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# Flows of constant mean curvature tori in the 3-sphere: The equivariant case

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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. We conclude with an instability result.
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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 ̺=ησ=ση, 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
µiσ=µ1
iµiη= ¯µiµi̺= ¯µ1
i.
(3) The forms dln µiare meromorphic diﬀerentials of the second kind with double poles at y±.
The singular parts at y+respectively yof 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 λ1S1denote the
value λ(y1) = λ(y2), and λ2S1the 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
λ2λ1
.
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 aCg[λ] is a polynomial of degree gand
¯
λ2g¯a(λ) = a(¯
λ1) and λga(λ)0 for |λ|= 1.
The involutions of the spectral curve are
σ(λ, ν) = (λ, ν),
η(λ, ν) = (¯
λ1,¯ν¯
λg1),
̺(λ, ν) = (¯
λ1,¯ν¯
λg1).
(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
λ
with polynomials biCg+1[λ] of degree g+ 1 and ¯
λg1¯
bi(λ) = bi(¯
λ1).
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 ciCg+1[λ] of degree g+ 1 and ¯
λg1¯ci(λ) = ci(¯
λ1).
Integrability 2
ln µi=2
(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 σΩ = Ω, we conclude that
Ω = π2C1
(λλ1)(λλ2)
λλ1λ2
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)(λλ1)(λλ2)
λ1λ2
,
and combining the above gives
(1.7) (tln µ1)dln µ2(tln µ2)dln µ1=π2C1
(λλ1)(λλ2)
λλ1λ2
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 ¯
λ2g¯a(λ) = a(¯
λ1)
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 G1.
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
p1h2.
FLOWS OF EQUIVARIANT CMC TORI IN S35
The following identities will be used below:
2pq21 = pλ1λ21/pλ1λ2,2ph21 = pλ12pλ21,
4(q2h2) = λ1λ1
1λ2λ1
2,4 q pq21 = λ1λ2λ1
1λ1
2
4 h pq21 = λ1λ1
1+λ2λ1
2,4 q ph21 = λ1λ1
1λ2+λ1
2
4 h ph21 = λ1λ1
2λ1
1λ2.
(2.4)
The derivatives of q,h with respect to the ﬂow parameter are
(2.5) ˙q = pq21
2 ˙
λ1
λ1
+˙
λ2
λ2!,˙
h = ph21
2 ˙
λ1
λ1˙
λ2
λ2!.
2.1. Spectral genus zero. In the spectral genus zero case, a1, 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 ¯
λ1¯
bi(λ) = bi(¯
λ1) 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 λ11 during the ﬂow. Then tln µj|λ=1 = 0,
which is equivalent to ci(λ= 1) = 0, or equivalently ˙
2˙
β2(β1λ+¯
β1)2˙
β1(β2λ+¯
β2) = C1
λ2
(λ2λ).
Evaluating this at λ=¯
βiigives
˙
βi=C1(βiλ1/2
2+¯
βiλ1/2
2)
2 ( ¯
β1β2β1¯
β2)
From (1.8) we get
˙
λ2
λ2
=c1
b1
=2˙
β1(1 λ2)
β1λ2+¯
β1
=C1
λ1/2
2λ1/2
2
(β1¯
β2¯
β1β2)
Set C1:= (β1¯
β2¯
β1β2). Then together with (2.4) and the fact that q = h when λ11, the ﬂow
equations (1.7) and (1.8) reduce to the single equation ˙
h = h21. The solution to this equation
is given by h(t) = tanh(t+C) with some constant of integration CR. 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= (λk)(λ1k).
Here k [1,1] is the real modulus. Then
(2.7) = k λ1λ
8ν
λ,
6 M. KILIAN, M. U. SCHMIDT AND N. SCHMITT
and the only roots of are at λ= k,k1. In addition to the three involutions in (1.2), the elliptic
curve (2.6) has a further holomorphic involution
(2.8) τ(λ, ν) = (λ1, 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 along the ﬁrst cycle vanishes. We assume that the integral of 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 µi2πZ, 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 1k
1+k given by
y2=1x21(1k
1+k )2x2.
Let Kand Edenote the complete integrals of the ﬁrst and second kind
K(k) := Z1
0
dx
(1x2)(1k2x2),E(k) := Z1
0s1k2x2
1x2dx.
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 + (1k
1+k )2(1 2x2)21k
1+k y
1(1k
1+k )2, λ1=1 + (1k
1+k )2(1 2x2) + 21k
1+k y
1(1k
1+k )2, ν =
1k
1+k x
1 + 1k
1+k
.
The integrals of the meromorphic diﬀerentials dln µialong all cycles of Y are purely imaginary.
Deﬁne
(2.10) := E(1k
1+k )K(1k
1+k )1(1k
1+k )2x22dx
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(p1k2) and E=E(k) = E(p1k2)
and the functions λand ν, and the formulas [1, 17.3.29 and 17.3.30]
E(1k
1+k ) = (1 + 1k
1+k )E+ (1 1k
1+k )K,K(1k
1+k ) = 2
1 + 1k
1+k
K,
the diﬀerential in (2.10) simpliﬁes to
(2.11) =2EkK(λ+λ1)
4πν
λ.
Proposition 2.2. The complete elliptic integrals Kand Esatisfy
12E
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=, we have that
Zλ=k1
λ=k
= 2 Zλ=1
λ=k
= 0 .
Hence the function 2EkK(λ+λ1) has a real root in λ(k,1), say at r(k,1), and thus
also at r1(1,k1). Then 2 < r + 1/r < k + 1/k, and together with 2EkK(r+r1) = 0
gives 2E>2 k Kand 2E<K(1 + k2). Finally, from 2 >k2+1 and E1 (13.8.11 in [9]), we
obtain 2E1 + k2.
For k (1,0), the function 2EkK(λ+λ1) has a pair of reciprocal real roots s, s1with
1< s < k1, and analogous arguments prove the assertion in this case.
Corollary 2.3. The diﬀerentials and have no common zeroes.
Proof. While in (2.7) has only roots at λ= k,k1, we saw in the proof of Proposition 2.2
that has only roots λ(k,1) (1,k1) when k (0,1), and λ(1,k) (k1,1) 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 (EqkKh)
1q2
1k2((1 + k2)E2 k2K)
k1h2
1k2(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 λλ1
8ν.
In agreement with the skew-symmetry of ωwith respect to τwe set
˙ω= C2
λλ1
4π ν .
From (2.9) we obtain tln µj=π( ˙xjν+xj˙ν+pj0˙ω) and dln µj=π(xj+pj0) 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 x2p20x1)( ˙ν ων˙ω)with
ν ν=k (λ1λ)
8λ, ν ω=2Ek (λ+λ1)K
4π λ ,
˙ν ων˙ω=C2k(λλ1)2+˙
k (λ2 k +λ1)(k (λ+λ1)K2E)
8π λ (λk)(λ1k) .
Note that while ˙ν ων˙ωhas simple poles at λ= k,k1, but Ω does not, we conclude that the
numerator of ˙ν ων˙ωmust vanish at λ= k,k1, giving
C2=˙
k(k2+1) K2E
k21.
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(k21)(k h KqE)
(x1p20 x2p10)(E2k2K2),˙x1x2˙x2x1=8 C1p1q2
k,
˙x1p20 ˙x2p10 =4πC1(k q(EK) + h(Ek2K))
E2k2K2.
Since ˙ω=kω˙
k, the above imply that
(2.14) ∂ω
k=(1 + k2)K2E
4π(1 k2)ν(λλ1).
Now setting
C1:= k (x1p20 x2p10)(E2k2K2)
4π(k21)
gives ˙
k = k(EqkKh) 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)νλ=λj
.
Using the above formulae as well as (2.4) we obtain
˙
λ1
λ1
+˙
λ2
λ2
=2pq21
h2q2k q(EK) + h(Ek2K)
1k2(2 k q h(k2+1)) + (q Eh k E)(q h k)
=2p(q21)
k21(1 + k2)E2 k2K,
˙
λ1
λ1˙
λ2
λ2
=2ph21
h2q2k q(EK) + h(Ek2K)
1k2((k2+1) q 2 h k) + (q Eh k E)(k q h)
=2ph21
k21k2E(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 Kand Eare 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 Kand Ethe 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)E2 k2Kand 2E(1 + k2)Khave zeros only at k = ±1,
and E= k Kholds 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
21
4(k 1) + 5
32 (k 1)27
64 (k 1)3+ O(k 1)4,
E(k) = π1
2+1
4(k 1) + 1
32 (k 1)21
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=λ1
2(rotational),h2= 1 λ1=λ2(H=),
q = 0 λ1=λ1
2,h = 0 λ1=λ2(H= 0) ,
q = h λ1= 1 or λ2= 1 ,q = hλ1=1 or λ2=1.
By the deformation equations (2.12) if q2(t0) = 1, for some t0, then q21 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 + 2cand min{1q2,1h2} ≥ 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 cR+. 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 + 2c
c.
On each integral curve k is either positive or negative. The second inequality follows from
min{1q2,1h2} ≥ (1 q2)(1 h2)c(1 sign(k) q h)2c(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 t0t, and if it is decreasing, it stays decreasing
for all t0t, 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(EqkKh) = |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} ∩ {qk h <0} ∩ Bto a point in {k = ±1} ∩ {qk 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
qsign(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/p1h2is 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 and 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 1h2is bounded away from zero. The maximal
interval of deﬁnition is of the form (−∞, tmax)with
lim
t↓−∞ k = 0 and lim
ttmax
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 |θ2θ1|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 ω2ω1=Rθ2
θ1is constant throughout the ﬂow. We claim that
|| ≤ 2||for θRand 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 Eπ
2[9, 13.8.(11)], Proposition 2.2 and (2.6) to obtain
||=
EkKcos(2θ)
πν || ≤
E(1 2 k
1+k2cos(2θ))
πν || ≤
2ν
1 + k2|| ≤ 2||.
When k cos(2θ)0, then again by Eπ
2, Proposition 2.2, and 2|ν| ≥ p1 + k21 (2.6) we get
||=
EkKcos(2θ)
πν || ≤
4E
πp1 + k2|| ≤ 2||.
Thus |θ2θ1| ≥ 1
2|ω2ω1|. Due to (2.11) and Proposition 2.2, the diﬀerential has no root
between θ1and θ2. Therefore |θ2θ1|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 λC×, where µ1and µ2are unimodular is the set S1∪ {k,k1}. 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]: νRif and only if λS1[k,k1]R, and λRand ωRif and only if
λ[0,k] [k1,). For k [1,0): νRif and only if λS1[k1,k] R+;λRand ωR
if and only if λ(−∞,k1][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 C2SL2Cwith
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=kpbe a Cartan decomposition. For a 1-form αon Σ, we will write α=αk+αpwhere
with αkkand αpp. 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 e0k,he0, e0i= 1
and [e0, e1] = e2, [e1, e2] = e0and [e2, e0] = e1, and set
ǫ1=1
2(e1e2), ǫ2=1
2(e1+e2).
Let f: Σ SL2Cbe a conformal immersion such that for the smooth function v: Σ Cwe 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=v2[fz, fw]. Further, there exists a unique pair of maps F, G : Σ SL2Cwhich
frame fin the sense that
(3.1) f=F G1, fz=vF ǫ1G1, fw=vF ǫ2G1, N =F e0G1.
Then α+1 := F1dF and α1:= G1dG, are smooth 1-forms on Σ with values in sl2Cgiven by
(3.2)
ασ=Aσdz +Bσdw where σ∈ {±1}and
2Aσ:= v1vze0+v(H+σ)ǫ1v12,
2Bσ:= v1vwe0+v11v(Hσ)ǫ2.
Integrability 2σ+ [ασασ] = 0 is equivalent to the Gauss-Codazzi equations
(log v)zw v2(H2+σ2) + v2QR = 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 G1in 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 =xy .
We say a smooth map F=F(x, y) : C2SL2Cis C-equivariant, or simply equivariant if it is
of the form F(x, y) = exp(x A)P(y) for some Asl2Cand smooth map P:CSL2C. What
characterizes an equivariant map is that F1dF =P1A P dx +P1Pdy 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
k+1
2[αpαp] = 0 ,
p+ [αkαp] = 0 ,
d(αp) + [αk(αp)] = 0
(3.5)
are α=g1g+g1dg, where gis a smooth x-independent map from Σinto the Lie group of k,
and Ω=Ω1dz + Ω2dw, where
(3.6) 21:= 2v1ve0+a21b1v1ǫ2,
22:= 2v1ve0+b2v1ǫ1a12,
with a1, a2, b1, b2, ν C, and v=v(y) : CCis deﬁned by
(3.7) (v1v)2+ (a1v+b1v1)(a2v+b2v1) = 4ν2,
(v1v)+a1a2v2b1b2v2= 0 .
(2) A solution to dF =Fis 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
2ν1v1v), χ2:= 2log(X1
1X2),
X1:= a1v+b1v1, X2:= a2v+b2v1, J1:= b1v1X1
1, J2:= b2v1X1
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 g1=be0dy, so the form g1αg +g1dg =ae0dx +g1αpgis 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=4v1ve0(dz +dw), where prime is diﬀerentiation with respect to y. Then
there exist functions a1, b1, a2, b2of ysuch that 2αp= (a21b1v1ǫ2)dz + (b21a1v1ǫ2)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 x:= v1ve0+ (a2v+b2v1)ǫ1
(a1v+b1v1)ǫ2and 2Ωy:= (a2vb2v1)ǫ1+ (a1vb1v1)ǫ2. To compute F1dF , we will ﬁrst
show that P(x) := exp 1
2χ0e0exp 1
2χ1e1exp 1
2χ2e0satisﬁes
νP 1e0P= Ωx,(3.9a)
P1P= Ωy.(3.9b)
We have P1e0P= cos χ1e0+eχ2sin χ1ǫ1eχ2sin χ1ǫ2and cos χ1=1
2ν1v1v, sin χ1=
1
2ν1X
1
2
1X
1
2
2,eχ2=X
1
2
1X1
2
2and thus 2 νP 1e0P=v1ve0X2ǫ1+X1ǫ2=2(Ω12) =
2x, proving (3.9a).
Now 2P1P= (χ0cos χ1+χ2)e0+eχ2(χ0sin χ1+χ1)ǫ1eχ2(χ0sin χ1χ1)ǫ2. With
C5:= a1b2a2b1we have
(3.10) χ0=2νC5
X1X2
, χ1=a1a2v2b1b2v2
X1X2
, χ2=C5v
vX1X2
.
Hence 2P1P= (a2vb2v1)ǫ1+ (a1vb1v1)ǫ2= 2Ωy, proving (3.9b).
Thus F1dF =F1Fxdx +F1Fydy =νP 1e0P dx +P1Pdy = 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),vv0with v4
0:= (b1b2)/(a1a2). The potential for the vacuum is V:= Ωzdz +
wdw, where 2z:= a2v0ǫ1b1v1
0ǫ2and 2 Ωw:= b2v1
0ǫ1a1v0ǫ2. Since [Ωz,w] = 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:R2SU2and two distinct sym points λ1, λ2S1, then
(3.11) f:= Fλ1F1
λ2
is a conformal immersion f:R2S3with constant mean curvature Hgiven in (1.1). For the
translation τγ:CC, p 7→ p+γwe write τ
γf=fτγ. If F1
λdFλis periodic with period γ, then
we deﬁne the monodromy Mλof Fλwith respect to γas
(3.12) Mλ(γ) = τ
γFλF1
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=µ1
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 pj1pj22Z.
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, λ1,1), 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(1+z)ǫ1(z+)ǫ2.
FLOWS OF EQUIVARIANT CMC TORI IN S315
Then also (up to isometry and conformal change of coordinates) any ﬂat cmc immersion f:R2S3
is of the form f=Fλ1F1
λ2with extended frame Fλas in (3.15) and two distinct sym points
λ1, λ2S1. The eigenvalues of Fλare µ±1with
(3.16) µ(z, λ) = exp(π(1
2+ 1
2)) .
Deﬁne
(3.17) hhx, yii := 1
2(xy+y x) = (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
1λ1/2
2ii ∈ 2Z.
The dual of a lattice Γ in Cis the lattice Γ={κC| hhκ, γii ∈ Zfor all γΓ}.
Proposition 3.4.
(i) A ﬂat cmc immersion f=Fλ1F1
λ2with extended frame Fλ(3.15) is closed with respect to a
lattice ΓCif and only if ΓΛ, where Λ := κ1Z+κ2Zand
κ1:= 1
2(λ1/2
1+λ1/2
2)and κ2:= 1
2(λ1/2
1λ1/2
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
γ12R. The corresponding periods γ1and γ2are dual to κ1and κ2and generate Λ. Clearly
Λ, and consequently Λare rectangular, since κ12R.
(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 Λto 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
λ2λ1
, γ2=λ1/2
1λ2p21 λ1λ1/2
2p22
λ2λ1
.
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 λ1λ2, 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 = 1dz + 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, λ1,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,S1su2and (λ, y)7→ Pλ(y),S1×RSU2. Now let
λ1, λ2S1be two distinct sym points. The immersion f:CSU2in (3.11) is then a conformal
equivariant cmc immersion with mean curvature (1.1).
Equivariance reduces the Gauss equation (3.7) to (v)2+ (v21)(v2k2) = 0. A solution to this
equation is the square root of the conformal factor v:RR+given by v(y) = dn(y, 1k2) where
dn(y, m) is the Jacobi elliptic function. All other solutions are of the form v(y+y0), y0R. 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 limk0v(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
(λ2λ1)2
4λ1λ2
v2(dx2+dy2),(3.20a)
(v2H+ Re(Q)) dx2Im(Q)dx dy + (v2HRe(Q)) dy2,(3.20b)
with Has in (1.1) and Hopf diﬀerential Q dz2with Q:= 4k(λ1
2λ1
1).
Proposition 3.6. A period of an equivariant extended frame is of the form
(3.21) γ=x π + 2 pK
where xR,pZ, and 2Kis 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:RR+, y 7→ dn(y, 1k2). Since vhas period 2K(k), the imaginary part of γin (3.21)
has to be of the form 2 pKfor some pZ. 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 y2KZ, and the monodromy with respect to a translation by γ=x π + 2 pKis
F(xπ, 2nK) = exp((π x ν +1
2χ0(2pK)) e0)
Clearly χ0(2pK) = 0(2K) so it suﬃces to show that χ0(2K,k, λ) = 2πω(k, λ) to conclude the
proof. Let λ=e , then a calculation yields
d(νJ1) =
d
(k(λ+λ1))
4νk(λ1λ)kλ1
J11
2
d2
dt2log J1(t)dθ ,(3.23)
d(νJ2) =
d
(k(λ+λ1))
4νk(λ1λ)kλ
J21
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
0=1
4ν4E2Kk (λ+λ1)= 2π dω .
FLOWS OF EQUIVARIANT CMC TORI IN S317
Further, since
χ0(2K,k, λ) = 2 νZ2K
0
k
v(λ1vkv1)k
v(λv kv1)dt
and ν(λ1) = ν(λ) it follows that χ0(2K,k, λ1) = 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 M1
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={gxIso(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={gIso(S3)|gk =kg for all kK}
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=ν1ν2
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(ν1ν2) + p(ω1ω2) 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=λ1
1.
Proposition 3.7. A spectral genus one cmc surface in S3is closed along two independent periods
if and only if there exists a TZ3\ {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 pj0KC×for some xjRand
pj0Z,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 TZ3\ {0}satisfying T·X= 0 and T·Y= 0. Let {P, Q}
be a basis for the lattice Λ = {PZ3|T·P= 0}. Then there exist x1, x2Rand p10, p20 Z
such that (3.28) holds. Then γj=xjπ+ 2 pj0KC×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!= s1ν1
k+s2∂ν2