Table Of ContentFORMAL CONSERVED QUANTITIES FOR ISOTHERMIC
SURFACES
3
F.E.BURSTALLANDS.D.SANTOS
1
0
2
Abstract. Isothermic surfaces inSn are characterised by the existence of a
n pencil ∇t of flat connections. Such a surface is special of type d if there is a
a familyp(t)of∇t-parallelsectionswhosedependenceonthespectralparameter
J t is polynomial of degree d. We prove that any isothermic surface admits a
3 family of ∇t-parallel sections which is a formal Laurent series in t. As an
application,wegiveconformallyinvariantconditionsforanisothermicsurface
] inS3 tobespecial.
G
D
. Keywords: special isothermic surfaces, polynomial and formal conserved quanti-
h
t ties.
a
m
MSC: 53A30, 53A05.
[
1
Introduction
v
7
Isothermic surfaces, that is, surfaces which admit conformal curvature line coordi-
4
4 nates, were intensively studied around the turn of the 20th century by Darboux,
0 Bianchiandothers[1,8,9,12]. Theseclassicalworksrevealedarichtransformation
. theory that has been revisited in modern times from the viewpoint of integrable
1
0 systems [2,10,16]. At the heartof the integrablesystems formalismis the observa-
3 tionthat there is a pencil offlatconnections ∇t =d+tη, t∈R,associatedto each
1 isothermic surface.
:
v Inourpreviouswork[4],wedistinguishedtheclassofspecialisothermicsurfaces (of
Xi typed)whicharecharacterisedbytheexistenceofapolynomial conserved quantity,
that is, a family p(t) of ∇t-parallel sections whose dependence on t is polynomial
r
a (of degree d). The existence of such a polynomial conserved quantity amounts to
a differential equation on the principal curvatures of the surface. For example, an
isothermic surface in S3 is special of type 1 if it has constant mean curvature with
respecttoaconstantcurvaturemetricon(anopensubsetof)S3 andspecialoftype
2 if it is a specialisothermic surface in the sense of Darboux andBianchi[1,11]—a
class of isothermic surfaces that originally arose in the study of surfaces isometric
to a quadric.
The purpose of the present paper is to answer a question posed to one of us by
NigelHitchin: doesanyisothermicsurfaceinSnadmitaformal conservedquantity,
thus a solution p(t) of ∇tp(t) = 0 with p(t) = p ti a formal Laurent series?
i≤0 i
We give an affirmative answer locally, away from the (discrete) zero-set of η, and
P
globally when n = 3. In particular, any isothermic 2-torus in S3 admits a formal
conservedquantity. This is an analogue of the existence of formal Killing fields for
harmonic maps [3] although the method is somewhat different since here we deal
with nilpotent rather than semisimple gauge potentials.
Since polynomialconservedquantities arealso formalconservedquantities, our ar-
gumentsallowustogiveconformallyinvariantconditions,intermsoftheSchwarzian
derivative and Hopf differential introduced in [6], for an isothermic surface to be
1
2 F.E.BURSTALLANDS.D.SANTOS
special of type d. We illustrate these results with the case of surfaces of revolu-
tion and other equivariantsurfaces where these conditions amount to a differential
equation on the curvature of a profile curve.
Some of the following results can also be found in the second author’s doctoral
thesis [15], using a different approach.
1. Preliminaries
1.1. The conformal sphere. We will study isothermic surfaces in the n-sphere
from a conformally invariant view-point and so use Darboux’s light-cone model of
the conformal n-sphere. For this, contemplate the light-cone L in the Lorentzian
vector space Rn+1,1 and its projectivisationP(L). This last has a conformalstruc-
ture where representative metrics g arise from never-zero sections σ of the tauto-
σ
logical bundle π :L→P(L) via
g (X,Y)=(dσ(X),dσ(Y)).
σ
Then Sn ∼= P(L) qua conformal manifolds. Indeed, for non-zero w ∈ Rn+1,1, let
E(w) be the conic section given by
E(w)={v ∈L:(v,w)=−1}
with(definite)metricinducedbytheambientinnerproductonRn+1,1. Thenπ
|E(w)
is a conformal diffeomorphism onto its image. In particular, when w is unit time-
0
like, we have an isometry x7→x+w from the unit sphere in hw i⊥ to E(w ) and
0 0 0
thus a conformal diffeomorphism from that sphere to P(L). More generally, E(w)
has constant sectional curvature −(w,w).
1.2. Invariants of a conformal immersion. Let Σ be a Riemann surface and
Λ:Σ→Sn ∼=P(L) a conformal immersion. We view Λ as a null line subbundle of
the trivial bundle Rn+1,1 =Σ×Rn+1,1.
The central sphere congruence assigns, to each x ∈ Σ, the unique 2-sphere S(x)
tangent to Λ at the point Λ(x), which has the same mean curvature vectoras Λ at
x. Having in mind the identification between 2-dimensional subspheres of Sn and
(3,1)-planes of Rn+1,1 via
V 7→P(L∩V),
the central sphere congruence of Λ amounts to a subbundle V of Rn+1,1, with
signature (3,1).
Fix a holomorphic coordinatez =u+iv onΣ and take the unique (up to sign) lift
ψ ∈ΓΛ such that
|dψ|2 =|dz|2.
Then
V ⊗C=hψ,ψ ,ψ ,ψ i.
z z¯ zz¯
Consider now the unique section ψˆ∈Γ(V) such that
(ψˆ,ψˆ)=0, (ψˆ,ψ)=−1 and (ψˆ,dψ)=0,
which provides a new frame for V ⊗C, namely
ψ,ψ ,ψ and ψˆ.
z z¯
According to [6], we have
c
ψ + ψ =κ,
zz
2
for a complex function c and κ ∈ Γ(V⊥⊗C). These latter invariants are, respec-
tively, the Schwarzian derivative and Hopf differential of Λ with respect to z and,
FORMAL CONSERVED QUANTITIES FOR ISOTHERMIC SURFACES 3
together with the connection D on V⊥ given by orthoprojectionof flat differentia-
tion, determine Λ up to conformal diffeomorphisms of Sn.
Our frame satisfies:
c
ψ =− ψ+κ
zz
2
1
ψ =−(κ,κ¯)ψ+ ψˆ
(1.1) zz¯ 2
ψˆ =−2(κ,κ¯)ψ −cψ +2D κ
z z z¯ z¯
ξ =2(ξ,D κ)ψ−2(ξ,κ)ψ +D ξ,
z z¯ z¯ z
for each ξ ∈Γ(V⊥⊗C). The corresponding structure equations are the conformal
Gauss equation:
1
c =3(κ,D κ¯)+(κ¯,D κ);
z¯ z z
2
the conformal Codazzi equation:
1
Im(D D κ+ c¯κ)=0
z¯ z¯
2
and the conformal Ricci equation:
D D ξ−D D ξ−2hξ,κiκ¯+2hξ,κ¯iκ=0.
z¯ z z z¯
See [6] for more details.
1.3. Isothermic and special isothermic surfaces. Classically, an isothermic
surface is a surface in Sn that admits conformal curvature line coordinates but we
shall follow [7,13] and adopt the following conformally invariant formulation:
Definition 1.1. An immersionΛ:Σ−→Sn ∼=P(L), Λ is an isothermic surface if
there is a non-zero closed 1-form η ∈Ω1⊗o(Rn+1,1) taking values in Λ∧Λ⊥ 1.
One makes contact with the classical formulation by defining q ∈Γ(S2T∗Σ) by
q(X,Y)σ =η d σ,
X Y
foranyσ ∈ΓΛ. Thendη =0ifandonlyifq isaholomorphicquadraticdifferential
which commutes with the second fundamental form of Λ. Now q and hence η van-
ishes only on a discrete set and, off that set, we can find a holomorphic coordinate
z such that q = dz2. In terms of the corresponding lift ψ of Section 1.2, we then
have
η =−ψ∧(ψ dz+ψ dz¯)
z¯ z
which commutes with the second fundamental form if and only if κ is real. In this
case, z =u+iv where u,v are curvature line coordinates.
The conformal Gauss and Codazzi equations are now given simply by
1
c =4(κ,κ) and Im(D D κ+ c¯κ)=0,
z¯ z z¯ z¯
2
while the conformalRicci equation amounts to the familiar assertionthat the con-
nection D on V⊥ is flat.
The key to the integrable systems theory of isothermic surfaces is the observation
that the family of metric connections d+tη, t∈R on Rn+1,1 are flat and so have
a good supply of parallel sections. In [4], we considered isothermic surfaces which
admitted parallel sections with polynomial dependence on t and so introduced the
notion of special isothermic surfaces of type d∈N :
0
1Recall the isomorphism V2Rn+1,1 ∼= o(Rn+1,1) via (u∧v)w = (u,w)v−(v,w)u, for all
u,v,w∈Rn+1,1.
4 F.E.BURSTALLANDS.D.SANTOS
Definition 1.2. An isothermic surface (Λ,η) in Sn is a special isothermic surface
of type d ∈ N if there is a polynomial p(t) = d p ti ∈ Γ(Rn+1,1)[t] of degree d
0 i=0 i
such that (d+tη)p(t)≡0. We call such a p(t) a polynomial conserved quantity of
P
(Λ,η).
Animmediateconsequenceof(d+tη)p(t)≡0isthatdp =0sothatp isconstant
0 0
and therefore, if non-zero, defines a conic section as in Section 1.1. We therefore
refine our definition:
Definition 1.3. An isothermic surface (Λ,η) in Sn is a special isothermic surface
of type d ∈ N in E(w) if (Λ,η) admits a polynomial conserved quantity p(t) =
0
d p ti of degree d with p ∈hwi.
i=0 i 0
TPhe condition that an isothermic surface (Λ,η) be special of type d amounts to
a differential equation on the principal curvatures of Λ. For example, generically,
(Λ,η) is a special isothermic surface of type 1 in E(w) if and only if the lift F :
Σ −→ E(w) of Λ is a generalised H-surface (which, in codimension 1, amounts to
the mean curvature H being constant) ([5], see also [4]). Again, in [4], we show
that (Λ,η) is special of type 2 in E(w) if and only if there are real constants A, B
and C such that
H +θ H −θ H − 1Mk −Ak −Be−2θ+C− 1L(w,w)=0
(1.2) uu u u v v 2 1 1 2
(Hvv−θuHu+θvHv+ 12Mk2+Ak2−Be−2θ−C+ 21L(w,w)=0,
where z =u+iv is a holomorphic coordinate for which η =−ψ∧(ψ dz+ψ dz¯),
z¯ z
I =e2θ(du2+dv2) and II =e2θ(k du2+k dv2)
1 2
are,respectively,the firstandsecondfundamentalformsofthe liftF :Σ−→E(w)
ofΛ,H = k1+k2 isthemeancurvatureofF,L=e2θ(k −k )andM =−HL. This
2 1 2
condition amounts to the surface being a special isothermic surface in the sense of
Darboux and Bianchi [1,12], at least when H H is non-zero (see [4]).
u v
2. Formal conserved quantities
Denote by Γ(Rn+1,1)[[t,t−1]] the vector space of the formal Laurent series in t
with coefficients in Γ(Rn+1,1), i.e., the series p tk, for some s ∈ N , with
k≤s k 0
coefficients in Γ(Rn+1,1).
P
We define a R[[t,t−1]] valued inner product on Γ(Rn+1,1)[[t,t−1]] by
p(t),q(t) := (p ,q )tk,
i j
k≤s+ri≤s,j≤r
(cid:0) (cid:1) X i+Xj=k
for all p(t)= p tk,q(t)= q tk ∈Γ(Rn+1,1)[[t,t−1]].
k≤s k k≤r k
Definition2.P1. Let(Λ,η)beanPisothermicsurfaceinSnandletp(t)= p ti ∈
i≤0 i
Γ(Rn+1,1)[[t,t−1]]suchthatp 6=0. Wesaythatp(t)isaformal conserved quantity
0 P
of (Λ,η) if (d+tη)p(t) is the zero series.
Proposition 2.2. Let (Λ,η) be an isothermic surface in Sn. If p(t) = p ti
i≤0 i
is a formal conserved quantity of (Λ,η), then
P
(1) p is a D-parallel section of V⊥;
0
(2) the series p(t),p(t) ∈R[[t,t−1]] (thus independent of x∈Σ).
(cid:0) (cid:1)
FORMAL CONSERVED QUANTITIES FOR ISOTHERMIC SURFACES 5
Proof. Item (1) is proved in [4, Proposition2.2].
That the coefficients of p(t),p(t) are constantfollows from the fact that d+tη is
a metric connection:
(cid:0) (cid:1)
d p(t),p(t) = d(p ,p )tk
i j
k≤0 i,j≤0
(cid:0) (cid:1) Xi+Xj=k
= ((d+tη)p ,p )+(p ,(d+tη)p ) tk
i j i j
k≤0 i,j≤0
Xi+Xj=k(cid:0) (cid:1)
=((d+tη)p(t),p(t) + p(t),(d+tη)p(t) =0.
(cid:3)
(cid:1) (cid:0) (cid:1)
The condition that p(t) be a formal conserved quantity amounts to a recursive
systemof equationson its coefficients which we now describe in terms ofthe frame
ψ,ψ ,ψ ,ψˆ of V associated with a holomorphic coordinate z as in Section 1.3.
z z¯
Proposition 2.3. Let (Λ,η) be an isothermic surface in Sn and let z =u+iv be
a holomorphic coordinate on Σ such that η =−ψ∧(ψ dz+ψ dz¯).
z¯ z
Let p(t)= p ti ∈Γ(Rn+1,1)[[t,t−1]] and, for i∈Z−, write
i≤0 i 0
P p =α ψ+β ψ +β¯ψ +γ ψˆ+q ,
i i i z i z¯ i i
where each α ,γ is a real function, β is a complex function and q ∈Γ(V⊥). Then
i i i i
p(t) is a formal conserved quantity if and only if, for all i∈Z−,
0
(2.1a) β =−2γ
i i,z¯
(2.1b) α =2γ +2(κ,κ)γ
i i,zz¯ i
(2.1c) D q =2γ κ−2γ D κ
z i i,z¯ i z¯
(2.1d) γ =2γ +cγ +2(q ,κ).
i−1 i,zz i i
Proof. For each i∈Z−, we have, using (1.1),
0
cβ β
pi,z +η∂ pi−1 = αi,z − i −β¯i(κ,κ)+2(qi,κz¯)+ i−1 ψ
∂z 2 2
+ (cid:0)α +β −2(κ,κ)γ ψ + β¯ −cγ −(cid:1)2(q ,κ)+γ ψ
i i,z i z i,z i i i−1 z¯
+(cid:0)β¯i +γ ψˆ (cid:1) (cid:0) (cid:1)
i,z
2
+(cid:0)β κ+2γ(cid:1)D κ+D q .
i i z¯ z i
The vanishing of th(cid:0)e ψˆ coefficient is equi(cid:1)valent to β = −2γ since γ is real
i i,z¯ i
and then (2.1b) and (2.1d) amount to the vanishing of the coefficients of ψ , ψ
z z¯
respectively while (2.1c) is the same as the vanishing of the normal component.
We are left with the ψ component but its vanishing is a differential consequence of
(2.1). Indeed:
β =−2γ =−2(2γ +c γ +cγ +2(q ,κ)+2(q ,κ ))
i−1 i−1,z¯ i,zzz¯ z¯ i i,z¯ i,z¯ i z¯
=−2(α −2(κ,κ) γ −2(κ,κ)γ +c γ +cγ +2(q ,κ)+2(q ,κ ))
i,z z i i,z z¯ i i,z¯ i,z¯ i z¯
cβ
=−2(α +2(κ,κ) γ +(κ,κ)β¯ − i +2(q ,κ)+2(q ,κ ))
i,z z i i i,z¯ i z¯
2
while
(q ,κ)=(−β¯κ−2γ κ ,κ)=−β¯(κ,κ)−γ (κ,κ) .
i,z¯ i i z i i z
(cid:3)
6 F.E.BURSTALLANDS.D.SANTOS
The equations (2.1) amount to a recursive scheme for constructing a formal con-
servedquantitystartingwithγ solongaswecanbeassuredthateachγ defined
0 i−1
by (2.1d) is real and that (2.1c) is solvable for each i. For this, we need:
Lemma 2.4. Let (Λ,η) be an isothermic surface in Sn and let z = u+iv be a
holomorphic coordinate on Σ such that η =−ψ∧(ψ dz+ψ dz¯).
z¯ z
Let γ be a real function and q ∈Γ(V⊥) such that
D q =2γ κ−2γD κ.
z z¯ z¯
Define γˆ by
γˆ =2γ +cγ+2(q,κ)
zz
and suppose γˆ is real. Then 2γˆ +cγˆ is also real.
zz
Proof. We compute:
2γˆ +c¯γˆ =4γ +2c γ+4c γ +2cγ +4(q,κ) +c¯ 2γ +cγ+2(q,κ) .
z¯z¯ zzz¯z¯ z¯z¯ z¯ z¯ z¯z¯ z¯z¯ zz
The conformal Gauss equation says that cz¯ = 4(κ,κ)z that(cid:0)cz¯z¯ = 4(κ,κ)zz¯ is r(cid:1)eal
and we readily conclude that
Im(2γˆ +c¯γˆ)=Im 4c γ +4(q,κ) +2c¯(q,κ)
z¯z¯ z¯ z¯ z¯z¯
=Im(cid:0)16γz¯(κ,κ)z+4(Dz2¯z¯q,κ)+(cid:1)8(Dz¯q,Dz¯κ)+2(q,2Dz2¯z¯κ+c¯κ)
=Im(cid:0)16γz¯(κ,κ)z+4(Dz2¯z¯q,κ)+8(Dz¯q,Dz¯κ) , (cid:1)
thankstotheconform(cid:0)alCodazziequation. Nowdifferentiatet(cid:1)hecomplexconjugate
of the equation for q and substitute in to get, after a short computation:
Im(2γˆ +c¯γˆ)=Im 8γ (κ,κ)+12 γ (κ,κ) +γ (κ,κ) =0.
z¯z¯ zz¯ z z¯ z¯ z
(cid:3)
(cid:0) (cid:0) (cid:1)(cid:1)
Theorem 2.5. Let (Λ,η) be an isothermic surface in Sn. Then locally, away from
the zeros of η, (Λ,η) has always a formal conserved quantity.
Proof. We workona simply connectedopenset with holomorphiccoordinatez for
which η = −ψ∧(ψ dz +ψ dz¯). We inductively construct a formal power series
z¯ z
p(t)= p ti with
i≤0 i
P p =α ψ+β ψ +β¯ψ +γ ψˆ+q ,
i i i z i z¯ i i
with α ,γ real, β complex and q a section of V⊥ satisfying (2.1). Then, by
i i i i
Proposition 2.3, p(t) will be a formal conservedquantity.
Webeginbytakingγ =0andq anon-zeroparallelsectionofV⊥ sothatp =q .
0 0 0 0
The issue is to define γ and q for then α ,β are given by (2.1a) and (2.1b).
i i i i
Suppose now that we have γ ,q , j >i∈Z− with
j j
D q =2γ κ−2γ D κ
z j j,z¯ j z¯
γ =2γ +cγ +2(q ,κ),
j j+1,zz j+1 j+1
and each γ ,q real. Define γ to be 2γ +cγ +2(q ,κ) and note that
j j i i+1,zz i+1 i+1
Lemma 2.4 (with γˆ =γ ) tells us that γ is real. Since D is flat, equation (2.1c)
i+1 i
for q is integrable when Im D (γ κ−γ D κ) =0 however,
i z¯ i,z¯ i z¯
Im D (γ κ−γ D κ(cid:0)) =Im γ κ−γ (cid:1)D2 κ =Im (γ + c¯γ )κ ,
z¯ i,z¯ i z¯ i,z¯z¯ i z¯z¯ i,z¯z¯ 2 i
bythecon(cid:0)formalCodazziequ(cid:1)ation,a(cid:0)ndthisvanishes(cid:1)thanks(cid:0)toasecondapp(cid:1)lication
of Lemma 2.4 with γˆ = γ . Thus, by induction, γ ,q are defined for all i ∈ Z−
i i i
satisfying (2.1c) and (2.1d) and we are done. (cid:3)
FORMAL CONSERVED QUANTITIES FOR ISOTHERMIC SURFACES 7
Theorem2.5isnotcompletelysatisfying: theresultisonlylocalandthequadrature
that determines each q means that we lack an explicit formula for the p . More,
i i
thesequadraturesintroduceaninfinite numberofconstantsofintegration(parallel
sections of V⊥).
However, the following simple observation allows us to control the constants of
integration: if p(t)= p ti is a local formal conserved quantity for (Λ,η) with
i≤0 i
r(t) = r ti = p(t),p(t) then r(t) is constant by Proposition 2.2. Thus, for
i≤0 t P
all i,
P (cid:0) (cid:1)
(2.2) 2(p ,q )=2(p ,p )=r − (p ,p ).
0 i 0 i i k l
k,l<0
X
k+l=i
Thus the component of eachq along p is completely determined up to a constant
i 0
by the p for 0≥j >i.
j
We use this to refine Theorem 2.5:
Proposition 2.6. Let (Λ,η) be isothermic and r(t):= r ti a formal Laurent
i≤0 i
series with coefficients in R, such that r >0. Then, locally, away from zeros of η,
0
P
there is a formal conserved quantity p(t) of (Λ,η) such that p(t),p(t) =r(t).
Proof. WerevisittheinductionofTheorem2.5. Beginwithγ(cid:0)=0and(cid:1)takep =q
0 0 0
to be a parallel section of V⊥ with (q ,q ) = r . For the induction step, suppose
0 0 0
we have defined γ ,q and so p , for 0≥j ≥i, with (p ,p )=r for j >i.
j j j k+l=j k l j
Wethenhavethatdp +ηp =0,forj >i,whiledp ⊥V⊥ sinceq solves(2.1c).
j j−1 Pi i
It follows that (p ,p ) is constant so that, replacing p by p +s p , for a
k+l=i k l i i i 0
suitable constant s , we may ensure that (p ,p )=r also. (cid:3)
P i k+l=i k l i
In codimension 1, we can say more: in thPis case, p frames V⊥ so that each q is
0 i
completely determined via (2.2) by p , j > i and r . It follows at once that p(t)
j i
is uniquely determined in this case on the domain of the holomorphic coordinate z
by p and r(t) (and so determined up to sign by r(t) alone). We use this to patch
0
together the local solutions provided by Proposition 2.6 to give a global formal
conserved quantity away from the zeros of η.
Theorem 2.7. Let (Λ,η), Λ : Σ → S3 be an isothermic surface in the 3-sphere
and let Z ⊂Σ be the (discrete) zero-set of η. Let r(t)= r ti ∈R[[t,t−1]] be a
i≤0 i
formal Laurent series with r >0.
0
P
Then there is a formal conserved quantity p(t), unique up to sign, defined on Σ\Z
with p(t),p(t) =r(t).
Proof(cid:0). Since Σ(cid:1)is orientable, V⊥ is orientable and so has a global section p with
0
(p ,p )=r . WenowuseProposition2.6tocoverΣ\Z withopensetsU onwhich
0 0 0 α
are defined formal conserved quantities pα with pα = p and pα(t),pα(t) =r(t).
0 0
If x ∈ U ∩U , then we may find a holomorphic coordinate z near x for which
α β
(cid:0) (cid:1)
η =−ψ∧(ψ dz+ψ dz¯). Thus, near x, pα and pβ are uniquely determined by p
z¯ z i i 0
and r(t) and so must coincide. (cid:3)
Sinceholomorphicquadraticdifferentialsona2-torusareconstant,weimmediately
conclude:
Corollary 2.8. Let (Λ,η) be an isothermic 2-torus in the 3-sphere and let r(t) =
r ti ∈R[[t,t−1]] be a formal Laurent series with r >0.
i≤0 i 0
Then there is a globally defined formal conserved quantity p(t), unique up to sign,
P
with p(t),p(t) =r(t).
(cid:0) (cid:1)
8 F.E.BURSTALLANDS.D.SANTOS
3. Special isothermic surfaces of type d in Sn
In this section, we fix an isothermic surface (Λ,η) and a holomorphic coordinate z
with η =−ψ∧(ψ dz+ψ dz¯).
z¯ z
As an application of the ideas of Section 2, we ask when (Λ,η) is special of type
d. Our starting point is the simple observationthat q(t) is a polynomialconserved
quantityofdegreedthenp(t)=t−dq(t)isaformalconservedquantitywithp =0,
i
foralli<−d,andconversely. Moreover,wecanchoosetheconstantsofintegration
so that p = 0, for all i < −d, precisely when γ = 0. Thus, in view of (2.1d),
i −d−1
we have:
Theorem 3.1. Let d ∈ N . (Λ,η) is a special isothermic surface of type d if and
0
only if there exists a formal conserved quantity p(t) of (Λ,η) such that
2γ +cγ +2(q ,κ)=0.
−d,zz −d −d
In this situation, if γ 6=0, then (Λ,η) is a special isothermic surface of type d in
−d
E(p ).
−d
In this case, with r(t) = p(t),p(t) , we have r = (p ,p ) and r =
−2d −d −d −2d+1
2(p ,p ) and so, by Proposition 2.2, these latter inner products are constant.
−d −d+1
(cid:0) (cid:1)
Perhaps surprisingly, a converse is available. First a lemma:
Lemma 3.2. Let p(t) be a formal conserved quantity for (Λ,η). Then, for i,j ≤0,
(3.1) (η p ,p )=γ γ −γ γ
∂/∂z i j i,z¯ j i j,z¯
Proof. Recall that η =−ψ∧ψ while
∂/∂z z¯
p =α ψ−2γ ψ −2γ ψ +γ ψˆ+q
i i i,z¯ z i,z z¯ i i
sothatη p =γ ψ −γ ψ. Nowwritep intermsofγ todrawthe conclusion.
∂/∂z i i z¯ i,z¯ j j
(cid:3)
With this in hand, we have:
Proposition 3.3. (Λ,η) is locally special isothermic of type at most d∈ N if and
only if it admits a formal conserved quantity for which either (p ,p ) or, in case
−d −d
d>1, (p ,p ) is constant.
−d −d+1
Proof. We have already seen necessity of the condition on inner products, so we
turntothesufficiency. Wesuppose,withoutlossofgenerality,thatγ isneverzero
−d
(otherwise (Λ,η) is special of type k < d) and observe that (p ,p ) is constant
−d −d
if and only if (p ,p ) =0. However,
−d −d z
1(p ,p ) =−(η p ,p )
2 −d −d z ∂/∂z −d−1 −d
γ
=−γ γ +γ γ =−γ2 −d−1 ,
−d−1,z¯ −d −d−1 −d,z¯ −d γ z¯
−d
(cid:0) (cid:1)
by Lemma 3.2. Thus, when (p ,p ) is constant, γ /γ is constant also
−d −d −d−1 −d
and we have γ = sγ , for some s ∈ R. Now define a new formal conserved
−d−1 −d
quantitypˆ(t)=p(t)−st−1p(t) andobservethatγˆ =γ −sγ =0so that
−d−1 −d−1 −d
(Λ,η) is special of type at most d.
When d>1 and (p ,p ) is constant,we arguesimilarly,assuming that γ
−d −d+1 −d+1
is non-zero and using
(p ,p ) =−(η p ,p )−(p ,η p )=−(η p ,p ),
−d −d+1 z ∂/∂z −d−1 −d+1 −d ∂/∂z −d ∂/∂z −d−1 −d+1
to conclude that there is a constant s such that γ = sγ and then work
−d−1 −d+1
with p(t)−st−2p(t). (cid:3)
FORMAL CONSERVED QUANTITIES FOR ISOTHERMIC SURFACES 9
Letusnowrestrictattentiontocodimension1wherewecancarryouttherecursions
ofSection2explicitly. Soassumethat(Λ,η)isisothermicinS3 andletN beaunit
(hence D-parallel) section of the line bundle V⊥. Let p(t) be a formal conserved
quantity with p(t),p(t) =r(t) and write κ=kN,
(cid:0) pi(cid:1)=αiψ+βiψz +βiψz¯+γiψ+δiN,
where we have introduced real functions k and δ .
i
We take, without loss of generality, p =N so that γ =0 and δ =1 whence
0 0 0
(3.2a) γ =2k
−1
(3.2b) δ =r /2
−1 −1
(3.2c) p =4(k +k3)ψ−4k ψ −4k ψ +2kψˆ+ r−1N
−1 zz¯ z¯ z z z¯ 2
(3.2d) γ =4k +(2c+r )k.
−2 zz −1
As an immediate consequence of Theorem 3.1, we have:
Proposition 3.4. [6] An isothermic surface has constant mean curvature in a
3-dimensional space-form if and only there is a constant H ∈R such that
(3.3) 2k +ck =Hk.
zz
Here, the space-formis E(p ) and H =−r /2. In particular,in this setting p
−1 −1 −1
and so (p ,p ) is constant, that is,
−1 −1
16(−k k−k4+k k )+r2 /4
zz¯ z z¯ −1
is constant. In view of Proposition 3.3, we deduce the following result of Musso–
Nicolodi:
Proposition 3.5. [14] An isothermic surface has constant mean curvature in a
3-dimensional space-form if and only
(3.4) k k+k4−k k
zz¯ z z¯
is constant.
Similarly, Theorem 3.1 for d=2 gives:
Proposition 3.6. (Λ,η) is special isothermic of type 2 if and only if there are
constants s ,s ∈R such that
1 2
4k +4ck +4c k +(2c +c2)k+8(k k+k4−k k )k
zzzz zz z z zz zz¯ z z¯
(3.5)
+s (2k +ck)+s k =0.
1 zz 2
4. Surfaces of revolution, cones and cylinders
We illustrate the preceding theory by applying it to surfaces of revolution, cones
and cylinders in S3. These surfaces are automatically isothermic surfaces and we
will find necessary and sufficient conditions for them to be special isothermic.
4.1. Surfaces of revolution and cones. We take a uniform approach to cones
and surfaces of revolution by viewing them as extrinsic products. For this, let
W ≤R4,1 be a 3-dimensional subspace and contemplate the direct sum
R4,1 =W ⊕W⊥.
Let Σ = I ×I be a product of intervals and suppose that Λ : Σ → S3 = P(L) is
1 2
of the form
Λ:=hφ +φ i
1 2
where φ : I → W and φ : I → W⊥ are curves with (φ ,φ ) = −(φ ,φ ) = C,
1 1 2 2 1 1 2 2
for a non-zero constant C ∈R.
10 F.E.BURSTALLANDS.D.SANTOS
Hereisthegeometryofthesituation: ifW hasindefinitesignature(sothatC <0),
then φ takes values in a hyperboloid while φ is circle-valued. Taking the half-
1 2
plane model of the hyperboloid, we see that Λ is a surface of revolution. Similarly,
if C >0, φ is a curve on a 2-sphere while φ takes values in a half-line so that Λ
1 2
is the cone over φ .
1
Allsuchsurfacesareisothermic: oneeasilychecksthatη =(dφ −dφ )∧(φ +φ )is
1 2 1 2
closed. Thecorrespondingholomorphiccoordinateisalsoeasytoidentify: withu,v
the arc-lengthparameters on I ,I respectively, set z =u+iv. Then ψ :=φ +φ
1 2 1 2
is satisfies |dψ|2 =|dz|2 and η =−ψ∧(ψ dz+ψ dz¯).
z¯ z
Set S ={x∈W :(x,x) =C} so that φ :I →S, let n be a unit normal to φ in
1 1 1
S and k the corresponding curvature so that
1
φ′′ =kn− φ .
1 C 1
We also have:
1
φ′′ = φ .
2 C 2
Using these, we compute:
(4.1a) ψ = 1(φ′ −iφ′)
z 2 1 2
k
(4.1b) N =n+ (φ +φ )
1 2
2
(4.1c) ψˆ= 1(k2 − 1)φ + 1(k2 + 1)φ + kn
2 4 C 1 2 4 C 2 2
with N a unit section of V⊥, for V the central sphere congruence of Λ. Then
1 1 k 1 k2 k
ψ = (φ′′−φ′′)=− (φ +φ )+ n=(− − )ψ+ N,
zz 4 1 2 4C 1 2 4 4C 8 4
so that the Schwarzianderivative and Hopf differential of Λ are given by
1 k2 k
c= + κ= N.
2C 4 4
In the notation of section 3, k = k/4 so that, in the current setting, Proposition
3.4 reads
Proposition 4.1. (Λ,η) is special isothermic of type 1 if and only if, for some
constant α,
k k3
(4.2) + +k′′+αk=0;
C 2
Note that equation (4.2) means exactly that φ is an elastic curve in S, and then
1
α=0 if and only if φ is a free elastic curve.
1
Similarly, Proposition 3.6 reads:
Proposition 4.2. (Λ,η) is special isothermic of type 2 if and only if there exist
real constants α and β such that
k k3 3k5 2k′′ 5 k k3
+ + + + (kk′2+k2k′′)+k(iv)+α( + +k′′)+βk=0.
C2 C 8 C 2 C 2
We now prove that the constant term p of a polynomial conserved quantity of
−d
(Λ,η) lies W. For this, we recall the notations of the previous sections and begin
with a lemma:
Lemma 4.3. Let p(t) be a formal conserved quantity for (Λ,η). Then, for each
i≤0, γ =−(ψ,p ) is independent of v.
i i
Proof. We induct, noting that both c and k =k/4 are independent of v. (cid:3)
