoke several languages. This was the atmosphere in which Carath6odory grew up. In 1895 he graduated from the Ecole Militaire de Belgique (modeled on the l~cole Polyt6chnique in Paris). He worked as an engineer in Turkey and Egypt, including a year and a half on the Aswan and Asiot dams. He writes that many nights he slept on the dry bed of the Nile when the water was diverted for the construction.

Carath6odory wanted to s tudy mathematics. He writes (p. 398):

My family and my old Greek friends . . . found my plan, to give up a secure position with many possibilities for the future to satisfy a romantic inclination, more than comical. I myself was not at all convinced that my plan would suc- ceed and bear fruit. However, I couldn't overcome the ob- session that only the unrestrained occupation with mathe- matics would give my life its content. I wanted to live for awhile somewhere where I could be free of external influ- ences; Berlin seemed better for this than Paris, where I had many friends and relatives.

He enrolled at the University of Berlin at the begin- ning of May 1900. Three full professors, Ferdinand Georg Frobenius, Hermann Amandus Schwarz, and Lazarus Fuchs, and three associate (= ausserordent- liche) professors, Kurt Wilhelm Sebastian, Alfred Hettner, and Johannes Knoblauch were on the mathe- matics faculty. Fuchs was elected rector just then, and Carath6odory had little contact with him. He writes (p. 399):

Also with Frobenius I had very little personal contact, though I followed his lectures with the greatest enthu- siasm. They must have been among the most beautiful lectures anywhere in Germany at that time. Their only fault was that they were too smooth and complete, and

20 THE MATHEMATICAL 1NTELLIGENCER VOL. 10, NO. 1, 1988

because of this the existence of unsolved problems was not even hinted at.

Ca ra th6odory benef i ted mos t f rom Schwarz, and not on ly f rom his official lectures. Schwarz ran a "Col- l oqu ium" twice a mon th to go through his collected works! (Schwarz was born in 1843 and, amazingly, his col lected works--Gesammelte Mathematische Abhand- lungen--were publ ished by J. Springer in 1890.) Here C a r a t h 6 o d o r y was e x p o s e d to confo rmal m a p p i n g (Schwarz had been the first to prove the Riemann

"I couldn't overcome the obsession that only the unrestrained occupation with mathe- matics would give my life its content."

ma p p ing theorem for Jordan domains with piecewise analytic boundaries; his "a l ternat ing p rocedure" for solving the Dirichlet problem had become a s tandard technique). After the seminar s tudents and professor wen t to a beerhall (the Franciscan, below the elevated train on Friedrichstrai~e) to cont inue the discussion. Cara th6odory learned from Schwarz that the best way to unde r s t and general theories is to master special ex- amples f rom the g round up.

A l m o s t m o r e i m p o r t a n t was tha t t h r o u g h the Schwarz seminar Cara th6odory met m a n y ta len ted s tudents , including F. Hartogs, P. Koebe, E. Hilb, and (from the USA) O. D. Kellogg. From his first days in Berlin Cara th6odory had become close friends with L. Fej6r a n d Erha rd Schmidt ; Schmid t especia l ly re- ma ined a close fr iend for life.

In 1901 Schmidt left for G6tt ingen; he re turned for a Chris tmas visit and told such enthusiastic tales of the mathemat ica l a tmosphe re that Cara th6odory trans- ferred there in April 1902. At that time there were two full p rofessors in G6t t ingen: Felix Klein and David Hilbert. Klein was an extremely polished lecturer, and also very conscientious. He could often be seen in the library, an hour or more before his lecture, reviewing the l i terature references he was going to give in the lec ture . Hi lbe r t ' s l ec tures w e r e less po l i shed , b u t "wi th their weal th of insights they were the most orig- inal and beautiful that I have ever heard ." Hilbert and Klein h a d labored to build up mathemat ics at G6t- t ingen, and th rough their influence a third Ordinariat in pure mathemat ics was created. To fill this chair they called H e r m a n n Minkowski, in the fall of 1903. The "Ex t rao rd ina r i a t " in appl ied mathemat ics , fo rmer ly occupied by F. Schilling, was conver ted in 1904 to an Ordinariat , and C. Runge was called.

In the fall of 1903 Gustav Herglotz and Hans H a h n came to G6tt ingen. H a h n had just finished his disser- tation wi th G. von Escherich, on the calculus of varia- tions, and short ly before Chris tmas he gave some lec- tures on Escherich's theory of the second variation for Lagrangian problems. Cara th6odory writes (p. 405):

We were all astonished that, according to this theory, there existed exceptional cases in which apparently no so- lution to the variational problem existed. I attempted to find a simple, geometrically clear example of this phenom- enon. After several days I found the following: a l~mp is surrounded by a half-spherical globe, and the points of the globe are thus centrally projected onto the floor. One seeks a curve of given length on the globe, joining two given endpoints, such that the length of the shadow on the floor shall be as long or as short as possible. One would have guessed that this shadow must consist of two segments that meet at an angle.

On the 22 of January I found myself in Berlin, where Schwarz had invited us to celebrate his 200th Colloquium. Afterward, in a few hours in a coffee shop on the Pots- damer Platz, I was able to calculate the Weierstrass E- function for my problem. A few weeks later I had the gist of my work on discontinuous solutions, and I was able to work it out in detail in Brussels during the Easter vacation. In the last few months in G6ttingen I had become attached to Minkowski; it was only natural that it was to him, and not to Hilbert or Klein, before whom I felt somewhat shy, that I handed my work as a dissertation. [This dissertation was reprinted in [1954-57], Vol. I, pages 3-79.] In July I passed the Rigorosum; in the cognate areas I was exam- ined by Klein in applied mathematics, and by Schwarz- schild in astronomy . . . .

Up to this point I had not for a moment thought of re- maining longer in Germany. The Saturday before my exam was the traditional annual outing of the mathemati- cians, in which both the professors and their students took part . . . . There Klein suggested to me that I should stay on in G6ttingen for my Habilitation. [This was an ad- ditional degree, beyond the doctorate; after this one could become Privatdozent and give lectures.] This conversation decided the fate of the rest of my life.

Cara th6odory a t t ended the Internat ional Congress of Mathematicians in Heidelberg that summer (1904).

Returning to GOttingen I came into closer contact with Hilbert, who insisted that I should write my Habilita- tionsschrift without delay. He acquainted me with the dissertation of his American student Noble, and I was able to make very good use of it. In the middle of December I returned home to Brussels to write it all up. On Hilbert's proposal the faculty gave me permission to submit the Habilitationsschrift immediately after the receipt of the doctorate; thus on March 5, 1905, I acquired the venia le- gendi.

Ca ra th6odory gave his first course of lectures in G6tt ingen that summer . At the end of the semester P. Boutroux (Poincar6's nephew) came to G6tt ingen for a few days.

During a walk he told me of his ideas for simplifying Borel's proof of Picard's theorem, which was then very popular. Boutroux had noticed that this proof only worked because of a curious rigidity in conformal map- pings, which, however, he had been unable to capture in a formula. This discovery of Boutroux gave me no peace, and six weeks later ! could prove Landau's sharp- ening of Picard's theorem in just a few lines by using the theorem nowadays called the Schwarz Lemma. I had proved this lemma using the Poisson integral; when I showed it to Erhard Schmidt he informed me that the re-

THE MATHEMATICAL INTELLIGENCER VOL. 10, NO. 1, 1988 21

sult was already in Schwarz, and that it could be proved by very elementary methods. And indeed, the proof that he showed me cannot be improved.

Cara th6odory remained in GOttingen as Privatdo- z e n t t h r o u g h Easter 1908. His accoun t b reaks off abrupt ly just after this, and we conclude with material f rom Schmidt 's obi tuary in the collected works [1954- 57], Vol. V, pages 411-419. Cara th6odory w e n t to Bonn and later, as Ordinarius, to the technical schools in H a n n o v e r and Breslau; he was finally called to G6t- t ingen in 1913 as Klein's successor. In 1918 he wen t to Berlin. He left after two years at the reques t of the Greek g o v e r n m e n t and was given full power to or- ganize the newly p lanned Greek universi ty in Smyrna. After some initial success, these efforts came to an abrup t end w h e n Turkey marched in and occupied the area. He spent two more years in Athens at the uni- vers i ty and technical school. In 1924 he went to Mu- nich as professor, where he finished his academic ca- reer. He died there February 2, 1950. After describing C a r a t h ~ o d o r y ' s v a r i e d m a t h e m a t i c a l i n t e r e s t s , Schmidt concludes:

His picture speaks to us: Willst du ins Unendliche schreiten, geh nur im Endlichen nach allen Seiten.

The mathematical world lost with him one of its leading figures, German science one if its most esteemed persona- lities, and I, if I may speak of myself, my best friend.

A p p e n d i x

The Schwarz Lemma states that if f is analytic in D, wi th f(D) C D and f(0) = 0, then ~(z)l ~ Izl for z E D. If equa l i ty ho lds for one n o n z e r o va lue of z t h e n f(z) = cz, where Ic[ = 1.

M o n t e l ' s t h e o r e m (also called the Monte l -Vi ta l i theorem) states that if ft,} is a uniformly b o u n d e d se- quence of analytic funct ions in a domain G, then {f,} contains a subsequence that converges uni formly on each compact subset of G. Corollary: If the sequence {f,(z)} converges for an infinite set of points having a limit point in G, then the sequence converges at every point in G, and uniformly on each compact subset.

R e f e r e n c e s

Abbreviations: MR = Mathematical Reviews, JFM = Jahrbuch ~iber die Fortschritte der Mathematik, ZBL = Zentralblatt ffir Mathematik.

L. Bierberbach [1913], Uber einen Satz des Herrn Carath6o- dory, Nachr. Kgl. Gesell. Wissen. G6ttingen, 552-560. JFM 44, 760.

C. Carath6odory [1912], Untersuchungen fiber die kon- formen Abbildungen von festen und ver~inderlichen Ge- bieten, Math. Ann. 72, 107-144. JFM 43, 524-525. [1954- 57], VoI. I.I.I, 362-405.

- - [19131], Uber die gegenseitige Beziehung der Rander

bei der konformen Abbildung des Inneren einer Jor- danschen Kurve auf einen Kreis, Math. Ann. 73, 305-320. JFM 44, 757. [1954-57], Vol. IV, 3-22.

- - [19132], Uber die Begrenzung einfach zusammenhan- gender Gebiete, Math. Ann. 73, 323-370. JFM 44, 757-758. [1954-57], Vol. IV, 23-80.

- - [1914], Elementarer Beweis ffir den Fundamentalsatz der konformen Abbildungen, Schwarz-Festschrift, J. Springer Verlag, Berlin, 19-41. JFM 45, 667-668. [1954- 57], Vol. III, 273-299.

- - [1936], Bericht iiber die Verleihung der Fieldsmedaillen, Congr6s Internat. Math. Oslo, 308-310. JFM 63, 816. [1954-57], Vol. V, 84-90.

- - [1954-57], Gesammelte Mathematische Schriften, Vols. I-V, C.H. Beck'sche Verlag, Mfinchen. MR 16, 435, 485; 17, 446; 18, 453; 19, 108.

E. F. Collingwood and A. J. Lohwater [1966], The theory of cluster sets, Cambridge Tracts in Math. 56, Cambridge Univ. Press. MR 38 #325.

R. Courant [1914], Uber eine Eigenschaft der Abbildungs- funktionen bei konformer Abbildung, Nachr. Kgl. Gesell. Wissen. G6ttingen, 101-109. JFM 45, 668. See also: Be- merkung zu meiner Note "Uber eine Eigenschaft . . . ," ibid. [1920], 69-70. JFM 48, 1235.

O. J. Farrell, [1934], On approximation to an analytic func- tion by polynomials, Bull. Amer. Math. Soc. 40, 908-914. JFM 60, 1010; ZBL 10, 348.

P. Koebe [1913], Randerzuordnung bei konformer Abbil- dung, Nachr. Kgl. GeselI. Wissen. G6ttingen, 286-288. JFM 44, 759-760. [1914], Zur Theorie der konformen Abbildung und Uni-

formisierung, Leipziger Berichte 66, 67-75. JFM 45, 670. E. Lindel6f [1914], Sur la representation conforme, C.R.

Acad. Sci. Paris 158, 245-247. JFM 45, 665. A. I. Marku~evi~ [1936], Sur la repr6sentation conforme des

domaines h fronti~re variables, Mat. Sbornik 1 (43), 863-886. JFM 62, 1215; ZBL 16, 310-311.

W. F. Osgood [1900], On the existence of the Green's func- tion for the most general simply connected plane region, Trans. Amer. Math. Soc. 1, 310-314. JFM 31, 420.

- - [1901], Allgemeine Theorie der analytischen Funk- tionen a) einer und b) mehrerer komplexen Gr6t~en, En- cykl. Math. Wissen. 22, i-114. JFM 33, 289.

W. F. Osgood and E. H. Taylor [1913], Conformal transfor- mations on the boundaries of their regions of definition, Trans. Amer. Math. Soc. 14, 277-298. JFM 44, 758.

G. Piranian [1958], The boundary of a simply connected do- main, Bull. Amer. Math. Soc. 64, 45-55. MR 20, #6526.

H. Poincar6 [1883], Sur un th~or~me de la th6orie g6n6rale des fonctions, Bull. Soc. Math. France 11, 112-125. JFM 15, 348.

- - [1884], Sur les groupes des 6quations lin6aires, Acta Math. 4, 201-312. JFM 16, 252-257.

Chr. Pommerenke [1975], Univalent Functions, Vandenhoeck & Ruprecht, G6ttingen. MR 58 #22526.

L. A. Rubel and A. L. Shields [1964], Bounded approxima- tion by polynomials, Acta Math. 112, 145-162. MR 30 #5104.

D. Sullivan [1985], Quasiconformal homeomorphisms and dynamics. 1. Solution of the Fatou-Julia problem on wandering domains, Annals Math. 122, 401-418. MR 87i:58103.

J. L. Walsh [1924], On the expansion of analytic functions in series of polynomials, Trans. Amer. Math. Soc. 26, 155-170..JFM 50, 239.

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