In the Course of the Investigations It Became Necessary to Devpise New and More General Versions of the Douglas-Dirichlet Functi

42 MATHEMATICS: H. E. RAUCH Paoc. N. A. S.

5 By choosing z to be a complex number, it is easily shown that (4) also holds. The result stated here includes the theorem of Euler concerning the sum of divisors of a positive integer n. That is, a(n) - (n -1) - (n -2) + oa(n -5) + o(n-7) - a(n - 12) - or(n - 15) + . . . = 0. Relation (4) may also be obtained using the Newton relation between sums of powers of the roots and the coefficients of this polynomial. 6 Prace Mat.-Fiz., 53, 13-23, 1936. 7Cf. Bachmann, op. cit., 2, 118, 1909. 8These PROCEEDINGS, 39, 963-968, 1953, Theorem 6. 9 Bachmann, op. cit., 2, 232. 1n These PROCEEDINGS, 40, 825-835, 1954, Theorem 3.

ON THE TRANSCENDENTAL MODULI OF ALGEBRAIC RIEMANN SURFACES BY H. E. RAUCH DEPARTMENT OF MATHEMATICS, UNIVERSITY OF PENNSYLVANIA Communicated by Hermann Weyl, November 12, 1954 1. INTRODUCTION AND PRELIMINARIES In this note I present the first by-product of a series of investigations, begun in 1951, whose purpose (as it developed) was to evaluate and apply the techniques of the calculus of variations used by Jesse Douglas' and others2 on the Plateau problem to certain other problems of analysis and mathematical physics of a potential-theoretic nature. In addition to the present note, for example, these methods yield a simple theoretical treatment of the subsonic plane flow of an ideal com- pressible fluid about a profile. In the course of the investigations it became necessary to devpise new and more general versions of the Douglas-Dirichlet functional, using ihlie Cartan's repere mobile, but this distinctive feature is not needed here. It may be remarked that Riemann's original application of potential theory to function theory met the urgent demand for a guide to the hitherto unperspicuous calculations in the theory of the higher transcendentals. Today that motive is lacking, and a theorem in Riemannian function theory is academic per se. If it is to have any vitality at all, it must be sought in the reverse direction-that the function theory sheds light on the potential theory, and, indeed, I hope to show in the future that this is the case. Riemann in his memoir, "Theorie der Abel'schen Funktionen,"3 observed that, while two closed (associated with an algebraic curve) Riemann surfaces of the same genus p ('/2 the number of nonseparating closed curves or "retrosections"4) cannot in general be mapped conformally on one another, the totality of classes of conformal equivalent surfaces depends on 3p - 3 (p > 2) parameters, which, since the modulus of the elliptic integral of first kind serves this function, he called "moduli." A brief r6sum6 of his reasoning will show, however, the vagueness of this dependence. Let each such surface be represented over the z-sphere by a surface of n sheets for fixed n. Then the only variable parameters are the branch points, of which there are 2(n + p - 1) as is easily established by Euler's polyhedron theorem. On the Downloaded by guest on September 29, 2021 VOL. 41,y 1955 MATHEMATICS: H. E. RAUCH 43 other hand, not all of these are essential, since a given surface may be mapped conformally on another n-sheeted surface by a function having polar multiplicity n. But the theorem of Riemann-Roch states that for the poles in general position there are n - p + 1 linearly independent functions with these poles, whence 2(n + p - 1) branch points minus n poles minus n - p + 1 functions make 3p - 3 parameters. For p = 1 there is only 1 and for p = Onone. That is manifestly too vague a specification; one would like numerical moduli- a set of numbers associated with each surface whose equality would guarantee conformal equivalence between two surfaces. Various attempts to construct algebraic invariants of the coefficients of the equa- tions of the associated algebraic curves have been made. But, quite apart from the incompleteness of these attempts, they are in the wrong direction, because the modulus question arises in all of conformal mapping theory, for example, in the mapping of plane multiply connected domains, where such invariants clearly have no meaning. A more promising approach, therefore, is one that deals with the transcendental invariants which are provided by the periods of the Abelian integrals of the first kind (everywhere finite) and, to be more precise, of the normal integrals of the first kind, which are defined as follows. On an algebraic Riemann surface S of genus p consider a canonical system r of 2p retrosections, 'y, 5f, i = 1,... , p, where -y, and bf are conjugate, i.e., 'yj joins the two sides of , and vice versa. Then the ith normal integral of the first kind, denoted henceforth by fdri, is an everywhere finite integral on S such that fy dp; = 5q. This specifies the integrals and their normal differentials, dri, completely. The remaining periods, wir = f b did, are thus de- termined by the normalization and form a symmetric matrix (one can prove) of p(p + 1)/2 elements; and it seems natural to assume that they are moduli in the sense that two surfaces with the same period matrices are conformally equivalent. That is in fact the case6, but, unfortunately, when p . 4, p(p + 1)/2 > 3p - 3, so that the -rij are superfluous in number. Therefore, a reasonable and long-standing conjecture is that a suitable subset of 3p- 3 of the rij is a set of numerical moduli; and the confirmation of this conjecture is the main theorem-Theorem 1 below.

2. QUADRATIC DIFFERENTIALS AND NOETHER'S THEOREM The key and unifying element is the application of the concept of quadratic differential to conformal mapping by Teichmueller.7 An everywhere finite analytic quadratic differential (henceforth denoted simply by "quadratic differential"), dD2 = a(z) dz2 (not necessarily the square of a di), is the product of dZ2 with an everywhere finite, complex analytic covariant tensor density a(z) of weight 2, i.e., is formally invariant under change of local parameter z. Teichmueller was led to conjecture a relationship of the quadratic differentials to the modulus problem by A. The number of linearly independent quadratic differentials on a Riemann surface Sof genus p 2 2 is 3p-3, 1 when p = 1, and Owhen p = 0. Proof: A differential of the first kind has 2p - 2 zeros (Tninus the Euler char- acteristic), and its square, dr2, a quadratic differential, has Up - 4 zeros, whence any dr12 has 4p - 4 zeros, since the quotient is a function on S with as many zeros Downloaded by guest on September 29, 2021 44 MATHEMATICS: H. E. RAUCH PROC. N-T. A. S. as poles. The same reasoning shows that the number of independent d42 is the number of independent functions with 4p - 4 given poles, i.e., by the theorem of Riemann-Roch, 4p - 4 - p + 1 + 0 = 3p - 3, when p > 2, since no dr can vanish at 4p -4 places. For p = 0 there are no finite differentials of any degree, while for p = 1 the square of d.1 is the only d¢2. A is merely a heuristic guide not needed here. Instead, the actual point of de- parture is the following special case of a theorem due to Max Noether.8 B. On a Riemann surface of genus p which is nonhyperelliptic, if p > 3 one can find a basis for the quadratic differentials among those of the form di,- dR1. Referring to A, I note that by explicit computation one can prove the following complement: C. On a hyperelliptic surface the number of linearly independent quadratic differentials of form d1i d~j is 2p - 1. As an adjunct to B, I introduce the followsing notation: if the index i belongs to a certain subset I of 1, . .. , p, and j belongs to a similar J, then I will say that (i, J) C (I, J). Thus B asserts the existence of I and J such that the d1i dVp (i, J) e (I, J , are a basis for thed,2.

3. MAIN THEOREM THEOREM 1. Let the normal integrals fd~i of the first kind of the Riemann surface S of genus p, which is nonhyperelliptic if p > 3, have periods irj over the bi of a canonical basis, and let dti dtj, (i, j) e (I, J) form a basis for the quadratic differentials on S.9 If another Riemann surface S' of genus p with normal integrals of periods 7rij over 6i' is such that Vrij = 'rts; (i, j) E (I, J), (1) then S' is conformally equivalent to S. For the present the hyperelliptic case must be dealt with separately (see Sec. 4). Proof of Theorem 1: According to Jordan and M6bius, one may assume that both S and S' have b.een represented on the canonical model of a surface of genus p, the sphere with p handles in such a way that the canonical bases have the same representatives. This gives rise to a homeomorphic mapping 4 of S on S' such that at goes into -yi' and Si into 6i'. The periods aice and ari/ then correspond under 4. One may assume 4 differentiable as often as one pleases, and, accordingly, after introducing a conformal metric, X dw div (w a local parameter), on S', one may form the Douglas-Dirichlet integral J(4) = '/2 ffs (E + G) dx dy, where, as a result of X), X dw de = E dx2 + 2F dx dy + G-dy2 z = x + iy being a local parameter on S. Now it follows from recent work of Morrey'0 and -others on parametric double integral problems in the calculus of variations that there exists a twice-differentiable co, which minimizes J(') among all homeomorphisms ,which are deformable into 4. Downloaded by guest on September 29, 2021 VOL. 41, 1955 MATHEMATICS: H. E. RAUCH 45

Then .the "method-of variation of independent variables"1' shows that (E - G)/2 - iF is an analytic function of z; and, if one writes7 X dw diD=b dz + 2Re (a dz2) (2) where b = (E + G)/2, a = (E - G)/2 - iF, then one sees immediately that a dZ2 must be invariant under change of parameter z and is, therefore, a quadratic differential on S. It is also clear from (2) that, if a dz2 is identically zero, then 4o is a. conformal mapping ofS on S'. Hence I must show that (1) implies that a dZ2 is identically zero. For p = 0, condition (1) is vacuous, and there are no quadratic differentials; therefore, the conclusion is automatic. When p 2 1, I resort to the following device. According to (2), one may imbed S and S' in a one-parameter (real) family S(t) of Riemann surfaces whose conformal metrics are given by X, dwt di4 = b dz de + 2tRe(a dz2), (3) where 0 < t < 1, so that S(0) = S and S(1) = S'. Observing that (3) may be written

XI dwt dwi = b dzdz 1 + 2tRe( d12)]' (4) one introduces with Teichmueller the idea of infinitesimal quadratic differential, written dt2/1 drl| 2, as the quotient of a dr2 by a differential with the indicated formal transformation properties. Hence a dzI/bI4l2 is an infinitesimal quadratic differential. From a formal point of view, the product of a quadratic differential and the conjugate of an infinitesimal quadratic differential transforms like the volume element dz - di -(exterior product). Therefore, following Tei6hmueller, one introduces their inner product,

(dR2, d,/ Id2 d22 Bdz - de, (5) where Bdz/d2 =dr,2/Id 12. Then one has the vital LEMMA 1. If the infinitesimal quadratic differential dD2/1 d¢| 2is orthogonal to every member dr2,, v = 1, ... ., 3p -, 3, of a basis for the quadratic differentials on S. then it is identically zero. Proof of Lemma 1: dD2 may be written as a. dr,2. Then 2 dz = drl 2) = a, I 2) = 0, J s dz|2 b (dr2, dr2/l (dr2%, dr2/ld where b = IdI /ldZl 2 > 0, whence- dP2-= . To, conclude the proof of Theorem 1,. it will now be sufficient to, show that (1) implies that a dz2/bI dzl 2 iS orthogonal to the dri dr,, (i, j) e (1, J). Downloaded by guest on September 29, 2021 46 MATHEMATICS: H. E. RAUCH Pisoc. N. A. S.

To that end, I note that as a consequence of (4) one may say that S(t + e) is obtained from S(t) by deforming its metric by the infinitesimal deformation ea dz2/' b |dzJ 2; in other words, in the limit S(t) is obtained from S = S(O) by the operations of a one-parameter group. There remains only the task of computing the effect of this group on the sit, and that is the substance of the VARIATIONAL THEOREM. If the periods of the normal integrals of the first kind of S(t) are denoted by rE(t), then r e(t)= eX~irf (6) where the infinitesimal transformation Xi, id {tj b/irij (not summed) and t arjj(o) 1 (' dri dr, a dz di. (7) be 2wiJ sdz dz b In this theorem the wq are treated as variables subject to the transformations of a one-parameter group expressed in the familiar symbolic form et'si. The form of Xt, is also standard.12 I defer the proof until Section 5. In particular, one has, using (1) and (6),

Irt = rJO(l) = erict} = rt; (i,j(I, J). (8) If one could conclude from (8) that Xq, = 0, then, by (7), Lemma 1, and the hypothesis of Theorem 1 on (I, J), the proof of Theorem 1 would be complete. This somewhat delicate conclusion is a consequence of the fact that a one-parameter group on one variable is equivalent to a group of translation (see pp. 32-34 of Eisenhart, op. cit., for what follows). That is to say, there exists a function f such. that k(Tt,(t)) = 0(irq) + t, so that, according to (8), when t = 1, one has a contra- diction unless Xe,, i.e., at} is zero. This concludes the proof of Theorem 1.

4. REMARK1S ON THE HYPERELLIPTIC CASE One sees that C implies the existence on a hyperelliptic surface of genus p of p - 2 quadratic differentials which are not of the form dri d, or a linear combination there- of. Hypothesis (1) of Theorem 1 would lead by the above reasoning only to the conclusion that a dz2/bI dz 2 is orthogonal to all the d~j dri, which conclusion, after orthogonalizing a basis of the dr2, would lead to the conclusion that a dz2 must be a linear combination of the aforesaid p - 1 dr2. If, then, one could prove that the deformation along such a dt2 as in (3) would change'a hyperelliptic surface into a nonhyperelliptic one, then one could state a complement to Theorem 1, where the hypotheses would be changed to demand that both S and S' be hyperelliptic and the equality of the periods iri = rijl, (i, j) e (I, J), where dri dr, form a basis for 2p - 1 dr2. I do not dwell further on this point here. See footnote 16. 5. PROOF OF THE VARIATIONAL THEOREM The only statement needing proof is (7). First, I need another lemma. As a preface,. it may be remarked with Teichmueller7 that, if one sets t = e in- (4) and assumes that w. = Z + ew(Z, Z) for e sufficiently small, then, on expanding both sides of (4) and comparing coefficients of e, one finds that Downloaded by guest on September 29, 2021 VOL. 41, 1955 MATHEMATICS: H. E. RAUCH 47 aU 6W= (9) this being a local equation only, i.e., co is not a tensor. But one can prove LEMMA 2. If S be canonically dissected to obtain a simply connected surface bounded by the contour y consisting of all 2p retrosections run through twice in opposite directions, then (9) can be solved for a contravariant tensor (infinitesimal displacement) in S-y taking on well-defined but, in general, different values at coincident points on 'y. Proof: The proof uses a device that will be needed later, namely, the uniformiza- tion of S leading to a map of S on a non-euclidean polygon in z < 1 when p > 2, or the familiar parallelogram in the z-plane if p = 1. One may assume that the boundary of the polygon, P(O), corresponds to y, and I will retain the same letter for it. In and on P(0), a/b is a function of z. To solve (9) I remark that bw/l = Wvbz and then differentiate with respect to t, to obtain - 4Acw = - I-I which is merely Poisson's equation and is readily solved for w. Integrating again, one obtains (9). To prove conclusion (7) of the variational theorem, I need to introduce the elementary integral of the third kind, ',(z),13 which has a positive logarithmic pole of mass 1 at x and a corresponding negative one at y, and which has zero periods over the yi. One can then prove by contour integration and differentiation that 1 dz = 27ri Jsi bya2Oz 71xy(z) -(Y)dy (10) Consequently, one derives _f J _ati7(z) dz dy = 7rq. ( 1) Letting a stand for a small change, one obtains = _ J X 6 71,y(z) dz dy. (12) arij 27r Si i by 6z Now I denote the polygon obtained by uniformizing S(e) by P(e) and its boundary (as well as the corresponding dissecting curve on S(e)) by zyE Also, let ?XV(z) be the elementary integral of the third kind on S(e). Then the Cauchy residue theorem implies

boX-(Z)-qxy(W) = (TVE(Z) - 1XV(Z)) - (nxE(w) - vXy(w)) = (u) J -7y'(u) du, (13) since flyE(u) -tlx(u) is an analytic function2-ri ofE[lxyV(u)u without exception. Now, if the integral in (13) is split into two parts, then the latter part is equal to the same integrand integrated over oy, thanks to Cauchy's theorem, and- this integral is zero, as may be seen by remembering that one can refer back to S where y Downloaded by guest on September 29, 2021 48 MA THEMA TICS: H. E. RA UCH PROC. N. A. S.

contains every retrosection twice in opposite directions and recalling the period normalization of % zv) Furthermore, at the points of y' which correspond to the "two sides" of one point on S(e), ti*f(u) will have the same difference (because of its definition) as 77v(u) has at corresponding points on y, while 6q/bu du will have the same values. Therefore,

xyZy(W) = 'xy' (i)7 (u) du = r (u - au) au (u) du

- E Jw a8U t~71(U)W(U) air (u) du + 0(62) =

- Ef a nI(u) tiZ,,, (u) w du + O(O2) (14)

where 5u = ecw(u), as follows from Lemma 2, and the estimate o(,2) is retained in the last step, since the two integrals, by Stokes' theorem, differ only by an integral over an area of order e. Substituting the extreme members of (14) in (12), one obtains

- . X w du + O(e2) tbrij 27rn J du dui Using Stokes' theorem in the form JQ = ifs8 where d2 is the exterior differential, one obtains ____0(0) _ 1 I dot d , du =l-= wdu- ahC- 2ri J du du 27ri s du du idff dtj 1 dudu(bujdiid 27rI s du dud do + 2rJ df dua by (9). Q.E.D. I should mention the fact that these calculations were patterned after analogous considerations due to Hadamard, Douglas,'4 and Schiffer.15"16 1 Am. J. Math., 61, 545, 1939. 2 Cf. R. Courant, Conformal Mapping and Dirichlet's Principle (New York, 1950). 3 Gesammelte Werke. 4 For general information on Riemann's theory of functions I refer here once and for all to (i) E. Picard, TraitM d'analyse, Vol. 2 (Paris, 1926); (ii) H. Weyl, Die Idee der Riemannechen Flache (Leipzig and Berlin, 1923); and (iii) R. Nevanlinna, Uniformisierung (Berlin, 1953). 6 A. von Brill and M. Noether, "'ber die algebraischen Functionen und ihre Anwendung in der Geometrie," Math. Ann., 7, 269, 1874. See also K. Hensel and G. Landsberg, Theorie der algebraischen Funktionen (Leipzig, 1902). 6 Cf. F. Severi, Vorlesungen uber algebraische Geometrie (Leipzig, 1921), Anhang I. 7Abhandl. Preuss. Akad. Wiss., Jahrgang 1939, No. 22, 1940. 8 Cf. Hensel and Landsberg, op. cit., p. 502. 9 I use B implicitly here. Downloaded by guest on September 29, 2021 VrOL.VMA41, 1955 THEMA TICS: R. SALEM449

10 Ann. Math., 49, 807, 1948. X may be chosen as differentiable as one pleases. In order to insure equi-continuity of a minimizing sequence of the.,'s introduced below one must fix the im- ages of three points when p = 0 and one point when p = 1. This is essentially due to the existence of 3 and 1 parameter groups of eonformal self-transformations in these cases. '1 T. Rado, On the Problem ofPlateau (Berlin, 1933), p. 88. 12 Cf. L. P. Eisenhart, Continuous Groups of Transformations (Princeton, 1933). 3 For an elegant summary of these matters see M. Schiffer, Am. J. Math., 68, 444, 1946. 14 Ann. Math., 40, 205, 1939. 5 Courant, op. cit., Appendix. 16 The hyper-elliptic case will be dealt with in a separate forthcoming publication.

ON MONOTONIC FUNCTIONS WHOSE SPECTRUM IS A CANTOR SET WITH CONSTANT RATIO OF DISSECTION By R. SALEM

DEPARTMENT OF MATHEMATICS, MASSACHUSETTS INSTITUTE OF TECHNOLOGY Communicated by E. Hille, November 16, 1954 Let P be the perfect set of the Cantor type and of constant ratio of dissection constructed on the interval 0 < x < 27r with the ratio of dissection (t, 1 - 2t, t), where 0 < ~< 1/2. We write t = 1/0 and denote the set P by P(0). Then, if x is a point of P, x =2 r [e1(1-t + IE24(l1 - 0) + ...+ 1EXk- (I1 - 0) + I I- 1 1 = 27(0- 1) L0- + -+02 ]I=i 2r EEtkrk, where ,k = 0 or 1 and rk = (0 - We- By the "Lebesgue function" having P as spectrum we denote, as usual, the function defined by

2 22 whenever x e P, continuous, and constant in every interval contiguous to P. Let S be the set of the algebraic integers w larger than 1 and such that all the conjugates of a, except w itself, have their moduli strictly inferior to 1. It is known1 that the Fourier-Stieltjes coefficient

2v eft df(x) of the Lebesgue function having spectrum P(0) tends to zero if and only if 0 does not belong to S. The question of knowing whether, if 0 belongs to S, the Fourier-Stieltjes coefficient (2r)'-1 fO2w enfX dF(x) does not tend to zero for every function of bounded variation F(x) having spectrum P(0) (or part of P) is open2 (except when 0 is ra- tional or quadratic). We propose to bring here a contribution to~the solution of this problem. Downloaded by guest on September 29, 2021