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LEMMA 1.4 J1u(C)112 < X(C) bIPFEREPTIALS Ai1 EXTRE2VEAL EAGTHIO RIEMANN SURPARSJ* BY ROBERT D. M. ACCOLA DEPARTMENT OF MATHEMATICS, HARVARD UNIVERSITY Communicated by Lars V. Ahlfors, January 14, 1960 1. Introduction.-Let rh(W) be the Hilbert space of square integrable harmonic differentials on a Riemann surface W. The main theorem, Theorem 1, relates the algebraic structure of rh(W) to the geometric structure of W. With this result we are able to give geometric significance to the fact of isomorphism between subspaces of rh for two different surfaces. The geometric structure of W will be given in terms of the extremal lengths2 of Various homology classes of cycles. If I{c is a family of rectifiable curves and p is a linear density on W, let L(p, {c}) = inf f, and let A(p) = cc IC) pIdzj if p2 dxdy. Then the extremal length of the family 4c} is defined to be the W supremum of the numbers L(p, 14c)2/A (p) as p ranges over all lower semi-contin- uous linear densities on W. Let rh6(W) be the exact differentials in rh(W) and let r1'he(W) be the semiexact differentials.' Let r1.(w) - rIh*(W)± and let rhm(W) = rAaC*(W)±. If c is a cycle we denote by a(c) the unique real differential in rI. such that f. C = (C, ov(c) *) for all w e rh. The o(c)'s are complete in rP.(W) as c runs through all cycles. We shall denote by 4c} and I4cl the families of curves in W and Q re- spectively which are homologous to c. By X(c) and XQ(c) we shall mean the ex- tremal lengths of 1c4 and 1c40. If W is a bordered surface ?W will denote the boundary of W. To say that w e r1(W) is zero along a, a contour of aW, will mean that w can be extended to be harmonic on W u a and that the extension is zero along a. 2. The Principal Result.-We list several lemmas that lead up to the main theorem giving one proof. LEMMA 1.4 J1u(c)112 < X(C). LEMMA 2. Let W be a finite Riemann surface. )W = a U 0, where a and IB are unions of contours of6W. Let u be harmonic on Wsuch that u = 1 on aandu = 0 on ft. Let c be a dividing cycle of W homologous to a. Then du = v(c). LEMMA 3. If W is a closed or finite surface then II0(c)l112 = X(c). Proof: We make essential use of the fact that the periods of a(c) over cycles and cross cuts are integers. We define-pi and p2, points of W, to be equivalent if there exists a path y connecting pi to p2 such that fJ, 0(c) is an integer. Since the periods of a(c) are integers f, a(c) is an integer for any path Sy connecting equiva- lent points. If E is an equivalence class, then the arc components of E are unions of analytic arcs, over any subarc of which a(c) has zero integral. Locally, then, E is a collection of level curves of a function of which 0(c) is the, differential. To show that E is closed we assume for p. e E that p. --a p # E. We can, by passing to a subsequence, join p. and Pn+l by a path lnso that 0 * Jn 0(c) 0. But this contradicts the fact that 0(c) has whole number integrals over paths connecting equivalent points. Let Eo be an arbitrary equivalence class if W is closed, and let Eo be the equiva- 540 Downloaded by guest on October 1, 2021 VOL. 46, 1960 MATHEMATICS: R. D. M. ACCOLA 541 lence class of oW if W is finite. It follows that W - Eo is a union of finite surfaces. If R is a component of W - Eo then v(c) restricted to R is exact and is the differ- ential of a function on R of the type considered in Lemma 2. For u(c) restricted to R still has integral periods. If such a period were different from zero we could construct a path starting on OR and ending in the interior of R over which cr(c) would have an integral of one. This is a contradiction. n Let W - = U Ri. a(c) (restricted to Ra) = dui. Ri = aj u 83i and u =- 1 i = 1 n on ai and ui = 0 on fi. If w e rPh(W) we have: (co, o(c)*) = Z (co dui*) n E fLi = (XX, o(> ai)*). Thus c is homologous to E ai. Let ci8 be the curve i = 1 in Ri, ui = s. Let c8 = ci,. If p is a linear density on W, then using a(c) + io-(c) * as a local parameter we have: L2(p, Ic}) < (fJ, pa(c)*)2 . f,. p2l(C) * o.(C)* = la(c)112 C p(c)*-p2 Integrating between s = 0 and s = 1 gives L2(p, {c}) < A(p)IIo(c)112 for any p. It follows that X(c) < I|a(c) 112. By Lemma 1 we have the desired equality. LEMMA 4. Let Q be an analytically imbedded, relatively compact subset of W con- taining a cycle c. Consider {Q, the set of all such El, ordered by inclusion. Let fu(c) denote the period reproducer in. rh(Q). Then Iao(c) IIQo-J(c) jj asQ2- W. LEMMA 5. If Q2 c W thenX(c) > X(c). THEOREM 1. If W is an arbitrary surface jju(c) 112 = X(c). Proof: Using the previous lemmas we have: X(C) . I|a(c)112 = lim 1I1o(C)IIQ2 = lim AD(C) > X(C). COROLLARY 1. Xa(C) X(C). COROLLARY 2. Let W be a bordered surface; a and ,3 two disjoint contours of 6W. Let c be a curve connecting a to A, and let {Ic be the family of curves weakly homologous to c. Let v(c) reproduce the integral along c of harmonic differentials that vanish along a and f3. Then IIa(c)II2 = X(c). Proof: Double W along a and /, and then apply Theorem 1. 3. Applications. THEOREM 2: Let W and W' be homeomorphic surfaces. Let c and c' bo corresponding cycles. Then the following two conditions are equivalent: (1) 3M ) 1 such that M-'X(c) < X(c') < MX(c); (2) 3X, a continuous linear isomorphismfrom rho(w) onto rho1(W') such that 4ua(c) = o-(c') and fc a = £'c Tfor all c, c'. Proof: Theorem 1 makes it obvious that (1) follows from (2). For the con- verse we first observe that linear combinations of the a(c)'s with real rational co- efficients are dense in the subspace of real differentials of rho. Using a common P denominator for the coefficients we see that elements of the form m-' E nkey(ck) = p k =1 m-' (Z nkck) are dense; that is, the real multiples of the u(c)'s are dense. Via k = 1 Theorem 1 the inequality of (1) can be extended to this dense subset. Thus a linear map 4 of the type considered in (2) is seen to be uniformly continuous on a dense subset of the real differentials in rIhO(W) and the range of 0 is a similar subset of rho(w'). (2) now follows easily. Downloaded by guest on October 1, 2021 542 MATHEMATICS: R. D. M ACCOLA PROC. N. A. S. COROLLARY 3. (w, o*) = (OC, (0a)*). Definition: Two surfaces satisfying the conditions of Theorem 2 will be called ho-equivalent. COROLLARY 4. Quasiconformally equivalent surfaces are ho-equivalent. A simple example shows that the converse of Corollary 4 is false. If W f OHD then rFo(w) = rF(W) and so r,,o(w) = rPo*(w). This observation leads to the following theorem. THEOREM 3. If W and W' are homeomorphic surfaces of class OHD then the follow- ing three conditions are equivalent: (1) W and W' are ho-equivalent; (2) 3M ) 1 such that N(c') < MX(c) for all c and c'; (3) 30, a linear map from rh(W) into rh(W') such that fS a = fc Oca. Proof: (3) implies (2): By the closed graph theorem 4 is continuous. (2) implies (1): Since rh = rh*, lwil = max I(w, a*) Since the real multiples of the o(c)'s are dense in the real differentials and since (a(c), a(cl)*) is the intersection number, c X cl, of c and cl, we have: =l(ci) 12 =max (u(cD, a(c)*)2 = max (ci X c)2 c IVc12 - c X(c) M max (cil X c')2 Xc) = MX(c11). THEOREM 4. Two closed surfaces, W and W', of the same genus are conformally equivalent if and only if N (c) = X (c') for corresponding cycles. A property that holds simultaneously on two ho-equivalent surfaces will be called ho-invariant. The most immediate ho-invariant properties are related to the structure of rho. In stating properties that depend on a particular canonical homology basis, the ho-invariance must, of course, be referred to corresponding canonical homology bases on the two surfaces involved. We list several ho- invariants: (1) The validity of the generalized bilinear relation.5 (2) Generalizations of the bilinear relation of the type considered by Kusunoki for parabolic surfaces.6 (3) rho(W) n rhe(W) = rhm(W). (4) For surfaces of class OHD that analytic differentials are determined by their A-periods.
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