(5.2) (5.3) where R U(l, zo),
contains
5. EXAMPLES
ExANIPLE 5.2. If u = 0 in (5.1). (5.2), and (5.3) (ge, ..rio, [col) becomes a Weierstrass data for a. Scherk' Hence, W(T, zo) is nonenrpty and contains a complex
niension not less than —1, which is not useful.
REMARK 5.3. In [7], we can see a family of one
which contains a Scherk's saddle tower.
98 , then the pair s saddle tower.
variety with
di-real pararneter
M inima
CHAPTER .3
1 annuli in Euclidean 3-space
29
1. INTRODUCTION3~1
1. Introduction
The purpose of this chapter is to give a moduli space of a certain class of nonplana.r algebraic minimal surfaces in R3 and in a translation
space R3/T(v). A translation space R3/T(v) is defined by the quotient space of R3 by the discrete group of isometrics T(a) generated by a translation by -v = (z'i, v9, v.r) E R3. A possibly branched complete conformal minimal immersion X : Al --> 1 3/T(r) from a R.ienlanIi sur-face Al is called an algebraic minimal sursur-face if it is of finite total
curvature.
The notable property of an algebraic minimal surface is that the
Riemalln surface Al is compactified conformally (cf. [18] and [22]).
Hence. this surface is denoted by (X. 11, R'1/T(u)) in this chapter,
where a,l is the Rieinann surface compactified from Al.
There are two kinds of methods to study the moduli space of al-gebraic minimal surfaces. One is to investigate certain operators char-acterizing minimal surfaces. The other is to investigate the moC1u11 space of the data corresponding to minimal surfaces. For example,
J. Perez and A. Hos [27_, 128_ proved the moduli space of
nondegen-crate, unbranched, properly immersed algebraic minimal surfaces in R3 with embedded horizontal ends becomes a real analytic manifold by applying the implicit function theorem to a certain real analytic
map. IIi [411, the Mo's construction of the moduli space of algebraic
minimal surfaces in R3 with fixed topology is appeared. This is doiie by investigating the moduli space of t.heir Weierstrass data. R.
Kus-ner and N. Schmitt [12. proved the moduli space of algebraic minimal surfaces of genus zero with embedded planar ends becomes a certain complex space by investigating the moduli space of their spinor data of the spinor representation.
The interesting problems about the study of moduli space are to had geometric structures defined on the moduli space and to obtain results about existence or characterization of the surfaces by the geometric properties. The above studies obtain such results.
To investigate the geometric property of the moduli space in detail, it is necessary to clarify the precise parametrization of the moduli space.
This is possible under the formulation by X. Mo when the compactified Riernarin surface is (CP1.
In this chapter, the moduli space W = W(13, P, (1.~%Z, v) is studied by the X. Mo's method where W = W(13, P, d l z, 1') is consisted of the Weierstrass data of algebraic minimal surfaces (X, CP',R1/T(.i.'))
which have 0 and oc, as puncture points of order I', and whose Gauss map ramifies at tire puncture points with the same order and the total
2. THE WEIERSTRASS R.EPRESENTATION31
order of branch points is B. The terminologies used above are described the following section. Let U be an open subset of R4'1+2 defined by
(1.1) U{ (eeeecc) 1,1~1,2~,2J,1,2J,2~1,2 E R4'/-
e2J,1 + e23,2 0, C1 + C2al'
Then, the set W is considered as a subset of U. Moreover, the following is proved:
THEOREM 1.1. If the moduli space W = W (B, P, dz/z, v) is non-empty, it is a nonsingular real algebraic variety of U whose dimension is4J-1 ifk<J-1,4J+1 if k > J + 1, and 4J if k = J 0.
2. The Weierstrass representation
In this section, a brief explanation of the Weierstrass representation
for algebraic minimal surfaces used in this chapter is given (cf. K. Yang [41], D. Hoffman and H. Karcher [6], and K. Moriya [18], [22]).
Let (X, CP1, R3/T(v)) be an algebraic minimal surface and M a
Riemann surface which is the domain of X. In the following, when
v (0, 0, 0), assume that there exists the minimal immersion X : M -*
R3 from a holomorphic covering M of M which make the following diagram commutative:
3X
~1 ? R3
3~
JyH
~1 ---> R3/T(v)
where II is the natural projection. In the case where v = (0, 0, 0), assume that X = X.
Let z be a local holomorphic coordinate of M. Then, local holo-morphic differentials 41i := i = 1, 2, 3 are extended to a merornorphic 1-forms on CP1. Hence, klfz := is denoted by W :_ (3X7/3z)dz in the following, where z is a local holomorphic
coordinate of M. The original surface is obtained by integration:
(2.1)Xi(z) := Re ~i+ Xj(zo), i = 1, 2, 3,
z()
where z0 is a point on M. From the conformality of X, the relation
(411 ®>P1) + (4'2 ®4'2) + (4'3 ®W3) = 0 holds. For 111 _ (411, , 3),
2. THE WEIEHSTR.ASS REPHESENTA'I'ION32
the divisor (tP) of T is defined by
(2.2)(kP) :_ min ord1,; • p.
pE Pi
Assume tha.t. (T)~j1B~• b •~11P;,p;,13,> 0, I> 0.
DEFINITION 2.1. A point b1 is called a. branch point of order B,
and a point p.; a puncture point of order Pi. The sum B :_ E .l3.- is called the total order of branch points and P : 1 Pi the total order of puncture points.
The total curvature of X is equal to 27-(2 + B — P) (cf. K. Nloriya [18, Corollary 2.2]) .
In the following, only the case where r = 2 is considered. The 1-
dimensional complex projective space (CPi is identified with C U {oc}.
The two puncture points may be assumed to be 0 and oc since any two points in CPI is transformed to 0 and x. simultaneously by a Mobius transformation. Hence, it may be assumed that (T)
13 - • b - — J) • 0 — P,•(pc. Letiabesimple closed curve around
-1J J
0 whose orientation is chosen as the origin lies in the left side. The
number s; := L = 277\/-1 Res(0; kli;) is called the period of kPi
around 0, 1 = 1, 2, 3, where Res(0; T;) is the residue of T; at. 0. Then, (Re sl, Re s2, R.e 53) is equal to v = (v1, iv2, v3).
DEFINITION 2.2. The condition
(2.3)(P1 (. 'P1) + (Wi +J 22) + ('P3 '3) = ~) is called the conforrnality condition. The condition
(2.4) (T)=>2,R1•hi—[1•0—P •'x,
is called the divisor condition,. The condition
(2.5)(Re .s , Re s.,, Re s3) = (i'1, v2, v3), is called the period condition.
Conversely, if three lneronlorphic 1-forms ;, i = 1, 2, 3, satisfy-ing the conformality, the divisor, and the period condition, that is
(2.3), (2.4), and (2.5), are given, then an algebraic minimal surface (X, C'1'), R3/T(c)) is obtained by integration (2.1). If the other base
point. ZO in the integral is chosen, the image of the immersion shifts by a translation.
2. THE WFIERSTRASS RLPRESENTA'TI()N33
Let Q be a. meromorphic 1-form dz/z, such that. (c) _ —1 • 0 — 1 • oo. Then, three tneromorphic 1-forms ~~~, i = 1, 2, 3, satisfying the confornlalit.y, the divisor, and the period condition is equivalent to a pair of rational functions (q, h) via the following
til/3 'P3 (2.6) (q, Ii) _ --- — ----
1 1 1~.
N/- y -Q.
There is the following relation among CO, (Ii), (CO, and (kit):
(2.8) ('P) = —(g)0 — (g) + (h) + (Q).
Since the relation (2.1) holds, the relation (2.8) becomes as follows:
+ (h)
(2.9) _~
B1 •b.7—(1(1— 1)•0—I)•oG .
j 1
DEFINITION 2.3. The pair (g, h) of meromorphic functions on CPI associated to an algebraic minimal surface (X, CP1, R3/T(c)) is called
the tifeierstrass data. of X.
Fix a. holomorphic coordinate z on C centered at. the origin. If q ramifies at each puncture point with ramification index k — 1, then
it may be assumed that q = a zk for some a E (C_ :_ C — {0} and k>0,keZ. Then, (g)=k 0—k•oo.
Assume that P, = Px. Then from the divisor condition (2.4), the divisor of h become as follows:
(2.10) (h.)=YB.;•bj—(P—k-1)•0—(P—k-1)•pc,
where P :_ Po. Hence, = c~;1(z — b )/ 1' k t for some c E C' . Since h. is a meromorphic function on CP1, the degree of (h) is
equal to 0. Hence, B — 2P + 2k +2 = 0. Thus, A is even. Let. B = 21.
Thenk=P—J-1>0 and P—k;-1=J.
The Weierstrass data
/ 2.1
(2.1 1)(q,/i.)_ ((az)1)
-.1 1,ca Ha-1N./b /a)
3. TIHE N1ODUI,I SPACE OF WEIERSTRASS 1):VI'1A3-i produces the same algebraic minimal surface as
~1
(2.1 ~)(y, h) =tih-.~-1('IZri(tihi)
since the difference of these data is caused by only the difference of the choice of a hololnorphic coordinate on C' fixing 0 and x.
Let A = A(13. P, v) be the set {(X, CP', R'3/1 (r;))} of algebraic minimal surfaces whose puncture points are 0 and ,DO of order P, whose total order of branch points is 13, and whose Gauss map ramifies at, the puncture points with the same ramification index. Let, .1' 1 for X and Y in A mean that X + _ I. for sonic :r e R3. Let 1N = W(B, be the set of Weierstrass data corresponding to algebraic minimal surfaces belonged with A by (2.Ii) and (2.7). Sum-marizing the above discussion, the following leninia has been proved.
LEMM.A 2.1 There e:iists a bijective corresponding between A(13, P, t'),/ and 1/1)(13, P, dc,l c, r). The set W(13. P, (L; c. r') is considered as the set {(g, h.) }of pairs of rational functions such that
(2.13) g—c1'-.1 r,h.=c'f,-1(-b;).
(2.11) Re Tr, Re 1119,Re 41:3 = (i's. c>, r'3).
where e, 01, ... .1.)B e .
HEiAIth 2.5. 'the rational function q of a Weierstrass data (g. h) of f is the stereographic projection of the extended normal Gauss map of f . Hence, Weierstrass data (exp[/-101g, h), 0 e R produce the same algebraic minimal surface as (q, Ii) rotated by an angle 0 around the ,r:3-axis. The Weierstrass data (y, nth), in e R` := R— {0} produce the similar algebraic nriiiiiiml surface as (y, h) transformed similarly by the center of' similitude the origin of R3 and the ratio of similitude iii.
3. The moduli space of Weierstrass data
In this section, the proof of Theorem 1.1 is given.
PRooF O1" 'THEOREM 1.1. Let the set 7Z = 7Z(13, P) be the set of pairs of rational functions (.q, h) where
C (3.1) y—c1.1- I ,lr= fl,1(- b )
3. THE MOl)iJI.I SPACE and 13 = 21 Then, 7Z is identified ide11t lhcat ion
(3.2)(9, h)
where t he space of posit ive divisol I)iv2:1(C' ). The space 1)iv21(C1
identi heat ion
2./
>=s,
(3.3)(1 • bi) ~ (al
.i1
where a; (bi) is the elementary sy i = 1.... , 21. After this, ai(bi) is ficat ions. 7Z is considered as (C2./ 1
On the other hand, the set 7Z R" 2 defined by
(3.1) {(e'1.1,f'I.2.... .C'9.1,C'2.9,C'r,(
Thus, 7Z is identified with U.
Let T: t C3 he the period r
(3.5)T(9, h) _ 4
I,llen
I-te Ti (9, It) = — 7T(c1 (c.1 k.
(:3.6) R.eT9(y,Ii)_—<<(c1(('a k.
Re T3 (11, h) = — Z7r(c1 C'.1.2 -+
tivhere k = 1' — .1 — 1 > () and and e;l = 1,~:,.2_0ifi-0. '1 the element (9, It) of U satisfying these equations the following e(lna
<< l := 27(c9('.i hr + c (3.7) C,i)•2 := 2 c,c.1 + C if k I — 1,
(~)l:— +t•,=0 ( 3.8)
(532ir(C•2e., 1
-+-OF \VEIF1iS'1'UASK with the set, I)iv2,,.'
DATA
( C)>C by
35 the
livisors of degree 2.1 on C is denoted by (C x) is considered as C2I 1x C by the
(ai (bi), (72.1(bjn,
try symmetric polynomial ‘vith degree hi) is denoted by pi. By the above identi-C2./ 1 x
set R is considered as an open subset of c,)eR•11'2
9 929
2.I.1+(1/.9 0,CI 0}.
riod map defined by
•
/412, qj3i
C,1 k.2 ( .1 k.2) C2(C.1 ('.1 c1 k.1)), C .1 ('.2 k.2) (C .1 k.1 + C./ k.1 )) C3,9 ,L1),
= Oil 1 > 21 or i < 0 0. The set 1/V(13, P, dz / e) consists of Lying Re T(g, h) P2, c3). Simplifying
equations are obtained:
.1 + cic j_ k..)) + c1 + 12= 0, + 'i.J f k.2) + 111 0, cle.1‘2) + -1'3 -= 0,
= 0,09 := p9 — = 0.
--)Jj ('ie3.9) + c3 = 0,
3. THE MODULI SPACE OF WEIER.STRASS DATA 36
if k > J + 1 and if J 0,
(3.9).-v1+V2=0,02:=v2-v1=
03 := 27c2 + v3 - 0,0,
if k > J + 1 and if J = 0, and
01 := 27rc2 + vl + v2 = 0,02 := U2 - v1 = 0, (3 .10) 0
3 := 27r(c2e.7,1 + cleJ,2) + v3 = 0, if k = J O. Hence, the following is easily obtained:
LEMMA 3.1. The set W(.13' (B, P, dz/z, v) is nonempty if and only if k>J+1 and v1=v2=0, k = J and v1 = v2, or k < J — 1.
From the above equations, the following are obtained:
01= 27r(eJ-k,idc2 + c2de,J_k,l + eJ-k,2dc1 + c_ldeJ_k,2), (3.11) d02= 27r(eJ+k,ldc2 + c2de.J+k,1 + eJ+k,2dc1 + cldeJ+k,2)
453= 27r(e,Lldc2 + c2dej,1 + eJ,2dc1 + cideJ.2),
ifk<J-1,
(3.12)d03 = 27(e,j,idc2 + c2de,7,1 + e,7.2dc1 + cide,/,2),
if k > J + 1 and if J 0,
(3.13)d03= 27rdc2i
if k > J + 1 and if J = 0, and dq1 = 27rdc9, (3.14) d0
3= 27r(e,j,1dc2 + c2dej,1 + e j,2dc1 + cideJ.2),
if k = J 0. Hence, if W (B, P, dz/z, v) is nonempty, then it is a
nonsingular real algebraic variety of dimension 4J — 1 if k < J — 1, 4J + 1 if k > J + 1, and 4J if k = J O.^
Part 2
The moduli spaces of algebraic
minimal surfaces as linear systems
CHAPTER . 4
Minimal surfaces in Euclidean 3-space
38
I. INTR.ODUCTION39 1. Introduction
The purpose of this chapter is to discuss the moduli space of alge-braic minimal surfaces in R3 and the Puncture Number Problem.
An algebraic iillilinia1 surfaces in R3 is a possibly branched com-plete minimal surface X: R3 of finite total curvature. In this case, we can consider Al a compact Rieiilann surface Al with finitely many points, called puncture points, removed. Thus, we may obtain a pair consisting of a Ineroiilorphic function on 11I and a Ineroinorpllic differential on Al satisfying certain conditions. 'I'he pair is called the
ll'eicr'.stra,ss data of X. We will discuss the moduli space of \\'eierst.rass data.
We can consider a Vveierstrass data as a pair of global sections of a holoiuorphic line bundle Oil Al. The space of global sections of' a holo-I11orphic line bundle on a compact, Riemann surface has been studied as a linear system. We will fix a component of' a meroinorphic function in the \\ eierstrass data and study the moduli space of \\ eierst rass data as a real subvariety of a linear system. We will show that the moduli space becomes R" with the origin removed, where n. is a certain positive integer.
We will discuss the Puncture Number Problem for algebraic
mini-mal surfaces which is began by Kichoon Yang in his papers ([38], [39_, [40] ). The account of the problem is given in his book [41]. too.
When we fix a compact Riemann surface Al,, of genus I , we will call a positive integer r a puncture number of Ht. if AI, with exactly r, punctures can be conformally immersed in R3 as an algebraic Illiiiinlal surface. We will denote by P( Al,) the set of all puncture number of
A!1.. Then we will state the Puncture Number Problem as follows:
THE Pt:N(" l'URE NUMBER PROBLEM. (;n'eo a compact Rienranrr su.rtace of genus F, determine the set P(Al,).
The results on the Puncture Number Problem for un.branch.ed alge-braic minimal surfaces are as follows.
If (' = 0, then _ Z r > 0} since the Jorge-Meeks surface exists (cf. [10]). thence the Puncture Number Problem has
been solved.
In the case that I'. > 0, Yang showed the following theorems.
THEOREM 1.1 ([41]). FO r a compact. Riemann surface Al, of genus I. I > 0, the set. P(AI,) contains the set {r E Z r > 40.
Let us denote by tl,(Al/) the 1111111InuIn of the set, of' puncture numbers of' Alf. From the above theorem, we can see that. tc(A/,,) < 4.j for any
~11F I' > 0. On the other hand, we can see that, there exists a compact.
2. PRELlMINARIES
Rielilann surface ;\11of genus f, f > 0 such that t.l(A1f) is strictly slimaller than /if.
TllnonEM 1.2 ([41]). If M, is a h.yper•elliptic curve of genus 1. i/rer) 1i('I1r) < ;~E' + 2.
Let us denote by P(f) the intersection of P(111F) over ill compact Rieniann surface A.!1 of genus I'. Yang made the following conjecture.
CONJECTIIRE. For any compact Rienrauri surface Alp of genus I, P(111) — RV).
We will discuss the Punct ure Number Problem for possibly branched algebraic minimal surfaces in R3 and show that P(A1) _ {i' E r > O}
for any colnpact, Rietnann surface Al. Hence we may see the conjecture is true for possibly branched algebraic minimal surfaces.
2. Preliminaries
In this section, we will give definitions and lemmas needed later.
Let Al be an open Rienlann surface. X : Al R3 he a possibly branched colnplete conformal minimal surface; and +i the holoniorphic one-form on Al defined by 1I)i = (dXi / ).. )d L. i = 1. 2, 3, where is a local holoniorphic coordinate of Al. Since ,Y is conformal, the relation (2.1 )(+1 ) + (+9 ' ~~~) + (+3 <l>3) = O.
holds. \Ve will define the map [(I)]: Al C P2 by [+' := [+1, +2, +:;]
in the homogenous coordinate on (CP2. \\'e will call the map [4] the
generali. ed Gauss null) of X. We can see the following theorem hold
(cf. [18]).
TIIEottEM 2.1. .-1 possibly branched conformal minimal surface X is of finite total curvature if and only if the image of the yen(,r'uliced
Gauss map [4)] becomes an algebraic curve.
D1':FINITION 2.2. We will said a possibly branched conforrnal min-imal surface to he algebraic if the generalized Gauss map is algebraic.
I'roIn the above theorem, we can see the following:
COROLLA RY 2.3. Each holom.orphic one-form j)i, i = 1. 2, 3 ca'-tends rrrcrom,orph.icall,y on a compact Iliemann surface AI such that A/
is hiholom.orphic to Al removed finitely many points.
In the following, we will denote by (X, A1, R3) a possibly branched
colnp1ete conformal 1Uiniinal surface
2. I'R.ELIMINARII,a II
DEFINITION 2.4. For <h = ((DI, (1?2, ~3), we will define the divisor (II)) of fi by
(2.2)(4))min ord7, (I)i p.
1) /11 Assume tha.t.
.~r
(2.3) ( )=~13.ih.i—I,~t~, Bi>0, ;>0.
where {bi: pi} is S + r distinct points.
DEFINITION 2.5. We will call a point. bi a brunch point of order 13; and a point p; a MITI CI llre poi.rlt of order P,. we will call the sum B :_ Bi the total order of branch points and P 1 1; the
total order of puncture points.
DEFINI'TION 2.6. We call the divisor 'BO)) :_ ~ .J -1 B.ib_i the branch dicisor of and 43(4)) :_ ~i. 1 1;pi the puncture divisor of (4)).
We can see t hat the puncture points coincide with points removed from I1 in Corollary 2.3.
Conversely. from a triple of Inerornorphic one-fonu s (1);, i = 1, 2. 3.
sat isfving the condition (2.1), we can obtain a possibly multi-valued algebraic minimal surface .V : Al(I) IR3 by integration:
(2.4)X (:;) := Re q> ,
where Al(4)) is the Rieiria.nIt surface Al with the puncture points re-moved and is a non-puncture point on Al. If we choose another base point in the integral, then the image of the map shifts by a. translation in R3.
Let B(I) he the set of bases for Hi (11(4)), Z). Then we can see that.
X is well-defined if and only if there exists -} _ (,I , .... ~,1 .,. I) E 13 (4) ) such that
(2.5) Re = (0,0,0), 1= 1,... ,2[+,' where l is the genus of Al.
DEFiNI'I'ION 2.7. 'W'e will call the condition (2.5) the period condi-tion.
Iin the following, we will assume tha,t. X is well-defined. We can obtain the total curvature of X from 13 and P.
2. PR1?LI?'-1INAIII} 5 •12
LEMMA /111.al surface (2.6)
where \(AI)
2.8 ([18]). 1 he total curvature r(X) of an algebraic min-(.V Al, R3) is
T(.l') — 27r(\(lll) + Ii — 1');
is the Euler number of Al.
We can oht~1111 a pair (q, ')'t) consisting of a Illeroniorplllc function on 11 and a Inerolnorphic differential on Al called 11 eierstrass data , of _V which is equivalent to by
= +3
— (2.7)(g>.1/) ~ >3
I—v/-14)9
(2.8)0)1,9CD3)
=(__1q,1ft
+q2gy
It is well-known that. q is the stereographic projection of the Gauss map of X. Let. us denote by (q), and CO the divisors of q and respectively . We can see the following relation holds among (q). (ft), and ((k):
(2.9)(CD) _ —(g)0 — (gl).y + (it)•
where (g)0 and (.q) are the zero divisor of q and the polar divisor of q respectively.
DEI-'lNe 1'IoN 2.9. We will call the condition (2 .9) the divisor con-dition of (D.
Iiv considering the
(2.10) R
degree of both side of (2.9), we may see that .
-1'=-2 deg g+21 -2 ..
Hence we can see that,
(2.11)r(X) _ —47r deg q,
by (2.6).
Fix a conlpact. Riemann surface Al, of genus 1, a positive divisor
t ~~, = ~t I 1 uipi on Alf, and a Iuerolnorphic function q on Alt. Let W )'V(, S' O) he the set of \V'eierstra.ss data corresponding to algebraic minimal surfaces (X, Ate, R3) such that the Gauss map is g and 1311 — (t~)u - (i) ~ < `43(I)) < To. Let A = A(y,' o) he the set. of algebraic minimal surfaces corresponding to VV. We will denote by X Y. for two elements X and Y in A if X + .r = Y for a. certain :v E R3. Then we can see the following lellnna. holds.
LE_M:MA 2.10. There exists a bijective correspondence between W(q . To) and A(q, 130)/
3. THE EXISTENCE OF ALGEBRAIC MINIMAL. S(TH A('ES.13 3. The existence of algebraic minimal surfaces
We will discuss W(9, To). We will prove the following theorem.
THEOREm 3.1. If 3(2k + r — 1) < deg 9 and if deg — 4 deg g >
21 — 2. then W(9, q3 )) becomes (t non-zero real vector .space u'ith. the
°ere vector removed.
We may assume that
7'1 i I',717
(3. f)(~~) - ----`ihi + Qk(/k,
1k 1
where { pi; qk } are r + in distinct points. We may assume a'1 ... , V,., >
0, N,'1-.-1, • .. Yr1_.,.., < 0, (21, . . • , > 0, and (2711.1 1, . .. , QM] -,711, 0, where rrr1 + 111•7 = n. We will define divisors 1)(1, i') and TO(., v) on
al and a positive integer (1(11, e) by
7'17'1r1',
1)(a.1') :=\j'(1!~jp;+ } 11
i 1r.=ri +1
(3.2)1111711 1 • 171_
+ } 21Qkgk ±('I' .(,)k gk.
k Ik uli1
(3.3) ' ;(a. 0) := 20 — I)(u, v), d(u•, i) : deg' 3(a. v),
We may see that (g)O — T)(1, 0) and (9) = I)(0. 1). Hence, (g)), + D(u, r) = I)(u + 1, e) and (q), + D(u. (') = 1)(u. v + 1).
Let us denote by G( L)) the complex vector space of nleronlorpllic functions such that G(D) :_ { f (f) -t- I) > 0}. We will assume the following theorem ([1]).
THEOREM 3.2. If 1) is a divisor of degree (1 011 :II, then 0 ifd<0. (
3.4) (1i111 G(1)) _ d
-1'+1 ifd>2f-2 . We will define G(11, v) and l(lt, •t') h~~
(3.5) G(rt, i) := G(13(a, v)), l(u, v) := dim G(u. t ).
In the following, we will assume that deg t 30 - 4 degg > 2 - 2. Then we can see that l(-u, e) — d(0, 0) — (u + c) deg 9 — 1! + 1 if a + 1' < 1, rl>0,andv>0.
We will discuss G(•u, v) where 2 < a -I e < 4, a > 0, and c > 0. We can easily see the following lemmas hold.
LF:AI NIA 3.3. G(n, v) is a complex vector subspace of G(w, .r') if and only if rl > a' and V > :r.
3. THE EXISTENCE OF ALGEBRAIC MINIMAL SURFACESII
LEMMA 3.4. G(u, v) n G(w, x) = G(S, [J ), where 3 = Illax{u,, w}
and T = nla.x{v, .r}.
Specially, the followings holds.
1. r(3, 0) c G(2,0), G(2, 1) c G(2,0).
2. G(2,1) c £(1, 1), G(1,2) c G(1, .1.).
3. G(1,2) c G(0, 2), G(0,3) c G(0,2).
LEMMA 3.5. If d(3, T) > 2I' - 2, then
(3.6) G(u, v) + G(w, x) = G(s, 1),
where + means a su.m of rector spaces, S = max-{u, ~a'}, 'F = Inax{v, x}, s = min{u, w}, and t = miii{v, x}.
PROOF. Since s = min{u, w} and t = minty, n{2y, x}, we may see that G(u, v) c G(s, 1) and G(w, x) C G(s, t). Hence, G(u, L') + G(2(', x) C G(s.t).
Since s < S and t. < I', we may see d(s. 1) > d($. T). Hence d(s, t) > 2[ - 2, in fact 1(s, t) = d(0, 0) (s + t.) deg g - (' + 1. Since G(u. r') n G(w..r) = G(S.. T), we may see
dini(G(u., r) 4. G(w, .r)) = 1(u, v) + 1(w, .r) - 1(5, T)
(3.7) =d(0,0)-( a+ +.r-3-T) deg g-t'+1
= d(s. t).
Therefore t(u, r) + G(w, x) = r(s, t). ^
By Lemma 3.4 and Lemma 3.5, we can consider the vector space G(u, v). n + v = 2, a > (), v > 0 as the direct sum of vector
sub-spaces:
(3.8) G(a, v) = G(u + 1, v + 1) =f? IC(u, v), where 4) means a direct sunl of vector spaces and (3.9) 1C(u, r) := M(u, + 1) (+N(u + 1, c), (3.10) G(u + 1, v) = G(u + 1, v + 1) + 1, v), (3.11) G(u,w+1)=G(a+1,v+1)( M(a,r+1).
Then we can see dim .w[(a, v + 1) = dim./T(a + 1, v) = deg g.
LEMMA 3.6. The set 1C(1, 1) contains the set W(g, q30).
3. THE EXISTENCE OF ALGEBRAIC MINIMAL SURFACES-15
PRooF. By the divisor condition (2.9), we can see
tiTIl1?11.1+1112
(q) =/' B,1 b, lQ k g k + >: Qk r 1k
j-1k1k--7111 + 1
(3.12) 1'1T17Tr7'1 t7'2 $7'3
J J (r - )Pi > I (I )L N )1) >2, ` i t )'L
--1L 1'1-I-11=7'1 1-7'2 i-1
where B — P + 2 deg g = 2f — 2. Since To — (g)o — (g), < T(>) < 13u, we ca.11 see that the set IC(1, 1) contains the set Mg, To). ^
LEMMA 3.7. The following holds.
1. If r > x, then N. (u -- 1, v) n G(u1. x) = N(,5', L'), where _ max{u + 1, u~}.
2. If v < x, then. N(u, v) n G(w, a:) _ {0}.
PROOF. By (3.10), we can see
G(u + 1, v) n G(w, ,r)
(3.13) = (G(u + 1, r + 1) n G(w, x)) E (N(21. + 1. e) f (w, x)).
If v > X, then
(3.1,1) G(rx + 1,v) n G(u~, .x) = G(S, v), (3.15) G(u + 1, v + 1) n G(w, x) = G(, v + 1).
Hence, by (3.10), we can see .Nf(u + 1, r) n G(w, a) = N(S, v).
If v < x, then
(3.16) G(a + 1, r) n G(w, x) := G(S, a'), (3.17) G(u -I- 1, v + 1) n G(w, x) = G(S, x).
Hence, we can see .A./(u + 1, r) n Lew, x) _ {0}.^
In a similar fashion we can prove the following lenulla.
LEMMA 3.8. The following holds.
1. If u > w, then AI (u, v + 1) in G(w, x:) _ 1,1(u., T), where T = max{v + 1, x}.
2. If u < u, then M (u, v + 1) n G(tx~, a:) _ {0}.
LEN'1IA3.9.Ifu+v=2 and 1<a<2,0<v<1, then 1C(u,v)n 1C(a — 1, v + 1) = {0}.
3. THE EXISTENCE OE ALGEBRAIC MINIMA!, SURFACESl(;
PROOF. By (3.8), we may see G(i, v) n G(u - 1, v + 1)
(G(u+1,1.'+1)nG(u,rH-2))
(3.18) + (G(a + 1, e + 1) n 1C(u 1. r - 1))
~a (G(u, r' + 2) n 1C( u, v)) (1C(u - 1,v+ 1)n1C(a,r)).
By Leiiinla 3.4, Lemma 3.7. and Lemma 3.8, the following holds:
(3.19) G(u, v) n G(u. - 1, v + l) = G(u, 1),
(3.20) G(i + 1, -+- 1) n G(u, v + 2) = G(u + 1, c p 2).
G(u.+ 1,v+ l)nJC(u - 1,e+ 1) ( 3.21) {
0} r' + 1),
G(u, +2)n/C(u,r) ( 3.'2)
=JVI(u, v + 2).1) {0}.
By (3.8) , we may y see 1C (a , c) n IC( u - 1, 1) _ {01. (~
LEMMA :3.10. The sets {grt rt E G(u . el} and {rt%g rt E G(u. 't,)}
are equivalent to G(a + 1, - 1) and G(a - 1, + 1) respectuel,r/.
PROOF. It is clear since ' z(u, v) - (y), - (g), _ 4.3(i1 + 1. v - 1) and 'V(a. r') + (g)0 + (g) _ (u - 1, r' ± 1). ( 1
From t lie Lemma :3. 10, we Can prove t he following lelnlna immedi-ately.
LEM MA 3.11 . The sets {grt rl E 1C(u., v)} and {rt/y rt E /COI, r)}
are equivalent to 1C(u + 1, v - 1) and IC(a -- 1, r ± 1) respectively.
From the above discussion, we can prove the theorem .3.1.
PRooF, OF THEOREM 3.1. By Lenima :3.9 and Leiiinia 3.11, we can take a basis for 1C(1, 1), k(0, 2), and k(2 , 0) satisfying the following.
conditions:
1. c(u, c, i), r = 1, ... , deg g, is a basis for KT (a., c) where Zr + v = 2, rr>0, and v>0.
2. .qf(1, 1, i) _ (.(2,0,i). i. = 1,... , deg q.
3. r(1 1, i)/g = f(0, 2, i), i = 1, ... , deg g.
13v Lenilna 3.6, we can assume that, deg rr
(3.23) rl = r'ir (1, 1, z),
-1
3. THE EXISTENCE OF ALGEBRAIC AIINI\IAL SURFACES17 where c'~i, i = 1, ... , deg g are complex numbers such that
0. Then 4)1, ?9, and .4)3 are as follows:
c1c 9
(3.2,1)(Di = ci (E(0,'2, 1) — (2, 0. 1)) ,
1 cleg!r
(3.25) (c (0, 2, 'i) + ((2, 0, 1)) ,
1 dig!J
(3.26)4)3=>2,c•E(1,1,z).
i=-1
Let us fix a ' C 8(4))and denote by cr(i, j, k,1) the periods of f (i, j, k) along -r, 1 = 1.... ,21+i'— 1, where (1, j) _ (1, 1), (0, 2), or (2, 0), and
= l , ... ,deg q. Then, we may obtain
1eg,r
(3.27) Re (Di = Re(ei(at(0.2,i,1) o(2,0,1,1))),
•c1c.g g
(3.28) Re (1)2 _ Re (\/ lci (a(0, 1) } a(2, 0, 1,1))) ,
1 ckg!/
(3.29)Re (1'3 — >_..; Re (cia,(l. 1, i,1)) .
-1 i ]
\\Te ca.11 consider each element in the set W(q, T11) as an element (e1 . . G1,,g q) C"1`" such that,
(3.30) Re 4;=0,1=1,...,21+r-1,j=1,2,3,
(1,12,(/
(3.31 )c-0.
i. 1
Since 3(21 + r — 1) < deg g. W(g,' Z„) is a real vector space with the
zero vector removed.
Finally we will discuss the Puncture Number Problem for possibly branched algebraic minimal surfaces.
THEOREM 3.12. The set P(111) is equivalent to the set of positive integers for any compact llieurr(Ln,n, surface M.
PR.c)OF. Let it! be a compact, R.iemann surface of genus 1' and r a posit ive integer. We can take a ineromorpllic function g on such that 3(21'+ r — 1) < 2 deg g. For this g, we can take a. positive divisor 1311 such
3. THE EXISTENC_'E OF ALGEBRAIC MINIMAL SURFACES18
that (leg i3 — 4 deg g > 2f — 2, the support of To is r distinct points, am! ' 3n -- (g)u — (g) > 0. Then the set W(g, q3u) is nonenipty by The-orem 3.1. Let (X, Al, R3) e W( ,g, g330) and (+1, cl).~, 4)3) the associated triple of ineromorphic one-forms. Since 0 < (g)u - (y) < 13(4'), (X, Al, R3) is an algebraic minimal surface with r puncture point. E
CHAPTER 5
Minimal annuli in Euclidean 3-space
49
1. INTH(DUCTION~ill 1. Introduction
The purpose of this chapter is to discuss the moduli space of a certain class of minimal annuli and its geometric properties.
The moduli space of minimal surfaces in Euclidean space has been
studied from several viewpoints (cf. F. Tollli and A. J. TroIni)a [34], J. Perez and A. R.os [27, 28], A. Ros [31], K. Yang [41], R.. Kusner and
N. Schmitt [12], G. P. Pirola [29, 30], and K. Moriya [18]). Except for [34], they are studying the moduli space of complete conformal
minimal immersions X: Al R3 or R1 from a Rieulann surface Al to Euclidean space of finite total curvature.
The Riemann surface Al can be cornpactified conforulally in this case. This fact holds also in the case where the immersion has branch
points (cf. [18]) or the target space of the immersion is R3/T(u) (cf.
Lemma 2.2), where T(e) is the discrete group of isolnetries generated by a. translation by v E V. The Euclidean space R3 is considered as Il /T((0, 0, 0)). A branched complete conformal minimal immersion from a Riemann surface Al to R3/T(v) of finite total curvature will be denoted by (.Y, AI, R3/T(v) ), where Al is the Riema.nn surface corn-pactified from Al. They are called minimal suifaces of algebraic type, or simply, algebraic minimal surfaces in this chapter
\iaokang Mo studied the moduli space of \Yeierstrass data, by algebro-geometric methods and and Kichoon Yang introduced it. (cf.
[41] ). When Weierst rass data is considered, it is possible for the
corre-sponding minimal immersion to have branch points. The moduli space of branched minimal immersions was also treated by them. In their study, a lower bound of the dimension of a complex analytic variety contained in the moduli space is obtained. Unfortunately, this is nega-tive in sonic cases, for example when the genus of the Rieman II surface
is high.
Therefore, to advance the study of the moduli space, concrete ex-amples should be examined.
In this chapter, a moduli space of a class of algebraic minimal
an-nuli (X, CP1, R3/I'(.L,)) is investigated. In K. Moriya [20], the defining
equations of moduli spaces of algebraic minimal annuli are given by Mo's method in the case where two puncture points are of the same fixed degree and whose Gauss map ramifies only at the puncture points.
The difference between [20] and this chapter is that the totality of the
rnn.)duli spaces of Weierstrass data for minimal annuli with two punc-ture points of possibly different degrees is investigated by a method different from that of Mo.
1. INTRODUCTIONin
The minimal annuli are classified easily in this moduli space by t,he orders of branch points and puncture points. Each moduli space, given by fixing the orders of branch points and puncture points, also becomes a smooth subma.nifold of the total moduli space.
The moduli space given by fixing a. translation symmetry of minimal annuli becomes a. real 3-dimensional algebraic smooth submanifold of a, real 6-dimensional smooth manifold. We introduce a Rietilaliiliari met-ric on the 6-dimensional manifold and an integrable complex structure compatible with the metric.
The curvature of the Riemannian metric on the 6-dimensional man-ifold and the curvature of the induced metric. on the moduli space given by fixing a. translation symmetry of minimal annuli are calculated. The helicoid and cat enoid are characterized by the lat.ter curvature. The following other characterizations are known:
THEOREM 1.1 ([33]). Let (X, M. R3) hare two ends, each.
embed-ded. 7'hen, it is a, ca.tenoid.
THE0RI-.'M 1.2 ([14] and [26]). Let (X, CP1, Il'`i) be unbranched
and properly embedded. Then it is a. plane or a eate110id.
THEOREM 1.3 ([35]). Let (.Y, CPI,R3/T(c)) be unbranched, prop-erly embedded and with two puncture points of iion::er•o finite total cur-vature. Therr, it is a helicoid.
THEOREM 1.4 (_26]). Let (X, CPI, R3% T(r)) be unbranched and properly embedded with finite number of helicoidal type ends. Then it is a helicoid.
It can he seen from the inonotonicity formula., if an immersion X :.l
R3 is of finite total curvature with exactly two embedded ends. then X(1.1) is embedded (cf. [32]). Moreover, algebraic minimal surfaces
(X, (Cl'.1 R3) and (X (CP1, R3/T(v)) are proper (cf. [6]).
An integrable complex structure compatible with the Riemannian metric is defined on the 6-dimensional manifold. The moduli space given by fixing a translation symmetry of minimal annuli becomes a 3-dimensional totally real subnia.nifold with respect. to the Hermitian
metric associated to the R.iemannian metric: of the 6-dimensional man-ifold. This result may be considered as an analogue of Theorem 8.1 in
[27].
The following is a brief summary of this chapter. In Section 2, a fundamental result about minimal annuli is described. The classifica-tion of branched minimal annuli via Weierstrass data is also given. In Sect ion 3, the moduli space of unbranched mminial annuli of algebraic