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ProQuest Information and Learning 300 North Zeeb Road, Ann Arbor, Ml 48106-1346 USA 800-521-0600 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. GENERALIZATIONS TO SPACE OF THE CAUCHY AND MORERA THEOREMS Maxwell Reade A thesis presented to the faculty of The Rice Institute in partial fulfillment of the requirements for the degree of Doctor of Philosophy. June 1S40. Hf-¥T93 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 3079848 _ __ _® UMI UMI Microform 3079848 Copyright 2003 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O . Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. GENERALIZATIONS TO SPACE OF THE CAUCHY AND MORERA THEOREMS INTRODUCTION. Consider the function CD iipa - Xi CU,0-}1 XVCtblhh defined in a simply connected domain D .* A neces sary and sufficient condition that -ft*) be analytic in f) is that the Cauchy-Riemann equations be satisfied there x1 - K Sir * - ttr 2Ky > 2)u that is (2 ) ^ur-o, where is a differential operator. From (2) we obtain ( 3 > which can be written in the form ( 4 ) E j=/ Conversely, j T ^ ' r 3 ” ° • =t (3) implies either (2) or A ur-o , *We shall consider only finite simply connected do* mains. 1. See the theorem of Looman-Menchoffs S. Sales, Theory of the Integral, sec. rev. ed., (1937), p. 199. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2where 7 _ *5 • 7) A = 'ZCL i tlT CTf ' If (3) holds, then we say that and form a couple of con.iugate harmonic functions. The real functions (5) Xj-z. defined and continuous in a simply connected domain D , will he said to define a surface S . if the first partial derivatives of the functions (5) are continuous and satisfy (6 ) PCU,U^« CrCU.U^ PCtWl^s-O, where ■4 - r if*>\x ^ = 2 1 ^ S-\ (7 ) F= Sif-*’ Z_. ^u. ^ ‘ J _l (La T" (£fj ' r~ bo- * * are the coefficients of the first fundamental dif ferential quadratic form,1" c/s*"- EduJ'*- ZFjiudif 4 Grd<f‘%' of the surface S , then the parameters «,ir are said 3 to he isothermic parameters, and the surface is said to he given in isothermic representation. P on 5 is conformal except where The map of . Now (6 ), which is a generalization of (4), can he written in the form (8 ) JLCA j-i O* 2. W. C. Graustein, Piffe-rantial Geometry, p. 82. 3. Graustein, op. cit., p. 131. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -3which is a generalization of (3). The following is a combination of the Cauchy and tj Morera theorems. If the function 4-c.fc} in (1) simply connected domain continuous in a , then a necessary and suf ficient condition that ^ analytic ip D is that for each closed rectifiable Jordan curve ft in D lying , (9 ) ^ k w ^ - o . We may write (9), when we separate its left-hand member into its real and imginary parts, in a form analogous to the Cauchy-Riemann equations: But the relations (10) imply j'-i j =l w which are analogous to the relations (4); and (11) may be written in the following form analogous to (3) s (12) t\ L S-i * A generalization of (12), analogous to (8 ), is (13) Z [ k <;tu',rt‘ii f = o , j-i 4. E. J. Tovmsend, Functions of a Complex Variable. p. 82. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -4where £ is an arbitrary closed rectifiable Jordan curve lying in f) , In this paper we shall study (13) and J—I where Kyiuuirtifcj - fin.*) , ^L is the circle in J) with variable radius A and fixed center Clu,^ , and where is a quan tity (not always the same quantity) such that A—■>O A.x /u ~ If the functions (5) are harmonic and satisfy (6 ) In a simply connected domain b , they have been called s a triple of conjugate harmonic functions. In terms of this definition, a theorem of tfeierstrass^may be stated as follows. A necessary and sufficient condition that the func tions (5), defined in a simply connected domain, be the coordinate functions of a minimal surface given in isothermic representation is that they form a triple of conjugate harmonic functions. We recall the following theorem. Theorem A. 5. i A necessary and sufficient condition S. F. Beckenbach and T. Radd, Subharmonic Func tions and Minimal Surfaces. Trans. Am. Math. Soc., v. 35 (1S33), pp. 648-661. 6 . Loc» ext., p. 649. 7. Graustein, op. cit., p. 98. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -5that the functions (1 ), having continuous partial de rivatives of the third order in a simply connected do main. he the coordinate functions of a spherical sur face that there exist a constant 06 such that where e _ \ - J fin Xavrl » i* lu u - X-L.U- - f W I ^ j^ J J- —---------- , H H H are the coefficients of the second fundamental quadratic form of 5 t , and where V e f r - F1- , ^ ' Here we have used the notation I Q-v. =■ Ou <W 0.* bi k ^ \ C.v C. C* It follows immediately that a necessary and suf ficient condition that the functions (5) be the coor dinate functions of a spherical surface given in iso thermic representation is that both (6 ) and (14) e' ~ l > hold. If the functions (5) have continuous partial deri vatives of the main 0 awv. th order in a simply connected do , then about each point ftia.vTo) of 0 we have a finite Taylor expansion for each functions S. If q , then 'S is a plane surface; loc. cit., p. 97. 9. Loc. cit., pp. 93-94. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where a - U o - a toi © > iT-tro -fi * where t^O- 2 - -f £cr is a differential operator, and where the partial de rivatives are evaluated at Ctu, tf6) . We establish the following result for functions (5) with Taylor expansions (15). If C/v is. the circle in D with center at c«.o, if radius a. , then IC (16) H[J j‘-_j lls?'*1' KTa-O ^)=o CA where ^ t+i ^ * "V;. -fe- where 0tt*' air-.’V where * r ^ ,c • ’ - / 3t dC-sl' and where ifi the greatest integer not greater than <,(IV*— Proof. We make use of the following equalities. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -7. , , r t ^ ) r c ^ j , PM Ar /A - + ----------------t&i?& de-^i AV\+Wf-1_\ 2_ TT /-.7 \ '' » - V C Jfc ' 1L ( SuA^fi C.O&*© J.&- - f Jo Jd + / C-i^(0sw*0 cos-4© dd t ( ^ 9 m se i© ^ [ i t - C - O ^ U ^ Jo o 0 <***« ^ Prom (17) we obtain 'sux*® e0-s»M<9ci© = I?/aaja<*. f (18) (b) - o where (a) holds for v* holds for ma and m both even and where (b) and At not both even, and where r t N g j r t ^ p^ w f u f l . a. =- Aj ' r »1------------W > r(s*i> --- , Ojljl,' ■ rcf^+o where w^j.^ and % . But- 10. W. F. Osgood, Advanced Calculus. p. 485, ex. 2. 11. These results follow from the following pro perties of the Gamma function (loc. cit., pp. 482-483)s T C r o d , for O o , rcO-a-i and rc^=Jir. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -8therefore (18), (a), may be written as (19) ( vx ' Jo -311— fcffK O p , 7* ? ' <£•' CP + P ' * -f’i " . Consider where the partial derivatives are evaluated at (u0>cr0) . We obtain (2° ) f | ( w e x siii l H - Js -0; =.4V '' ^ 0o Here we have used (18), (b). Similarly we obtain (21) J \(co*©£- -KtMa-^) /f/lsoiS de -o, o *> to dt*-' -i Consider the expression ( 22) ^^ * =<Vj' ' j -ljz.j3. Applying (18), (b), to (22) we obtain f^ T - 1 *** s & « ]« * /» % ,£ % _ je which, by (19), may be written in the form LIT _ _ 3-=,tl , . s i o \C e g & o -t-tux© ^ S K: hm£> = m t O / JiS *\I ^.j L * L _ 1 sTa^oi &Uir^u'w t ] (23) H ^ l l l L 2 -a.V 2.15 S( C£+/i' So. j -> Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. J . -9Similarly we obtain 3 = -0 , (24) 'ixs 5{«-t-or a#- J* From (20), (21), (23) and (24) we obtain, for the functions (15), 1*4-1 1 U;iu,^At-rr^«r^r from which we obtain (16). For , (16) is displayed in (26), (72) and (58) respectively. 1. CHARACTERIZATION' OF ISOTHERMIC MAPS. Theorem 1. If the functions (1) have continuous partial derivatives of the first order in a sinrolv connected domain 0 , then a necessary and sufficient condition that they map 0 S ig, that for each point (25) isothermicallv on a surface CMo,^) D , t i l * . j- i where Cv 0 jj=[ the circle in and radius Proof. with center at CUa>fo) A If the first partial derivatives of the functions (5) are continuous in j) , then we obtain a finite Taylor expansion for each function about an arbitrary point C(u,ir0) of j) If C/l is the circle in 0 by setting /*v- ( in (15). with center at C U o and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -1 0 - radius a. , then upon setting in (16) we ob tain (26) From (26) it follows that a necessary and sufficient condition that (8 ) hold in 0 is that (26) hold. This completes the proof of Theorem 1. 2. CHARACTERIZATION OF THOSE ISOTHERMIC SPHERI CAL MAPS THAT DO NOT MAP CIRCLES ON CIRCLES. Lemma i. If the functions (5) are not identical- IX constant in a simply connected domain D they map P on a sphere is_othermically on a surface J Qf finite radius (X , and if 3> that lies , such that circles are mapped on circles, then the functions (5 ) have the representation where the are real constants. where has one of the following forms. frirtf 1 frt+S and where & $1%) ia the real part, and the imaginary Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. part, of . Proof. Since the functions (5) are not identical ly constant in D sphere. he , it follows that ^ is not a point - Let the coordinates of the center CO-i* d*., 0.-*) . Q of $ Since the functions (5) map D thermically on 5 iso- such that circles are mapped on cir cles, the stereographic projection of aJ on the - plane, the coordinates of the pole of projection be ing (CL•>ft A<-) j where X and /-^I mum of the two quantities ICU*&I duces an isothermic map of D is the maxi and I on a domain , in- D of the $,■6 -plane whereby circles are mapped on circles. Here the 3,-6 positive S - and Xi - and -plane is so placed and oriented that the & -axes coincide with the positive X^ -axes respectively. ^ P on D The function mapping I^ must be linear, and therefore of the form Ca) = S b+S (6) + ft* S' where (a) holds if the map of P on P is directly con- formal and where (b) holds if the map of inversely conformal. D* P on P is The stereographic projection of is given by 12. This follows immediately from Theorem 25, p. 32, of L. R. Ford, Automorphic Functions. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12 ^l Ou |Jtl (S-tf.i) C s-ao1-* Q v+ (27) *«^l « ^ > ----- c s-aY’ +tt-flo'vlv <3.(2.li-l *• t+Lir-Qj'+X1' ■ CS-CU It follows that the map of D - On on S is given by litI <R ft-*-) ------------ * (28) where Vur)~ P-ifc)- t A . + c a J , and hence it follows that the functions (5) have the required form. We note that if it is given only that one non J) null circle in is mapped on a circle on S by the isothermic functions (5), then, as in the above discussion, O is mapped isothermically on such that one non-null circle in P circle in linear; D . D is mapped on a Therefore the mapping function is hence it follows that the functions (5) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -13have the representation (28). Lemma 2. If the functions (5) are not identical ly constant in a simply connected domain 0 , and if they man P S that lies isothennically on a surface &ZL a gffhere J gf finite radius 0. , then a necessary and sufficient condition that they man circles on cir cles that the first quadratic form of 5 have the representation (29) cU*«. Uu-uo^ur-u^ + b* \ Necessity. If the center Q l>o, of d is at then, by Lemma 1, the functions (5) are given by (28). From (27), a computation shows that we obtain £ H e l" = X- > I* where m $ Z ( £') . J =1 But lt t /*r-ptr l^'COI * — T 1 and Cs-atf-w* -a*y- - 1kttl"--1 ^ ; therefore 13. Compare with L. P. Eisenhart, A Treatise on thg Differential Geometry of Curves and Surfaces, p. 109. 14. Compare with Ford, op. cit., p. 118. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. li<t+piVJl'-|C«.+Av ] *■ which may he written as follows: (30) 0^ E - [tu.'U0)u+(.iMrfl)v+ ^ \ x where -K a o - -iUo = c^ifw-^,)+iv((rir1. - ^ <r,)J bv - (f|V+ A M l T , CttflT-nT^, N H<sl'u£-wr f\ - where the ^ >Q% ifcV U l V > &> , fir; and Fj are real constants, such that oc - a^-tt aTi., ^ -i + c ^v, <Sr"=- 'fi■+<■ Tv-j S' S, 4 * ' JL , Here we have used the representation (a) on page 11 for Ftfc) in order to derive the relation (30). Simi larly, we can derive (30) when F64-) has the form (b) on page 11 , the values of the constants (T0 and b*~ in (30) being altered accordingly. it The Gaussian (total) curvature of o (31) K - ^ 15. ze is given by J E. F. Beckenbach and T. Rado, Subharmonic Func- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -15which, with (30), yields k - ' Cv But jb fore is a sphere of finite non-null radius; >o . Under the hypotheses, meters. u>iT are isothermic para From (6 ) and (30) it follows that the first quadratic form of S Sufficiency. D there is precisely (29). Let the stereographic projection of on the sphere be with center at , where ^ b- is the sphere and radius and where the pole of projection is at (.CUtTo, b) . This projection is given by CtU-Uo^ * v- - U o + - ---------------------------- CU-Uo^V CVT-UaiV bV ^0+ (32) ^ ^\} > Ca-.u.oit'+ctr-tr0)V(>w I _ The first quadratic form of o * is found to be (33) \ (U -U o ) V ClT-{To)1' + b X which is identical with that of S 1 . It follows If- from (29), (31) and (33) that 5 and S tions and Surfaces of Negative Curvature. lie on Trans. Am. Math. Soc., v. 35 (1933), pp. 662-674. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -16spheres with equal radii. We shall show that to each circle in D there corresponds, by means of the functions (5), a circle on S' . Consider the non-null circle and let the interior of Q functions (3A) map R on yf S be denoted by R is the image of map R on a surface H C ^ , • The isothermically on a surface 27 ^ which is bounded by a non-null circle where curve in Co on S .According to the C "if, , and the functions (5) which is bounded by a reasoning in the proof /c of Lemma 1, it is sufficient that we take the non- null circle Co so\small that the spherical cap 21 ■jp is contained on a hemisphere. * There is a pole of i S'* . Let Pi* be a point of £ , and let Pa* P* be the arc of the n p ^ q ^ geodesic passing through ra and and lying on jf. ^ . Let the image of P» image of P/ in D P/ of in D on S be - in P be V 0 » the and the image of the arc be the arc . Let the image be Po , the image of <£>,on S be P, 16. See the bottom of page 12. 17. The geodesics on spherical surfaces are arcs of great circles; Graustein, op. cit., p. 149. is on a hemisphere; hence through each pair of points -4c there passes exactly one geodesic that lies on 2 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. • -17and the image of on 5 be P*P/ . Now f<sP, t* is a geodesic. Keep "*> fixed and allow r, to de'*r~^ scribe the circle C . is of fixed length ^ ^ ^ As P/ describes £ The geodesic arc for all points P' ^ Pa P> on ^ . Q , • From (29) and (33) it follows that the geodesic arc also m describes is of the same fixed length ^ for all points P» C . Moreover, as P» describes d arc ?& in An on S describes ^ exactly once around, exactly once around. D For, let the be the image of the geodesic arc and let the image of the geodesic arc on S , so that P6 ^ are corresponding geodesic 18. are on C arcs. Let P&*A * and A» and 4v are is a geodesic arc. and geodesic arc P*P/ corresponding curve on C Here . Since For, the geodesic arcs are arcs of shortest distance. not a geodesic arc on 21 -be and P6^' be another pair of corresponding geodesic arcs. Ai and on If PoP, were i then there would be a through Po and Pi on Z7 P#*Pf* would be of the same length. through P» and P' Since P» ft are of equal length and since. P»Pi and . The on Pa* P'* is shorter than Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -18the functions (5) and (32) are isothermic, it fol lows that the angle formed by and a, is equl to that between 11> A/ and a H P<a Aw . Hence there is a one-to-one correspondence between the points of C and ^ Hence the curve at ?0 • Co is C as P/ on S describes C a is once around. circle with center Hence we have shown that the non-null circle mapped on a circle C on S by the functions (5) Hence the functions (5) map all circles in therm! cally on circles on Theorem 2. ^ iso S If the functions (5) are not identi cally constant in a simply connected domain 0 , and if they have continuous partial derivatives of the third order in D , then 3 necessary and suffi cient condition that they map D a surface ^ isotherm!callv on that lies on a sphere of finite radius. such that circles are not mapped on circles, is that for each point vUa.iTo) in D j ii. (34) where Ca. is, the circle in 0 and radius ji , and that there exist a point CUi>(/Y) it follows that This is not so; with center at hence ft is not a geodesic arc, is a geodesic arc. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. in D such that «6) r \ I ^ X/CU,fF) -4(ftft*)i J -I where is the circle in j) with center at and radius jv. . Necessity. map D Under the hypotheses, the functions (5) isothermically on a e^Qface S that lies on a sphere of finite non-null radius. Hence (6 ) and (14) hold, and there exists a constant * (36) e ^ of.F, For the present representation of of Gauss <? such that gs. S , the formulas become *>uu “ R u-XjY - RirXjV- f>e 2. Ru“Xj'a 4- Ru. x ^ Xjwr- - where and where (38) in cyclic order, are the direction cosines of the normal to S We shall need the following relations. 19. Graustein, op. cit., pp. 135-137. 20. Loc. cit., p. 92. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. . -20- 4 A *; - (39) *; ^ i-\ (40) T inHif ^ j'il , < * ( i f ) v, To obtain (39) and (40), we shall use a method which depends upon the existence and continuity of the partial derivatives of order higher than three of the functions (5). These functions map thermically on a spherical surface. 0 iso- Therefore, as in the proof of Lemma 1, we can consider an intermedi ate stereographic projection to show that the func tions (5) have the representation (28), where either •?(.■£) or is analytic in D . From this rep resentation, it follows that the functions (5) have continuouspartial derivatives of all orders. To obtain (39) and (40), we shall need the follow ing equalities; (4 1 ) (42) Ax; 5 a.(e \ 3 (43) £ S ; < U ; =^o, 5-{ 3 (44) Z A<jiC=0* JV'l 3 (45 ) H av£; /U; ^o, Szi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. + -213 (46) 21 j =i A.X-i & Kj — & > 3 (47) XL J=( AX] fcA-Tj -so. It follows from (37) that -xACe zj) > and hence it follows that (41) holds. Operating on (41) with the operator A (48) A5; -f-Ae AC; * ^ -=9.(e&?; and operating on (48) with Recall the equalities: we obtain A we obtain (42). %■i (49) (50) j-I (51) 71 Z; t -\ i i (52) j -* i (53) +* - Z 3 = - H KjV , We note that (43) and (51) are identical. From (8) and (51) it follows that X I Cu. 21. - Z _ A;u ^ = Z.X- Graustein, op. cit., pp. 93-94. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -ot Hence the vectors whose direction components are , <)V, < ; . respectively, are mutually perpendicular.3'*' From (50) and (53) we obtain i 3 ^ju. ~ j -i j"-( ~ and hence the vectors whose direction components are respectively, are mutually perpendicular. (64) ija 22. 23. Hence V , “ Loc. cit., p. 5. Zju. and . K>‘u are proportionals *>a t Hence from (7), (36) and (52) we obtain s— oc . Here f - o only at isolated points. For if not, then there would exist a sequence of points of • • , and a point *© of & following properties* ao, , with the ~ and But, as has been noted on page 20, the functions (5) have the representation (28), where either or i's analytic in . 0 Then, as on page 14, we obtain £r- _--- 4d.v Jta' t where and + 1 1 ) 1 have the same meaning as in the proof of Lemma 2. hence i W /)Jc 4C> -t* ' , . Therefore -f'CW’o , It follows that -TCtlso and in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. D , From (50) and (53) we obtain YL - T 1 <;*;u J -I J-l and therefore the vectors whose direction components are respectively, are mutually perpendicular. Hence C55) We can write (54) and (55) in the following forms (56) *.- 0Lkti , *46, From (56) we obtain £ Ax; A?; --«X<:Ax;>\ Jit which, with (8), yields (44). j's| From (44) we obtain A Z.A<j A£j ^ E 4=1 «f=t /-< *“ * which, with (56), establishes (45). From (44) we obtain _ i> XI1Aj(;K; - T I m J /=< i-( from which it follows that -k*) is identically constant in 0 . But if foi) were identically constant in D , then it would follow from the representation (28) for the functions (5) that the functions (5) are identi cally constant in D • From this contradiction of the hypothesis, it follows that £ isolated points of D 24. can vanish only at . See footnote 23. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -24which, with (56), establishes (46). From (41), (43) and (44) we obtain 3 21 Aitj a Ax:; ~ o 3 i-1 which, with (56), yields (47). From (42)-(47), inclusive, it follows that (57) H A<i - a A \ 21 AKy A 5 . j =i From (36), (56) and (57) we obtain (39). From (8) and (56) we obtain I (A<j)v= o, /-t which combines with (36), (41), (49) and (50) to yield (40). Since the functions (5) have continuous partial derivatives of all orders in 0 , the following ex pression, obtained from (16) by setting /vk - S' , is valid: (58) ZL\ { -- -TSV v 3 -jr^5E l * Ax; a*A x;{-3C6Ax;)v]-e<r6 0 , «f-( where the partial derivatives are evaluated at L(io,(Js) and where Ca, is the circle in 0 and radius with center at Cilicia) is an arbitrary point in 0 Applying (8) and the relation Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. . to the relation (58),J we obtain I (59) x;tu,(rtcU ^3L J"-' cfc ™ Therefore (34) holds. a A.k ;+*<*AX£>v1f<T(^#), s-A Let us now suppose that (35) does not hold, i.e., for each point (U.O(iT<>) in ^ , (60) , j“-l Ca where C*, is the circle in D and radius A. . C6i) with center at (UojU*) From (59) and (60) it follows that Yl[ t/ic; W x / t i c a / l x / H - ° M holds. From (39), (40) and (61) we obtain (62) But, by (36), ? an(i therefore (62) yields (63) ~o, From (63) we obtain \e r which may be written in the following form: 2l A( 25. ^ A (Ijf . This relation follows from (47) and (56). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -26Hence (64) if'-f e ) where (65) \ &<.*) =-<D , Prom the imaginary part of (63) we obtain Pvr fr F which imply ( ) n E a '- f 66 where 4«.^) tion of tr . e 4 it a ) j £V - £ is a function of cl From (64), (65) and e 4Uo) j and 4vtl>) is a func (66) it follows that the function e (67) QLfc) ^ e 4du> , u is an analytic function of t-=-u-t-tir ^ . o) From the Cauchy-Riemann equations for the function (67) it fol lows that £ where uto , ou 4du.) =. ldoU--t-Gu , 4vtu-> ^ . 2S U o ^ +Av and Q-v are real constants. may be written in the form which yields (68) ^ci^-CKloU.-HLOdLuL, e^ij. gU-L Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. , Hence (64) -27From (68) we obtain AoCuN:1^)^ouu-hk(a-ku, -=. where is a real constant. (69) Hence e - y OloLU^^) +■ On.U-VdatT td>,|X Substitution in (31) shows that (70) _ which, since S implies that do^-0 . is on tjuobi ~ d a sphere of non-null radius, Hence (69) yields I where (71) ” dv H « *£O From (70) and (71) it follows that since $ But is on a sphere of finite non-null radius, it follows that form of K ’s-Cfco'M . 5 . Therefore the first quadratic may be written as JUS- _ where Cu> Therefore the first quadratic form of 5 has the repre Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -28sentation (29). It follows that the functions (5) map all circles in 0 on circles on 5 made use of Lemma 2. This is a contradiction of our hypothesis. here we have Therefore (35) holds. Sufficiency. By setting * j-l ; -iv in (16) we obtain ' c* * “ where Q<\, is the circle in 0 /=( with center at the arbi trary but fixed point CU&itfb) and radius A , and where the partial derivatives are evaluated at . From (34) and (72) it follows that (8) and (73) Z A<j dJ*! J-l must hold. Operating on (8) with \ , we obtain j-i Operating on (74) with the operator X , and applying the relation (73) to the result, we obtain (75) X & k; =-o . j-i The four real linear homogeneous equations in HQ' j^.i, inulled by (74) and (7S) are ( 76) i Z Z .*ja /=.t * /=< 4_ Atfj = lr — ^ KjutT J~( Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4“ ‘^iuir ^ j'-( , -29One solution of the system (76) is (77) which, by (8) and the theorem of Weierstrass, implies that the functions (5) map D mal surface. isothermically on a mini From (8), (16) and (77) it follows that (78) J^ [ holds for all circles Cholds. in ^ , Therefore (60) But this is a contradiction of (35), and therefore the functions (5) do not satisfy (77). It follows that the given functions, which satisfy (76), do not satisfy (77), and therefore the rank of the natrix JCiu- fr-u* UuO“ Kvuu* tv.a'AT' ~Jla-lMr X'Vuti”’ (/■(/“ Urn is less than three. Hence, in particular, we obtain [ J flu Kw Therefore(14) holds. ^ V io T - I jf lU L tvr From (8), (14) andTheorem A it follows that the functions (5) map & _ * on a surface 3 that lies on a sphere isothermically J of finite ai 26. M. Bocher, Introduction to Higher Algebra, p. 47, 27. If the functions (5) defined a plane surface in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30non-null radius. Suppose tliat the functions (5) map D cally on the spherical surface *5 are mapped on circles. in D is a circle, be 0 . such that circles Then the map of the circle C* ,on graphically on the plane ^ isothermi- S . Project of C stereo- and let the map It follows that P C of has been mapped iso- thermically on the plane surface . By the theorem of Weierstrass, the mapping functions form a triple of conjugate harmonic functions: ^ CU.\0-\ J *■>* * where (6) and hold. Asabove, it follows (79) ^ that IU.iA holds for the arbitrary fixed circle for tu»o* on C in ^ . But C , which with (79) implies that (78) holds. that (60) holds. It follows But this is a contradiction of (35). Hence the functions (5) map P isothermically on 3 such that circles are not mapped on circles, isothermic representation, then (77) would hold. have shown that (77) cannot hold; We therefore the sphere is one of finite radius. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I -313. CHARACTERIZATION OF THOSE ISOTHERMIC SPHERI CAL MAPS THAT MAP CIRCLES ON CIRCLES AND OF MINIMAL SURFACES IN ISOTHERMIC REPRESENTATION. Theorem 3. the functions (5) have continuous partial derivatives of the third order in a simply con nected domain ]) , then a necessary and sufficient con dition that they either map D isothermically on & sur face that lies on a sphere of finite non-null radius such that circles are mapped on circles, or be the coordinate functions of a minimal surface given in isothermic representation, is that for each circle in 0 C , <8 °> Necessity. We have already shown that if the functions (5) are the coordinate functions of a mini mal surface in isothermic representation, then i.e., (78), holds. We have also shown that if the functions (5) map 0 isothermically on a surface S that lies on a sphere of finite non-null radius such that circles are mapped on circles, then (80) holds. Sufficiency. If (80) holds, then (60) holds. It follows that (8) and (73) hold, and therefore, as in the proof of Theorem 2, we obtain (76). 28. See p. 29. 29. See p. 30. Then, as be- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I -32fore, it follows that the functions either map P isothermically on a surface that lies on a sphere of finite non-null radius such that circles are mapped on circles, or map P minimal surface. isothermically on a Here we have used the results of the second part of the proof of Theorem 2. But in the first part of the proof of Theorem 2 we showed that (60) implies that the spherical surface S , of which the coordinate functions are the func tions (5), has its first quadratic form in the re presentation (29); circles inP therefore, by Lemma 1, all are mapped on circles on S Corollary 1. . If the functions (5) have con tinuous partial derivatives of the third order in a simply connected domain D , then a necessary and sufficient condition that —° hold for all circles C point CUajtte) irj [) is !) that for each , j={ where in the circle in D with center at and radius Jt . 30. If the radius of the sphere were either null or infinite, (5) would be a triple of conjugate harmonic functions. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -33CoroXlary 2. £f the functions (5) have continu ous partial derivatives of the third order in a sim ply connected domain D , then a necessary and suf ficient condition that they either man 0 isothermi cally on a surface that lies on a sphere of finite non-null radius such that circles are mapped on circles. or map 0 isothermically on a minimal sur face. is that for each point ((JUjiTo) in is, the circle in P and radius A 4. with center at . CHARACTERIZATION OF ISOTHERMIC PLANE MAPS. Lemma curve: , x;cu>irt<il|=<rCA*) J r lL where C/v D Let T fee a closed rectifiable Jordan denote the simply connected domain inside T , and let R be a closed region that lies in D> is analytic in Q closed region Oi-T tive number and continuous in the , then for any arbitrary posi there exists a positive number which is independent of ^ in ft j for which (so) for all , i' “ and in R that satisfy the in equalities (si) it'-tsur. 31. iv-touJ, t V t ',51 Clearly, by the uniform continuity of the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -34Proof*. -V If £ and t are any two points of ^ , _W , then "by the Cauchy integral formula we obtain (82) r 4'-4“ If 4<Qj? UTc Jr C4-r)v (£-?) is an arbitrary pre-assigned positive number, then it follows from (82) that I pi I* e I - ~ .:p - - * < * " > r T - . (83) if t84) |i- 4 " U J , * where is the maximum of length of r R and f and d . is the , \ on T , L is the minimum distance between Here c/> O . The f u n c t i o n i s uniformly continuous in hence there exists a positive number (85) R such that in ^ I ^ C 4 10~Vt«ro>) I < ^2., when (86) | *"-%oU «TX . If we choose T - min(x^i 9 $>. )» then the inequalities (81) imply (84) and (86). Hence, if (81) holds, then (83) and (85)hold. Prom (83) and (85) we obtain (80). function Vtt} in ft , this last restriction can be removed by defining ^ . ; t'st! Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Lemma 4. and the function If J ) 2. Let the function be schlicht. analyticT in the circle i a an arbitrary positive number. there exists a positive number . , then such that the function = *5, (87) is schlicht in the circle ifcl c , for all ^ such that Proof. If the lemma does not hold, then there exists a triple of sequences, (88) L £/* (89) [ 4 m \*. (90) I v; , > I " H f * *t> *i'. with the properties (91) /w-*oo ~ 0> (92) (93) ^ where (94) 32. The analytic function £,(■£> is said to be schlicht in if '<$(*“) implies * Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -36From (92) it follows that (88 ) and (89) have cluster points in the circle . Hence there exist subsequences» and 1 ’ such that ft 1Iu m (95) i i ik/wp ~ io , ^-*QD (96) -fiuftA P'fcoo I lLi*p x , II where fce. and £e> are cluster points of (88) and (89) respectively. Then (97) l4?Uf-S, and from (91) it follows that (98) Hx m *| s° P -» c e From (93)-(96) and (98) it follows that J.Ct'.ir 3, (fc'O holds. Since is schlicht in the circle l£l it follows that (97) and (99) imply (100) From (93)-(100) and Lemma 3 we obtain (101) --- t‘ ^ — — P-*oo Since ttol£ - rJ<(«L a o- ^ , (101) implies that the function is not schlicht in the circle l^«£- . From this Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. $t ^ -37contradiction of our hypothesis| it follows that Lemma 4 must hold. Theorem 4. If the functions (5) have continuous partial derivatives of the third order in a simply 0 9 then a necessary and sufficient connected ^amain condition that they map 0 isothermically on ^ plane surface is that for all closed rectifiable Jordan curves ^ lying in D C102) , Z [ fft a Necessity. ’ If the functions (5) map thermically on a surface ^ 0 iso- that lies on a plane *1, *v»^I coincides with the plane X3 a-a then we make a rigid transformation of the space such that and the positive normal at the image of an arbitrary coincides with the positive X3 - point C ^ i T ^ of D axis. Let tivj.s transformation be given by (103) 3 Xj - ZL Aic/ Kk where + ^ Z Astc AjV d°4) *c*i s-j > and where the Q.J and . S',) are real constants There fore the functions 33. C. H. Sisam, Analytic Geometry, p. 267. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. , -383 X;la.lh) - H tH , J ^ I,z, % (105) x‘*oi.<rt SO, map 0 isothermically on a plane surface; hence, by the theorem of Weierstrass, they form a triple of jugate harmonic functions. From (3), (6 ), (7), (104) and (105) it follows that the functions and con x/ (tt>o^ are a couple of conjugate harmonic func tions. Hence by Cauchy's theorem, analogous to (12), we obtain t ia | ® which, with (105), implies (106) -o> From (9), (103) and (104) we obtain 3 I (i°7) I* “ Z l U | x';cu,vr:)J%\ * hi j"-> From (106) and (107) we obtain (102). Sufficiency. If (102) holds for all closed recti fiable Jordan curves K lying in P , then, by Theorem 3, the functions (5) either are the coordinate functions of a minimal surface given in isothermic representation, or they map D isothermically on a sur face that lies on a sphere of finite non-null radius such that circles are mapped on circles. Part I. We first consider the case when the func- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39tions (5) are the coordinate functions of a minimal surface given in isothermic representation. he an arbitrary point in in 0 0 Let and let the circle have its center at CU.&,, tTj>). If ^ Ca is an ar bitrary closed rectifiable Jordan curve lying in D which contains terior of 'i , in its interior, then the in can be mapped conformally on the in terior of the circle CI such that the image of (.Uo>irV) in the S, "t -plane is the center of C . The expansion of the inverse of this mapping function has the representation (108) where nra.s+t t~ , io -U-ott <T0 , Since the functions (5) are the isothermic coor dinate functions of a minimal surface, it follows, from the theorem of V/eierstrass, that these functions may be written in the form (109) where and X;cu,o-wV[^J^^ <£;(.•& is a function that is analytic in its conjugate function. '.Ve may write 4>JC*> = I I V ^-0 then Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. D -40e» _ , 3^ m-o where the series for 4 ; ^ and 4^ 1*) are absolute ly convergent in the interior of and on C 0 , and where 4->,* and and are conjugates of respectively. (no) Therefore - i'*>0 +XL ^ C a,>Ma *x /M*l since 34. This is a Taylor expansion for it converges absolutely in and on Co ; <^C%) and hence likewise for <§(&, 35. Since (108) is absolutely convergent for lunc^ f M the series for is absolutely convergent in the circle vergence . p. 75. , /vt-viji,*** , See W. L. Ferrar, Con Moreover, the Taylor series for 4;(-*5 is uniformly convergent in £ o , and therefore it is uniformly convergent in and on . Hence the series qft TAivkHiVirt-Wf* •■m t all ^ . ( uniformly convergent, | u H ^ , for Hence by the Weierstrass double series theorem (K. Knopp, Funktionentheorie. I, p. 89), the series ZL /M-i the analytic function Zt>Y* is uniformly convergent to j=A,V>3. Since a similar discussion is valid for the right-hand member of (110* it follows that (110) must hold. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -41oo where C (u.tMA ~ 21 bu;. »•* *b t<* * - * • (1 X 1 ) v k v+ M » * ** » M>AW* If C r us R of 0^ is the circle concentric with C , then the function (108) maps the interior conformally on the interior of a closed rec tifiable .Jordan curve £ of & and of radi ; that lies in the interior then from (9), (108) Cii2) f and(109) we obtain * M rR c rt From (108) we obtain oo ( 113) -& W ) - 21 ^ Ww u r * * " 1 . 4*3-1 I After setting UTsRe^ in (108), (110), (112) and (113), it follows, since the resulting series are uniformly convergent, that c (114) >2 1 I ®R must hold. ^ (lis) g£_ P-1-W4 --1ft £ a;,* M-l ^ K , From (114) it follows that r r T -\( J-t K oo op i ao t i. *l*| S-( J'l oo , LL. *M=I holds, where (116) $ __ = '“ L C»,«. a..***'.*,— Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. . -42Note that (117) ^ K 1 JL. ** KjA Therefore it follows from (102), (115) and (117) that UlL (us) Z Z I I A m > s S M)/v* i \ *o» <•*.* *»*t ^ i | S=.| where (119) AyUjJ** As,(VV “ ' Jv - ( But recall that the circle was an arbitrary circle concentric with and interior to the circle Therefore the relation (118) is independent of R i.e., (118) holds for all R * - l l-M* (1 2 0 ) , o^R . C , Hence IH ZLZLZ1 Aiw>s v * **-» s-( ,*^1 Prom (120) we obtain, by an induction, (12!) For, when ■(■*-* in (120) we obtain (122) At>| ■ From (111), (116) and (122) we obtain C123) A w C t ; = o . 36. This follows from (111) and (116). 37. See theorem on p. 170 of K. Knopp, Theorie und Anwendung der Unendlichen Reihen. 2nd ed.. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -43But the function whose inverse is (108) maps the interior of & on the interior of C 3? one manner. Therefore (124) in a one-to- b(* 0 , From (123) and (124) we obtain (125) Ai.t - O • Now suppose that A/w,s (126) where M , and p are positive integers. We shall show that (127) holds. M*3=-p, For , (120) yields P—I (128) P - aM w 2 Z X I ZZa*>s8*>** **-1 s-/ Avm ~ o, From (126) and (128), ?-l (129) H A k >v_k ?>*,* & P^ p _ K ^ 0 ' K^-l r But (111) and (116) yield - S k, k,S , ***' and hence (129) may be written in the form 38. A consequence of Darboux’s theorem. See W. F. Osgood, Functions of ^ Complex Variable. p. 167. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -44- (130) b p Z L KC^ K b kr=-l A ^ O . 1 From (124) and (130) we obtain (131) teCj?-.obfcL £ ^.p-k'0 - tc=l By Lemma 4, for a fixed R , the func tions (132) Gr^urt = + lv|^ur ' are schlicht in the circle iur\ where 6-i,g provided is a positive constant whose existence was established in Lemma 4. We further restrict &Jg to satisfy the condition for S £n p and - ^ l 9 where S , in order that the map by (132) shall lie inside is inside (133) is the distance between . if . £* If of in $>R Moreover, C(U,<To) Hence, if ocjiV^l^d^ , Y r *]» I-'iV - , P-l, then we obtain the following result which is analo gous to (131) for the functions (132)s (134) P-» (a) L*<.P-*>b|cl- A ^ + i J C K ) U Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. h n =0, -45- (134) c b ) E « ^ A w ~ °‘ where (a) holds for all % , /4 S 4 P*< g-*p/i , and where (b) holds for p(* . * except From (131) and (134) we obtain (a) (135) to) S^C1^ where (a) holds for all % + a-b £ ^ ^ , o> (<* g^-p-l , except %-tli , and where (b) holds for Since ^ . is an arbitrary constant which is sub ject only to the restriction (133), it follows from the relation (135), (b), that (136) *|£,? ~ 0 - must hold, provided p positive integer, if is even. |c g ( £. p-| ‘ If g, is a fixed , and , and , then it follows from (135), (a), that 39. The statement has meaning only when p an even positive integer because I integer. is a positive Hence when we state "except for we mean that if p is an even integer, then J not take on the integral value & . " does In a simi lar manner, we say that (134), (b), holds for Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A is -46(137) A SlJ(> If £v is a fixed positive integer, /a g*4fM and if ^ * \>„ a -° , then we consider the function "'Ok (138) H ^.UJrt * <vf where , Vp-g,. a ur^u > constant, The function (138) is schlicht in the circle and maps on a closed rectifiable Jordan curve which lies inside side iwi - R , . . Moreover, (Ua,ir6) is in From (108) and (138) we obtain (139) where If we apply (139) as we have applied (108) in the earlier part of this proof of Theorem 4, then we ob tain the following result, which is analogous to the relation (135), (a): (m o ) 40. aiv(p-gv)4 *l‘ s * C * °> 8v,P-Jv f »- This holds because IW I c d« ? •P'S* ’ Ov was defined in (133). , where d Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. l-iu -47where m Zv is an arbitrary constant, (14D o i^ gv« Ox- Here the constant d|w is in the same relation to as is to . Since (y|' and , we * iw obtain the following result from (140) and (141)s (142) A f„f-S. = ° - From (120), (136), (137) and (142) it follows that (127) holds provided (126) holds. Since (125) holds, our induction is now complete. From (119), (121) and the substitution we obtain the following relations 3 2— ^ Jit ^ ^»»s ) =•O, (143) 2 - where the W)\<w e , ■+ °(*» i ^ > **0 = and ^ ** efficients in the expansion of about the point CU.6,iTo ) • 00 (144) K;tu.o^ = *:10+ 7~n.*"( AHil 41. /V\> S i I j ^ are the Fourier co Xycu»o^ , jitjX,'* , These expansions are V * %tu ivhs),i -I. ^ ^ , Compare with «J. W. Hahn and E. F. Beckenbach, Triples of Conjugate Harmonic Functions and Minimal Sur- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. —48— We transform the axes in the -space such that the new origin is at the image of Ctlo.tTo') , the plane is tangent to the surface there, and the positive nor mal to the surface there coincides with the positive axis. — This transformation is given by the equations in (103), where a; Ak ; K-l and where (104) holds. Then the relations (143) are invariant uncer this rigid transformation. For, since the functions (5) form a triple of conjugate har monic functions, it follows that the new coordinate functions (145) 3 H I A*; Xfetu,*) - ca*,(r0)L > * 1> z,i, i ¥»forma triple of conjugate harmonic functions. Hence we may write faces. Duke Math. Journ., v. 2 (1936), Lemma 1, p. 699, and footnote on p. 700. 42. Since the functions (5) are harmonic in P follows that the functions (145) sire harmonic in 0 From (104) and (145) we obtain '~ Hence the functions (145) are isothermic. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. » , , it -49(146) co x}itt,o*3. H AMS.O Since the X'yiUoXi>) eot»uA w am &) , oi ^ \J*d , , J > J. are equal to zero, it follows from the relation (146) that v (147) «*'. -Oj *>o From (144), (145) and (146) we obtain 4~ l . a where ' 4- which, with (104) and (143), imply C14S) J ,jV"> ^ V vt 1" (“is ^iw + ^ J,m ?ii) J ' and s are positive integers. /M>S— I Therefore the relations (143) are invariant under the rigid trans formation defined by (103) and (104). To prove the functions (5) define a plane surface it is sufficient that we show that *'1,0 (149) holds. Let (a) “ J'vm Hi be the positive integer for which IT ~ o, (150) a.) 43. k < t- 4 0 . ¥ f J-I ,c If (149) holds, them. zo and- ^ is a plane surface. 44. If for all''4* , then it follows from the relation in (148), for , that 2LP* £ |-V for all • * vvi does not vanish for SM Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. i -50From (146), using formulas analogous to (38), we obtain (a) (151) (b) - *'„* ?*ct- (c) *\,t < t + « C since the positive normal to 3 - < * ?U - < * $U , at the point C0,0,o) coin- (Vi\-i-66>o , then there is a least positive integer * for which (150), (b), holds; 45. Since the functions (145) are harmonic, we may write )C»ca»o^ tain (146). 3«- therefore (150) holds. i m*o Therefore ®] from which we ob- 4 T . v/henever 21 j-y *U (MiO , then ?[.CXfcttXto-“X!cwXlj<|) ^ l c | ^ ) \ A in cyclic order, are the direction cosines of the normal to S tions (145) are harmonic, then Z ! . Since the func- s° near CUo^d) only if these functions are identically constant in If the functions (145) are identically constant in P then (149) holds, since the functions have . , expansions (146). If the functions (145) are not identically constantin 0 then for A, , sufficiently small 3 . Therefore < =[(<* ?t.±~4.t whereOtn) denotes a quantity (not always the same quan- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1+0(0, -5 1 - cides with the positive X 3 -axis. Equations (151), (a) and (b), are linear and homogeneous in ^\ct and ^\(t j since by (150), (b), and (151), (c), their determinant of coefficients is not zero, it follows that (152) o « \ tt - f a * From (148), for (153) , (150), (b), and (152) we obtain 06'(X * " &•* ^0, Cl64) ^tVh* + < * From (151), (c), (152), (153) and (154) we obtain ( 1 5 5 ) $ ' » * • From (148), for s=-fc£/u. , (152) and (155) it follows hold. The elimination of from the relations (156) yields. (157) and the, elimination of $'■»■><* from (156) yields (158) + tity) <€10 such thatl^/w/hl ly small. at t o ,6 ,0) is bounded for /u sufficient Then (151) follows, since . Note that remains continuous at ^ -I , thus insuring the existence of a normal to S even at ^ 3)C*v ^ 5X<> t points where 2T(-g-J) = 0 when ZIC ^ 0 . See Hahn and ■»=* ^ /-i Beckenbach, loc. cit., pp. 701-703. Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. -52Prom (153), (157) and (158) it follows that (159) From (148),- for , and (159) we obtain ** ^?jV) , from which it follows that (160) Prom (148), f tain (149), o r , (150), (a), and (160) we ob This completes the proof for Part I. Corollary. If the functions (5) are harmonic in a fl-fmrnv connected domain D expansions about the such that their Fourier point LUfyiTo) o£ 0 are given by (144). and ££ (143) holds« then the functions (5) are isothermic coordinate functions of a plane surface. Part II: We now consider the case when the func tions (5) map ^ isothermically on the surface S which lies on the sphere , whose radius ^ is finite and non-null, such that circles are mapped on circles. Since the functions (5) have the representa tion (28), it follows that the functions (5) may be continued analytically to map the entire closed d»U'plane isothermically on on circles. such that circles are mapped Let the point P point £- od , and l e t b e ponding to jection P as a pole. on'P , thermic map of the Ui<r on 5 correspond to the the equatorial plane corres Then the stereographic pro with? as pole, induces an iso -plane on such that circles are mapped on circles and such that the point at infini- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -53ty in the -plane corresponds to the point at infi nity in the plane ^ of (S J . Here we have taken the center as the origin in a system of coordinates on , such that the positive axis and the ray Q P 3(.-axis, the positive - » in that order, have the same dis position as the coordinate axes ( » Xk* . The mapping function must have one of the following repreWC. sentations: (a) (L61) or- (b) -=>-oCi-f-F ^ 6^7 - C * . U-J itr,--s.Y-c^y, where (a) holds if the map of the U)V -plane on ^ is directly conformal and where (b) holds if the map is in versely conformal; u, and are constants^ Consider the system of axes is at Q XI1,K%) whose origin , where the positive Xr » xl - and X* -axes have the directions of the positive Si -axis, the positive axis and the ray Q? transformation of the respectively. *i> origin into the point ^ There exists a rigid -space which carries the , such that the positive X, -, X-w - and X* -axes coincide with the positive X\K -, xj I and X) -axes respectively. This transformation has the representation (103), where (104) holds. coordinate functions of 5 If the new are 46. See L. R. Ford, loc. cit., p. 32, Theorem 25. 47. When we refer to (161) we shall refer to that part of (161) that effects the mapping noted in the proof. i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -54(162) x';~ xi‘tu.>(h\ then it follows from (9), (103) and (104) that (107) holds, where is an arbitrary closed rectifiable Jor dan curve lying in D . Since the functions (5) map circles on circles, it follows that the functions (162) map circles on circles. Hence it follows from Lemma 1 that the functions (162) have the representation (28); moreover, in (28), the function ft*} (163) x /iu .o -W is given by (161). — ^ g|— Therefore » a'-*s,V*«v xaV.. 1 1 a ^ ^ vl « j| From (9), (107) and (163) we obtain (164) X * where & is the map of if on the plane by (161), and where (165) Since <i66) 48. ' %t-— , <Pv£s./&)- — » fv^ ) - --- ---- , it follows from (102) and (164) that s*~i r Since the functions (5) are not identically constant in j) , it follows that ActO. in (161). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. « Without any loss of generality we may assume that *■ £the line w -© passes through D , where 0 la the ¥■ map on curve T 1) by (161). Therefore there exist a with the following properties; the vertices of T have the following polar coordinates^ A* «(A.+u5«>-'c,) , 2) T AV«0ii+u>»,Ti>, o^r,c1YHj ot-f\t, oztaV, is composed of two arcs of circles and two straight-line segments: circle ti’" - , A%A*f A<AV is an arc of the is an arc of the circle MU>i)v , each arc subtending an angle zr, at the origin, and A-vA-s and A-«*Ai are on rays through the origin* For the closed rectifiable Jordan curve r we obtain i -j_ Y~ j j ^(.Si^dur.js /-i k.(AbU),)~4scui'-& ICM'M) , r where 49. For, consider the rotation defined by v (6 UJV=e or, 4 where & has been so chosen that the line -o passes through D . If <f,-i avsNC* and 'K - ----------, then ' WtC Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ?>»-1 L a. t v q '-+(\,'-11 av4iAi-«i»ov i *• OMUrHtiiY' and where the 1 ±_ d''"+A«v a^i*' ■+rs*- 1 ^ (jv* toi)v 1 ’ a ^ w ^ - -a ^ W m ^ were obtained from (165). 1*. < • . * . & - Since - u ^ . r L o N iU i it follows that Fix |\-i and u)i so that v Since (^cViu^i » . and no^ proportional, 6» y K . SrtU ^ * ■ * • * > there exists a ^ < such that (IM) 2 1 1) j =( r Now Ct67^ is a contradiction of (166). functions (5) do not map D Therefore the isothermically on a sur face that lies on a sphere of finite non-null radius. This completes the proof of the theorem. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -1 -575. 5.1. MEAN-VALUE SURFACES. Mean-value surfaces. If the coordinate functions (168) of a surface S are continuous in a domain D , then the circular averages (169) where A ^ is a positive constant, will be said to define a mean-value surface associated with 5 . We define A^lo u u » ~ Xj cu,«rt , J -l> *-j i . 7/e note that the functions (169) are defined and have continuous partial derivatives of the first or der in an open set of points Dp to D which is interior .so Theorem 5.1.1. If the functions (168) are con tinuous in a simply connected domain D > then a necessary and sufficient condition that (170) J~l 50. Since set of points, c. is not necessarily a connected may consist of several pieces. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -58hold for each circle C in £ is that all mean- associated with 3 value surfaces 32® given in isothermic representation bv (169). Proof. The first partial derivatives of the func tions (169) are given by the relations (171) £ i '-rr > a T' - 'v r which are valid for points of D ^ , From (171) we obtain s-i which, with (8) and (170), yields our theorem. From Theorems 3 and 6.1.1 we obtain the following result. Theorem 5.1.2. If the functions (168) have con tinuous partial derivatives of the third order in a simply connected domain p , then a necessary and sufficient condition that they map 0 isothermically either on a surface that lies on a sphere of finite non-null radius such that circles are mapped on circles, or on a minimal surface, is that all meanvalue surfaces 51. associated with the surface S See T. Radd, Subharmdnic Functions. p.11. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. „ -59defined bv the functions (168) be given in isothermic representation bv (169). 5.2. axes. Mean-value surfaces and transformations of In §SA we shall make use of the following ob servations. 5.2.1. is invariant under rigid transforma tions in the -spaces if 3 Xj - Xk ^ i ^ * where the are constants and where (104) holds, then where 5.2.2. is invariant under each of the reflec tions a-u, u-'z-IT ana <r'~v. If, for example, then where Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -60- (172) rn. f - J J x'tu'rtjPH,) </{</«(,1 j J-M. 5.2.3. Similarly, ^ rigid transformations in the 5.2.4. is invariant under the U-> -plane. Under the transformation (173) the mean-value surface mean-value surface is transformed into, the whose coordinate functions are given by (172), where Hence the family of mean-value surfaces ciated with a given surface S' S^ asso- is invariant under the transformation (173). 5.3. Conformal mean-value surfaces. Since, by Theorem 5.1.2, the only surfaces in isothermic represen tation for which all associated mean-value surfaces are given in isothermic representation by (16S) are spherical maps, in representation whereby circles are mapped on circles, and minimal surfaces, the ques tion arises as to the nature of the mean-value surfaces in these two cases. Theorem 5.3.1. connected domain ^ face 5 If the functions (168) map a simply isothermically on a minimal sur » then each mean-value surface Sa associated Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -61vrith 5 is a minimal surface given in isothermic repre sentation by (169) and coinciding with S for in t>^ . Proof. By the theorem of Y/eierstrass, the func tions (168) are harmonic in D ; consequently, as is well known, the functions (169) coincide with the func tions (168), A J)fliktA 3 in the open set . Hence all mean-value surfaces associated with a given minimal surface are themselves minimal surfaces given in isothermic representation by (169); moreover, coincides with S for tUiuO in P Theorem £.2,.2. nected domain D I£ the functions (168) a simply con isothermically on a surface S lies on a sphere ^ that of finite non-null radius (L , such that circles are mapped on circles, then each meanvalue surface of revolution tation by (169). associated with S lies on a surface and is given in isothermic represen Further, for , pp is not a sphere. proof. fining S hi As has been shown before, the functions de > (168), may be continued analytically to map the entire closed Uyir -plane isothermically on the 52. See the proof of Theorem 4, Part II, p. 52. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -62whole ofjS . Farther, the functions (168) then have the representation (28) where is given by (161). From 5.2.1-5.2.3 it follows that we may assume (Xv- o > and (174) aoa. in (28). point This is equivalent to assuming that the P on corresponding to the point i-2-00 is the point ( the point Pl site P , that on ^ corresponds to that is diametrically oppo , and that the £ -point corresponding to P;CQjb#d) iS reai and positive. Since we are in vestigating all mean-value surfaces associated with it follows from 5.2.4 that we may take in (174), in which case the functions (168), as given by (28), have the following familiar representation: t - xa^ct. UH (175) *1 - is the circle +41’ ^ t i a (,— 1 V If <!* kv avnrwe<*t ? u ‘u*o>vs.A'w , and if U 4 o r = . e td , then (169) and (175) yield Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -63- C176) • Ai^(a»»)+t4a., =■e tdf i [4i>pC\oVtAt^pOuo)!. /»*■ From (176) it follows that the map C v of the mean-value surface sphere on associated with the ^ defined "by (175)^ is a circle in a plane perpendicular to the ter of is * Ci 0A< is on the 4*>p -axis. A-axis The cen and its radius nf Since S lies on Tp tions (169) map D p , it follows that the func isothermically on a surface that lies on a surface of revolution, .namely In i 5.4, it will appear that sphere for o c ^ < oo . . is not a We shall call a mean- value sphere. 5.4. Mean-value spheres. The mean-value sphere is a surface of revolution about the Accordingly, to investigate Tp -axis. it is sufficient to •V. study the intersection T ^ At,p — O . of Tp with the plane since, by (175), it follows that U . A = oj Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -64hence, by (176), the intersection of Tj» with the plane can be obtained from (169) by setting lT«.o ing in (175). A computation yields the follow results: gu * S > ‘> ‘ ^ "T" Jo _ \ f^ *iv+ 2<L'-(U+‘flc*5&)nJaj£ “ “ C&S6 ““ » A- QW ~ ^TL a ^ -f ( a ^ ^ *j # and - rv r ^ - a T r*° 2^l,u n~<jxd&_____ d ^ /jjf a ^ ^ 1 f sv V*** ia» v . from which it follows that the coordinate functions r>* of are AX (177) ' 1, 4 - i - 2 £ /„[ y p Q 1- , / _ We make the following observations. 5.4.1. The curve Tp has horizontal tangents at the points Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4-.,^ ^ At»e-«. -a* “65“ A^s.0.- tej — -I , and vertical tangents at the points ^ i ’T o ^ - ^ l ^ > a-h£% A * >e^ v . ^ i j Here we have made use of the following expression for v . the slope of s 1-a CL ^.•4- ------a v-uv+^v The horizontal tangents occur for il -zo, vertical tangents occur when a. - A-ij(* and the . Moreover, zT™^' from which we can obtain the result that the curve Tj> is symmetric about the line Am - ta a - . In addition, from the representation (177) fori? , V it follows that the curve about the Ai,(j -axis. is also symmetric From these symmetries, and from the following relations for the slope m\ it follows that 5.4.2. -i ^ o ^ + r T q , Ma. u - ttkaV^ is convex. The height ^ of of is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. —66— 4 and the width is \ v Therefore, for circle. a c ^ < c© , the curve is not a Moreover, t aco; x so that „ flattens out while approaching the point Co*4*) , as ^ becomes infinite. Each member of the family 1V i 5.4.3. through the point (0, (O passes and is tangent there to every other member of the family. Further, for [-0 , Tp is the circle IU . . V +• A and for , Tp - Q* .> is the point C°» ^ . From 5.4.1-5.4*3 it follows that the family of surfaces of revolution L T^\ consists of convex surfaces each of which passes through the point CO,0,<0 in the -apace and is tangent there to every other member of the family. Since the ratio of width to height becomes infinite with ^ , it _ * •« follows that Fp flattens out as ^ becomes infi nite. For ^ —o , Tp is the sphere with radius & and center at the origin, and for point SJLl.Sl 53. For e--i , T(» is the _From_(177) we obtain the following and the values is a p(g£ture of the curve Tp •* , there on the page following page 68. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ! -67- Tp : representation for * 5.5. ^ lUl (v T u t'-uv- T * f )v +H-a H u . ^ ) J *\ r Homothetic mean-value surfaces. If the func tions (168) are analytic in the M.CT -plane, a compu tation shows that the circular averages (169) have the representation 00 (178) Aj U l ^ I I ^ ' * *■' Mil « , ■».1,1,4. t Two surfaces ^ and ^ , whose coordinate functions are Kj'CUjir) and respectively, will be said to be homothetic provided there exists a constant A * o such that 3y (178) it follows that the problem of determining analytic surfaces S s Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. j such t -68that all corresponding mean-value surfaces are homo thetic to S is equivalent to determining solutions of (179) 21^* f*i*0 — • A Xj , In particular, sill solutions (180) c.^ x; > ’j*'*1**, satisfy (179). For , we see that sill mean-value surfaces associated with harmonic surfaces 5 , (surfaces whose coordinate functions are harmonic), are homo thetic to 'S , and, in fact, coincide with 5 , The functions /,-i 0L,^IkK IL SOA IT > (181) XV CL Xv *»- QrtPWL } which map the -plane on an infinite sheeted sphere, also satisfy (179^ with C ^ 2 -I . 'Ue note that the surface defined "by the functions (181) is not given in isothermic representation by (181). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I p.o Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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