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arXiv:1011.2875v1 [math.DG] 12 Nov 2010

FLOWS OF CONSTANT MEAN CURVATURE TORI IN THE 3-SPHERE:

THE EQUIVARIANT CASE

M. KILIAN, M. U. SCHMIDT AND N. SCHMITT

Abstract. We present a deformation for constant mean curvature tori in the 3-sphere. We show

that the moduli space of equivariant constant mean curvature tori in the 3-sphere is connected, and

we classify the minimal, the embedded, and the Alexandrov embedded tori therein.

Introduction

The euclidean 3-sphere S3admits compact embedded minimal surfaces of any genus [19, 15, 14]. For

simple examples like a great 2-sphere or the (minimal) Cliﬀord torus there is a smooth deformation

through constant mean curvature (cmc) surfaces with the same topology, which can be expressed

in terms of changing radii. For other minimal tori, as well as higher genus compact embedded

minimal surfaces in S3it might be useful to have a general deformation technique. We do this for

the case of cmc tori, and present a smooth topology preserving deformation for cmc tori in S3in

Theorem 1.2. By this theorem the moduli space of cmc tori in S3is locally one dimensional. To get

an idea of the global structure of the moduli space we turn to equivariant cmc tori in S3[12, 28, 4],

which are either ﬂat or a truncation of some member of an associated family of a Delaunay surface

[7]. We show that the moduli of equivariant cmc tori in S3is a connected inﬁnite graph whose edges

are parameterized by the mean curvature, and by ﬂowing through this moduli space of equivariant

cmc tori we classify the minimal, the embedded, and the Alexandrov embedded tori therein.

Amongst harmonic maps [24, 29, 30] there is an important class consisting of harmonic maps of

ﬁnite type [3, 11, 23]. To a harmonic map of ﬁnite type there corresponds an associated algebraic

curve whose compactiﬁcation is called the spectral curve. The genus of the curve is called the

spectral genus, and denoted by g. The crucial fact that makes it possible to adapt the Whitham

deformation technique [20, 18, 10] to the case of cmc tori in S3, is that a cmc torus in S3always has

ﬁnite spectral genus [11, 23]. The spectral curve of a cmc torus is a double cover of the Riemann

sphere with 2g+ 2 many branch points. During the deformation the spectral curve changes such

that two branch points remain ﬁxed, while the other 2gbranch points may move around. The

closing conditions involve a choice of two double points on the real part of the spectral curve: we

call these the sym points, and to ensure that the topology of the surface stays intact during the

deformation, the ﬂow of the sym points has to be controlled.

In the smooth deformation family of the Cliﬀord torus there is a Z-family which allow a deformation

into absolute cohomogeneity one rotational embedded cmc tori. Such a deformation is possible

when in addition to the two sym points there is a further double point on the real part of the

spectral curve. By opening up this additional double point and moving the resulting two branch

points oﬀ the real part, the spectral curve becomes a double cover of the Riemann sphere branched

now at four points: It has spectral genus g= 1 and is the spectral curve of a Delaunay surface, and

the corresponding cmc torus is a truncation of a Delaunay surface in S3. We show that at the end

of the ﬂow the new branch points pair-wise coalesce with the two ﬁxed branch points. Hence in the

limit the coalescing pairs of branch points disappear, and the limit curve is an unbranched double

cover of the sphere: the spectral curve of a bouquet of spheres. In the rotational case, see also [13],

2000 Mathematics Subject Classiﬁcation. Primary 53A10, 53C42; Secondary 58E20. November 15, 2010.

1

2 M. KILIAN, M. U. SCHMIDT AND N. SCHMITT

our deformation corresponds to pinching the neck of a Delaunay surface, starting at a ﬂat torus and

continuing through to a bouquet of spheres. Thus the connected component of the Cliﬀord torus

is an inﬁnite comb: The spine (g= 0) consists of embedded ﬂat cmc tori parameterized by the

mean curvature, and each tooth (g= 1) of embedded Delaunay tori ends in a bouquet of spheres.

By considering covers of Cliﬀord tori the moduli space of rotational cmc tori is a Z2–family of

such combs. It turns out that each bouquet of spheres occurs exactly twice in this moduli space

(Theorem 4.4), so that we may glue the two families together there. Thus the moduli space of

rotational cmc tori in S3is an inﬁnite connected graph.

A similar picture transpires in the non-rotational case. In each isogeny class there is a sequence

of g= 0 tori that can be deformed into g= 1 tori. In the non-rotational case a g= 1 defor-

mation family stays away from bouquets of spheres, and we prove that every g= 1 deformation

family begins and ends at a g= 0 torus (Theorem 4.5). The above results combined give that

every deformation family of absolute cohomogeneity one cmc tori ends at a ﬂat cmc torus. The

classiﬁcation of absolute cohomogeneity one cmc tori is thus reduced to that of spectral curves of

ﬂat cmc tori with a double point on the real part; this initial data is classiﬁed and interpreted

geometrically. We classify the equivariant minimal tori, as well as the embedded and Alexandrov

embedded equivariant cmc tori, and prove that the minimal Cliﬀord torus is the only embedded

minimal equivariant torus in the 3-sphere. We also show that the spectral curve of an equivariant

cmc torus has no double points oﬀ the real part (Theorem 2.9), which implies that there can not

be a Bianchi-B¨acklund transform of an equivariant cmc torus into a cmc torus. We conclude the

paper by proving that the moduli space of equivariant cmc tori in S3is connected. Throughout

the text we provide graphics of some of the surfaces under discussion. More images and videos of

deformation families can be viewed at the website [25].

1. Spectral curve

We start by recalling the description of cmc tori in terms of spectral curves and abelian diﬀerentials

[2, 11, 21, 16]. We then adapt a deformation technique from [10] and prove that any generic cmc

torus in S3lies in a smooth family of cmc tori in S3.

Let Y be a hyperelliptic Riemann surface with meromorphic function λof degree two and with

branch points over λ= 0 (y+) and λ=∞(y−). Then Y is the spectral curve of an immersed cmc

torus in S3if and only if the following four conditions hold:

(1) Besides the hyperelliptic involution σ, the surface Y has two further anti-holomorphic in-

volutions ηand ̺=η◦σ=σ◦η, such that ηhas no ﬁx points and η(y+) = y−.

(2) There exist two non-zero holomorphic functions µ1, µ2on Y \{y+, y−}such that for i= 1,2

µi◦σ=µ−1

iµi◦η= ¯µiµi◦̺= ¯µ−1

i.

(3) The forms dln µiare meromorphic diﬀerentials of the second kind with double poles at y±.

The singular parts at y+respectively y−of these two diﬀerentials are linearly independent.

(4) There are four ﬁxed points y1, y2=σ(y1), y3, y4=σ(y3) of ̺, such that the functions µ1

and µ2are either 1 or −1 there.

Deﬁnition 1.1. Given the spectral curve of an immersed cmc torus in S3, let λ1∈S1denote the

value λ(y1) = λ(y2), and λ2∈S1the value λ(y3) = λ(y4)at the four ﬁxed points of ̺where µ1and

µ2are either 1or −1. We call λ1and λ2the sym points.

For =√−1, the mean curvature of the corresponding cmc torus in terms of the sym points is

(1.1) H=λ2+λ1

λ2−λ1

.

FLOWS OF EQUIVARIANT CMC TORI IN S33

We shall describe spectral curves of cmc tori in S3via hyperelliptic surfaces of the form

ν2=λ a(λ)

where a∈Cg[λ] is a polynomial of degree gand

¯

λ−2g¯a(λ) = a(¯

λ−1) and λ−ga(λ)≥0 for |λ|= 1.

The involutions of the spectral curve are

σ(λ, ν) = (λ, −ν),

η(λ, ν) = (¯

λ−1,−¯ν¯

λ−g−1),

̺(λ, ν) = (¯

λ−1,¯ν¯

λ−g−1).

(1.2)

Making use of a rotation of λand a rescaling of νwe may assume that ais a polynomial with

highest coeﬃcient one. The meromorphic diﬀerentials dln µihave the form

(1.3) dln µi:= πbi(λ)

ν

dλ

λ

with polynomials bi∈Cg+1[λ] of degree g+ 1 and ¯

λ−g−1¯

bi(λ) = bi(¯

λ−1).

We next describe a one parameter family of deformations of the spectral curve, that depends on a

deformation parameter t. We view all functions on the corresponding spectral curves as functions

in the variables λand t.

Since the path integrals of the diﬀerentials dln µialong all cycles in H1(Y,Z) are integer multiples

of 2π, these diﬀerentials do not depend on the deformation parameter t. Further, see [22] (ch.3,

Prop. 1.10), a meromorphic function fon a hyperelliptic Riemann surface given by ν2=h(λ)

is of the form f(λ) = r(λ) + ν s(λ) with rational functions r, s. Hence both ∂tln µiare global

meromorphic functions on Y with only possible poles at the branch points of Y. More precisely

these meromorphic functions are of the form

(1.4) ∂tln µi=πci(λ)

ν

with polynomials ci∈Cg+1[λ] of degree g+ 1 and ¯

λ−g−1¯ci(λ) = ci(¯

λ−1).

Integrability ∂2

tλ ln µi=∂2

λt ln µireads ∂tλ−1ν−1bi=∂λν−1ciwhich yields

(1.5) 2a(λ)˙

bi(λ)−˙a(λ)bi(λ) = (2λa(λ)c′

i(λ)−λa′(λ)ci(λ)−a(λ)ci(λ)).

Here dash and dot denote the derivatives with respect to λrespectively t.

The diﬀerential

Ω=(∂tln µ1)dln µ2−(∂tln µ2)dln µ1

is a meromorphic 1-form on Y with only poles at most of order three at λ= 0,∞, and roots at the

sym points λ1, λ2. Further, since η∗¯

Ω = Ω, ̺∗¯

Ω = Ω and σ∗Ω = Ω, we conclude that

Ω = π2C1

(λ−λ1)(λ−λ2)

λ√λ1λ2

dλ

λ

with a real function C1= C1(t). Using (1.3) and (1.4) we obtain

(1.6) b1(λ)c2(λ)−b2(λ)c1(λ) = C1a(λ)(λ−λ1)(λ−λ2)

√λ1λ2

,

and combining the above gives

(1.7) (∂tln µ1)dln µ2−(∂tln µ2)dln µ1=π2C1

(λ−λ1)(λ−λ2)

λ√λ1λ2

dλ

λ.

4 M. KILIAN, M. U. SCHMIDT AND N. SCHMITT

To preserve the topology during the ﬂow, the values of ln µishould be ﬁxed at the two sym points

λ=λ1and λ=λ2. Consequently ∂tln µi|λ=λj=(∂λln µi)˙

λ+∂tln µi|λ=λj= 0, which gives

(1.8) ˙

λj=−∂tln µi

∂λln µiλ=λj

.

Theorem 1.2. Let Ybe a genus gspectral curve of a cmc torus in S3. If the two diﬀerentials dln µi

for i= 1,2have no common roots, then Yis contained in a unique smooth one dimensional family

of spectral curves of cmc tori in S3.

Proof. If the roots of b1and b2are pairwise distinct, then the 2g+2 values of equation (1.6) at these

roots uniquely determine the values of c1and c2there. Therefore the equation (1.6) determines the

polynomials c1and c2uniquely up to a real multiple of b1and b2. The choice c1=b1and c2=b2

in (1.4) and (1.5) corresponds to a rotation of λ. For given c1and c2the equations (1.5) determine

uniquely the derivatives of all roots of the polynomials aand bi. The condition ¯

λ−2g¯a(λ) = a(¯

λ−1)

together with the assumption that the highest (and lowest) coeﬃcient of the polynomial ahas

absolute value one determines the polynomial ain terms of its roots up to multiplication with ±1.

We conclude that these conditions on adetermine ˙a,˙

b1and ˙

b2in terms of c1and c2. We remark

that due to equation (1.6) the solutions ˙aof both equations (1.5) coincide. Finally the condition

that the highest (and lowest) coeﬃcient of ais qual to one, ﬁxes the rotations. Therefore there

exist unique solutions c1and c2of (1.6).

We expect that one can pass through common zeroes of the b′

is, therefore making the deformation

global, but this will be considered elsewhere. Below we will restrict to the cases of spectral genera

g= 0, where this is trivially true, and g= 1 where the roots of b1and b2turn out to always be

distinct throughout the ﬂow (Corollary 2.3).

2. Flows of equivariant cmc tori in the 3-sphere

The double cover of the isometry group of S3∼

=SU2is SU2×SU2via the action P7→ F P G−1.

A surface is equivariant if it is preserved set-wise by a one-parameter family of isometries. To

an equivariant surface we associate two axes: These are geodesics which are ﬁxed set-wise by the

one-parameter family of isometries. An equivariant surface is rotational precisely when one of

its axes is ﬁxed point-wise. Equivariant cmc surfaces have spectral genus zero or one [4]. By a

theorem of DoCarmo and Dajczer [7], an equivariant cmc surface is a member of an associated

family of a Delaunay surface, so up to isometry determined by an elliptic modulus and its sym

points. Hence an equivariant cmc surface in the 3-sphere is parameterized by its elliptic modulus,

its mean curvature, and its associated family parameter. We will express the diﬀerential equations

(1.7) and (1.8) in terms of the three coordinates

(2.1) (k,q,h) ∈[−1,1]3

where k is the elliptic modulus, and q,h are deﬁned in terms of the sym points as

(2.2) q := 1

2pλ1λ2+ 1/pλ1λ2and h := 1

2 rλ1

λ2

+rλ2

λ1!.

The mean curvature Hin (1.1) can then be expressed as

(2.3) H=h

p1−h2.

FLOWS OF EQUIVARIANT CMC TORI IN S35

The following identities will be used below:

2pq2−1 = pλ1λ2−1/pλ1λ2,2ph2−1 = pλ1/λ2−pλ2/λ1,

4(q2−h2) = λ1−λ−1

1λ2−λ−1

2,4 q pq2−1 = λ1λ2−λ−1

1λ−1

2

4 h pq2−1 = λ1−λ−1

1+λ2−λ−1

2,4 q ph2−1 = λ1−λ−1

1−λ2+λ−1

2

4 h ph2−1 = λ1λ−1

2−λ−1

1λ2.

(2.4)

The derivatives of q,h with respect to the ﬂow parameter are

(2.5) ˙q = pq2−1

2 ˙

λ1

λ1

+˙

λ2

λ2!,˙

h = ph2−1

2 ˙

λ1

λ1−˙

λ2

λ2!.

2.1. Spectral genus zero. In the spectral genus zero case, a≡1, the spectral curve is ν2=λ, and

the elliptic modulus k ≡ ±1. The functions biin (1.3) for i= 1,2 are polynomial in λof degree

one, and from ¯

λ−1¯

bi(λ) = bi(¯

λ−1) we conclude that bi=βiλ+¯

βifor some smooth complex valued

functions t7→ βi(t). Thus dln µiintegrates to

ln µj=2π(βjλ−¯

βj)

ν,

and consequently the functions ci, i = 1,2 in (1.4) are ci= 2 (˙

¯

βi−˙

βiλ). We may ﬁx one of the sym

points, and assume without loss of generality that λ1≡1 during the ﬂow. Then ∂tln µj|λ=1 = 0,

which is equivalent to ci(λ= 1) = 0, or equivalently ˙

βi∈R. Thus equation (1.7) reads

2˙

β2(β1λ+¯

β1)−2˙

β1(β2λ+¯

β2) = C1

√λ2

(λ2−λ).

Evaluating this at λ=−¯

βi/βigives

˙

βi=C1(βiλ1/2

2+¯

βiλ−1/2

2)

2 ( ¯

β1β2−β1¯

β2)

From (1.8) we get

˙

λ2

λ2

=c1

b1

=2˙

β1(1 −λ2)

β1λ2+¯

β1

=C1

λ1/2

2−λ−1/2

2

(β1¯

β2−¯

β1β2)

Set C1:= (β1¯

β2−¯

β1β2). Then together with (2.4) and the fact that q = h when λ1≡1, the ﬂow

equations (1.7) and (1.8) reduce to the single equation ˙

h = h2−1. The solution to this equation

is given by h(t) = −tanh(t+C) with some constant of integration C∈R. Consequently, the

argument of the sym point arg[λ2] = 2 arccos(−tanh(t)) can vary over all of the interval (0,2π),

and is strictly monotonic. Hence also the mean curvature Hby equation (2.3) is strictly monotonic,

and given by H(t) = −sinh(t). In summary, we have proven the following

Theorem 2.1. Every ﬂat cmc torus in the 3-sphere lies in a smooth R–family of ﬂat cmc tori. Each

family is parameterized by the mean curvature.

2.2. Spectral genus one. In the spectral genus one case, the meromorphic diﬀerentials dln µiare

linear combinations of the derivative of a meromorphic function and an elliptic integral. We write

these meromorphic diﬀerentials in terms of Jacobi’s elliptic functions. We deﬁne an elliptic curve

by

(2.6) 4ν2= (λ−k)(λ−1−k).

Here k ∈[−1,1] is the real modulus. Then

(2.7) dν = k λ−1−λ

8ν

dλ

λ,

6 M. KILIAN, M. U. SCHMIDT AND N. SCHMITT

and the only roots of dν are at λ= k,k−1. In addition to the three involutions in (1.2), the elliptic

curve (2.6) has a further holomorphic involution

(2.8) τ(λ, ν) = (λ−1, ν ).

We decompose ln µi

πinto the symmetric and skew symmetric part with respect to τ. The symmetric

part is a real multiple of the single valued meromorphic function ν, and the skew symmetric part

is a real multiple of a multi valued function ωwith real periods. The ﬁrst homology group of the

elliptic curve (2.6) is generated by a cycle around the two branch points at λ= k±1and the cycle

S1={λ| |λ|= 1}. The ﬁrst cycle is symmetric with respect to τand the second skew symmetric.

Therefore the integral of dω along the ﬁrst cycle vanishes. We assume that the integral of dω along

the second cycle is equal to 2. Together with the ﬁrst order poles of ωat y±and the skew symmetry

with respect to τthis normalization determines ωuniquely. Further, since RS1dln µi∈2πZ, we

conclude that the functions ln µiare linear combinations of νand ω, and given by

(2.9) ln µ1=π(x1ν+p10 ω),ln µ2=π(x2ν+p20 ω) for p10, p20 ∈Z.

We shall express ωas a linear combination of complete elliptic integrals of the ﬁrst and second

kind. First we shall relate the curve (2.6) to the elliptic curve Y in Legendre’s form with elliptic

modulus 1−k

1+k given by

y2=1−x21−(1−k

1+k )2x2.

Let Kand Edenote the complete integrals of the ﬁrst and second kind

K(k) := Z1

0

dx

√(1−x2)(1−k2x2),E(k) := Z1

0s1−k2x2

1−x2dx.

Since Ehas ﬁrst order poles at the two points over x=∞, these two points correspond to y±. The

involution τcorresponds to (x,y)7→ (x,−y). Therefore we set

λ=1 + (1−k

1+k )2(1 −2x2)−21−k

1+k y

1−(1−k

1+k )2, λ−1=1 + (1−k

1+k )2(1 −2x2) + 21−k

1+k y

1−(1−k

1+k )2, ν =

1−k

1+k x

1 + 1−k

1+k

.

The integrals of the meromorphic diﬀerentials dln µialong all cycles of Y are purely imaginary.

Deﬁne

(2.10) dω := E(1−k

1+k )−K(1−k

1+k )1−(1−k

1+k )2x22dx

πy.

The cycle around the two branch points λ= k±1corresponds to the real period and the cycle S1

to the imaginary period of Jacobi’s elliptic functions. The integral along the real period vanishes

and due to Legendre’s relations [1, 17.3.13] the integral along the imaginary period is equal to 2.

In summary, we have found a linear combination of elliptic integrals of the ﬁrst and second kind

which obeys the conditions that characterize ω.

In terms of the complete elliptic integrals K′=K′(k) = K(p1−k2) and E′=E′(k) = E(p1−k2)

and the functions λand ν, and the formulas [1, 17.3.29 and 17.3.30]

E(1−k

1+k ) = (1 + 1−k

1+k )E′+ (1 −1−k

1+k )K′,K(1−k

1+k ) = 2

1 + 1−k

1+k

K′,

the diﬀerential dω in (2.10) simpliﬁes to

(2.11) dω =2E′−kK′(λ+λ−1)

4πν

dλ

λ.

Proposition 2.2. The complete elliptic integrals K′and E′satisfy

1≤2E′

1 + k2<K′<E′

|k|for 0<|k|<1.

FLOWS OF EQUIVARIANT CMC TORI IN S37

Proof. Assume ﬁrst that k ∈(0,1). Since τ∗dω =−dω, we have that

Zλ=k−1

λ=k

dω = 2 Zλ=1

λ=k

dω = 0 .

Hence the function 2E′−kK′(λ+λ−1) has a real root in λ∈(k,1), say at r∈(k,1), and thus

also at r−1∈(1,k−1). Then 2 < r + 1/r < k + 1/k, and together with 2E′−kK′(r+r−1) = 0

gives 2E′>2 k K′and 2E′<K′(1 + k2). Finally, from 2 >k2+1 and E′≥1 (13.8.11 in [9]), we

obtain 2E′≥1 + k2.

For k ∈(−1,0), the function 2E′−kK′(λ+λ−1) has a pair of reciprocal real roots s, s−1with

−1< s < k−1, and analogous arguments prove the assertion in this case.

Corollary 2.3. The diﬀerentials dν and dω have no common zeroes.

Proof. While dν in (2.7) has only roots at λ= k,k−1, we saw in the proof of Proposition 2.2

that dω has only roots λ∈(k,1) ∪(1,k−1) when k ∈(0,1), and λ∈(−1,k) ∪(k−1,−1) when

k∈(−1,0).

Theorem 2.4. Every spectral genus one cmc torus in S3lies on an integral curve of the vector ﬁeld

(2.12)

˙

k

˙q

˙

h

=

k (E′q−kK′h)

1−q2

1−k2((1 + k2)E′−2 k2K′)

k1−h2

1−k2(2E′−(1 + k2)K′))

.

The vector ﬁeld (˙

k,˙q,˙

h) is analytic on the set D:= {(k,q,h) ∈R3|k6= 0}, and its zero set is

{k2= 1} ∩ {q = k h} ∩ D.

Proof. We shall calculate the diﬀerential equations (1.7) and (1.8) in terms of the coordinates (2.1).

From (2.6) we compute the derivative of νwith respect to t, and obtain

(2.13) ˙ν=˙

k2 k −λ−λ−1

8ν.

In agreement with the skew-symmetry of ωwith respect to τwe set

˙ω= C2

λ−λ−1

4π ν .

From (2.9) we obtain ∂tln µj=π( ˙xjν+xj˙ν+pj0˙ω) and dln µj=π(xjdν +pj0dω) for j= 1,2.

Putting all this together, the diﬀerential Ω = (∂tln µ1)dln µ2−(∂tln µ2)dln µ1reads

Ω = π2( ˙x1x2−˙x2x1)ν ν′+ ( ˙x2p10 −˙x1p20 )ν ω′+ (p10 x2−p20x1)( ˙ν ω′−ν′˙ω)with

ν ν′=k (λ−1−λ)

8λ, ν ω′=2E′−k (λ+λ−1)K′

4π λ ,

˙ν ω′−ν′˙ω=C2k(λ−λ−1)2+˙

k (λ−2 k +λ−1)(k (λ+λ−1)K′−2E′)

8π λ (λ−k)(λ−1−k) .

Note that while ˙ν ω′−ν′˙ωhas simple poles at λ= k,k−1, but Ω does not, we conclude that the

numerator of ˙ν ω′−ν′˙ωmust vanish at λ= k,k−1, giving

C2=˙

k(k2+1) K′−2E′

k2−1.

8 M. KILIAN, M. U. SCHMIDT AND N. SCHMITT

The unique solution of equation (1.7) can now be computed, and is given by

˙

k = 4πC1(k2−1)(k h K′−qE′)

(x1p20 −x2p10)(E′2−k2K′2),˙x1x2−˙x2x1=8 C1p1−q2

k,

˙x1p20 −˙x2p10 =4πC1(k q(E′−K′) + h(E′−k2K′))

E′2−k2K′2.

Since ˙ω=∂kω˙

k, the above imply that

(2.14) ∂ω

∂k=(1 + k2)K′−2E′

4π(1 −k2)ν(λ−λ−1).

Now setting

C1:= −k (x1p20 −x2p10)(E′2−k2K′2)

4π(k2−1)

gives ˙

k = k(E′q−kK′h) as required.

The deformation equations of the sym points (1.8) read

˙

λj(x1ν′+p10 ω′) = −( ˙x1ν+x1˙ν+p10 ˙ω)

˙

λj(x2ν′+p20 ω′) = −( ˙x2ν+x2˙ν+p20 ˙ω)

Multiplying the ﬁrst equation by p20 and the second equation by p10 and subtracting gives

˙

λj=−( ˙x1p20 −˙x2p10)ν+ (x1p20 −x2p10) ˙ν

(x1p20 −x2p10)ν′λ=λj

.

Using the above formulae as well as (2.4) we obtain

˙

λ1

λ1

+˙

λ2

λ2

=2pq2−1

h2−q2k q(E′−K′) + h(E′−k2K′)

1−k2(2 k q −h(k2+1)) + (q E′−h k E′)(q −h k)

=2p(q2−1)

k2−1(1 + k2)E′−2 k2K′,

˙

λ1

λ1−˙

λ2

λ2

=2ph2−1

h2−q2k q(E′−K′) + h(E′−k2K′)

1−k2((k2+1) q −2 h k) + (q E′−h k E′)(k q −h)

=2ph2−1

k2−1k2E′−(1 + k2)K′

and putting these into (2.5) gives the equations for ˙q and ˙

h, and concludes the proof of (2.12).

The elliptic integrals K′and E′are analytic on k ∈R×, and at k = ±1 equal to π

2. Therefore the

right hand sides of (2.12) extend analytically to k ∈R×. Due to Proposition 2.2, for q,h∈(−1,1)

we have

˙q >0 for 0 <|k|<1,˙

h>0 for −1<k<0,˙

h<0 for 0 <k<1.(2.15)

By the properties of K′and E′the vector ﬁeld ( ˙

k,˙q,˙

h) is analytic in k on R×and has simple zeros

at k = ±1. Thus the vector ﬁeld is analytic on D. The zero set statement follows from the fact

that on {k6= 0}, the functions (1 + k2)E′−2 k2K′and 2E′−(1 + k2)K′have zeros only at k = ±1,

and E′= k K′holds only at k = 1. That k = ±1 is a simple root follows from the series expansions

at k = 1 (similarly at k = −1), given by

K′(k) = π1

2−1

4(k −1) + 5

32 (k −1)2−7

64 (k −1)3+ O(k −1)4,

E′(k) = π1

2+1

4(k −1) + 1

32 (k −1)2−1

64 (k −1)3+ O(k −1)4.

FLOWS OF EQUIVARIANT CMC TORI IN S39

Figure 2.1. Equivariant (2,1, n)cmc tori (n= 3,4,5). By Proposition 5.3, there are no twizzled

tori with one or two major lobes.

2.3. Global behaviour of the integral curves. The proof of the following Proposition 2.5 is deferred

to section 3.

Proposition 2.5. An equivariant cmc torus in S3is rotational if and only if the sym points are

reciprocal.

Special values of q and h include

q2= 1 ⇐⇒ λ1=λ−1

2(rotational),h2= 1 ⇐⇒ λ1=λ2(H=∞),

q = 0 ⇐⇒ λ1=−λ−1

2,h = 0 ⇐⇒ λ1=−λ2(H= 0) ,

q = h ⇐⇒ λ1= 1 or λ2= 1 ,q = −h⇐⇒ λ1=−1 or λ2=−1.

By the deformation equations (2.12) if q2(t0) = 1, for some t0, then q2≡1 throughout the ﬂow.

Hence rotational tori stay rotational during the ﬂow. We begin the qualitative analysis of the ﬂow

by ﬁrst considering equivariant cmc tori which are not rotational (q2<1): we call these twizzled

tori. The tori of revolution (q2= 1) are treated subsequently in Proposition 2.8.

The ﬂow will be investigated in the open solid cuboid

B:= {(k,q,h) ∈(−1,1)3|k6= 0}.

Due to (2.15) q −sign(k) h is strictly monotonic on Bwith the locally constant function sign(k).

Proposition 2.6. Deﬁne the set

Lc:= {(k,q,h) ∈ B| (1 −q2)(1 −h2) = c(1+k2

2 k −q h)2}, c ∈R+.

(1) Then every integral curve in Blies in Lc∩ B for some c∈(0,1).

(2) The following uniform estimates hold on the integral curve through (k0,q0,h0)∈ Lc:

|k| ≥ √c

2 + 2√cand min{1−q2,1−h2} ≥ c(min{1− | q0|,1− |h0|})2

(3) The function ˙

khas at most one zero along any integral curve in B.

Proof. (1) From (2.12) we compute Wt[(1 −q2)(1 −h2),(1+k2

2 k −q h)2] = 0, where Wt[X, Y ] :=

X˙

Y−˙

XY is the Wronskian with respect to the ﬂow parameter t.

Since (1 −q2)(1 −h2) and (1+k2

2 k −q h)2are strictly positive in B, then every integral curve in Blies

in Lcfor some c∈R+. In B, we have (1 −q2)(1 −h2)≤(1 −q h)2<(1+k2

2 k −q h)2, so c∈(0,1).

10 M. KILIAN, M. U. SCHMIDT AND N. SCHMITT

(2) For (k,q,h) ∈ Lcthe ﬁrst inequality of follows from

1

|k|≤1 + k2

|k|≤2s(1 −q2)(1 −h2)

c+ 2|q h | ≤ 2 + 2√c

√c.

On each integral curve k is either positive or negative. The second inequality follows from

min{1−q2,1−h2} ≥ (1 −q2)(1 −h2)≥c(1 −sign(k) q h)2≥c(max{1− |q|,1− |h|})2.

In fact, due to (2.15) either sign(k) q h ≤0 or one of the functions 1 − |q|and 1 −| h|is increasing

and the other one decreasing. Furthermore, if at some point twith sign(k) q h >0 one of these

functions is increasing, it stays increasing for all t0≤t, and if it is decreasing, it stays decreasing

for all t0≥t, since the derivative of these functions can change sign only at the maximal value 1.

(3) If ˙

k(t0) = 0 for some value of the ﬂow parameter t=t0, then due to (2.12) and (2.15)

sign(k)¨

k = sign(k) d

dt k(E′q−kK′h) = |k|(E′˙q −kK′˙

h)) >0

for t=t0. Thus sign(k)˙

k is increasing at each of its zeros, and hence can have at most one zero.

The next result shows that one endpoint of each integral curve corresponds to a ﬂat cmc torus.

Proposition 2.7. Every maximal integral curve of (2.12) in Bis deﬁned on a ﬁnite interval and

passes from a point in {k = ±1} ∩ {q−k h <0} ∩ ∂Bto a point in {k = ±1} ∩ {q−k h >0} ∩ ∂B

with the same sign of k. Furthermore, the mean curvature t7→ H(t)is a diﬀeomorphism of the

maximal interval of deﬁnition onto a ﬁnite interval.

Proof. Due to (2.15) both functions q and h are strictly monotonic. Since ˙q and ˙

h have no roots

in B, the integral curve must hit the boundary of Bat the end points (either inﬁnite or ﬁnite)

of the maximal interval of deﬁnition. Due to Proposition 2.6 (2), k takes the same value ±1 at

both endpoints and the vector ﬁeld (2.12) does not vanish there. Hence the maximal interval of

deﬁnition is ﬁnite. Due to Proposition 2.6 (3) the sign of ˙

k changes once and is proportional to

q−sign(k) h at the end points. Due to (2.15) q −sign(k) h is strictly increasing and the ﬁrst claim

follows. Since h 7→ h/p1−h2is strictly increasing on (−1,1), the mean curvature (2.3) is strictly

monotonic, and by Proposition 2.6 (2) it is bounded.

2.4. Global behaviour of the integral curves on the boundary. The boundary ∂Bconsists of the

three parts with k = ±1 or k = 0, q = ±1 and h = ±1. The ﬁrst set contains the ﬂat cmc tori

and the bouquets of spheres, which we consider later. The third set corresponds to the inﬁnite

mean curvature limit, that is cmc tori in R3. In this limit the two sym points coalesce and the

diﬀerentials dln µihave zeroes there. Since dν and dω do not have common zeroes by Corollary

2.3, there are no such examples with spectral genus zero or one. Therefore we treat only q2= 1

and h ∈(−1,1).

By Proposition 2.5 the tori of revolution appear in the two-dimensional boundary {q2= 1}of the

moduli space of equivariant cmc surfaces in S3. Since ˙q ≡0 along q2= 1, tori of revolution stay

tori of revolution throughout the ﬂow. The ﬂow (2.12) thus describes one parameter families of

tori of revolution. Later in Theorem 5.9 we describe the range of mean curvature for the families in

the ﬂow, and exhibit those that contain a minimal torus of revolution. A consequence of the next

result is that spectral genus g= 1 tori of revolution lie in 1-parameter families with one endpoint

at a ﬂat cmc torus and the other endpoint at a sphere bouquet. In the process we also show that

the mean curvature stays bounded during the ﬂow.

Since we often need to evaluate the functions νand ωat the two sym points we set

(2.16) νk=ν(λk) and ωk=ω(λk) for k= 1,2.

FLOWS OF EQUIVARIANT CMC TORI IN S311

Proposition 2.8. On the integral curve of (2.12) through (k0,q0,h0)∈∂Bwith q0=±1and

k0,h0∈(−1,1) the function qis equal to ±1and 1−h2is bounded away from zero. The maximal

interval of deﬁnition is of the form (−∞, tmax)with

lim

t↓−∞ k = 0 and lim

t↑tmax

k = ±1.

The mean curvature t7→ H(t)is a diﬀeomorphism from (−∞, tmax)onto a ﬁnite interval.

Proof. By (2.12) q ≡q0is constant throughout the ﬂow. Let λi=e2θifor i= 1,2 be the two sym

points. If |θ2−θ1|is bounded away form zero, then 1 −h2is bounded away from zero. For q = ±1

the values of νat the sym points coincide.

By (1.8) we have that ω2−ω1=Rθ2

θ1dω is constant throughout the ﬂow. We claim that

|dω| ≤ 2|dθ|for θ∈Rand k ∈[−1,1].

We will need to consider the two cases k cos(2θ)≤0 and k cos(2θ)≥0 separately. In the ﬁrst case

we use E′≤π

2[9, 13.8.(11)], Proposition 2.2 and (2.6) to obtain

|dω|=

E′−kK′cos(2θ)

πν |dθ| ≤

E′(1 −2 k

1+k2cos(2θ))

πν |dθ| ≤

2ν

1 + k2|dθ| ≤ 2|dθ|.

When k cos(2θ)≤0, then again by E′≤π

2, Proposition 2.2, and 2|ν| ≥ p1 + k2≥1 (2.6) we get

|dω|=

E′−kK′cos(2θ)

πν |dθ| ≤

4E′

πp1 + k2|dθ| ≤ 2|dθ|.

Thus |θ2−θ1| ≥ 1

2|ω2−ω1|. Due to (2.11) and Proposition 2.2, the diﬀerential dω has no root

between θ1and θ2. Therefore |θ2−θ1|and 1 −h2are bounded away from zero.

Due to Proposition 2.2, the function k2is strictly increasing for 0 <|k|<1. Therefore the maximal

integral curve passes from k = 0 to k = ±1. The vector ﬁeld is at the left end point of order O(k)

and at the right end point not zero. Therefore the maximal interval of deﬁnition has the form

(−∞, tmax).

Since h is strictly monotonic, and 1 −h2is bounded away from zero, the mean curvature (2.3) is a

diﬀeomorphism from (−∞, tmax) onto a ﬁnite interval.

2.5. Regularity of spectral curve. A corollary of Theorem 2.9 below is that equivariant cmc tori

cannot be dressed to tori by simple factors [17, 27]. While there exist large families of cmc cylinders

which can be dressed to cylinders with bubbletons [26], it is still an open question raised by Bobenko

[2], whether there are cmc tori with bubbletons. A double point is a point on the spectral curve at

which both logarithms µ1and µ2are unimodular, but the spectral curve is not branched.

Theorem 2.9. The spectral curve of an equivariant cmc torus has no double points in C×\S1.

Proof. At a double point both logarithms µ1and µ2are unimodular. Therefore it suﬃces to prove

that the set of all λ∈C×, where µ1and µ2are unimodular is the set S1∪ {k,k−1}. This set

coincides with the subset of (λ, ν )∈Y, such that νand ωare real. Due to (2.6) and (2.11) we

have for k ∈(0,1]: ν∈Rif and only if λ∈S1∪[k,k−1]∪R−, and λ∈Rand ω∈Rif and only if

λ∈[0,k] ∪[k−1,∞). For k ∈[−1,0): ν∈Rif and only if λ∈S1∪[k−1,k] ∪R+;λ∈Rand ω∈R

if and only if λ∈(−∞,k−1]∪[k,0]. With k = ±1 we cover the genus zero case.

12 M. KILIAN, M. U. SCHMIDT AND N. SCHMITT

3. Equivariant extended frames

To obtain a more geometric description of the above deformation we compute the frames of the

surfaces. We do this in a slightly broader context by studying equivariant maps C2→SL2Cwith

complex mean curvature as in [8]. This generalized framework for complex equivariant surfaces is

then applicable to not only cmc immersions into 3-dimensional space forms , but also pseudospher-

ical surfaces in R3,cmc surfaces in Minkowski space, and other integrable surfaces in many others

spaces by performing an appropriate reduction.

Let Σ ⊂C2be an open and simply-connected domain with complex coordinates (z, w), and deﬁne

∗on 1-forms on Σ by ∗dz =dz,∗dw =−dw extended complex linearly.

Let sl2C=k⊕pbe a Cartan decomposition. For a 1-form αon Σ, we will write α=αk+αpwhere

with αk∈kand αp∈p. If we equip the matrix Lie algebra sl2Cwith a multiple of the Killing form

hX, Y i:= −1

2tr(XY ), then we may choose a basis e0, e1, e2for sl2Csatisfying e0∈k,he0, e0i= 1

and [e0, e1] = e2, [e1, e2] = e0and [e2, e0] = e1, and set

ǫ1=1

2(e1−e2), ǫ2=1

2(e1+e2).

Let f: Σ →SL2Cbe a conformal immersion such that for the smooth function v: Σ →C∗we have

hfz, fzi= 0 = hfw, fwiand 2hfz, fwi=v2. The remaining invariants of fare smooth functions

H, Q, R : Σ →Csuch that 2hfzz , N i=Q, 2hfzw, N i=v2Hand 2hfww, N i=R, and the normal

map is N=−v−2[fz, fw]. Further, there exists a unique pair of maps F, G : Σ →SL2Cwhich

frame fin the sense that

(3.1) f=F G−1, fz=vF ǫ1G−1, fw=vF ǫ2G−1, N =F e0G−1.

Then α+1 := F−1dF and α−1:= G−1dG, are smooth 1-forms on Σ with values in sl2Cgiven by

(3.2)

ασ=Aσdz +Bσdw where σ∈ {±1}and

2Aσ:= v−1vze0+v(H+σ)ǫ1−v−1Qǫ2,

2Bσ:= −v−1vwe0+v−1Rǫ1−v(H−σ)ǫ2.

Integrability 2dασ+ [ασ∧ασ] = 0 is equivalent to the Gauss-Codazzi equations

(log v)zw −v2(H2+σ−2) + v−2QR = 0 ,(3.3a)

v2Hw=Rz, v2Hz=Qw.(3.3b)

Conversely, given smooth functions v, H, Q, R : Σ →Cwith vnon-vanishing, satisfying the

integrability equations (3.3), a conformal immersion fwith these invariants can be recovered by

integrating (3.2) to obtain maps F, G : Σ →SL2Cwhich are unique up to transforms (F, G)7→

(AF, BG), A, B ∈SL2C. By the form of ασ, the maps Fand Gframe f:= F G−1in the sense

of (3.1), and fhas the speciﬁed invariants. The map fis unique up to f7→ AF B −1,A, B ∈SL2C.

3.1. Equivariance. Let us ﬁrst introduce new coordinates (x, y) on Σ such that

(3.4) z=x+y , w =x−y .

We say a smooth map F=F(x, y) : C2→SL2Cis C-equivariant, or simply equivariant if it is

of the form F(x, y) = exp(x A)P(y) for some A∈sl2Cand smooth map P:C→SL2C. What

characterizes an equivariant map is that F−1dF =P−1A P dx +P−1P′dy is x-independent [4]. We

ﬁrst compute the logarithmic derivative of an equivariant map, and then integrate this to obtain

the equivariant map itself.

FLOWS OF EQUIVARIANT CMC TORI IN S313

Theorem 3.1. (1) The x-independent solution α=αk+αpto

dαk+1

2[αp∧αp] = 0 ,

dαp+ [αk∧αp] = 0 ,

d(∗αp) + [αk∧(∗αp)] = 0

(3.5)

are α=g−1Ωg+g−1dg, where gis a smooth x-independent map from Σinto the Lie group of k,

and Ω=Ω1dz + Ω2dw, where

(3.6) 2Ω1:= −2v−1v′e0+a2vǫ1−b1v−1ǫ2,

2Ω2:= −2v−1v′e0+b2v−1ǫ1−a1vǫ2,

with a1, a2, b1, b2, ν ∈C, and v=v(y) : C→Cis deﬁned by

(3.7) (v−1v′)2+ (a1v+b1v−1)(a2v+b2v−1) = 4ν2,

(v−1v′)′+a1a2v2−b1b2v−2= 0 .

(2) A solution to dF =FΩis given by

(3.8)

F(x, y) := exp (x ν +1

2χ0)e0exp 1

2χ1e1exp 1

2χ2e0,

with χ0:= 2 νZy

0

(J1(t)−J2(t))dt ,

χ1:= arccos(−1

2ν−1v−1v′), χ2:= 2log(X−1

1X2),

X1:= a1v+b1v−1, X2:= a2v+b2v−1, J1:= b1v−1X−1

1, J2:= b2v−1X−1

2.

Remark 3.2. The second equation in (3.7) is the derivative of the ﬁrst, and eliminates spurious

constant solutions which would appear if only the ﬁrst equation were present.

Proof. (1) Write αk=ae0dx +be0dy for some functions a=a(y) and b=b(y), and let g=

exp(−(Ryb(t)dt)e0). Then dg g−1=−be0dy, so the form g−1αg +g−1dg =ae0dx +g−1αpgis a

multiple of dx =1

2(dz +dw). Then αk=4f(dz +dw)e0for some function f=f(y). Let v=

exp(Ryf(t)dt), so αk=4v−1v′e0(dz +dw), where prime is diﬀerentiation with respect to y. Then

there exist functions a1, b1, a2, b2of ysuch that 2αp= (a2vǫ1−b1v−1ǫ2)dz + (b2vǫ1−a1v−1ǫ2)dw.

By a calculation, (3.5) is equivalent to the second equation of (3.7), along with a′

1=b′

1=a′

2=

b′

2= 0. Hence αis of the required form, and concludes the proof of (1).

To prove (2), we have Ω = Ωxdx + Ωydy in (3.6), where 2 Ωx:= −v−1v′e0+ (a2v+b2v−1)ǫ1−

(a1v+b1v−1)ǫ2and 2Ωy:= (a2v−b2v−1)ǫ1+ (a1v−b1v−1)ǫ2. To compute F−1dF , we will ﬁrst

show that P(x) := exp 1

2χ0e0exp 1

2χ1e1exp 1

2χ2e0satisﬁes

νP −1e0P= Ωx,(3.9a)

P−1P′= Ωy.(3.9b)

We have P−1e0P= cos χ1e0+e−χ2sin χ1ǫ1−eχ2sin χ1ǫ2and cos χ1=−1

2ν−1v−1v′, sin χ1=

−1

2ν−1X

1

2

1X

1

2

2,eχ2=X

1

2

1X−1

2

2and thus 2 νP −1e0P=v−1v′e0−X2ǫ1+X1ǫ2=−2(Ω1−Ω2) =

2Ωx, proving (3.9a).

Now 2P−1P′= (χ0′cos χ1+χ2′)e0+e−χ2(χ0′sin χ1+χ1′)ǫ1−eχ2(χ0′sin χ1−χ1′)ǫ2. With

C5:= a1b2−a2b1we have

(3.10) χ0′=−2νC5

X1X2

, χ1′=a1a2v2−b1b2v−2

√X1√X2

, χ2′=−C5v′

vX1X2

.

Hence 2P−1P′= (a2v−b2v−1)ǫ1+ (a1v−b1v−1)ǫ2= 2Ωy, proving (3.9b).

Thus F−1dF =F−1Fxdx +F−1Fydy =νP −1e0P dx +P−1P′dy = Ωxdx + Ωydy = Ω.

14 M. KILIAN, M. U. SCHMIDT AND N. SCHMITT

Figure 3.1. Two views of an equivariant cmc (2,1,13) torus in S3.

Example 3.3. The vacuum is the case in which the function vin (3.7) is constant. By the second

equation in (3.7),v≡v0with v4

0:= (b1b2)/(a1a2). The potential for the vacuum is ΩV:= Ωzdz +

Ωwdw, where 2Ωz:= a2v0ǫ1−b1v−1

0ǫ2and 2 Ωw:= b2v−1

0ǫ1−a1v0ǫ2. Since [Ωz,Ωw] = 0, the

extended frame of the vacuum is F= exp(Ωzz+ Ωww)with eigenvalues exp±2(rz +sw)where

r, s ∈Care determined by r2=a2b1,s2=a1b2,rs =a1a2v2

0. This diﬀers from the frame in

Theorem 3.1 by left multiplication by a z-and w-independent element of SL2C.

To describe equivariant cmc immersions into S3= SU2we specialize the above formulas. We ﬁrst

make the reduction in (3.4) that w= ¯z, which is equivalent to x, y ∈R. Given an extended frame

Fλ:R2→SU2and two distinct sym points λ1, λ2∈S1, then

(3.11) f:= Fλ1F−1

λ2

is a conformal immersion f:R2→S3with constant mean curvature Hgiven in (1.1). For the

translation τγ:C→C, p 7→ p+γwe write τ∗

γf=f◦τγ. If F−1

λdFλis periodic with period γ, then

we deﬁne the monodromy Mλof Fλwith respect to γas

(3.12) Mλ(γ) = τ∗

γFλF−1

λ.

The closing condition τ∗

γf=fwith respect to a translation is equivalent to

(3.13) Mλ1(γ) = Mλ2(γ) = or Mλ1(γ) = Mλ2(γ) = −.

If µ1and µ2=µ−1

1denote the eigenvalues of the monodromy, then (3.13) reads µj(λk) = ±1, or

equivalently that there exist four integers pj k ∈Zsuch that for j, k = 1,2 we have

(3.14) ln µj(λk) = π pj k and pj1−pj2∈2Z.

The torus is embedded if and only if the winding numbers pjk all have absolute value equal to one.

3.2. Flat cmc tori in S3.Integrating Ω (3.6) with (a1, b1, a2, b2) := (λ, 1, λ−1,1), v ≡1 yields an

extended frame Fλof a ﬂat cmc surface in S3. By Theorem 3.1 and Example 3.3, the extended

frame of any ﬂat cmc surface (up to isometry and conformal change of coordinates) is

(3.15) Fλ= expπ(zλ−1+z)ǫ1−(z+zλ)ǫ2.

FLOWS OF EQUIVARIANT CMC TORI IN S315

Then also (up to isometry and conformal change of coordinates) any ﬂat cmc immersion f:R2→S3

is of the form f=Fλ1F−1

λ2with extended frame Fλas in (3.15) and two distinct sym points

λ1, λ2∈S1. The eigenvalues of Fλare µ±1with

(3.16) µ(z, λ) = exp(π(zλ−1

2+zλ 1

2)) .

Deﬁne

(3.17) hhx, yii := 1

2(xy+y x) = ℜ(xy).

Then by equation (3.14) the immersion ffactors through the lattice Γ = γ1Z+γ2Zif and only if

(3.18) hhγj, λ1/2

kii ∈ Zand hhγj, λ1/2

1−λ1/2

2ii ∈ 2Z.

The dual of a lattice Γ in Cis the lattice Γ∗={κ∈C| hhκ, γii ∈ Zfor all γ∈Γ}.

Proposition 3.4.

(i) A ﬂat cmc immersion f=Fλ1F−1

λ2with extended frame Fλ(3.15) is closed with respect to a

lattice Γ⊂Cif and only if Γ⊆Λ∗, where Λ := κ1Z+κ2Zand

κ1:= 1

2(λ1/2

1+λ1/2

2)and κ2:= 1

2(λ1/2

1−λ1/2

2).

(ii) The torus is rectangular and embedded if and only if Γ = Λ∗.

(iii) Every ﬂat cmc torus is isogenic to a rectangular embedded ﬂat cmc torus.

Proof. (i) Since λ1/2

k=κ1±κ2the condition (3.18) is equivalent to Γ ⊆Λ∗.

(ii) The torus is embedded if and only if hhγj, λ1/2

kii ∈ {±1}, and rectangular if and only if

γ1/γ2∈R. The corresponding periods γ1and γ2are dual to κ1and κ2and generate Λ∗. Clearly

Λ, and consequently Λ∗are rectangular, since κ1/κ2∈R.

(iii) From (i) we know that a lattice Γ of any torus is a sublattice of Λ∗, which by (ii) is the lattice

of the embedded rectangular torus. Hence there is an isogeny taking Λ∗to Γ.

Proposition 3.5. The lattice of an embedded ﬂat cmc torus is square if and only if it the mean

curvature is zero. Swapping the sym points does not aﬀect the period lattice of a ﬂat cmc torus.

Proof. Solving the four equations (3.14) for the periods gives

(3.19) γ1=λ1/2

1λ2p11 −λ1λ1/2

2p12

λ2−λ1

, γ2=λ1/2

1λ2p21 −λ1λ1/2

2p22

λ2−λ1

.

In particular, setting p11 =p12 =p21 =−p22 = 1, a computation shows that the generators of a

lattice of an embedded ﬂat cmc torus satisfy

|γ1|

|γ2|=±p1 + H2+H.

Thus |γ1|=|γ2|if and only if the mean curvature is zero. Setting λ1=, λ2=−we obtain the

generators γ1= 1/√2 and γ2=/√2 of the square lattice of the Cliﬀord torus.

Swapping the sym points λ1↔λ2, results in the integers pjk swapping second indices: p11 ↔

p12, p21 ↔p22. This does not change the periods γ1, γ2in (3.19).

16 M. KILIAN, M. U. SCHMIDT AND N. SCHMITT

3.3. Spectral genus one. Consider the 1-form Ω = Ω1dz + Ω2d¯z, where Ωjare given in equa-

tion (3.6) with νas in (2.6) for some 0 <|k| ≤ 1 and (a1, b1, a2, b2) := (λ, −k, λ−1,−k). Inte-

grating the 1-form Ω with these choices then yields an R-equivariant extended frame Fλ(x, y) =

exp(x Aλ)Pλ(y) for smooth maps λ7→ Aλ,S1→su2and (λ, y)7→ Pλ(y),S1×R→SU2. Now let

λ1, λ2∈S1be two distinct sym points. The immersion f:C→SU2in (3.11) is then a conformal

equivariant cmc immersion with mean curvature (1.1).

Equivariance reduces the Gauss equation (3.7) to (v′)2+ (v2−1)(v2−k2) = 0. A solution to this

equation is the square root of the conformal factor v:R→R+given by v(y) = dn(y, 1−k2) where

dn(y, m) is the Jacobi elliptic function. All other solutions are of the form v(y+y0), y0∈R. The

function vis even and has no zeros on R. The range of vis [min(1,|k|),max(1,|k|)]. The function

satisﬁes v(0) = 1 and v′(0) = 0, and limits to lim|k|→1v(y) = 1 and limk→0v(y) = sech y. The

period of vdepends on the parameter 0 <|k| ≤ 1 and is equal to 2K′(k). Now a straightforward

calculation shows that the fundamental forms of such an equivariant conformal cmc immersion are

−(λ2−λ1)2

4λ1λ2

v2(dx2+dy2),(3.20a)

(v2H+ Re(Q)) dx2−Im(Q)dx dy + (v2H−Re(Q)) dy2,(3.20b)

with Has in (1.1) and Hopf diﬀerential Q dz2with Q:= 4k(λ−1

2−λ−1

1).

Proposition 3.6. A period of an equivariant extended frame is of the form

(3.21) γ=x π + 2 pK′

where x∈R,p∈Z, and 2K′is the period of v. The monodromy (3.12) with respect to such a

period is

(3.22) Mλ(γ) = exp(π(x ν +p ω )e0).

Proof. The imaginary part of a frame period has to be a period of the square root of the conformal

factor v:R→R+, y 7→ dn(y, 1−k2). Since vhas period 2K′(k), the imaginary part of γin (3.21)

has to be of the form 2 pK′for some p∈Z. From (3.8) the extended frame of an equivariant cmc

torus is of the form F(x, y) := exp (x ν +1

2χ0)e0exp 1

2χ1e1exp 1

2χ2e0. The middle and right

factor are periodic in yand do not depend on x. Hence both these factors have trivial monodromy

if y∈2K′Z, and the monodromy with respect to a translation by γ=x π + 2 pK′is

F(xπ, 2nK′) = exp((π x ν +1

2χ0(2pK′)) e0)

Clearly χ0(2pK′) = pχ0(2K′) so it suﬃces to show that χ0(2K′,k, λ) = 2πω(k, λ) to conclude the

proof. Let λ=eiθ , then a calculation yields

d(νJ1) = −

d

dθ (−k(λ+λ−1))

4νk(λ−1−λ)−kλ−1

J1−1

2

d2

dt2log J1(t)dθ ,(3.23)

d(νJ2) =

d

dθ (−k(λ+λ−1))

4νk(λ−1−λ)−kλ

J2−1

2

d2

dt2log J2(t)dθ .(3.24)

Since J1and J2are functions of v, they are also periodic with period 2K′, so

Z2K′

0

d2

dt2log Jkdt =

d

dt Jk

Jk

2K′

0

= 0 .

Subtracting (3.24) from (3.23) and integrating over the interval [0,2K′] gives

dχ0=1

4ν4E′−2K′k (λ+λ−1)dθ = 2π dω .

FLOWS OF EQUIVARIANT CMC TORI IN S317

Further, since

χ0(2K′,k, λ) = 2 νZ2K′

0

k

v(λ−1v−kv−1)−k

v(λv −kv−1)dt

and ν(λ−1) = ν(λ) it follows that χ0(2K′,k, λ−1) = −χ0(2K′,k, λ). Thus χ0shares the properties

of ωwhich determine it uniquely. Hence χ0(2K′,k, λ) = 2πω(k, λ).

By (3.22) we have τγf=Mλ1(γ)f M−1

λ2(γ) = exp(π(x ν1+n ω1)e0)fexp(−π(x ν2+n ω2)e0), so

translation by a period γ(3.21) induces an ambient isometry. The equivariant action is the action

of the 1-parameter group Kof isometries of S3deﬁned by

(3.25) K={gx∈Iso(S3)|gx(p) = exp(x ν1e0)pexp(−x ν2e0), x ∈R}.

Since ν16= 0 6=ν2the commutator of the equivariant action (3.25)

(3.26) ˆ

K={g∈Iso(S3)|gk =kg for all k∈K}

in the group Iso(S3) of orientation preserving isometries of S3is a two-dimensional torus. With the

exception of two geodesics the orbits of ˆ

Kare two-dimensional embedded tori. These geodesics,

which we call the axes of the surface, are linked, and are situated so that every geodesic 2-sphere

through one is orthogonal to the other. Every orbit of the equivariant action (3.25), with the

exception of the two axes, is a (m, n)-torus knot in the corresponding orbit of ˆ

K, with

(3.27) m

n=ν1−ν2

ν1+ν2

.

If we identify

S3= SU2=a b

−¯

b¯a|a|2+|b|2= 1and choose e0=0

0−,

then the equivariant action extends to an action on (a, b)∈C2given by R(s, t)(a, b) = (esa, e tb),

called the extended action. In particular, the translation τγby a period γ(3.21) induces the ambient

isometry R(s, t) with s=x(ν1−ν2) + p(ω1−ω2) and t=x(ν1+ν2) + p(ω1+ω2).

Proof of Proposition 2.5. For a ﬂat cmc torus this can always be achieved since we have not used

the freedom of the M¨obius transformation. A spectral genus one torus is a surface of revolution, if

and only if the equivariant action is the rotation around a geodesic, or equivalently if (3.25) ﬁxes

point wise one geodesic of S3. The generator of the extended action has eigenvalues (ν1±ν2).

Thus there exists a zero eigenvalue, if and only if ν2=±ν1, which is equivalent to λ2=λ−1

1.

Proposition 3.7. A spectral genus one cmc surface in S3is closed along two independent periods

if and only if there exists a T∈Z3\ {0}with T·X= 0 = T·Yfor X:= (0, ν1, ν2)and

Y:= (1, ω1, ω2).

Proof. Suppose we have two R-independent periods γj=xjπ+ 2 pj0K′∈C×for some xj∈Rand

pj0∈Z,j= 1,2. Let Mλ(γj) be the respective frame monodromies with eigenvalues µ±1

j. Then

there exist four further integers pjk ∈Zas in (3.14) for j, k = 1,2. Hence pjk =xjνk+pj0ωk, and

we write this system as

(3.28) x1p10

x2p20X

Y=p10 p11 p12

p20 p21 p22:= P

Q.

Hence T:= P×Qis in Z3\ {0}and satisﬁes T·X= 0 and T·Y= 0.

Conversely suppose that there exists T∈Z3\ {0}satisfying T·X= 0 and T·Y= 0. Let {P, Q}

be a basis for the lattice Λ = {P∈Z3|T·P= 0}. Then there exist x1, x2∈Rand p10, p20 ∈Z

such that (3.28) holds. Then γj=xjπ+ 2 pj0K′∈C×generate a lattice with respect to which the

surface is doubly periodic.

18 M. KILIAN, M. U. SCHMIDT AND N. SCHMITT

Figure 3.2. A sampling of (k, 13) cmc tori of revolution in S3.

Remark 3.8. The closing conditions in Proposition 3.7 can be used to describe the intersection of

the zero sets of two functions on the parameter space (k,q,h). The curve forming the intersection

of two level sets then integrates to the vector ﬁeld (2.12). For X, Y as in Proposition 3.7, there

exists s= (s0, s1, s2)∈Z3such that s·X= 0 and s·Y= 0. The closing conditions are thus

F= 0, G = 0, where F:= s·Xand G:= s·Y. If we set λk=e2θk, then the system of implicit

ﬂow equations is ∂F

∂k

∂F

∂θ1

∂F

∂θ2

∂G

∂k

∂G

∂θ1

∂G

∂θ2!

˙

k

˙

θ1

˙

θ2

= 0 ,

of which we next compute the matrix on the right hand side. Up to scale, (˙

k,˙

θ1,˙

θ2) is a cross

product of its rows. Hence

∂F

∂k

∂F

∂θ1

∂F

∂θ2

∂G

∂k

∂G

∂θ1

∂G

∂θ2!= s1∂ν1

∂k+s2∂ν2