Rend. Sem. Mat. Univ. Pol. Torino Vol. 57, 1 1999
G.P. Pirola
∗
A REMARK ABOUT MINIMAL SURFACES WITH FLAT EMBEDDED ENDS
Abstract. In questo lavoro si prova un teorema di ostruzione all’esistenza di super- ficie minime complete nello spazio Euclideo aventi soltanto code piatte lisce. Nella
dimostrazione confluiscono tecniche di geometria algebrica spinori su supeficie di Riemann compatte e differenziale herisson e formula di monotonicit`a. Come
corollario si ottiene che una superficie minimale di genere due avente tre code piatte lisce e tipo spinoriale pari non `e immergibile minimalmente in R
3
.
1. Introduction
In this paper we prove a non-existence result for certain complete minimal surfaces having em- bedded flat ends and bounded curvature. The obstructions have been found by means of an alge-
braic herisson see 11 and [14]. In particular we show see Theorem 5.2 the non-existence of untwisted genus 2 minimal surfaces having 3 embedded flat ends.
We systematically use the theory developed by Kusner and Schmitt cf. [5]. A short ac- count of it is given in § 1. The minimal surfaces are studied by means of holomorphic spinors on
compact Riemann surfaces. This fits very well with D. Mumford’s previous work cf. [8], more- over some new hidden geometry appears see Remark 5.2. The results of § 1, except Proposition
5.2, are due to the previous mentioned authors.
In § 2 we recall the spin representation of a minimal surface and prove our result. The main tool is a well known singularity or monotonicity formula cf. [4] and [3]. Theorem 5.2 works
out a heuristic argument of Kusner and Schmitt cf. [5] § 18 in the easiest non trivial case.
2. Spin bundles on a compact Riemann surface
Let X be a compact connected Riemann surface of genus g. References for the basic facts we need can be found in the first chapter of [1] or in [9]. Let F be an abelian sheaf on X and denote
by H
i
X, F , i = 0, 1, its coomology groups, H
X, F is the space of global sections of F . Set h
i
F = dim H
i
X, F .
Let O
X
and ω
X
denote respectively the structure and the canonical line bundle of X . Let D
= P
d i
=1
p
i
, d = degD, be an effective divisor with distinct points. Fix a spin bundle of X,
i.e. a line bundle L on X such that L
2
= ω
X
cf. [5] and [8]. The isomorphism φ : L ⊗L → ω
X
defines a spin structure of X . We say that L is even odd if h L is even odd. Consider the
line bundle LD = L ⊗ O
X
D, LD
2
= ω
X
2D. We identify LD with the sheaf of the meromorphic sections of L having simple poles at D. The Riemann-Roch and Serre-duality
∗
Partially supported by 40 project Geometria Algebrica M.U.R.S.T., by G.N.S.A.G.A. C.N.R. Italy and by European project Sci. “Geom. of Alg. variety”.
59
60 G.P. Pirola
theorems give: h
LD − h
1
LD = d and h
1
LD = h
L −D .
Here L −D is the sheaf of the holomorphic sections of L vanishing at D. The sheaf inclusion
L −D ⊂ LD defines the quotient sheaf L = LDL−D. Set V = H
X, L, dimV =
h L
= 2d. Fix coordinates
{U
i
, z
i
} centred in p
i
, z
i
p
i
= 0, we assume that p
i
∈ U
j
if and only if i
= j . We trivialise the previous line bundles on U
i
. If s and t are sections of LD defined on U
i
, we write: 1
s ≡
a
i, −1
z
−1 i
+ a
i,0
+ X
n0
a
i,n
z
n i
ζ
i
; t
≡
b
i, −1
z
−1 i
+ b
i,0
+ X
n0
b
i,n
z
n i
ζ
i
where ζ
2 i
= φζ
i
⊗ ζ
i
= dz
i
is provided by the spin structure. The truncated expansions: s
′
= a
i, −1
z
−1 i
+ a
i,0
ζ
i
, t
′
= b
i, −1
z
−1 i
+ b
i,0
ζ
i
define elements of ŴU
i
, L . Multiplication provides st
∈ ŴU
i
, ω
X
2D, a meromorphic differential on X with double pole at D. The residues:
2 a
i, −1
b
i,0
+ b
i, −1
a
i,0
= Res
p
i
st are well defined. Accordingly we set:
3 s, t
p
i
= a
i, −1
b
i,0
. The form 3 defined on ŴU
i
, LD is intrinsic. To see this cf. [5] introduce the meromorphic
function h =
s t
, h p
i
=
a
i, −1
b
i, −1
as b
i, −1
6= 0. We have: s, t
p
i
= 1
2 h p
i
Res
p
i
t
2
if b
i, −1
6= 0 and s, t
p
i
= Res
p
i
st if b
i, −1
= 0 . The symmetric part of 3 is 2 and both vanish on ŴU
i
, LD
− 2 p
i
. Then 4
Bs, t =
X
i
s, t
p
i
= X
i
a
i, −1
b
i,0
. defines a bilinear form on V . The symmetric and the anti-symmetric part of B are respectively:
Qs, t =
1 2
Bs, t + Bt, s =
X
i
Res
p
i
st 5
s, t
= 1
2 Bs, t
− Bt, s . 6
We have three maximal isotropic space of Q: I V
1
= {a
i, −1
= 0}
i =1,...,d
= {images of local holomorphic sections}. II V
2
= {image of global section of LD}. III V
= {s ∈ V, a
i,0
= 0, i = 1, . . . , d} = {s ∈ V : Bt, s = 0 for any t of V
1
}.
A remark about minimal surfaces 61
Observe that V
1
and hence V are intrinsic. The exact sequence
→ H X, L
−D → H X, LD
τ
−→ LDL−D identifies H
X, LDH X, L
−D and V
2
= imageτ . By Riemann-Roch dimV
2
= d. The isotropy of V
2
follows from the global residues theorem. We identify V
1
∩ V
2
= V D with H
X, LH X, L
−D. Set SD = V ∩ V
2
and K D
= τ
−1
SD =
s
∈ H X, LD : s
≡
a
i, −1
z
−1 i
+ X
n0
a
i,n
z
n i
ζ
i
near p
i
.
Note that K D ∩ H
X, L = H
X, L −D. This gives an isomorphism K DH
X, L
−D ∼ = SD.
R
EMARK
5.1. If h L
−D = 0, e.g. if d g − 1, the spaces SD and V D can be identified respectively with K D and H
X, L. In particular, following Mumford cf. [8], we identify H
X, L with V D which is the intersection of two maximal isotropic spaces: V
1
∩ V
2
= V D. From this it follows that the parity of h L is invariant under deformation.
The forms B and were introduced in [5]. Next we consider . The restriction of to any Q-isotropic space equals B, in particular
B = : V
2
× V
2
→ C:
s, t =
X
i
a
i, −1
b
i,0
= − X
i
b
i, −1
a
i,0
. Define, composing with τ , the anti-symmetric form
′
on H X, LD. Set
7 ker
= {s ∈ V
2
: s, t = 0 for any t ∈ V
2
} ker
′
= n
s ∈ H
X, LD :
′
s, t = 0 for any t ∈ H
X, LD o
We may identify ker and ker
′
H X, L
−D. P
ROPOSITION
5.1 T
HEOREM
15
OF
[5]. We have: i h
LD = dimker
′
mod 2 and d = dimker mod 2;
ii ker = V D ⊕ SD and ker
′
= K D + H X, L;
iii dimSD + dimV D = d mod 2.
Proof. i and
′
are anti-symmetric. ii Clearly V D = V
2
∩ V
1
and SD = V
2
∩ V are contained ker. Conversely let s
= s + s
1
∈ ker, s
i
∈ V
i
, i = 0, 1. One has
Bs
1
, v = 0 = Bv, s
for any v ∈ V . Take t ∈ V
2
then Bs, t = s, t = 0:
= Bs + s
1
, t
= Bs ,
t + Bs
1
, t
= Bs ,
t + Bt, s
= Qt, s .
Hence Qt, s = 0 for all t ∈ V
2
. Since V
2
is maximal isotropic s ∈ V
2
, in the same way s
1
∈ V
2
. iii It follows from i and ii .
62 G.P. Pirola
R
EMARK
5.2. If h L
= 0 and d is odd, Proposition 5.1 implies the existence of s ∈ H
LD such that s ≡
n a
i, −1
z
−1 i
+ P
n0
a
i,n
z
n i
o ζ
i
near all the points p
i
. R
EMARK
5.3. One has dimSD ≤ dimV
1
= d and the Clifford inequality 2 dim V
2
∩ V
1
≤ 2h L
≤ g − 1 cf. [9]. We can prove the following:
P
ROPOSITION
5.2. If D =
P p
i
and dim K D = d = degD then d ≤ g + 1. Assume
d = g + 1 and g 1 or d = g and g 3. Then X is hyperelliptic and the points p
i
of D are Weierstrass points of X . If d
= g + 1 and g 1 then L is isomorphic to O
X
D − 2 p
i
. If d = g
and g 3 L is isomorphic to O
X
D − B where B is a Weierstrass point distinct from the p
i
. Proof. If d
≥ g we have h L
−D = 0 and hence d = dim SD = dim K D. This holds if and only if
′
= 0, i.e. H X, LD
= K D and then ker = SD. Now ii of Proposition 5.1 implies V D
= H X, LH
X, L −D = 0. It follows that h
L = 0.
From Riemann Roch theorem we get therefore h L p
i
= h L
− p
i
+ 1 = 1 for all i. The sheaf inclusion L p
i
⊂ LD provides sections σ
i
∈ H X, L p
i
⊂ K D, σ
i
6= 0 . If i
6= j the σ
i
are holomorphic on p
j
, then σ
i
p
j
= 0 σ
i
∈ K D. Therefore the zero divisor, σ
i
, of σ
i
contains D − p
i
, hence g = degL p
i
≥ d − 1: d ≤ g + 1. We write:
8 σ
i
= D − p
i
+ B
i
, where B
i
is an effective divisor of degree g + 1 − d. For any i and j , j 6= i. We obtain:
D − 2 p
i
+ B
i
≡ D − 2 p
j
+ B
j
, ≡ denotes the linear equivalence. We get
B
i
+ 2 p
j
≡ B
j
+ 2 p
i
.
If d = g + 1 we have B
i
= B
j
= 0 and 2 p
j
≡ 2 p
i
: X is hyperelliptic and the p
i
are Weierstrass points. The 8 shows that L is isomorphic to O
X
D − 2 p
i
.
Assume d = g and g 3. The B
i
are points of X . The first 3 relations: B
1
+ 2 p
2
≡ B
2
+ 2 p
1
; B
1
+ 2 p
3
≡ B
3
+ 2 p
1
; B
3
+ 2 p
2
≡ B
2
+ 2 p
3
give 3 “trigonal” series on X . If X were non-hyperelliptic only two of such distinct series can exist one if g 4 see [1]. Then two, say the firsts, of the above equations define the same
linear series: B
1
+ 2 p
2
≡ B
2
+ 2 p
1
≡ B
1
+ 2 p
3
≡ B
3
+ 2 p
1
. We obtain 2 p
2
≡ 2 p
3
: X is hyperelliptic, which gives a contradiction. Now X is hyperelliptic and let ϕ be its hyperelliptic involution. Assume that p
1
is not a Weierstrass point, i.e. ϕ p
1
6= p
1
. Set B
1
= B. Any degree 3 linear series on X has a fixed point. Since p
1
6= p
i
for i 6= 1 and B + 2 p
i
≡ B
i
+ 2 p
1
it follows that B = p
1
. From 8 we obtain
L p
1
≡ σ
i
≡ D − p
1
+ B = D , then L
≡ O
X
D − p
1
≡ O
X
P
i1
p
i
. This is impossible: L has not global sections. It follows that all the p
i
are Weierstrass points. From D − 2 p
1
+ B ≡ L we get ω
X
≡ O
X
2D −
4 p
1
+ 2B, then B is a Weierstrass point: L = O
X
D − B, 2B ≡ 2 p
i
, and B 6= p
i
for all i .
A remark about minimal surfaces 63
3. Spinors and minimal surfaces
