APPENDIX SARlO POTENTIALS ON

RIEMANN SURFACES by

MITSURU NAKAI

Three kernels are considered in classical potential theory: the hyperbolic kernellog[ (1 - as-) / (S- -a) J for the disk, the parabolic kernel log (l/Is- -aJ) for the plane, and the elliptic kernel log(1/U,aJ) for the sphere. The first one can be extended to Green's kernel on hyperbolic Riemann surfaces; the corresponding potential theory has been given a comprehensive discussion in the monograph of Constantinescu-Cornea [1]. The second kernel generalizes to the Evans kernel on parabolic Riemann surfaces (Nakai [9J); the corresponding potential theory may be de­veloped in the same fashion as that of logarithmic potentials presented in Tsuji's monograph [5 J.

These two kernels, however, suffer from the restrictions im­posed upon their domain surfaces. In contrast, the third kernel can be extended to the Sario kernel, i.e. the proximity function in the terminology of Chapter V, on an arbitrary Riemann surface (Sario [21J, [25J, [26J). Thus the corresponding theory of the Sario potentials has the advantage of full generality.

In this appendix we shall systematically develop the theory of Sario potentials from the viewpoint of potential-theoretic princi­ples. We shall establish the continuity principle, unicity princi­ple, Frostman's maximum principle, capacitary principle (the fundamental theorem of potential theory), and energy principle (Nakai [7J, [8J, [1OJ). Their applications e.g. to value distribu­bution theory (cf. Nakai [7J, [8J) are not included.

305

806 APP. SARlO POTENTIALS ON RIEMANN SURFACES 11A

§l. CONTINUITY PRINCIPLE

We shall study Sario potentials sp(t) = fs(t,a)dJ,t(a) with respect to the Sario kernels(t,a). In this section we shall establish the continuity principle for these potentials (Nakai [7J). This will follow from the joint continuity of s(t,a). The unicity principle will also be established.

1. Joint continuity of s ({,a)

lAo Definition of s ({,a). We shall first review the definition of s(t,a). On an arbitrary Riemann surface S take arbitrary but then fixed points tj (j = 0,1) and disjoint parametric disks D j with centers tj (j = 0,1). Let to(t) = t(t,to,tl) be a harmonic function on S - {to,tIl such that to(t) +2 log It -tol and to(t) -2 log It -tIl are harmonic in Do and DI respectively. Moreover we require that to = (J)LltO in a neighborhood of the ideal boundary of S, with (1)Ll the normal operator of Chapter 1. Such a function to is unique up to an additive constant, and we assume that to(r) + 2 log It -tol-tO as t-tto in Do. The functions so(t) = 10g(1 +eto(f))

and so(t) +2 log It -tol are finitely continuous on S - Ito} and Do respectively.

For an arbitrary point a in S - Ito} we construct the function t(t,a) = t(t,a,to) in the same manner as t(t,to,td but this time we choose the normalization t(t,a) -2 log It -tol-tso(a) as t-tto in Do.

Let sl(t,a) = so(t) +t(t,a). Then the functions sl(t,a) and Sl (t,a) +2 log It -al are finitely continuous on S - la} and a parametric disk about a respectively. We also put Sl (t,to) = So (S), i.e. t(t,to) = O.

The function sl(t,a) thus defined on SXS is essentially the same as the proximity function constructed in V.l. By Theorem V.l.1F it is therefore bounded from below. We finally define s(t,a) = sl(t,a) +c where the constant c is so chosen that

(1) s(t,a) >0

for all (t,a) E S XS. By Theorem V.l.1D

(2) s(t,a) = s(a,r)

for all (t,a) in S XS. This function s was introduced and properties

Ie] §1. CONTINUITY PRINCIPLE 307

(1) and (2) were established by Sario [21J, [25J, [26]. From a po­tential-theoretic viewpoint, we shall refer to it as the Sario kernel on S. It is a positive symmetric kerneL

lB. Continuity outside the diagonal set. For efficient de­velopment of a potential theory for the Sario kernel a task of com­pelling importance is to prove the continuity of s on S XS. As the first step we shall establish the finite continuity of s on S X S out­side the diagonal set, i.e. we shall show that s(~,a) is finitely con­tinuous at (~',a') E S xS with ~'~a'. In view of (2) we may assume that ~'~ ~ o. Take parametric disks U and V about ~' and a'respectively, such that 0(\ V = )25 and ~o~ 0. Since (~,a)~ so(r) is finitely continuous on U X V and s (~,a) = so(t) + t(~,a) +c, we have only to show that (~,a) ~t(~,a) is finitely continuous on U X V.

We note that t(~,a) has the following properties:

(a) ~~t(~,a) is harmonic on U for fixed a E V, ((3) a~t (~,a) is finitely continuous on V for fixed ~ E U, ('Y) (~,a) ~t (~,a) is bounded from below on U X V.

Property (a) is a direct consequence of the definition of t (~,a) , and ((3) follows from (2) on observing that t(~,a) = s(a,t) -C -so (~). By virtue of this equality and (1) we deduce ('Y).

From (a), ((3), ('Y), and the Harnack inequality it follows that (~,a) ~t(~,a) is finitely continuous on U X V.

IC. Decomposition of s ({,a). Let Q be a regular region with a E Q, and go(~,a) be Green's kernel on Q. It is easily seen that go is continuous on Q XQ.

Consider the 2-form A2(~)dSr on S - {~o,~d defined by

etol grad t (>-)1 2

A2(0 = LlrSo(t) = 0 ~ , (1+e to(r»2

with A(~) 2::0 and the local Euclidean area element dSr on S. It is readily verified that A 2(t)dS)r can be continued to a nonnegative finitely continuous 2-form on S and that

(3)

on S - {a,~o,~d.

308 APP. SARIO POTENTIALS ON RIEMANN SURFACES [lC

We introduce the functions on 0 and 0 X 0

where

VIl(t,a) = 8(t,a) -2gll (t,a).

We shall prove the continuity of Gil and H Il. We start with Gil. Let t' E 0 and let U be a disk with center S' and radius 1 such that Geo. Denote by Ur the disk \b-t'\ <r in U with O<r<l, and by g Green's kernel on U. Then

Il-fbl 1 g(b,t) = log b-t ~loglb_tl+log2.

Since gll(b,t)-g(b,t»O is finitely continuous on UXU, its supremum M for (b,t) E U1/ 2 X U1/ 2 is finite. For 0 <e < t and tE U., gil (b,t) ~ -log Ib -tl +M' in Ib -tl <2e with M' = M + log 2.

On setting m = sup p,2 (b) Ib E U} < 00 we obtain

(4) f >.2 (b)YIl (b,t)dSb Ib-tl<2.

For any til E U. we have

IGIl (S') -Gil (til) I ~ f >.2 (b) Igll (b,t') -gil (b,t") IdSb Il-U,

In view of (4) the second term of the right-hand side is 0 (1)), and

lD] §1. CONTINUITY PRINCIPLE 309

since On(b,t')-70n(b,t") uniformly on fl-U. as S-"-7(, we see that

lim sup IGn«() -Gn(t") 1<0(0.). i"-i'

Thus lim Gn (t") = Gn (t'). i"-i'

We turn to the continuity of H n. By IB, (b,a) -7Vn(b,a) =

s(b,a) -20n(b,a) = s(b,a) is finitely continuous on aflXfl and the same is true of the coefficients of *dbOn (b,n as functions of the pair (b,t) of local parameters band t on afl and fl. Therefore we can easily conclude that H n (?;,a) is finitely continuous on fl X fl.

ID. Next we prove that

(5) Vn(t,a) = -Gn(n -Hn(t,a)

for all (t,a) E fl Xfl and consequently Vn is finitely continuous on flXfl.

For a fixed a E fl, the function b-7Vn (b,a) is bounded and con­tinuous on Q, of class Coo on Q-{a,to,td, and ~bvn(b,a) = A2(b). Assume that a is different from to and tl. Remove from fl dis­joint small closed disks with centers to and tl and radii lin such that the resulting region fln contains a. Let On be Green's kernel on fln and let U. be a disk about t E fln of radius e such that U.Cfln.

By Green's formula

f vn(b,a) *dbOn(b,n - f On(b,t) *dbVn(b,a) ann-au, ann-au,

= f (v!l(b,a) ~bOn(b,t) -On(b,n ~bVn(b,a) )dSb. nn-U,

On letting e-70 we obtain

(6) 211'vn(t,a) + f vn(b,a) *dbOn(b,t) + f vn(b,a) *dbOn(b,n an Pn

310 APP. SARIO POTENTIALS ON RIEMANN SURFACES [IE

where (3n = aa,,-a~2. We set gn(b,t) = 0 on a-a" and note that gD(b,r) -g,,(b,r);::::O converges to 0 uniformly on Q with respect to b as n~ 00. Therefore the coefficients of *dbg" converge uni­formly to those of *d"gD locally, and

converges to zero as n~ 00. Since *dg" >0 on (3n and IVn (b,a) I ::; K < 00 on a for a fixed a, we have

II vn(b,a) *dbg,,(b,r) I ::;K I *dbgn (b,t), I f3n I 13n

and the right-hand side tends to 0 as n~ 00. Thus on letting n~ 00 in (6) we obtain (5) provided r and a are different from ro arid rl. By the separate continuity of VD, H D, and GD on a, we deduce the validity of (5) for every rand a.

IE. We have arrived at the following conclusion (Nakai [7J):

Theorelll. The 8ario kernel s(r,a) is continuous on 8X8 and finitely continuous on 8X8 outside the diagonal set. Moreover, for every regular region a of 8 the decomposition

(7)

is valid, where gD is the Green's kernel on a and VD is a finitely con­tinuous function on a x a.

2. Sario potentials

2A. Potential-theoretic principles. Let I-' be a nonnegative regular Borel measure with compact support 81' in 8. Unless specified otherwise we consider only such measures 1-'. The Sario potential s" of the measure I-' is defined by

s,,(t) = f s(r,a)dl-'(a).

2C) §1. CONTINUITY PRINCIPLE 311

By (1) it is nonnegative, and positive unless )J. = O. As a con­sequence of Theorem IE, it is lower semicontinuous on Sand finitely continuous on S - Sp.. By virtue of (3), Sp. is subharmonic on S -Sp..

The object of potential theory is to find properties independent of p. of the family {Sp.} 1" Such properties are customarily called principles of potential theory.

2B. Local DlaxiDluDl principle. Since Sp. is subharmonic in S -Sp., its magnitude is determined by its behavior at the ideal boundary of S and at Sp.. Regarding the latter we shall show:

TheoreDl (local DlaxiDluDl principle). Let F be a closed sub­set of S containing Sp.. Then for any t' E F

(8)

For the proof take a parametric disk D about (, let p.' = p.ID, i.e. p.'(.) = p.(. (lD), and set JI." = p.- p.'. Then Sp. = Sp.' +s"" and Sp." is continuous on D. Thus if we can prove (8) for Sp.' it will follow for s.~. Therefore we may suppose that F CD. Let u(t,a) =

s(t,a) -2Iog(1/It-ai) on D XD. By Theoerm IE, u(t,a) is finitely continuous on D XD and a fortiori t--? Ju(t,a)dp.(a) is finitely continuous on D. Hence the proof of (8) is reduced to that of

lim sup f log ~I dp.(a) ~ lim sup f log ~[ ~ -I dp.(a), s<D-F .s-s' J a s<F .s-s' J a,

which is elementary (see e.g. Tsuji [5, p. 53J).

2C. Continuity principle. As a consequence of the local maximum principle we are now able to state:

TheoreDl (continuity principle). If sp.l Sp. is continuous (resp. finitely continuous) at t ESp. on SM then the same is true for

Sp. at t on S.

In fact, by the lower semi continuity of Sp. on Sand (8)

312 APP. SARlO POTENTIALS ON RIEMANN SURFACES [3A

But the last term is sl'(r) since sI'181' is continuous, and the as­sertion follows.

3. Unicity principle

3A. Uniqueness. The potentials SI' determine a linear operator p.~SI' from the measure space into a function space. We now see that this operator is injective (Nakai [10J):

Theorem (unicity principle). SI' = s. implies that p. = v.

We shall actually prove more: if SI' = s. +u, with u a harmonic function on 8, then p. = v.

3B. Let C02 (8) be the space of twice continuously differentiable functions with compact supports in 8. If C, is the clockwise oriented boundary of a disk D, about b E 8 with radius E>O, then Green's formula yields

(9) f (f(a) I:!.as(a,b) -s(a,b) I:!.aj(a) )d8a S-D.

= f (f(a) *das(a,b) -s(a,b) *daj(a)) c.

for every j E Co2(8). In view of (3), we obtain on letting E~O in

(10) 1 1

j(b) = - f j(a)}..2(a)d8a -- f s(a,b)l:!.a j(a)d8a. 4~ S 4~ S

Let u be the signed measure p.- v. Then for every j E Coco(8)

u(8) f 1 f (11) fj(b)du(b) = - j(a)}..2(a)d8a-- sq(a)l:!.aj(a)d8a, 4~ 4~

as is seen from (10) on integrating both sides with respect to u.

Since Sq == u,

(12)

and fSq(a)l:!.a j(a)d8a - fj(a)l:!.aSq(a)d8a = 0, (11) implies that

f j(b)(du(b) - U(8\2(b)d8b) = 0 4~

IB) §2. MAXIMUM PRINCIPLE

for every f E COoo(S). From this we conclude that

O"(S) dO"(b) = -}.2(b)dSb•

411"

313

If O"(S) were not zero, then 0" would be of constant sign and there­fore 8" ¢O, i.e. 8" ¢8., a contradiction. Thus O"(S) = ° and conse­quently 0" = 0, i.e. JJ. = v.

3C. Remark. In Theorem 3A the assumption 8" = 8. on Scan be weakened as follows: 8" = 8. on S except for a set on which there is no unit measure T with J8(a,b)dT(a)dT(b) < 00 (see 2.2A below).

In fact, by (7) we see that

8E(a) = lim ~ f 8E(a+ee i6 )d8 .... 0 211"

for ~ = JJ., v. From this it follows that 8,.. = 8. on all of S without exception.

§2. MAXIMUM PRINCIPLE

Frostman's maximum principle will be established for Sario potentials (Nakai [8J). As a consequence the fundamental theorem of potential theory will be proved. We shall also deduce the validity of the energy principle.

1. Frostman's maximum principle

lAo Global maximu:m. By the local maximum principle the behavior of 8" near S" is regulated by that of 8"IS". Moreover the construction suggests that maxs_s"8,, is ruled by the behavior of 8" at the relative boundary of S -S". Therefore it is plausible to state (Nakai [8 J) :

Theorem (Frostman's maximum principle). 8" I s" ~ M implie8 8" ~ M on S.

lB. Let K be a compact subset in S. We then have

(13) M(K) = sup lim sup 8(t,a) < 00

aclr r ... fJ

314 APP. SARlO POTENTIALS ON RIEMANN SURFACES [IC

where f3 stands for the ideal boundary of S. If f3 = f2f we under­stand that M (K) = O. For the proof of (13) take a regular region So containing KU{ro}. By 1.1B, (r,a)--'>t(r,a) is finitely con­tinuous on aSo xK and hence it takes a finite maximum Mo; t(r,a) :S;;Mo on aSo for any fixed aEK. Since LIt = ton S-So, we conclude that t(r,a) :S;;Mo for (r,a) E (S -So) XK. Clearly 80(r) is bounded on S -So, and therefore (13) follows from the definition of S (r ,a) .

Set B(M,p.) = max {M,p. (SI')M (SI') }. By the local maximum principle and the maximum principle for subharmonic functions we obtain the following partial result:

(14) 81'1SI':S;;M implies that sl':S;;B(M,p.) on S.

Ie. If S is closed then B(M,p.) = M and Theorem lA follows. If S is open but parabolic then, since SI' is bounded by B(M,p.) and subharmonic on S -SI"

sup SI' = sup lim sup sI'IS -SI" s-s~ .'€as~ .-.'

By the local maximum principle the right-hand side is dominated by

sup lim sup 81'ISI' :S;;M . • '€s~ .€s~ .• -.'

Thus Theorem lA again follows. Therefore the proof will be complete if we give it for a hyperbolic S.

ID. Hereafter in 1 we always suppose that S is hyperbolic, i.e. Green's kernel g(r,a) exists on S. In this case we conclude by the unicity of to on S that

(15)

where k is a suitable constant. Therefore

Similarly we obtain

(17) t(r,a) = 2g(r,a) -2g(r,ro) +so(a) -k.

IE) §2. MAXIMUM PRINCIPLE 316

We set

(18) u(t) = log (e-2g(I'.l'o) +e-2u(I'.1'1)+k) - log (1 +ek ).

Then Sarlo's kernel is expressed in terms of Green's kernel as follows:

(19) s(t,a) = 2g(t,a) +u(t) +u(a) +m

with a suitable constant m.

IE. We are to prove that s" 5:.M on 8 under the assumption s"18,, 5:.M. For this purpose we may assume without loss of gen­erality that p.(8) = 1. By (19) what we have to show is that the validity of

(20)

on 8" implies that on all of 8, where g,,(t) = fg(t,a)dp.(a) and

M' = M -m- f u(a)dp.(a).

We now assume that (20) holds on 8", and first prove that M'~O.

To this end take the unit measure v on 8 such that 8.C8" and

f g(t,a)dv(ndv(a) = Vu(8,,) = inf f g(t,a)do(ndO(a) 6

where the o's are unit measures with 8 6 C8". Such a measure v always exists and has the properties g.5:.Vg (8,,) on 8 and g. =

V g (8,,) on 8" except for a set which carries no measure 0 ¢O with f gdOdo < 00. For this well-known fact we refer the reader to e.g. Constantinescu-Cornea [1, p. 48J (see also the proof of Theorem 2B below).

Since fgdp.dp. < 00 in view of (20), the p.-measure of the subset of 8" on which g. ¢ V g (8,,) is zero, and hence fg.dp. =

fV g (8,,)dp. = V g (8,,). Fubini's theorem now yields fg"dv =

316 APP. SARlO POTENTIALS ON RIEMANN SURFACES [IF

Va (81'). On integrating both sides of (20) with respect to v we obtain

(21) 2Vo (81') + f uU;)dv(t) ~M'.

By (18) we see that

with

if;(~) = log(1 +eHk ) - log(1 +ek ).

Since if;(~) is a convex function, Jensen's inequality if;(f~dv(t)) ~ f if;(~)df.L (r) is valid for any v-integrable function ~ = Ht) and in particular for Ht) = 2g(r,ro) -2g(r,r!). Hence

and therefore

Since g.(ro) and g.(r!) are dominated by V o (81')' we now conclude that

This with (21) implies M' ~O, as desired.

IF. Consider the family 5' of sequences {rn} cS converging to the ideal boundary (3 of S, and the family 5'+C5' consisting of {rn} with liminfng(rn,a) >0 for some and hence for all aES. There exists a positive superharmonic function v on S such that limn v (r n) = 00 for {r n} E 5'+ (see e.g. Constantinescu-Cornea

2A) §2. MAXIMUM PRINCIPLE 317

[1, p. 27J). Observe that

is a superharmonic function on 8 -8" for any m = 1, 2, .... By the local maximum principle and (20),

(22) lim inf w",(r) ~O r<S-S~.i->i'

for every S' E a8". We also have

(23) n

for every Is .. } Eff. In fact, if liminf"wm(s,,) <0 for some Is,,} Eff then since gllo and u are bounded near {3, Is n} cannot be in ff+. Therefore we can find a subsequence Is,,'} of Is,,} such that lim" wm(s,,') <0 and lim" g(Sn',a) = 0 for every a E 8. Clearly this implies that

.. n

Thus we would obtain

n "

in violation of M' ~O. By (22), (23), and the minimum principle for superharmonic

functions we see that wrn(r) ~O on 8 -8110 , On letting m~ 00 we obtain (20) on 8 -8Jl and hence on 8.

The proof of Theorem 1A is herewith complete.

2. Fundalllental theorelll

2A. Capacity. We define a set function V(K) = V.(K) first for compact sets K C 8 by

V(K) = inf f 8(s,a)d~(s)d~(a) "

318 APP. SARlO POTENTIALS ON RIEMANN SURFACES [2B

where p. runs over all unit measures with S" eK. For a general set X eS we write

VeX) = sup V(K) K

where K runs over all compacta K ex. The quantity

1 c(X) = cB(X) ---

V(K)

will be referred to as the (inner) Sario capacity of X. From the definition it is easy to verify that for Borel sets X, c (X) = 0 is characterized by

(24) p.(X) = 0 for every p. with f s(t',a)dp.(ndp.(a) < 00.

Using this we can prove:

Theorem. A set X is of Sario capacity zero if and only if X is locally of logarithmic capacity zero.

For the proof we may suppose that X is compact. Take a parametric disk D. By virtue of (7), p. (DnX) = 0 for every p.

in D with JD log (l/It' -al)dp.(ndp.(a) < 00 provided (24) is valid. This means that XnD has logarithmic capacity zero.

2B. Capacitary measure. Let K be a compact set with c(K) >0. Since s is jointly continuous, by the selection theorem for a sequence of measures we can find a unit measure p. with S" eK such that J sdp.dp. = V (K). Such a measure p. is called the capacitary measure for K. We shall see in 3B that p. is unique. For this measure we prove the following capacitary principle:

Theorem (fundamental theorem of potential theory). Let K be a compact subset of S with positive Sario capacity, and p. its capacitary measure. Then s" ~ V (K) on S, and 8" = V (K) on K except for an Fq-set of Sario capacity zero.

2C] §2. MAXIMUM PRINCIPLE 319

2C. As the first step we shall prove that 81' 2:: V (K) on K except for an F .. -set of Sario capacity zero. Let A and An be the subsets of K on which 81'<V(K) and 81'~V(K) -lin (n = 1,2",,) respec­tively. Then An is compact and

This indicates that A is an F .. -set. We shall show that c(A) = O. Suppose that this were not the case. Then there would exist

an n with c(An) >0. This means that for a suitable e>O there exists a compact subset KI CK with

(25)

Note that fShdJ.L = V(K) implies the existence of a point foESp with 81' (fo) > V(K) -e. By (25), foEK I • Therefore we can choose an open disk U about fo with OnKI = )25 and

(26)

Moreover since foE Sp

(27) J.L( U) >0.

By virtue of c(Kr) >0 there exists a measure v with S.CKl such that

(28)

Using this v we construct a new signed measure VI by

Clearly J.Lt = J.L+tVI is a unit measure for every tE (0,1) with Sp, CK. Therefore

(30)

320 APP. SARlO POTENTIALS ON RIEMANN SURFACES [2D

On the other hand a simple calculation shows that

<2t(p,(U) (V -2e) -p,(U) (V -e)) +t2 f Sdll1dll1

The last member can be made negative by taking t sufficiently small. This contradicts (30) and we have c(A) = o.

2D. As the second step we shall show that splSp ~ V (K). Contrary to the assertion assume that Sp (rl) > V (K) for a rl ESp. Take an open disk U 1 about rl such that

Note that p,(U1 ) >0. By 2C we see that

a contradiction.

V(K) = f spdp,+ f spdp, Ul S-Ul

> (V(K) +e)p,(U1 ) + V(K)p,(S -U1 )

= V(K)+ep,(U]»V(K),

By the maximum principle splSp::; V (K) implies that Sp ~ V (K) on all of S. This with 2C proves Theorem 2B.

2E. Subadditivity. As an application of the fundamental theorem we prove the subadditivity of the Sario capacity:

Theorem. Let Xn (n = 1,2",,) be sets in S and X = U~Xn. Then

(31)

We may assume that VeX) < 00 and X and Xn are compact.

3A] §2. MAXIMUM PRINOIPLE 321

Let p. and P.n be capacitary measures for X and X n respectively. Then

(32) VeX) = f s(t,a)dp.(a) ~ f s(t,a)dp.(a) x Xn

for t E X n eX except for a set of Sario capacity zero. Since JSll.(a)dp.n(a) = V(Xn ), integration of both sides of (32) with respect to P.n and Fubini's theorem give

VeX) ~ f s!'.(a)dp.(a) = V(Xn)p.(Xn), Xn

I.e.

Therefore ro ro

c(X) = c(X) L p.(Xn):::; L c(Xn).

3. Energy principle

3A. Ninomiya's theorem. Let Q be a locally compact Haus­dorff space and k (x,y) a continuous positive function on Q XQ

with k(x,x) = 00 and k(x,y) = k(y,x). One can consider po­tentials k!'(x) = Jk(x,y)dp.(y). Ninomiya [1, Lemma 6J proved that if k!, satisfies Frostman's maximum principle and the unicity principle in the form of Remark 1.3C then

f k(x,y)du(x)du(y) >0

for any nonzero signed measure u.

Since the Sario potentials enjoy both Frostman's maximum principle and unicity principle, Ninomiya's theorem can be ap­plied to obtain:

Theorem (energy principle). For any measures p. and j/ with u = p. -j/ ~O,

(33) f s(t,a)du(t)du(a) >0.

322 APP. SARlO POTENTIALS ON RIEMANN SURFACES [3B

For signed measures u = JJ. -II set

Ilull = V(u,u).

IIul12 is referred to as the energy of u. Let e = {uiliull < oo}. The energy principle assures that e is a pre-Hilbert space with (UI,U2) inner product and Ilull norm. Thus we have the Schwarz inequality I (UI,U2) I :::; I hll II u211 and the triangle inequality II UI + u211 :::; II uI11 + II u211·

3B. Unicity of capacitary llleasure. We are now in a posi­tion to prove the unicity of the capacitary measure as anticipated in 2B.

Let JJ.I and JJ.2 be capacitary measures for a compact set K with positive Sario capacity so that IIJJ.II12 = 11JJ.2112 = V(K). By Theorem 2B

8,., (r) = V (K)

on K except for a set of Sario capacity zero, and hence

Observe that

= V(K) -2V(K) + V(K) = O.

The energy principle yields ILl = JJ.2.

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C. R. Acad. Sci. Paris 229 (1949), 1293-1295. [2] Quelques proprietes d la frontiere se rattachant d la classification des surfaces de

Riemann. Ibid. 230 (1950), 42-44. [3] Existence des integrales abeliennes sur les surfaces de Riemann arbitraires.

Ibid. 230 (1950), 168-170. [4] Questions d'existence au voisinage de l{l frontiere d'une surface de Riemann.

Ibid. 230 (1950), 269-271. [5] On open Riemann surfaces. Internat. Congr. Math. Cambridge, 1950. 398-399. [6] Linear operators on Riemann surfaces. Bull. Amer. Math. Soc. 57 (1951), 276. [7] Principal functions on Riemann surfaces. Ibid. 58 (1952), 475-476. [8] A linear operator method on arbitrary Riemann surfaces. Trans. Amer. Math.

Soc. 72 (1952), 281-295. [9] An extremal method on arbitrary Riemann surfaces. Ibid. 73 (1952), 459-470.

[10] Construction of functions with prescribed properties on Riemann surfaces. Con­tributions to the theory of Riemann surfaces, Princeton Univ. Press, Prince­N. J., 1953.63-76.

[11] Minimizing operators on subregions. Proc. Amer. Math. Soc. 4 (1953), 350-355. [12] Capacity of the boundary and of a boundary component. Ann. of Math. (2)

59 (1954), 135-144. [13] Functionals on Riemann surfaces. Lectures on functions of a complex variable,

Univ. of Mich. Press, Ann Arbor, Mich., 1955. 245-256. [14] Extremal problems and harmonic interpolation on open Riemann surfaces. Trans.

Amer. Math. Soc. 79 (1955),362-377. [15] Strong and weak boundary components. J. Analyse Math. 5 (1956/57), 389-398. [16] On univalent functions. 13 Scand. Congr. Math. Helsinki, 1957. 202-208. [17] Stability problems on boundary components. Seminars on analytic functions

II, Institute for Advanced Study, Princeton, N.J., 1958. 55-72. [18] Analytic mappings between arbitrary Riemann surfaces. Bull. Amer. Math. Soc.

68 (1962),633-637. [19] Picard's great theorem on Riemann surfaces. Amer. Math. Monthly 69 (1962),

598-608. [20] M eromorphic functions and conformal metrics on Riemann surfaces. Pacific J.

Math. 12 (1962), 1079-1097. [21] Value distribution under analytic mappings of arbitrary Riemann surfaces.

Acta Math. 109 (1963), 1-10. [22] Islands and peninsulas on arbitrary Riemann surfaces. Trans. Amer. Math.

Soc. 106 (1963), 521-533. [23] Second main theorem without exceptional intervals on arbitrary Riemann surfaces.

Mich. Math. J. 10 (1963), 207-219. [24] On locally meromorphic functions with single-valued moduli. Pacific J. Math.

13 (1963), 709-724. [25] Complex analytic mappings. Bull. Amer. Math. Soc. 69 (1963), 439-445. [26] General value distribution theory. Nagoya Math. J. 23 (1963), 213-229. [27] An integral equation and a general existence theorem for harmonic f1mctions.

Comment. Math. Helv. 38 (1964), 284-292. [28] Classification of locally Euclidean spaces. Nagoya Math. J. 25 (1965), 87-111. [29] A theorem on mappings into Riemann surfaces of infinite genus. Trans. Amer.

Math. Soc. 117 (1965), 276-284. [30] Cf. AHLFORS, L.; SARlO, L. [1]. [31] Cf. KLOTZ, T.; SARlO, L. [1]. [32] Cf. LARSEN, K.; NAKAI, M.; SARlO, L. [1]. [33-43] Cf. NAKAI, M.; SARlO, L. [1-11]. [44-45] Cf. RODIN, B.; SARlO, L. [1-2].

BIBLIOGRAPHY

SARlO, L.; FUKUDA, N. [1] Harnwnic functions with given values and minimum rwrmB in Riemannian

spaces. Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 270-273.

SARlO, L.; N OSHIRO, K. [1] Value distribution theory. D. Van Nostrand Co., Inc., Princeton, N.J., 1966.

236 pp.

SARlO, L.; SCHIFFER, M.; GLASNER, M. [1] The span and principal functions in Riemannian spaces. J. Analyse Math. 15

(1965), 115-134.

SARlO, L.; WEILL, G.

[1] Normal operators and uniformly elliptic self-adjoint partial differential equations. Trans. Amer. Math. Soc. 120 (1965), 225-235.

SAVAGE, N. [1] Weak boundary components of an open Riemann surface. Doctoral dissertation,

Univ. of Calif., Los Angeles, Calif., 1956. 44 pp. [2] Weak boundary components of an open Riemann surface. Duke Math. J. 24

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SCHIFFER, M. [1] The span of muUiply connected domains. Duke Math. J. 10 (1943), 209-216. [2] Cf. SARlO, L.; SCHIFFER, M.; GLASNER, M. [1].

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Los Angeles, Calif., 1957. 42 pp.

SEGRE, B.

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SORENSEN, W.

[1] Cf. BADER, R.; SORENSEN, W. [1].

SPENCER, D.

[1] A generalization of a theorem of Hodge. Proc. Nat. Acad. Sci. U.S.A. 38 (1952), 533-534.

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BIBLIOGRAPHY SS6

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SS6 BIBLIOGRAPHY

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458pp.

AUTHOR INDEX J stands for the Introduction to a chapt(fl' or section

Accola, 0.1.1L; II.6.J; II.6.3E; II1.4.4A

Ahlfors (cf. Ahlfors-Beurling, Ahlfors­Sario), 0.2.1C; 0.2.3F; I.J; 1.1.2A; 1.4.3A; 1.4.3B; 1.4.3D; 1.4.3E; I1.2.2B; I1.3.J; II.4.J; II.5.J; II.5.lE; II.5.2A; IV.2.1I

Ahlfors-Beurling, I1.6.J; III.J; IV.2.2E; V1.1.4L

Ahlfors-Sario, O.J; 0.1.1H; 0.1.3C; 0.2.1C; 0.2.ID; 0.2.lE; 0.2.1F; 1.4.4A; II.3.1C; IV.1.2G; VI.J; V1.3.2G

Akaza-Oikawa, II1.4.3B

Behnke-Stein, 11.5 .• J Bergman, J.7; II.3.J Besicovitch, II.6.1J Beurling (cf. Ahlfors-Beurling) Bieberbach, II.5.1D; II.5.lE Blatter, II.6.J; II.6.3E Bourbaki, 1.4.3B Breazeal, V1.1.4L Brelot (cf. Brelot-Choquet), VI.1.3D;

VIL1.J; VILl.lB; VILl.lC; VII.l.lD

Brelot-Choquet, II1.2.4G

Chevalley, 0.1.1L Choquet (cf. Brelot-Choquet), L4.3A Constantinescu-Cornea,0.1.2A;

L4.3B; VL2.3A; VI1.l.lC; App.J; App.2.lE; App.2.1F

Cornea (cf. Constantinescu-Cornea) Courant-Hilbert, VL1.3D

Duff (cr. Duff-Spencer), VI.l.lE; VL1.3D; VII.1.3B

Duff-Spencer, VI.2.3A

Evans, IIL4.4B

Feller, VI.l.lE Fuglede, II.6.J; II.6.2B Fukuda (cf. Sario-Fukuda)

Glasner (cf. Sario-Schiffer-Glasner), V1.1.4L

Goldstein, L4.3A; L4.3B; II.3.J Grunsky, I1.2.2A; I1.2.2D; III.4.J

Hilbert (cf. Courant-Hilbert) Hille, I.4.2F Hodge, VI.l.lB Hormander, VI.1.1E

Jenkins, II1.4.lE Johnson, VI.1.4L Jurchescu, III.4.3D

Kelley, 0.2.lE Kobori-Sainouchi, I1.5.J Korn, V1.1.1A Kuramochi, III.4.4B Kusunoki, II.5 .. J

Larsen (cf. Larsen-Nakai-Sario), VI.1.4L

Larsen-Nakai-Sario, VI.3.J; VI.3.3B; VI.3.3C; VI.3.4B

Lehto, IL2.2D Lichtenstein, V1.1.1A Loeb (cf. Loeb-Walsh), VII.l.lB;

VIL1.2A; VII.1.2B Loeb-Walsh, VILl.lC Lokki, IL2.2D

Maeda, lIl.2.4G Marden (cf. Marden-Rodin), II.6.J;

II.6.3G Marden-Rodin, II.6.J; III.J; III.l.lD;

III.l.lF; III.1.2C; II1.1.3A; III.1.3B; III.2.J; IIL2.2A; IIL2.3A; IIL2.3B; III.2.3C; III.2.3E; III.2.3F; III.2.3G; III .2.4B; III.2.4C; III.2.4D; IIL2.4E; III.3.J; IIL3.1C; III.3.1D; III.3.3B

Meehan, VI.1.4L Miranda, VI.1.lE Mizumoto,II.5.1E Mori,IV.2.1I Myers, VI.1.4L Myrberg, IV.2.1I

337

338 AUTHOR INDEX

Nakai (cf. Larsen-Nakai-Sario, Nakai­Sario), 1.4AB; IIA.3A; III.2AG; II1.4.4B; V1.1.4L; VI.3.2E; VI1.2.J; VII.2.3A; VI1.2.4C; VI1.2AD; VI1.3.2A; App.J; App.1.J; App.l.lE; App.1.3A; App.2.J; App.2.1A

Nakai-Sario, O.1.2A; LJ; 1.4.3A; 1.4.3B; I.4AA; I.4.4B; IV.J; VI.1.J; V1.1.4H; V1.1.4L; VI.2.J; V1.2.2B; VI.2.2F; VI.2.3C; V1.3.J; V1.3.2B; V1.3.2C; VI.3.2D; VI.3.2E; V1.3.2F; V1.3.4A; VI.3.5B; VI.3.5C

Nickel, II1.3.J Ninomiya, App.2.3A Noshiro (ef. Sario-Noshiro), II1.4.4B

Ohtsuka, I1.6.J Oikawa (ef. Akaza-Oikawa, Oikawa­

Suita), LJ; 1.l.lA; 1.4.2B; 1.4.2C; 1.4.2E; 1.4.2F; I.4.2G; II1.3.J; III.4.3B

Oikawa-Suita, II.2.2A; II1.3.J; III.4.J; II1.4.2C

Ow, V1.LJ; V1.1.4E; VL1.4F; V1.1.4G; V1.1.4L

Parreau, V1.L4L Phelps, 1.4.3A

Rado, O.2.1F Rao, V1.L4L Reich (cf. Reich-Warschawski),

II1.2.2A; II1.3.J; II1.3.2E Reieh-Warschawski, II1.3.J;

II1.3.2A; III.3.2B; II1.3.2C de Rham, V1.1.lB; VI.3.1A; VI.3.2D;

VI.3.2G Richards,O.L2C Rodin (ef. Marden-Rodin, Rodin­

Sario), II.3.J; II.3.2C; II.3.2F; I1.5.J; II.5.1B; I1.5.2A; I1.6.J; I1.6.3E; IV.2.2A

Rodin-Sario, 1.J; 1.2.3B; I.3.2A; 1.3.2B

Royden, 1.4.3A; 1.4.3C; I.4.4C; II.5.J; II.5.lE; IV.2.1I

Ryff, 1.4.3A; I.4.3E

Sainouchi (cf. Kobori-Sainouehi), I1.5.J

Sario (cf. Ahlfors-Sario, Larsen-Nakai­Sario, Nakai-Sario, Rodin-Sario, Sario-Fukuda, Sario-Noshiro, Sario-Sehiffer-Glasner, Sario­Weill), LJ; 1.1.lA; 1.L2A; 1.L2B; 1.L2G; 1.3.IC; I.3.2B; 1.4.1B; I1.1.lC; II.2.lE; I1.2.2A; I1.2.2D; II.2.2E; II.2.3B; II.3.J; II.4.J; I1.4.3A; IILJ; III.2.J; II1.2.4D; II1.3.2C; II1.4.J; II1.4.1B; III.4.3B; II1.4.3C; IV.L2B; V.1.J; V.L1D; V.1.lF; V.L2B; V.2.1B; V.2.2F; V.2.3B; V1.LJ; V1.L2B; V1.L3D; V1.L3G; VI.L3I; V1.L4A; VI.L4B; V1.L4E; V1.L4F; VI.L4G; VI.L4L; App.J; App.1.lA

Sario-Fukuda, V1.LJ; V1.L5B Sario-Noshiro, III.4AB; V.J; V.2.3A Sario-Sehiffer-Glasner, V1.LJ;

V1.L2B; V1.L2D; VI.L3B; VI.L3D; V1.L3G; V1.L3I; V1.L4A

Sario-Weill, V1.LJ; VI.L2B Savage, I.4.3A; 1.4.3E Schiffer (cf. Sario-Schiffer-Glasner),

II.2.2A; II.2.2D; IIIA.J Selberg, II1.4.4B Smith, V1.L4L Spencer (ef. Duff-Spencer) Springer,O.2.3E Stein (ef. Behnke-Stein) Strebel, IILJ; III.2.2A; III.3.J:

II1.3.2E Suita (cf. Oikawa-Suita)

Toki, V1.1.4L Tsuji, 1.4.3B; App.J; App.L2B

Virtanen, IV.L2B

Walker,O.LIL Walsh (ef. Loeb-Walsh), VI1.2.2D Warschawski (ef. Reieh-Warschawski) Watanabe, II.5.J Weill (cf. Sario-Weill), III.3.J Whitney, V1.3.2D

Yoshida, VI1.2.2A; VI1.2.2B

SUBJECT INDEX J stands for Introduction

Italicized section numbers refer to definitions.

A, IV.l.2A AB,IV.2.1I AD, IV.2.1D

- - null set, IV.2.2E Abel's theorem, IL5.2A Abelian covering space, VL1.4L Adjoint of a differential operator,

VL3.1A Admissible

- function, II.l.lB - mapping, V.2.SA; V.2.3B;

V.2.3C - meromorphic function, V.3.1B;

V.3.1D; V.3.IE - partition, III.S.IA

Ahlfors' problem, L4.3D Alexandroff compactification, 0.2.2B;

VIL1.2A Almost every, II.6.2B Analytic

- are, 0.1.1 I - differential (cf. r a), 0.2.2E - function with prescribed coef-

ficients, II.4.2E; IL4.3A Are, O.l.lA

analytic -, 0.1.11 - element, VI.l.lA closed -, 0.1.11 open -, 0.1.1 I simple -, 0.1.11 smooth -, 0.1.11

Area integral, 0.2.SB Argument principle, II.2.1B

B, IV.l.M; VL1.4K BD,IV.1.2A B(u), J.5 Barrier, O.2.1B Basis

- for conformal structure, 0.1.1 E canonical homology -, O.l.SD

339

homology -, O.l.SD homology - modulo dividing

cycles, O.l.SD Bergman kernel, J.7 Border, O.l.IH

- reduction theorem, VL2.3C Bordered

- boundary neighborhood, J.J; I.l.2B

- Riemann surface, 0.1.1 H interior of - Riemann surface,

O.l.lH - subregion, 0.2.1 D

Boundary - coordinates, VI.2.1A - operator, O.l.SB ideal - component, 0.1.2D Royden harmonic -, 1.4.4A Wiener harmonic -, 1.4.3B

Bounded compactness, VL1.1E B-principal form, VI.2.2B Branch point, O.l.lG

multiplicity of -, O.l.1G

C, IV.l.2D; IV.1.2G; IV.1.2F; IV.2.1G

C(·), I.l.M; VL1.2A Cp, C,,(, VI.l.4G; VL1.4L Canonical

- exhaustion, IIL1.1A - homology basis, O.l.SD - open set, VI.l.lA - partition, J.3; 0.2.1D - subregion, 0.2.1D

Cantor set, VL1.4L generalized -, VL1.4L

Capacitary - measure, App. 2.2B - principle, App.J; App.2.2B unicity of - measure, App.2.3B

Capacity, IL6.2C; III.2.4B; IIL4.4A; IIL4.4B; IV.1.ID; IV.1.2F; IV.2.1C

340 SUBJECT INDEX

- function, J.9; IIL4.4B; VI.LiE; VLL4G; VLL4L

- of a boundary component, J.9; IILL3B; III.4.3B; IIL4.3D; IV.L2D; VLL4E; VLL4G

- of ideal boundary, J.9; VLL4E generalized -, III.l.2C; III.2.4B Lo- -, III.2.4B; III.2.4C; IIL4.3C logarithmic -, App.2.2A Sario -, App.2.2A subadditivity of Sario -, App.2.2E

Chain,0.1.3A relative -, II.6.2E

Characteristic, VI.l.M - function, V.2.2B of bounded -, VI.l.M

Chordal distance, V.3.IA Circular slit mapping, J.4; II.2.3B Closed

- curve, 0.1.1 I - differential, 0.2.2E - harmonic form, VI.3.1B;VI.3.4B - surface, 0.1.1 L

Coclosed - differential, 0.2.2E - harmonic form, VI.3.1B;VI.3.4B

Coderivative, VI.l.l B Cofinal, 0.2.1E Compactification

Alexandroff -, 0.2.2B; VILL2A Kerekjart6-Stoilow -, 0.1.2C;

O.L2D; III.l.lA Compactness

bounded -, VI.l.lE monotone -, VI.LIE

Complete continuity, VILJ; VII.2.ID; VIL2.2A; VIL2.2B

Completeness, VI.1.1E; VI.L4H Component (cf. ideal boundary

component) strong -, J.9; III.4.3A; III.4.3C unstable -, J.9; III.4.3A; VI.L4L weak -, J.9; III.4.3A; IIL4.3B;

VLL4G Conformal

- equivalence, 0.1.1 D - mappings (cf. slit mappings),

J.4; J.1O; II.5.ID - metric, V.l.2A - structure, O.l.lA; O.LIE

Conjugate complex - of a differential, 0.2.2B - extremal distance, II.6.1C;

III.2.2A star - of a differential, 0.2.2B

Connected Hausdorff space, O.LIA; VII. 1.1 A

Connectivity,0.1.2C Consistent system of partitions,

0.2.1D; VLL3F Constant p-form, VI.2.2A; VI.2.2D Continuation, III.4.3D

compact -, III.4.3D essential-, III.4.3D

Continuity - lemma, IIL2.IB - principle, App.J; App.L2C

Contour sequence, III.2.ID Convex set, I.4.3A

extreme point of -, I.4.3A Countability axiom, O.2.IF; VI. 1.1 A Counting function, V.2.1B Covering surface

ramified -, O.l.lG universal -, O.2.IF

Curve, II.6.1 A closed -, 0.1.11

D, IV.1.2A; VLL4K D,,/, VI.3.1A Defect, V.2.3B; V.3.ID

- - ramification relation, V.2.3B; V.3.lD

Degree of a divisor, II.5.1A . Differential, 0.2.2B

analytic - (cf. ra), 0.2.2E closed -, 0.2.2E coclosed -, 0.2.2E coefficient of a -, 0.2.2B complex conjugate of a -, 0.2.2B - form, VI.J; VI.l.lB; VLl.lC - operator, VI.3.5A exact -, 0.2.2E; L4.4A; II.3.IA;

II.3.IC harmonic -, 0.2.3D inner product of a harmonic -,

II.3.1A norm of a -, O.2.2F; II.3.1A order of a -, O.2.2B pure _., 0.2.2E

SUBJECT INDEX 341

reproducing - (cf. !/Ie and !/It), J.7; II.3.M; IV.l.lA

semiexact -, II.3.1C smooth -, 0.2.2B star conjugate of a -, 0.2.2B

Direct sum, II.2.2E Directed

- family, VII.l.lB; VII.l.lC -limit,0.2.1E

Dirichlet - inner product, VI.l.lC - integral, J.5; 0.2.3F - norm, 0.2.3F; O.2.3G - operator, J; I.l.2D; VI.l.2C;

VI.2.3A; VI.2.3C - principle, I.4.4B - problem, O.2.IC; VI.l.lE generalized - problem, VI.2.3A

Distinguished coordinate neighbor­hood, VI.2.2A

Dividing cycle, 0.1.3D Divisor, II.5.1A

degree of a -, II.5.1A - of a meromorphic differential,

II.5.1A - of a meromorphic function,

II.5.1A integral -, II .5.1 A multiple of a -, II.5.1 A

Double, 0.1.1 K Dual

- operator, VII.2.2A - space, VII.2.2A

E, IV.l.M; VI.l.4K EP, VI.3.2A Efoo, VI.3.1A; VI.3.5A ETo, VI.3.5A Eigenvalue, VII.2.2B; VII.2.2C

multiplicity of an -, VII.2.2B Eigenvector, VII.2.2B Elliptic kernel, App.J Energy, App.2.3A

- principle, App.J; App.2.3A Equicontinuous, VII.l.IC Euler

- characteristic, V.2.2E - index, V.2.3B

Evans - kernel, App.J - potential III.!,..4B; V.3.2A

Exact - differential, 0.2.2E; I.4.4A;

II.3.IA; II.3.1C - form, VI.3.2D; VI.3.3C

Exhaustion, VII.l.2B canonical -, III.l.IA countable -, O.2.1F; VI.1.1A

Extreme point of a convex set, I.4.3A Exterior

- derivative, 0.2.2D; VI.l.lB - product, 0.2.2D

Extremal conjugate - distance, II.6.1C;

III.2.2A - circular slit annulus, III.3.2A;

III.4.tD - distance, II.6.1C; II1.2.2A;

IV.2.2A; IV.2.2E -length, J.lO; II.6.1B; II.6.1H;

II.6.3A; III.l.lF; III.2.1B; III.4.1D; IV.1.tD

- metric, III.2.3A - radial slit annulus, III.3.2A;

III.3.2D; III.4.1D - slit annulus, III.3.1A; III.3.1C;

III.3.1D; III.3.3A - slit disk, III.3.1A infinite -length, II.6.2A; II.6.2D;

III.2.4B; III.2.4C; III.2.4G; III.3.2F

main - theorem, II.1.1C; VI.1.3B; VI.1.3G

S:(aO, aI, ,,0, ,,1), S:*(aO, a1, ,,0, ,,1), III.2.1A; III.2.2A; IV.l.lD

F, VI.1.4I; VI.1.4K FD, VI.1.41 FH, VI.1.41 Fatou's theorem, I.4.3B First main theorem, V.J; V.2.J;

V.2.1B Fluid flow, VI.1.6A Flux, I.l.2A; VII.2.4B

- condition, J; I.1.2B L- -, VII.3.1C

Fredholm integral equation I.4.1A

SUBJECT INDEX

Frostman's maximum principle, App.J; App.2.IA

Function admissible -, II.l.lB capacity -, J.6; IIL4AB; VI.l.If E;

VLl.4G; VLl.4L characteristic -, V.2.2B counting -, V.2.1B Green's -, J.2; IIL2AF; IV.l.2F;

VI.l.lD; VLl.4C harmonic -, O.2.1A; VLl.ID;

VII.l.lB holomorphic -, O.l.lC mean proximity -, V.2.1 B principal-, J; I.l.2B; VLl.2B;

VLl.3A; VII.2.1B; VIL3.IA proximity -, J.11; V.LJ; V.1.1D;

V.l.2A; V.3.IA quasi-rational-, II.5.2A subharmonic -, O.2.1B; VLl.ID;

VIIol.IB superharmonic --, VL1.1D;

VII.l.1B univalent -, IV.l.IC; VLl.31

Fundamental potential, J.I

r, I.~.~A ra, II.S.IC; IL5.IC; II.5.IE

- interpolation problem, II.4.2B - reproducing differential, IL3.3A

(P)r ••• , II.S.IC; IL5.1B; IL5.2A - interpolation problem, IL4.2B;

IIA.2E - reproducing differential, II.3.3F;

IL5.2C r e, I.~.~A

reo, I.~.~A rh, II.S.IA

- interpolation problem, IL4.2B - reproducing differential, II.3.2C;

IL3.2F; IL6.3E r'<6, I.~.~A; II.3.1C

- interpolation problem, II.4.2B - reproducing differential, IL3.3D;

IV. 1.1 A rho, II.S.IC

- reproducing differential, IL3.3C; II.6.3G

(p)rhm, II.S.IC; IV.2.IC - reproducing differential, II.3.3B;

IV. 1.1 A (p)rh8e, II.SolC

- interpolation problem, IL4.2B - reproducing differential, II.3.3E;

II.6.3C; II.6.3E

Gaussian curvature, V.l.2C Generalized Cantor set, VI.l.4L Genus, Ool.2C Green's

- formula, O.2.3F; IV.1.2F; VL3.IC

- function, J.2; IIL2.4F; IV.I 2F; VI.l.lD; VLl.4C

- kernel, App.J; App.2.1D

H,IV.l.2A H(·), I.l.2B; VL1.1D; VII.1.1A HI (·), I.l.M; VLl.2A HB, IV.1.2F; IV.2.1E; VL1.1D

- -null set, IV.2.IE HD, IV.1.2F; IV.2.IA; IV.2.IC;

IV.2.1D; IV.2.1E; IV.2.1G; VI.l.1D; VLl.4A

- -null set, IV.2.IE HMq , VI.1.~I Hcf!, VI.l.~I HP, VI.S.SA HP, IV.2.1H; VI.l.lD H(R; a, b), VI.l.SI Hahn decomposition, VII.2.2E Harmonic

- diameter, I.~.2G - differential, O.2.3D; II.S.IA;

VI.J - field, VI.S.IB; VL3.2G; VL3.3D - form, J.12; VI.1.1B; VI.3.3A - function, O.2.1A; VL1.1D;

VII.l.1B - measure, J.2; IL3.IC; IV.l.lD;

IV.1.2F; VUAD - metric, I.~.2F - semifield, VI.S.IB - space, VII.J; VII.l.J; VII.l.lB - structure, VII.l.lB Royden - boundary, L4.4A Wiener - boundary, L4.3B

SUBJECT INDEX

Harnack's - function, VI.l.lE - inequality, 1.l.1B; VLl.1E

Hodge star operator, VI.l.1B Holomorphic

- function, 0.1.1 C - mapping, O.LlA; 0.1.1 D

Homologous, O.l.SB weakly -, II.6.2E

Homology canonical - basis, O.l.SD first - group, O.l.SB generalized -, II.6.SA - basis, O.l.SD - basis modulo dividing cycles,

O.l.SD - class, O.l.SB weak -, II.6.2E; II.6.3E

Horizontal slit mapping, J.4; II.2.1B; IV.LlC

Hyperbolic, VI.1.4C - harmonic space, VII.l.SB;

VII.2.4A - kernel, App.J - metric, I.4.2A; I.4.2E

IrJ, VI.l.4K Ideal boundary

capacity of -, J.9; VI.1.4E - component, 0.1.2D; IV.2.1H isolated -, IIL3.3C; IVA.lB removable -, III.S.SB

Identity partition, .L3; 0.2.1D Inclusion relations, IV.1.2G; VI.1.4B;

VI.1.4L Induced partition, III.l.lA Inner product

- of a harmonic differential, II.S.IA

point -, VI.2.1B Integral divisor, II.5.1 A Integration, O.2.3A; O.2.3B; O.2.3F;

II.6.2F area -, 0.2.SB line -, O.2.SA

Interior of a bordered Riemann sur­face, J.J; I.l.2B

Interpolation, J.8; II.4.1A; IIA.2B; VI.1.4L; VL1.5A

Intersection number, O.l.SC; I1.4.2C

Invariant measure, VII.2.2D - associated with TL, VII.S.IC

Jensen's inequality, App.2.1E

K, IV.1.2A; VI.1.4I; VI.1.4K KD, IV.2.1B; IV.2.1C k'Y' VI.1.4E; VL1.4G k.(r, a\ ')10, ')II), III.2.SC;

III.2.3G Kerekjlirt6-Stoilow compactification,

0.1.2C; O.1.2D; III.LlA Kernel

elliptic -, App.J Evans -, App.J Green's -, App.J; App.2.1D hyperbolic -, App.J parabolic -, App.J positive symmetric -, App.1.1A reproducing -, VI.1.3D Sario - (cf. s(r, a) and proximity function), App.J; App.l.1A;

App.2.1D Krein-Milman theorem, I.4.3A Kronecker delta, VI.1.1A

L, Vl.1.4J; VI.1.4K LC, Vl.1.4J; VI.1.4K LP, VI.1.4L Laplace-Beltrami operator, VI.l.lB Lindel6f property, O.2.1F Linear

- density, II.6.1A - independence, II.4.1B

Line integration, 0.2.SA meal

- maximum principle, App.1.2B - parameter, 0.1.1 B

Locally - compact, VII.1.1A - connected, VII.1.1A - Euclidean (= - fiat), VI.1.1A - fiat, VI.l.lA; VL1.3D; VI.2.2A

Logarithmic - capacity, App.2.2A - pole, IV.l.l A - potential, App.J

M(a, (3), I.4.SC

SUBJECT INDEX

Main Existence Theorem, J; 1.1.2B; 1.4.1A; VI.1.2B

generalized -, VI1.3.2A - by integral equation, 1.4.1A - by orthogonal projection,

1.4.4C - for principal fields, V1.3.2B - for principal forms, V1.2.2B;

V1.2.2F; V1.3.3B - for principal semifields, V1.3.4A - for principal tensor potential,

V1.3.4B - on harmonic spaces, VI1.2.3A;

VII.2.4D - with estimates, 1.1.2G

Main extremal theorem, I1.1.1C; V1.1.3B; VI.1.3G

Maximum principle, II1.l.lE; V1.l.lE Frostman's -, App.J; App.2.1A local -, App.1.2B

Mean proximity function, V.2.IB Measure preserving map, 1.4.3D Metric tensor, V1.l.1A Minimal

- circular slit annulus, III.S.2B - circular slit disc, III.S.2B - radial slit annulus, III.S.2G

Modulus, V.2.1A; V1.l.4L Monotone compactness, V1.1.1E Multiple of a divisor, II.5.IA Multiplicity

- of a branch point, O.I.IG - of an eigenvalue, VII.2.2B

N(a, (3), I.4.SB Nevanlinna theory, V.3.J; V.3.1C Ninomiya's theorem, App.2.3A Norm

Dirichlet -, O.2.SF; O.2.3G - of a differential, O.2.2F; II.S.IA point -, VI.2.IB supremum -, 1.1.2E

Normal family, O.2.SH Normal operator, J; I.I.2A; 1.4.3A;

V1.1.2A; V1.2.2D convergence of -, 1.2.3A; 1.3.2A direct sum of -, I.2.2E extreme -, I.4.SA integral representation of -, 1.3.1B

- for p-forms, VI.2.2D - Lo, J3; I.2.IA; 1.2.2B; IV.2.1A;

IV.2.1B; V1.1.2E ~ 4, J2; I.2.IB; 1.2.2D; 1.4.4C;

IV.2.1A; IV.2.1B; IV.2.1C; V1.1.2E

- Lr*, VI.S.5A quasi -, VII.S.IA space of -, 1.4.3A

Normal part, VI.2.IA Null set, IV.2.ID

AD -, IV.2.2E HB -, IV.2.1E HD -, IV.2.1E

OAB, IV.l.2C; IV.1.2E; IV.l.2G; IV.2.1I

OABD, IV.1.2G OAD, IV.l.2C; IV.1.2G OAE, IV.1.2C; IV.l.2E; IV.l.2G OG, IV.1.2F; IV.1.2G; IV.2.1E;

IV.2.1F; IV.2.IG; V1.l.4C; V1.2.2B; V1.2.3A; V1.3.2E; VI1.l.3B

OHB, IV.l.2B; IV.1.2G; V1.l.4B; V1.1.4L

OHBD, V1.l.2B; IV.l.2G; V1.l.4B OHD, IV.l.2B; IV.l.2G; IV.2.1A;

IV.2.1F; IV.2.1G; V1.l.4A; V1.l.4B

Om, V1.l.4L OHMq , V1.l.4L OHP, IV.l.2B; IV.l.2G; IV.2.1F;

V1.l.4B OUo, V1.l.4L OKB, IV.l.2G; V1.l.41 OKBD, IV.l.2G; IV.2.1B; IV.2.2A OKD, IV.1.2G; IV.2.1B; IV.2.2A;

V1.l.41 OKP, IV.l.2G OSB, IV.1.2D OSD,IV.1.2D Open set

canonical-, V1.l.IA regular -, VI.I.IA; VII.I.IA smooth -, VI.I.IA

Operator Dirichlet -, J; I.1.2D; V1.l.2C;

VI.2.SA; V1.2.3C dual -, VI1.2.2A

SUBJECT INDEX

Hodge star -, VI.1.1B Laplace-Beltrami -, VI.1.1B normal-, J; I.1.2A; 1.4.3A;

VI.1.2A; V1.2.2D - B, VII.1.SA; VII.2.1B - T, VII.2.1C; VII.2.1D;

VI1.2.2C - T L , VII.S.1B quasinormal-, VII.S.1A

Orientation, 0.1.1J Orthogonal

- complement, II.3.1C; V1.3.2A - decomposition, 1.4AA; I1.3.3A,

ff. - projection, 1.4AA; I.4AC;

I1.3.3A Outer regular set, VII.1 .2B

'ltc, J.7; II.S.2A 'ltt, J.7; II.S.2B P, IV.1.2A; VI.1.4K Po, P1, J.4; II.2.1A; I1.2.2A; II.2.2C;

I1.2.2D; I1.2.2E; III.4.1D; IIIA.2C

Pos, P1s, II.2.1A; III.4.1A; IIIA.1B; I1.4.2A

po, Pl, I.S.2B; II.l.1A; VI.1.3A; VI.1.3F; VI.1.6A

Pas, pl, II.2.1A; I1I.4.1A Pa, III.2.4A; III.2.4D Pkh, II .1.1 A ff 'PAl" VI.1.SA; VI.1.3B; VI.1.3F;

VI.1.3G; VI.1.5A PP., PPh POo, POh II.S.2B Pp,,, PPn II.S.2F; I1.5.2A p(t, a 1, 'Yo, 'Y1), II.1.2A; III.1.2C;

I11.1.3B; I11.2.3C; I11.2.3E; III.3.1A; IV.l.1D; IV.2.2D

Pairing, II.5.1B Parabolic, III.2.4B

Lo- -, III.2.4C - harmonic space, VII.1.SB;

VII.2.4A - kernel, App.J - Riemann surface, IV.1.2F - Riemann space, VI.1.4C

Parallel, VI.2.2A - coordinate covering, VI.2.2A - slit region, I11.4.1D; III.4.1E

Parametric disc, 0.1.1B

Parreau's decomposition, VI.1.4L Partition of boundary, J.3; 0.2.1D;

1.2.1C; III.l.1A canonical-, J.3; 0.2.1D consistent -, I.2.2C consistent system of -, 0.2.1D;

VI.1.3F identity -, J.3; 0.2.1D induced -, III.1.1A regular -, 0.2.1D

Period, 0.2.SA Perron's

- family, VII.1.1D; VII.1.2A - method, O.2.1C - theorem, VII.1.1D

Picard - point, V.2.2A; V.2.3C; V.3.1B -:- theorem, J.ll; V.2.J; V.3.1D

Pick's theorem, 1.4.2F Poincare

- diameter, I.4.2B - inequality, 1.4.4B - metric, I.4.2A; 1.4.2F

Point branch -, 0.1.1G - inner product, VI.2.1B - norm, VI.2.1B regular -, VII.1.2A

Potential double-layer -, VI.1.6A electrostatic -, VI.1.6A equilibrium -, VI.1.4L logarithmic -, App.J Sario -, App.J; App.1.2A. single-layer -, VI.1.6A tensor -, VI.S.1B; VI.3.4B velocity -, VI.1.6A

Presheaf, VII.1.1B Principal

B- -, VI.2.2B - field, VI.J; VI.S.2A ff - form, VI.J; VI.S.SA ff; V1.3.5A - semifield, VI.S.4A - tensor potential, VI.S.4B

Principalfunction, J; 1.1 .2B; VI.1.2B ; VI.1.3A; VII.2.1B; VII.3.1A

application of - to physics, VI.1.6A

bounds for -, I.1.2G; VI.1.2D

SUBJECT INDEX

convergence of -, L2.3B; L3.2A; VI.L3F

extremal property of -, II.LIB; 11.1.10; VLL3B; VI.L3I

meromorphic -, II.2.IA; II.2.3A - Po and Pl, I.S.2B; IL1.1A;

VLL3A; VLL3F; VLL6A Principle

argument -, II.2.IB capacitary -, App.J; App.2.2B continuity -, App.J; App.L20 Dirichlet -, L4.4B energy -, App.J; App.2.3A Frostman's maximum -, App.J;

App.2.IA local maximum -, App.L2B maximum -, III.1.1E; VL1.1E unicity -, App.J; App.L3A

Projection map, Ool.IG Proximity function, J.11; V.LJ;

VI.l.lD; V.L2A; V.3.IA; App.J; App.1.1A

boundedness from below of-, V.1.1F; App.1.1A

Pure differential, 0.2.2E

qa, III.2.lfA; IIL2.4E ql, ql, II.2.1A q-Iemma, L1.1A; VL2.2E; VII.J

estimates for -, L4.2B; 1.4.20; L4.2E; L4.2G

Quasiconformality, VLL4L Quasi-isometry, VLL4L Quasinormaloperator, VII.S.IA

first kind -, VII.S.IB second kind -, VII.S.IB

Quasirational function, II.5.2A

Rp-surface, V.S.2A Radial incision, III.3.2D; IIL3.2E Rado's theorem, O.2.IF Ramification index, V.2.SB; V.3.ID Ramified covering surface, O.l.lG Regular

outer - set, VII.l.2B - open set, VI.l.lA; VII.l.lA - partition, 0.2.1 D - point, VII.l.2A - subregion, 0.1.1 D

Relative - boundary, II.6.2E - cycle, Il.6.2E - n-chain, Il.6.2E

Removable, III.3.3B - singularity, J.6; IV.2.ID

Reproducing - differential, J.7; IL3.2Aj

IV. 1.1 A - kernel, VLL3D

Residue theorem, II.2.2B; 11.2.20 Riemannian space, VI.l.l A Riemann-Roch theorem, 11.5.10;

V.2.2B Riemann's mapping theorem, II.2.1D Riemann surface, O.l.lA; IV.L2A

bordered -, 0.1.1 H closed -, O.l.lL double of bordered -, O.l.lK open -, 0.1.1 L

Riesz - representation theorem, 1.4.30;

I.4.4C - Schauder theory, VILJ;

VIL2.2B; VII.2.3B Royden harmonic boundary, L4.4A

Se, SQ, IV.l.2A; IV.L2B; IV.L2G s(r, a), VI.l.lD; V.L2A; V.3.1A

joint continuity of -, App.1.lE Sario

continuity of - kernel, App.LIE - capacity, App.2.2A - kernel (cf. s(r, a) and proximity

function), App.J; App.l.lA; App.2.1D

- potential, App.J; App.l.2A subadditivity of - capacity,

App.2.2E Schwarz

- lemma, 1.4.20 - theorem, L4.3B

Schottky set, VLL4L Second main theorem, V.J; V.2.J;

V.2.2F Semiexact differential, II.S.le Semifield, VI.S.4A

harmonic -, VI.SolB principal-, VI.S.4A

SUBJECT INDEX 347

Sphere-like, VI.S.2D Simplex, O.l.SA Singularity function, .J; V II.2.1 B Slit mapping

annular circular -, III.3.2A annular radial-, III.S.2D annular -, III.3.IA circular -, J.4; II.2.3B horizontal-, J.4; II.2.1B; IV.l.1C radial-, J.4; II.2.3B vertical-, J.4; IV.l.IC

Span, J.5; II.1.20; IV.l.1B; VI.l.SO; VI.1.3E; VI.1.3H

Stream line, VI.1.6A Strong component, J.9; III.ft..SA;

III.4.3C Stokes' theorem, O.2.3C; VI.l.IC Subharmonic function, 0.2.1B;

VI.l.lD; VII.l.lB Subadditivity of Sario capacity,

App.2.2E Subregion

bordered -, 0.2.1 D canonical -, 0.2.1 D regular -, 0.2.1D; VI.1.3A regular adjacent -, V.2.IA

Superharmonic function, VI. l. 1 D ; VII.l.lB

Tangential part, VI.2.1A Temperature distribution, VI.1.6A

Tensor metric -, VI.1.IA - potential, VI.S.IB; VI.3.4B

Ua , III.2.4A; III.2.4B u(aO, 0'.1, ,,0, ,,1), II.6.ID; III.l.lO;

III.1.3A; III.2.2A; III.2.3A: III.3.IA: IV.l.lD

Unicity - of capacitary measure, App.2.3B - principle, App.J; App.1.3A

Uniquely continuable, III.4.SD Univalent function, IV.1.IC;

VI.1.31 Universal covering surface, O.2.IF Unstable component, J.9; III.4.SA;

VI.1.4L

Va, III.2.4A; III.2.4C Vague convergence, I.4.3B Volume element, VI.l.lA

Weak - component, J.9; III.4.SA;

III.4.3B; VI.1.4G - derivative, VI.S.IA - homology, II.6.2E; II.6.3E

Weyl's lemma, I.4.4A Weyl-Kodaira-de Rahm theorem,

VI.3.IA Wiener harmonic boundary, I.4.3B Winding number, II.2.1B; II.2.2C