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DISSERTATION FOR A DECREE OF

DOCTOR. OF SCIENCE

TOKYO IMIETROPOLITAN UNIVER.SITY

TITLE THE MODULI SPACES OF

ALGEBRAIC MINIMAL SURFACES

AUTHOR. Katsuhiro i\ I OR IYA

EXAMINED BY

Examiner in chief ^{7 C''} 7,4

Examiner f l ' n 1' '1

Examiner jA,./3

QUALIFIED BY THE GRADUATE SCHOOL OF SCIENCE

TOKYO METROPOLITAN UNIVERSITY

Dean

Date

..t.rt, „.:) 'j

THE MODULI SPACES OF

ALGEBRAIC MINIMAL SURFACES

IKatsullirc) MIORTYA

Preface

The purpose of this paper is to investigate the moduli spaces of possibly branched minimal surfaces via their Vveierstrass data..

A minimal surface in Euclidean space IR' is given by a branched conformal immersion from a Riemann surface to I1'' such that its mean curvature vanishes identically. The theory of minimal immersion has a. long history in the field of differential geometry.

On the other hand the theory of minimal surfaces is studied in sev- eral areas of mathematics. For example, the theory of minimal surfaces is studied in of t he theory of calculous of variations or partial differential equations since they are critical points of the area functional; in com- plex algebraic geometry since minimal surfaces can be represented by pairs of sections of holomorphic line bundles; and in complex function

theory (Nevanlinna theory) since a minimal surface has a holomorphic

Gauss map.

For these reasons, the theory of minimal surfaces continues to be studied actively.

However it is often necessary to assume that the minimal surface satisfies strict. conditions. Then, unfortunately, we can study only a small class of minimal surfaces. To advance the theory, we need to study more general minimal surfaces.

The author attacks this problem by constructing a theory of moduli spaces of minimal surfaces and by introducing new related concepts.

A moduli space of minimal surfaces is a set of minimal surfaces with the structure of a variety. The fundamental problem in the study of a moduli space of minimal surfaces is as follows:

I. Choose an appropriate set of minimal surfaces which becomes a moduli space.

2. Find geometric structures defined on the moduli space.

3. Identifying properties of minimal surfaces which are reflected by such geometric structures.

4. Investigate the topology of the moduli space.

Using a geometric structure on a moduli space, we can distinguish be- tween minimal surfaces. One of the author's goals is to obtain new

PREFACE

results on the (non-)existence, uniqueness, characterization, and coll- struct,ion Of minimal surfaces.

The author has studied the moduli space of minimal surfaces called algebraic. A minimal surface is algebraic if its generalized Gauss map is an algebraic curve. For example, a complete rninimal surface in a flat torus of finite total curvature is algebraic.

In this case, a pair of sections of holomorphic line bundles corre- sponding to a minimal surface, called its Weierstrass data, is a pair consisting of a meromorphic function and a meromorphic differential on a compact Rieman!) surface Al. The Weierstrass data satisfies a cer- tain condition on the divisors called the divisor condition. The divisor condition specifies the number, the order, and the position of branch points and puncture points, which are isolated points characterizing algebraic minimal surfaces. Any pair of a meromorphic function and a Illerolllorphlc differential on Al corresponds to a minimal surface from a universal covering of Al with finitely many points removed. It corre- sponds to an algebraic minimal surface from Al if and only if it satisfies the period condition. that is the real parts of the periods of n pairs of Illerolllorphlc different.ials on Al obtained from a pair of a meromorphic function and a meromorphic differential are certain coIlstants. Hence the moduli space of Weierstrass data is a subvariety of a space of pairs of a meromorphic function and a meromorphic differential on Al.

The subject of this paper is moduli spaces of Weierstrass data of algebraic minimal surfaces. It. consists of two parts. Part, 1 is about the moduli spaces as divisor spaces and Part. 2 is about the moduli spaces as linear systems.

In Part 1, we will discuss two general theories and one concrete construction of a moduli space.

In Chapter 1, we will discuss a. general theory of the moduli space of algebraic minimal surfaces in R4. When we fix a meromorphic differen- tial on a compact Rienlann surface Al,- we can consider the Weierstrass data as a triple of meromorphic functions on Al . We can consider the space .M of meromorphic functions with degree d on Al as a fiber space P of a space of principal divisors by projecting the zero divisor and the polar divisor from M to P. Hence we caall consider the spa.ce !/l as a complex analytic variety. Hence, we will study a moduli space of

Weierstrass data as a subvariety of triples of meromorphic functions.

We will prove that, the divisor condition and the period condition becomes a defining equation of the moduli space of Weierstrass data, and that, the moduli space becomes a real analytic variety when we fix the following data: a compact Rieman!) surface Al, the number and the order of branch points and puncture points, and the degree of the

PIIEFAC'Eiii

first. and the second Gauss map. We will prove that the moduli space contains a complex analytic variety and we will get a lower bound for the dimension of the moduli space.

Chapter 1 is based on the article [18].

In Chapter 2, we will discuss a general theory of moduli spaces of minimal surfaces invariant, under a screw motion. In this case, we can see that two meromorphic differentials correspond to the minimal sur- face by generalizing the \ eierstra.ss representa.tion. We will get. similar results to those above about the moduli space of data corresponding to the minimal surfaces from a universal covering of Al.

Chapter 2 is based on the article [21].

In Chapter 3, we will discuss a concrete construction of the moduli space of minimal annuli in R3 or R3/T, where T is a discrete group of isometrics generated by a translation. In the lat,ter case, we can see that, the generalized Gauss map is well-defined and hence an algebraic minimal surface is well-defined, too. A minimal annulus is an algebraic minimal surface with two puncture points such that Al = CCP1 = C U {oc}.

It is known that there does not, exist, an algebraic minimal surface in R3 or in R3/T without puncture point. and that any pair of a meromor- phic function and a meromorphic differential on CP' corresponding to a minimal surface from a universal covering of CP' with one puncture point must, satisfy the period condition. Hence we can consider the moduli space of minimal annuli to be one of the simplest examples of the moduli spaces.

We will obtain the defining equation of the moduli space of minimal annuli when we fix two puncture points 0 and x, the Gauss map z and its total curvature, where z is the standard holomorphic coordinate of C.

Chapter 3 is based on the article [19] and 120].

In Part 2, we will discuss one general theory and one concrete con- sttruct,ion of a moduli space.

In Chapter 4, we will discuss a, general theory of the moduli space of algebraic minimal surfaces in R3. We will prove that the space of triples of meromorphic differentials on Al which appear in the period condition becomes a. complex vector space when we fix I, the num- ber and position of puncture points the total curvature, and the Gauss nlap and restrict, the order of puncture points. We will prove that, the period condition becomes a system of linear equations and obtain a lower bound for its dimension. We will discuss the Puncture Number Problem, too. We call a positive integer r a puncture number for Al if

PREFACE iv A1 can be conformally immersed with exactly r punctures. The Punc- ture Number Problem is to determine the set of numbers of puncture points for Al. Let, us denote by P(Al) the set, of all puncture numbers for Al. 'I'hen it is known that P(11.1) D Q(q) :^ {r e Z r > q} for any g, where g is the genus of Al. We will prove that the set P(Ai) is strictly larger than Q(g).

Chapter 4 is based on the article [231.

In Chapter 5, we will discuss a concrete construction of the moduli space of minimal annuli in R3 or R3/7'. When we fix the total curvature to be —47r, the number of branch points to be 2 (counting multiplic- ity), and the puncture points to be 0 and oo, we will prove that the moduli space VV(v) corresponding to the moduli space of minimal an- nuli in R3/7'(i,) become a three dimensional totally real subrnarlifold of a certain six dimensional manifold. We will prove that a helicoid in W(0,0, t') is a point, which attains the maximal value of the scalar curvature of W(0, 0, v).

Chapter 5 is based on the article [221.

Acknowledgments. The author would like to thank Professor Y.

Ohnita for calling my attention to the study of geometric structures on moduli spaces. for his advice on Theorem 1.1 in C'hapter I, and for his encouragement. Iie also would like to thank Professor A.

Guest, Professor T. Kasai, and Professor R. tiliyaoka, for their advice and constantly encouragement. He would like to t hank Doctor NI.

Kokubu for a useful suggestion concerning the proof of Lemma 1.1 in Chapter 1, Professor K. Kenmotsu and Professor R. Kobayashi for their comments about, an early version of Chapter 5. and Professor NI. .1. i\licallef, Doctor N. Miyazaki, and Doctor A. Fujioka for their suggestions about Theorem 5.10 in Chapter 5, too.

Contents

Preface

Part 1. The moduli spaces of algebraic minimal surfaces as divisor spaces

Chapter 1. Minimal surfaces in Euclidean 4-space 1. Introduction

2. A modified Chern-Osserman theorem 3. Representation formula

4. Proof of the theorems

Chapter 2. Minimal surfaces invariant under a screw motion 1. Introduction

2. A multi-valued function and a meromorphic one-form 3. A representation formula

4. A variety of Weierstrass data 5. Examples

Chapter 3. Minimal annuli in Euclidean 3-space 1. Introduction

2. The Weierstrass representation

3. The moduli space of Weierstrass data

Part 2. The moduli spaces of algebraic minimal surfaces as linear systems

Chapter 4. Minimal surfaces in Euclidean 3-space 1. Introduct ion

2. Preliminaries

3. The existence of algebraic minimal surfaces Chapter 5. Minimal annuli in Euclidean 3-space

1. Introduction

2. A classification of Weierstrass data.

3. The moduli space of unhranched minimal annuli 4. A subnlanifold of Weierstrass data

2 3 10 12 18 19 20 21 25 27u

29 30 31 34

37 :38 39 40 43 49 50 52 57 60

5。(モeOlnetrV

Bibliograplly

oft,}ieillodul

CONrl'Elv'1'S ispaces

vi

Ei1 67

Part 1

The moduli spaces of algebraic minimal surfaces as divisor spaces

Minimal

CHAPTER. 1

surfaces in Euclidean 4-space

2

1. INTRODUCTION3 1. Introciuction

Let, M be a Riernann surface and f : R" a branched conformal

minimal immersion whose induced degenerate Riemannian metric d52 is complete in the sense that any locally rectifiable divergent path has

infinite length. Then, by modifying the Chern-Osserinan theorem [3, Theorem 1], we can prove that the total curvature is finite if and only if the Gauss map (1)./. is algebraic, i.e. Al is biholoinorphic to a compact, Rienlarin surface A19 punctured at a finite set. of points and (D .f. extends to a holomorphic map from My to Q„ _2(CC). We call a branched im- mersed II11Ililllal surface with finite total curvature an algebraic zfliru7/l(ll surface.

X. Mo constructed the moduli space of pairs of certain meromorphic functions on a compact Rienlann surface which give algebraic minimal surfaces in R3 by the \''eierstrass formula. Ile proved that the moduli space has the structure of a real analytic variety and that it contains a subset having the structure of a complex analytic variety. We can

see this work in the book written by Yang [41, Chapter 31. His idea of the proof is to clarify the conditions satisfied by the divisors of meromorphic functions.

In this chapter, by following this idea, we will construct the moduli space of the triples of certain meromorphic functions on a conlpa.ct Riema.nn surface which give algebraic minimal surfaces in R4, and give a lower hound for the dimension of the moduli space.

We fix a compact, Riernann surface AL, of genus q, a holomorphic (if g=0, then a meromorphic) 1-form Si on A1,,, integers k•, r (k > 0, r >

1), and an integer vector I31.,,. _ (J;; I;) e Tti >; Z' (J > 1, I; > 2).

We denote by AM = Al (211,,, I31,..r) the set of algebraic minimal surfaces f = (fl, f2, f3, f1): AI —> (1114, ds`') in R4 satisfying the fol- lowing conditions:

• The Riema.nn surface Al is biholornorphic to 111 — {puncture points };

• It is branched at k points with order J; (J = 1.... ,.) and punctured at r points with order I, (i = 1, ... ,r);

• fl + —1 f2 is not holomorphic.

We denote by FI) = f11)(A1,1, Si, 13k,,.. a, /3) the set of the triples (F, 1, (P2,) of meromorphic functions on AL, satisfying the following conditions:

(1.1) F 0, deg((P1)-x. _ (Y, deg(2)

(1.2) —(cP1). ((PAN.. + (F) + (Q) _ > , J1b1 — IiPi,

1i-1

1NTHOr)UCT10N 1

-1-

Re {/E11Fc1}= 0 for any -y E I11 (211„ {1~1,...,~~,.})

(1.3)

(a = 1, ... ,1);

where {h1; pi} are distinct points. We denote by (,,o) _ (Lp)„ -- ((p),,, the divisor of a Ineromorphic function on My where (:)11 is the zero divisor and (yo)- is the polar divisor. In particular, deg(c), = 0 for c E C*

and we define (0) = 0 and deg(0)- = cc for later use. Similarly, (w) is the divisor of a inerolnorpllie 1-form on Al y. We define

I;1 = 1 + 4 14p.', _ ^-(1 - , P2).

(1.4) F

3 = '1 -- 402, E1=--1(1+2).

We call the condition (1.2) divisor condition and the condition (1.3) period condition.

13y using t he result of Osserman [25], we will show the following lemma 03):

LEMMA 1.1. There is a bi,jecti e correspondence between A.\I( ,r, BA..r); and Li„„ FD(M71, S2 I3~.,. et. 3). where for (i and II E -1111(

.11,r, BA. ,. ), ( H means (3 and II are congruent by a parallel trans- formation in R.

The above lemma provides a correspondence between the space 1 \1~ ti of algebraic minimal surfaces in R_1 and the moduli space El) of triples of inerolnorphic functions on a compact Riemann surface.

We fix a, /3 e {0, 1, 2.... } U {ac} and let I be the number of 0 or ac in {a, t3}. Av.'hen r>,;'3 are finite, i.e. (F, y;l, )) E FI)(.1I, C2 BA,,., 3) are functions not identically zero, Our results are

THEOREM 1.2. If FI)(1111, S2, BA.,., rx, 3) is rronempt;!J, then it has the struct ure of a real analytic variety.

T1111',oftEm 1.3. If I)(lllr, Q, 13k.r, (.,/3) is rrorrernpty, then it 60/i- tains a subset which: has the structure of a complex analytic variety of complex dimension at least (k + 2a + 2j3 + 7) - { (11 - 1)g + 3r } .

When cY or /3 = 00 i.e. (pi or 2 — 0, our minimal surfaces are considered as branched hololnorphic curves in C2 which is identified with Ri in a certain manner (4). In this case, we can construct the

moduli space in a similar fashion above. Let in be the number of

0c E {r1r, /3}. Then, 1 U in, < l U 2. For ci E 7L U {oc}, we define cx' by

= 0 if ci = oo and by a! = rY otherwise . Using the results of Micallef

([16, Corollary 5.2] and [17, Theorem] ), we can prove

2. A MODIFIED CIIERN-Oss1,RMAN THEOREM

THEOREM 1.4. If ev or (3 — oo, then. the element of D(Itiq,

a, /3) corresponds to a branched complete stable minimal surfaces in 11V of finite total curvature. If I 'D is nonerrrpty, then it has the structure of a com.pler analytic variety of comple.x diinension at least {k + 2er' + 2/3' +2(3 — m)} — {(? — rrt)r + (9 — l — 27109}.

2. A modified Chern-Osserman theorem

In the theory of immersed algebraic minimal surfaces, the Cherli-

Osserman theorem [3, Theorem 1] plays an important. role. In this

sect ion, we shall modify it to apply to the theory of branched immersed algebraic minimal surfaces.

First, we shall define a singular Hermitian metric on a Rieman]]

surface (cf. [37], p.141). Let Al be a R,ieinann surface and (.1 a coor- dinate neighborhood of M. We define a (l, 0)-forni 71 of meroniorphic type on (' as a form rl = c-r~'hd for each p E (.', where z is a holomor- phic coordinate with (p) = 0, h. is a complex-valued smooth function with h(p) 0, and J1, is a.n integer. We call the integer .J, the order of rl at p and denote it by ordr,'ii. If' Jr, > 0 for each p e U, we call rta (1. 0)-form of holomorphic type. We write (ii) _ Epi i, ordr, rt and call it a divisor of rt. We say that ds2 is a singular Hermitian metric on AT if it is given locally as ds2 = 71 • 7-1, where rt - 0 is a (I., 0)-form of meroinorpllic type. We call ds2 degenerate at p if ordj, rt > 0, regular at p if ordr, rt = 0, and divergent at p if ordr, rt < 0. We note that ds2 is a Hermitian metric on Al if ds2 is regular for any p E At We call p E Al a singular point of ds2 if ds2 is degenerate or divergent at p.

We define the singular divisor S of a singular Hermitian metric ds2 as

the divisor of ri, i.e., S _ ~ ordr,(i) p.

Next, we generalize the Gauss-Bonnet theorem. Let Al be a H,ie- mann surface, ds2 = rt•ita. singular Hermitian metric on Al with finitely many singular points, d.1 the area element of ds2, and K the Gaussian curvature of (Ts'. We denote by U an open subset. of Al such that. its closure U is compact and that, the boundary of C consists of finitely iany smooth Jordan curves /3;(i = 1, ... , in) whose orientation is cho- sen as (' lies on the left-hand side. We define k,r.; to he the geodesic curvature of ;'j,. We assume that there is no singular point on each /34.

Then we state a generalized local Gauss-Bonnet theorem as follows:

LEMMA 2.1. Under the above situation, W( have

ur

(2.1)KdA = 27r (.(C,') + deg (S (`)) — k,r.ids,

r; -i.rjr

where x(U) is the Euler number of U

2. A MODIFIED (TIER N-OSSEIiMAN "I'HFOlil•;M(i

PROOF. Let. {q1, ... , q,.} he all the singular points of ds2 contained in U. Since ds 2 is a. singular Hermitian metric, we can write 9/ _ ..:Lh„(::)d.r On a neighborhood of q„((.1 = 1, ... , c), where is a hole- niorphic coordinate around (Jr, with :,-2(q„) — 0, h„(.~) is a complex- valued smooth function with h(0) 0, and .1„ is the order of rt at q„. We denote by 1J((h,, I?) the set {‘ti < R}. We choose a. suffi- ciently small R > 0 such that, 1)(qi,, I?) f 1)((if, II.) — 0 for I I. Let /1,,,,.r; _ 0I)(q„, H) where its orientation is chosen as I)(q„, H) lies on the left-hand side. \\'e denote by k(,,,,,,,r,> the geodesic curvature along 11(1,,.r,,, and I ' R = 1 ' \ U„. I I)(q„, R). Then, by the local (;a.iiss-f3onnet theorem, we have

(2.2)A”d.-1

.L.

.i1 .(L1

J~yn .li

We express k,1,,,,.r,ds explicitly. Let ds2 _ 01 ' : 01 + 02 `--;” 02 be the the singular Hermitian metric, where 01 5 02 is a oriented ort honorinal frame.

We define c; to be the dual of 0` (i = I.2). We denote by L,(.; the Levi- Civila connection form satisfying dO' _ -c.~ A0-1,,,,,i --;,,.'! (1, j = 1, 2).

Let. - be the curve in .1I such t.hat. (1-„,,/ ds = 1 c'1 + c12(-'9,where .5 is the arc-length parameter and v is the vect OF field normal to (h, /s denoted by 1/ _ —e2c1 + elc, We denote by k,, the geodesic curvature along -.

Then we have

(2.3) kgds = [(de + e2i.4).) )e1 + ((12 + 1L41 )C9_ .r'.

Introducing polar coordinates (r,11 to a neighborhood around q,,, we can express the singular Ilerrnitian metric ds2 in the form ds2 = 1,2.1,,

ir„ 2(dr :: dr + r22dt dl). We assume 01 = r” h„ dr and 02 _ r'- ' 1 h„ di. Then we have

r. .r„ 1) 1, (.1., , 1) 0 (2 .-I)c1 — ---,C9_ --- --- .

Ii.,,0rh„ Ot

In terms of polar coordinates, we can express the curve it in the form /1 y,,.1t = (H, I), t [0, 27r]. Hence, dp„,,,.n/ds = ('•). Since

(101 =_1 0 logh„ dr A 02, (2.5) I~~t

(102 _ — (,.L1. -A- 1) ± r' O

r--- di n 0 we obtain

(2.6)~,~_ 1

0 log

fr„---dr _ ((.J„ + 1) + r°log h” di.

2. A MODIFIED (;I-IER.N-OSSEIii\-IAN 'I'III•;()REM 7

`I,1111s.

_031(og/1,

(2.7)k'Ir 1,.1rd.5=(.I„+ 1) +H(It. Or Hence,

IIt.

h(I.•1=27r\(L'TIr) — >k'y.;ds

i

(`~ri _. 1 II'

c---

^ 27 0 log h„

+> _.., ((Ja+1)+R—

tit

(2.8)III

;i

=27r( ((I) + deg(~~' (r)) — Ek~,r •;(./s

J.1•,

•27 'fl ogh.„

+>Rdt. '~~Or

/I i

Since ) log h:„ /Or is bounded on I)(ct„, H), we have

(2.9)liiii rr-->u(I?~T r) log jhl~dt = 0. ,Or Thus, as R tends to 0. we obtain

II,

•

(2.10)It d.-I 27r (x(1.') ± (leg (SI(•)) —\k,,;ds.

I111nlediately. we also obtain

COROLLARY 2.2. If :II is a cornpact Ri(-'inann surface with a sin- gular Hermitian metric (/.522, then we have

(2.11)K(LI

J1

The following lemma is an analogue of the theorem of Huber [9.

Theorem 13] in the case where a singular I-Iernlitian metric with finitely

many degenerate 1)oint and no divergent point. is equipped on a R.ie- 1na1111 surface.

LEmmA 2.3. Let 11I be an infinitely connected Ri.errrann .su face, ds2 = 11 • fi a singular Hermitian metric on 111 with finitely many de- generate points and no divergent point. If ds2 is complete, then

K d:1 =+Do,

Al

where h = Inax{0, —K}.

2. A MODIFIED (II ERN-( )SSERMAN TIIFORFM8

PROOF. We prove that if .IA/ K < --oo. then, (Is' is not complete.

\Ve denote by { (-; } all exhaustion of Al, i.e., a sequence of open subsets of Al such that C (/, for i < j, that the closure ('; of each ('; IS cotnpact, that the boundary of Ili consists of' finitely Illau smooth Jordan curves d< = 1, ... ,'in.;), and that Lx I _ AI. A'Ve choose the orientation of each /3;i as (', lies on the left-hand side. Let

Al \ (1—) [Ti','S~whereS~—~jWe ^—tJ ,+ .r—r,l assume that all the singular points {b„}(a, = 1, ... , e) of ds2 on Al are contained in the (II. ft

Lemma 2. 1 , we have

nt,

(2.12) Kd.1 = 27r (.k((i;) + deg (s)) k',,.r;ds.

(' I •

lience,

+n ,

(2.13) — k d.-1 + 27r (i (('i) + deg (S)) — k

r,

.i I , ri,.r

As i tends to ,x:, the left-hand side of (2.13) tends to Therefore, for sufficiently large 1,

In 1

(2.11)h1.11d. < —2 K d;1.

.i- I ~,, v

I-lence, there exists J, c > 0 such that

(2.15)k'y.lr••s=

1/.11211

We Call choose a Jordan curve cS in S21,1 llomotopic to 'I1r.r and

(2.16)K ' dA < €,

where l' ' = max{0, K } , (311, 6) is a domain surrounded by 1.1 and 6. The following two lemmas are proved in [9, p.62, Lemma 6] and [9, p.23, LeInIlta 1:

LEN1 IA Under the above situation, there exists a number C” >

0 satisfies the following property:

• Per any integer i, there exists a rectifiable curve a; : [0, 1) -> 1I .such that cr;(0) 6, limt ,I cr;(t) E ui';, ft, ds < C.

LEMMA 2.5. l'i'e denote by Cl a doubly connected region in S2. Let 1', ; denote the two boundaries of SZ, and S20 the simply connected open set containing and surrounded by F. Assume that th.ere exist a.

sequence of rectifiable curves {cr„ }, a„: [0, 1) -f S1, a compact subset

2. A M'IODIF1I1_) C'IJERN-O5S11INIAN 'fHEOHFNI`1

K C C20 and a number (' > 0 such that they sa:ti.sfit the following conditions:

• For each a,,, Im{a,,.}iiK 0;

• For any compact subset L C S i U,, hn{cr„ } is not contained in

L;

• J ds < C for all 11.

Then there exists a locally rectifiable divergent path cr iii S2 such that J~ ds <•+oc(111(1that lilulr1a(t) F.

IRv Lemma 2.4, we obtain a sequence Of curves in C2/ J, compact, set ( and a number C > 0 satisfying the assumption of Lenlnla 2.5.

Ifence, t here is a locally rectifiable divergent path a : [U, 1) - 11 such that J' ds < Therefore, ds2 is not. complete.^

Now. we can modify the Chern-Osserman theorem as follows:

PrioPoSITIc)N 2.6. Let f : :11 > R" be a branched conformal mini- mal immersion such that the singular Riemannian metric ds2 induced by f is complete. Then. the total curvature is finite if and only if the Gauss map (I )f is algebraic.

PROOF. First, we observe that we can extend (1) f over all branch points. Indeed, a branch point. b is locally a common zero point of

holoinorphic functions p F--> (0f"/0z)(p). (a = 1.... . ii). Hence there exists the minimum of orders of their functions at a branch point b, which we denote by h•. AV'e define

(2. 17) (1) f(b)_111f 1 (b), 1~c2f 2(b)....,1kOf (b) .

'i hen f becomes holomorphic at b. AV'e also observe that ds2 is a singular Hermitian metric on M with no divergent point in this case.

Indeed, we have locally

dS2 _ 9~2 ~« 'd,d~ (2.18) O

R 1 z

since op " ,/Oz (a = 1, ... , TO is lloloinorphic, we have op ' 0.:

h„(, ), (a = 1, ... ,n), where 'u,, is a nonnegative integer and b„ is a holoinorpllic function not equal to 0 at 0. Thus, d.s' _u/1(z)dz, • dz, where u = nlln{'a,, a = 1, ... , 11 } is a ilonnegative integer and h is a local real-valued positive smooth function. AVhen we set rt z'” v (z)dz, we see that ds`' = 1t - rt is a singular Ilei'niitian metric with no divergent point_

We assume that the total curvature is finite. Then 11 is finitely connected by Lemma 2.3. Then, in the similar way as the proof of

3. P EPIiESEN'r! rIoN FORMULA I,A1 ~1

the C'hern-Osserman theorenl([3], Theorem 1), we can prove that Al

is bihololnorphic to a compact Hienlann surface A'q punctured at fi- nite points and that 4)j is extended to be holon-lorplric at all puncture points. Thus f is algebraic.

Conversely, we assume that (1)f is algebraic. Let !tLI be the com- pact Riemanll surface on which cl) f is extended to a hololnorpllic map,

{b1 , b .} the branch points, {pi, ... , p,. } the puncture points. Then ds' is a singular Ilermitian metric on A/1 degenerate at b (j = 1, ... ,k) and divergent at pi(i = 1, ... , r). By Corollary 2.2, we have

(2.19)K d.ld,2 _ 271 (x(11,1,1) + rleg(S)) .

A l,,

Since both (Al,,) and deg (S) are finite, the total curvature is finite.

3. Representation formula We shall prove Lemma 1.1.

We will write

(3.1)(() _ E IIIin(ord,, (" )p,

AL,

where the minimum is taken over those a for which (" is not identically zero. Let CD = C'D(M11. l3k,,_, o, 3) he the set of all the quadruples ((1, ; ^1 of Illeromor )hic 1-forms on Al,, satisfying the following conditions:

4

(3.2)(1 —\/-1(`' (3 J(U -.~IG=0;

"

(("+N/-1(-1s+_11

(3.3) deg=cr, —^l~r~1—v (leg—~v/_3; 1(2 x.

(3.4)(() = E J ibi — El;p;

1--1i. -1

Re (" = 1 l for each 7 E II l Ply ,, — { p 1, ... , p, . } ) ( 3.5)

and each a(a=1,...,4).

3. REPR.ESENTATION FORMULA Then, by the relations

(3.6)

CI 1' f2, f3, f4) f"((1, (2, (3, (4)(z) = Re

Of"

=dz;

az

("dz (a = 1, ... , 4),

11

we can define a bijective correspondence between AM(111q, Bk,r)/

and [L41 CD(Mq, Bk,,., ce, /3). Indeed, it is clear that an element of AM/ ti corresponds to an element of some CD, and an element of CD corresponds to a minimal surface branched at k points with or- ders Ji and punctured at r points with orders I. Let (f1, f2, f3, f4)

be a minimal surface corresponding to an element of CD. For a punc- ture point p of order I, we take a local holomorphic coordinate z such

that z(p) = 0. Then the singular Hermitian metric ds2 induced by f

becomes as follows:

(3.7)ds2 =h(~) dz •dz,

z~

where h(z) is a positive smooth function. Let a(t) = x(t) + ly(t) be

a smooth locally rectifiable curve tending to p as t tends to oo. Then, we have

(3.8)do2= dt( h(a(t)) • (dx/dt)2 + (dy/dt)2 x(t)2 + y(t)2)' •

Hence, 11 do-I dill tends to oo as t tends to oo. Thus a has infinite

length, and we see that the induced metric is complete. Therefore,

(fl, f2, f3, f 4) gives an element of AM. Hence, AM/ is in one-to- one correspondence with U CD through the relation above.

On the other hand, it is standard to check the relations

F'((1, (2, (3, (4)1/43—~—1c2

SZ '

(3.9)1((1,(2,(3,(4)(3 + V-1(4 (1 _ 1(2 ^ (1

(3.10)"(F',~pl,SPZ) =1E"FS2(a = 1,..., 4),

2 define a bijective correspondence between CD(Mg, Bk,,., a,13) and FD(

M9,1, Bk,,., cx, ,3) (cf. [25, Section 4] or [8, §3, Remark 4]) . Then, for each ((1, (2, (3, (4) E CD and its corresponding (F,1,2) E FD, we

4. PROOF OF THE THEOREMS ¹²

have

(3.11)(C) = —(V31)x — (70 2) + (F) + (SZ).

We have finished proving Lemma 1.1.

4. Proof of the theorems

We shall prove Theorem 1.2, Theorem 1.3, and Theorem 1.4. We

fix Mg, S2, k, r, Bk,,., a, and j3 as above. We denote by Div`+(Mg) the

space of effective divisors of degree d on Mg. We observe that D =

D(Mq, Bk,,., a,13) = Div (M9) x Div+(M9) x Div`'/(M9) x Div+'(Mg) x Div+ (M9) x (II 9), where J = jk— J; and I = ~~ Ii, has the

structure of a compact complex manifold of dimension J + I + 2a' +

20' (cf. [4, p.236]). Let L = L(Mq, Bk,r, a,13) be the open subset of M9 x • x M9 (k + r + 2a' + 2/3' times) consisting of the elements (bi; p; s6; td; XE; yE) such that {bi; pi} are distinct points and that {so}n {t6} = {xE} n {y,} = 0. We will see that {so; to}({Xe; yf}, respectively) corresponds to the support of the divisor of (,o1 (,p2, respectively). Let DAD'(Mg, ft B,," a, i3) be the set of D's defined by

(4.1) ^D=

where D1, D2, and L ditions:

(4.2)

(4.3) When ((I,

(D1i0,0) ifa'=l3'=0

(D1i D2, 0), if a' 0 and 0' = 0 (D1,0, D3), ifa'=0 and /3'0 (D1, D2, D3), otherwise,

are divisors on Mg satisfying the following con-

(YIjr

D1 = >ljbj —>Ijpj+t6 +EyE — (ci),

j=1i=16=1E=1

cr' cr'

D2 = ss — Et,, D3 = EX, — EyE,

6=16=1 E=1 E=1

where (bj; pi; s6; t6; xE; yE) E L.

, (4) E CD and (F, cpt, cp2) E FD are the elements cor- responding to each other such that (cm) = D2 and that ((p2) = D3, we

have

(4.4)(F) = D1,

by (4.4) and (3.11). We will prove the following lemma:

LEMMA 4.1. The set DAD' has the structure of a complex analytic subvariety of D with the complex dimension k + r + 2a' + 2/3'.

4. I'ROOF OF TI-IE TIIEOREMS13

PROOF. Let C — C(111,,, Bk..,., cr, /3) be the subset of D consisting of

the elements

krrec cY l~, Ii'

(.1.5) Jjbj li]~ia 5c~a tcy, C a ~~E

j: 1 i 1cS 1 cS 1 E 1 r 1

such that (bj: pi; s6; t6; xE; yE) E L. Then C is an analytic subva.riety of Dand

(4.6) din~~C—k+r+2cY'+2/3'.

Clearly, we can define a. bijective correspondence between C and DAD'.

We have thus proved Lemma 4.1.^

Let D.4D(Al9, Q, 13k..,., a, /3) be the subset of DAlY such that each element. consists of principal divisors on AI9.

LEMMA 4.2. 'The set DAD becomes (1 complex analytic s'ubvariety of DAD'. If DAD is nonempty, then

47) dim: DAD > k + r + 2a' + 2,3' — (3 — 1)g.

PROOF. Let J(111,,) be the Jacobian variety of /9 and 11: Div(A!9) J (AL,) the Jacobi map. We define u : D. I D - J (A1,, )'3 by

(4. 8) i(D)

We note that deg deg Di = 0. deg D2 p. 225), D E 1)ivu(

Thus, DAD v-' the complex struct subvariety of DAD'.

dimDA ( 4.9)

We assume a' .4D (AIq, Q, 131,,,1,

/3) he the set of triples (F, (pt, (,o2) of meromorphic functions that ((I' ), ((P1), ((p2)) E DAD,cleg(c21),= o, and deg((F2) define 7): AD-*DADby the projectiony~2)_((J

(p2)) and we set V = DAD — {all singular points}.

(u(D1). 0, 0), if o' =3' = 0 (u(Di), u(D,), 0), if a' 0 and;3' = 0 (u(D1), 0, u(D3)), ifa' = 0 and d' 0 (u(Di), u(D'), u(D3)), otherwise.

= deg D9 = de; D3 = 0. Indeed, by (4.1), leg D3=0 is clear. By Abel's theorem (see [4],

is a principal divisor if and onlyif u(D) = 0.

0, 0). Sinceholomorphic with respect to

ure ireinduced as above, DAD is a complex analytic By the definition of 1, we also have

din> ' DAD' — (bIn (J(1,1))3 -1

El

=k+r+2(Y+2%3'—(3-1)g.

similar. Let,

on x.

!,]

_ (Y1)

such

4. PROOF OF THE THEOREMS1.1

LEmmA 4.3. The set AD has the structure of a comple:r, analytic variety, and then 7t : 71-1(17) —> V becomes a holomorphic principal (C*) bundle. If AD is n.onem,pty, then

(4.10) dinm _ ,AD > k -b r + 2( + 20' — (3 — l)g + (3 — nn).

PROOF. Assume (F, y~i, (p2) E AD. Then ((F), (oi), (pp2)) _ ((w1 • F), (w2 • (pL), (w3 cp.))) for ally (wi, w2, w3) E (C:*)3. Hence (C*)3 acts on AD. Moreover, we can easily see that (C*)3 acts on rt 1(V)

To simplify the proof, we prove the claim for only one of the factors corresponding to the functions not vanishing identically. We locally induce a complex structure from DAD and prove that this complex structure is globally defined on AD.

First, we assume that, g > 1. Let 19 be the Riemann theta functions,

and D = ~li bi — pi a divisor of My with u(D) = 0. The following lemma is proved in [24, Chapter 2,

LEMMA 4.4. There exists a constant A in Cil depending only on the choice of the normaliz;ed basis for the space of holomorphic 1-forms on 14 and satisfying the following conditions:

• For a point v = (v1,— , .U, i) in (11I1)9 -1 such that {b1, , b,t, Pt, ... , p,l} n { vi , ... , vy - 1 } = 0, the mapping h,,: V x ALI

C U {:x} " CP1 defined by

lli-119(~—~jlu(rj) + u(ti) — u(bi))

(4.1.1) lr,,(D)(z) = ,~~` y 1

i9 (!, — LJj -1 u(vi) * u(s)11(pi))

is a m.erornorphic function on My such that (h,.,,(D)) = D.

We define h,,,(0) = 1. We fix such a v for each divisor B in V and denote it by 'V B. Then h.,,B(D)(z) is locally a holomorphic function with respect to D. Assume that UB is a sufficiently small neighborhood of

B in V. Then (h,,,, (D)) = D for D in L.

We define

(4.12)TUB : // 1 (Ur) Un x C*, f H (.f ),f

lr,,, ((f )))

Then this is a bijective map between 71-1 (UB) and (1B x C*. Hence we can give 71-1 (Li B)a complex structure c(cB). If H is another divisor and ('B n UH 0, then UB n UH has two complex structures C(vB) and

c(ji). But

(4.13) Tu o T1 1 ,W) = (I), (guB.ul, (D)) • w) , guB.0 h •

4. PROOF OF 'THE THEOREMS1 5 for each D E U13 n UH and .,L' E C*, and gr,13 (.T is holomorphic with respect. to D. Hence the two complex structures are compatible. in the same fashion as above, this complex structure is independent, of the

choice of {vm}. Therefore, we can induce the complex struct,ore c to rl_r(V), where rl: (7l i (V), c) V and TuB : (71 ' ((f 13), c') ('13 x C*

are holoniorphic and the following becomes a. commutative dia.gram.

f3~

it_1(un)>U13C

x (4.1'!)

(_' B---) (r13 We also have

(4.15) Tu (D, lt'l'u?2) = 'W• W9 • hopi — T1'1 (.1.), 1) • u'2•

Hence, we can give rl: (11 I (l'), c) - V a structure of a holomorphic principal (C* bundle. If 13' is a singular point, of DAL), then B' x CC*

is a singular locus of 13' h: (C*. Thus we can give AD the structure of a complex analytic variety. Since the number of components AD is 3 — r7z, we have

(4.16) dim AD > k+r+2o'+2F3' — (3 — l)g + (3— rrl).

In the case where g = 0, we can prove the lemma in a similar fashion above only by taking

~l (4.17)fl,1(—b) tt

[1(

instead of h,,B .^

Now, we shall prove our theorems. We note that D(11 , C2, Bk_1, o, 13) consists of all the triples of nieromorphic functions in AD satisfying the period condition.

PROOF OF THEOREM 1.2. We note that in. = 0, rx' = c~, 3' = '3 in this case. We fix ( , p1o, (p2o) E FD and denote —((pio) x — (90), ^{- +-}

Letl

'I/9g ,. 1 } be a basis for Hi PI {p1o, ... , pro}) such that {ryi, ... , -yzrr}

is a basis for Hn (Me,) and that 7994i is a simple closed curve around p.lo (i = 1, ... , r — 1). We denote by H o a neighborhood of (Fa, (pio, 20) in AD such that for (F, cp2) r is still a basis for H, (11,1,1 — {p1, ... , p1 }) where p;, are puncture points of (F, 4~1, (p2). We define holoniorphic functions A `: C (i = 1, ... , 2g+r-1, a = 1, ... , 4)