Table Of ContentRiemann Surfaces of In(cid:12)nite Genus
Joel Feldman
(cid:3)
Department of Mathematics
University of British Columbia
Vancouver, B.C.
CANADA V6T 1Z2
[email protected]
http://www.math.ubc.ca/ feldman/
(cid:24)
Horst Kno(cid:127)rrer, Eugene Trubowitz
Mathematik
ETH-Zentrum
CH-8092 Zu(cid:127)rich
SWITZERLAND
[email protected], [email protected]
http://www.math.ethz.ch/ knoerrer/
(cid:24)
This work is dedicated to Henry P. McKean.
Research supported in part by the Natural Sciences and Engineering Research Council of Canada, the
(cid:3)
SchweizerischerNationalfonds zurFo(cid:127)rderung der wissenschaftlichenForschung and the Forschungsinstitut
fu(cid:127)r Mathematik, ETH Zu(cid:127)rich
Preface
In this book we introduce a class of marked Riemann surfaces (X;A ;B ; ) of
1 1
(cid:1)(cid:1)(cid:1)
in(cid:12)nite genus that are constructed by pasting plane domains and handles together. Here,
A ;B ; , is a canonical homology basis on X. The asymptotic holomorphic structure
1 1
(cid:1)(cid:1)(cid:1)
is speci(cid:12)ed by a set of geometric/analytic hypotheses. The analytic hypotheses primarily
restrict the distribution and size of the cycles A ;A ; . The entire classical theory of
1 2
(cid:1)(cid:1)(cid:1)
compact Riemann surfaces up to and including the Torelli Theorem extends to this class. In
our generalization, compact surfaces correspond to the special case in which all but (cid:12)nitely
many of Aj ; j 1, have length zero.
(cid:21)
The choice of geometric/analytichypotheses was guided by two requirements. First,
thatthe classicaltheory of compact Riemann surfaces could be developed in this new context.
Secondly, that a number of interesting examples satisfy the hypotheses. In particular, the
heat curve (q) associated to q L2(IR2=(cid:0)) . Here, (cid:0) is the lattice
H 2
(cid:0) = (0;2(cid:25))ZZ (! ;! )ZZ
1 2
(cid:8)
where ! > 0; ! IR and (q) is the set of all points ((cid:24) ;(cid:24) ) C C for which there
1 2 1 2 (cid:3) (cid:3)
2 H 2 (cid:2)
is a nontrivial distributional solution (x ;x ) in L (IR2) of the \heat equation"
1 2 1loc
@ @2
+ q(x ;x ) = 0
@x (cid:0) @x2 1 2
(cid:18) 1 2(cid:19)
satisfying
(x + ! ;x +! ) = (cid:24) (x ;x )
1 1 2 2 1 1 2
(x ;x +2(cid:25)) = (cid:24) (x ;x )
1 2 2 1 2
The general theory is used to express the solution of the Kadomcev-Petviashvilli equation
with real analytic, periodic initial data q explicitly in terms of the theta function on the
in(cid:12)nite dimensional Jacobian variety corresponding to (q) . It is evident that the solution
H
is almost periodic in time. In [Me1,2], this result is improved to (cid:12)nitely di(cid:11)erentiable initial
data.
This book is divided into four parts. We begin with a discussion, in a very general
setting, of L2- cohomology, exhaustions with (cid:12)nite charge and theta series. In the second
part,thegeometrical/analyticalhypotheses areintroduced. Then, ananalogueoftheclassical
theory is developed, starting with the construction of a normalized basis of square integrable
holomorphic one forms and concluding with the proof of a Torelli theorem. The third part
is devoted to a number of examples. Finally, the Kadomcev-Petviashvilli equation is treated
in the fourth part.
1
We speculate that our theory can be extended to surfaces with double points, that
a corresponding \Teichmueller theory" can be developed and that there is an in(cid:12)nite dimen-
sional \Teichmueller space" in which \(cid:12)nite genus" curves are dense.
2
Table of Contents
Part I - Parabolic Surfaces
Introduction to Part I p 5
1 Square Integrable One Forms p 8
x
2 Exhaustion Functions with Finite Charge p 26
x
3 Parabolic Surfaces p 39
x
4 Theta Functions p 62
x
S Appendix: Summary of Results from Part I p 81
x
Part II - The Torelli Theorem
Introduction to Part II p 89
5 Geometric Hypotheses p 94
x
6 Pointwise Bounds on Di(cid:11)erential Forms p 110
x
7 Zeroes of the Theta Function p 145
x
8 Riemann’s Vanishing Theorem p 164
x
9 The Geometry of the Theta Divisor p 177
x
10 Hyperelliptic Theta Divisors p 200
x
11 The Torelli Theorem p 213
x
Part III - Examples
Introduction to Part III p 222
12 Hyperelliptic Surfaces p 226
x
13 Heat Curves: Basic Properties p 238
x
14 Heat Curves: Asymptotics p 258
x
15 Heat Curves: Veri(cid:12)cation of the Geometric Hypotheses p 285
x
16 Fermi Curves: Basic Properties p 291
x
17 Fermi Curves: Asymptotics p 295
x
18 Fermi Curves: Veri(cid:12)cation of the Geometric Hypotheses p 316
x
A Appendix: A Quantitative Morse Lemma p 322
x
Part IV - The Kadomcev Petviashvilli Equation
Introduction to Part IV p 328
19 The Formula for the Solution p 330
x
20 Approximations p 335
x
21 Real Periodic Potentials p 364
x
3
2
Part I: L {Cohomology, Exhaustions with Finite
Charge and Theta Series
4
Introduction to Part I
It is an elementary fact of topology that for any oriented, compact surface X of
genus g there is a \canonical" basis A ;B ; ;A ;B of H (X;ZZ) satisfying
1 1 g g 1
(cid:1)(cid:1)(cid:1)
A A = 0
i j
(cid:2)
A B = (cid:14)
i j i;j
(cid:2)
B B = 0
i j
(cid:2)
for all 1 i;j g, and that all canonical bases are related by Sp(g;ZZ), the group of
(cid:20) (cid:20)
integral symplectic matrices of order 2g. For any smooth closed forms !;(cid:17) on X the
Riemann bilinear relation
g
! (cid:17) = ! (cid:17) (cid:17) !
^ (cid:0)
ZX i=1 ZAi ZBi ZAi ZBi
P
is satis(cid:12)ed.
A A
1 2
B B
1 2
If, inaddition, X hasacomplexstructure , inother words, X is aRiemann surface,
then it is a basic result that the complex dimension of (cid:10)(X), the vector space of holomorphic
one forms, is g and that for every canonical basis A ;B ; ;A ;B of H (X;ZZ), there is
1 1 g g 1
(cid:1)(cid:1)(cid:1)
a unique \normalized" basis ! ; ;! of (cid:10)(X) satisfying
1 g
(cid:1)(cid:1)(cid:1)
! = (cid:14)
j i;j
ZAi
for all 1 i;j g. It is also easy to show that the \period matrix"
(cid:20) (cid:20)
= !
X j
R
(cid:16)ZBi (cid:17)
5
is symmetric and Im is positive de(cid:12)nite and that the associated theta series
X
R
(cid:18)(z; ) = e2(cid:25)i z;n e(cid:25)i n;Rn
X h i h i
R
n ZZg
2
P
convergestoanentirefunctionof z Cg. Theexistenceof ! ; ;! isasimpleconsequence
1 g
2 (cid:1)(cid:1)(cid:1)
of elementary Hodge theory. Uniqueness and the properties of the period matrix follow
directly from the Riemann bilinear relations.
Suppose X is an open Riemann surface marked with an in(cid:12)nite canonical homology
basis A ;B ;A ;B ; , and let (cid:10)(X) be the Hilbertspace of square integrable holomorphic
1 1 2 2
(cid:1)(cid:1)(cid:1)
one forms. In 1 we show, again by elementary Hodge theory, that there is a sequence
x
! (cid:10)(X); j 1, with
j
2 (cid:21)
! = (cid:14)
j i;j
ZAi
for all i;j 1. In general, however, one cannot determine whether the sequence ! ; j 1,
j
(cid:21) (cid:21)
is unique or whether it’s span is (cid:10)(X). Additional structure is required.
In 2weintroduce(De(cid:12)nition2.1)theadditionalstructure ofanexhaustionfunction
x
h with (cid:12)nite charge on a marked Riemann surface (X;A ;B ; ). More precisely, h is a
1 1
(cid:1)(cid:1)(cid:1)
nonnegative, proper, Morse function on X with
d dh <
(cid:3) 1
ZX
(cid:12) (cid:12)
(cid:12) (cid:12)
that is \compatible" with the marking. A basic result (Theorem 3.8) is that for every marked
Riemann surface (X;A ;B ; ) on which there is an exhaustion function with (cid:12)nite charge
1 1
(cid:1)(cid:1)(cid:1)
one can construct a unique sequence ! (cid:10)(X); j 1, that spans (cid:10)(X) and satis(cid:12)es
j
2 (cid:21)
! = (cid:14)
j i;j
ZAi
Furthermore, X is parabolic in the sense of Ahlfors and Nevanlinna. Roughly speaking, an
exhaustion function with (cid:12)nite charge allows one to derive enough of the Riemann bilinear
relations (Theorem 2.9 and Proposition 3.7) to make the dual sequence ! ; j 1, unique.
j
(cid:21)
In 3, we (cid:12)rst review the basic properties of parabolic surfaces. Next, the canonical
x
map from X to IP (cid:10) (X) is constructed and we show that (Proposition 3.26) that it is
(cid:3)
injective and immersive whenever X has (cid:12)nite ideal boundary and is not hyperelliptic. The
(cid:0) (cid:1)
Abel-Jacobi map is also discussed (Proposition 3.30).
In 4, we de(cid:12)ne formal theta series of the argument z C for all in(cid:12)nite matrices
1
x 2
R with Rt = R and ImR > 0. We prove
6
Theorem 4.6 Suppose that the symmetric matrix R and the sequence t (0;1) ; j 1 ,
j
2 (cid:21)
satisfy
(cid:12)
t <
j 1
j 1
X(cid:21)
for some 0 < (cid:12) < 1 , and
2
n;ImRn = n ImR n 1 logt n2
h i i i;j j (cid:21) 2(cid:25) j jj j
i;j 1 j 1
X(cid:21) X(cid:21)
and all vectors n ZZ with only a (cid:12)nite number of nonzero components. Let B be the
1
2
Banach space given by
B = (z = (z1;z2;(cid:1)(cid:1)(cid:1)) 2 C1 jlim jlojzgjtjjj = 0)
(cid:12) !1
(cid:12)
(cid:12)
with norm
z = sup jzjj
k k j 1 jlogtjj
(cid:21)
Then, for every point w B the theta series
2
(cid:18)(z;R) = e2(cid:25)i z;n e(cid:25)i n;Rn
h i h i
nnX2<ZZ1
j j 1
converges absolutely and uniformly on the ball in B of radius r = 1 2(cid:12) centered at w to a
(cid:0)
8(cid:25)
holomorphic function.
Suppose
= !
X j
R
(cid:18)ZBi (cid:19)
is the period matrix of the marked Riemann surface (X;A ;B ; ) on which there is an
1 1
(cid:1)(cid:1)(cid:1)
exhaustion function with (cid:12)nite charge. In Proposition 4.4, we put a geometric condition
on the cycles A ;A ; , that implies the hypothesis of Theorem 4.6 for . We also
1 2 X
(cid:1)(cid:1)(cid:1) R
derive some properties of these theta series that will be used in part IV to prove that all
smooth spatially periodic solutions of the Kadomcev-Petviashvilli equation propagate almost
periodically in time.
Part I is completed by Appendix S, which contains a summary of the results of this
Part.
7
1 Square Integrable One Forms
x
In this section we develop L2-cohomology for an open Riemann surface marked with
a canonical homology basis. Our goal (Theorem 1.17) is to derive a criterion for the existence
of a unique, normalized basis for the Hilbert space of all square integrable holomorphic
one forms. In the next section, the concept of an exhaustion function with (cid:12)nite charge is
introduced. It is ultimately shown (Theorem 3.8) that a marked Riemann surface on which
there is an exhaustion function with bounded charge satis(cid:12)es our criterion.
Let (cid:21) be a measureable one form on the Riemann surface X. Suppose z = x1+ix2
is a coordinate on U X and write
(cid:26)
(cid:21) = f dx1 +f dx2
U 1 2
(cid:12)
Now set (cid:12)
(cid:21) = f dx1 +f dx2
(cid:3) U (cid:0) 2 1
(cid:12)
We can use the Cauchy-Riemann equations to derive
(cid:12)
Proposition 1.1 Let (cid:21) be a measureable one form on the Riemann surface X. The local
one forms (cid:21) ; (U;z) a coordinate system on X, are consistent and de(cid:12)ne a global
(cid:3) U
measureable one form (cid:21) on X. Furthermore,
(cid:12)
(cid:12) (cid:3)
(cid:21) = (cid:21)
(cid:3)(cid:3) (cid:0)
and
(cid:21) (cid:21) = f 2 + f 2 dx1 dx2
^(cid:3) U j 1j j 2j ^
(cid:12) (cid:0) (cid:1)
(cid:12)
Proof: Change coordinates to w = y1+iy2 on V X . Then,
(cid:26)
(cid:21) = f @x1 +f @x2 dy1 + f @x1 +f @x2 dy2
U V 1@y1 2@y1 1@y2 2@y2
\
(cid:12) = g(cid:0) dy1 +g dy2 (cid:1) (cid:0) (cid:1)
1 2
(cid:12)
Applying the Cauchy-Riemann equations
(cid:21) = f @x1 +f @x2 dy1 + f @x1 +f @x2 dy2
(cid:3) U V (cid:0) 2@y1 1@y1 (cid:0) 2@y2 1@y2
\
(cid:12) = (cid:0) f @x2 f @x1(cid:1)dy1 +(cid:0)f @x2 +f @x1 d(cid:1)y2
(cid:12) (cid:0) 2@y2 (cid:0) 1@y2 2@y1 1@y1
= (cid:0) g dy1 +g dy2 (cid:1) (cid:0) (cid:1)
2 1
(cid:0)
8
Let L2(X;T X) be the Hilbert space of all measureable one forms (cid:21) on X satis-
(cid:3)
fying
(cid:21) 2 = (cid:21) (cid:21) <
k k ^(cid:3) 1
ZX
with inner product
(cid:21);(cid:22) = (cid:21) (cid:22)
^(cid:3)
ZX
and let (cid:10) (cid:11)
= (cid:21) L2(X;T X) (cid:21) d’ = 0 for all ’ C (X)
CX 2 (cid:3) ^ 2 01
n (cid:12) ZX o
= the orthogonal com(cid:12)plement of dC (X) in L2(X;T X)
(cid:12) (cid:3) 01 (cid:3)
be the closed subspace of all weakly closed forms. Here, dC (X) denotes the closure of
(cid:3) 01
dC (X) in L2(X;T X) .
(cid:3) 01 (cid:3)
Example 1.2 Recall that
dr
dlogr =
r
dr x dx +x dx
2 1 1 2
dlogr = (cid:3) = (cid:0) = d(cid:18)
(cid:3) r r2
where r = z on C 0 . If X is any open subset of C on which r is bounded away from
j j nf g
zero and in(cid:12)nity, then
dr d(cid:18)
dlogr dlogr = ^ <
^(cid:3) r 1
ZX ZX
To prepare for the construction of normalized, square integrable holomorphic one
forms on noncompact surfaces, we quickly review elementary Hodge theory.
Observe that
dC (X) = the closure of dC (X) in L2(X;T X)
01 01 (cid:3) (cid:26) CX
since
d d’ = d( d’) = 0
^
ZX ZX
for all ; ’ C (X) . By de(cid:12)nition, the Hodge-Kodaira cohomology space H1 (X) is
2 01 HK
the orthogonal complement of dC (X) in . That is,
01 CX
= H1 (X) dC (X)
CX HK (cid:8) 01
or, equivalently,
H1 (X) = dC (X)
HK CX 01
(cid:14)
9
