Links between different analytic descriptions of constant mean curvature surfaces

9 Pages·0.12 MB·English

by  Eugene V. Ferapontov

Journal of Nonlinear MathematicalPhysics 2000, V.7, N 1, 14–22. Letter
Links Between Different Analytic Descriptions
of Constant Mean Curvature Surfaces
E.V. FERAPONTOV † and A.M. GRUNDLAND ‡
0
0
0 † Department of Mathematical Sciences, Loughborough University, Loughborough,
2 Leicestershire LE11 3TU, United Kingdom

n e-mail: [email protected]
a
J Centre for Nonlinear Studies, Landau Institute for Theoretical Physics,

1 Academy of Sciences of Russia, Kosygina 2, 117940 Moscow, GSP-1, Russia

e-mail: [email protected]
]
G
‡ Centre de Recherches Mathematiques, Universit´e de Montr´eal, C.P. 6128 Succ.
D
Centre-Ville, 2910 Chemin de la Tour, Montr´eal, Qu´ebec H3C 3J7 Canada
.
h e-mail: [email protected]
t
a
m Received September 6, 1999; Revised October 5, 1999; Accepted November 5, 1999
[
Abstract

1
v Transformations between different analytic descriptions of constant mean curvature
9 (CMC) surfaces are established. In particular, it is demonstrated that the system
8
1 2 2
∂ψ1 =(ψ1 + ψ2 )ψ2
01 ∂¯ψ2 = |(ψ|1 2+| ψ|2 2)ψ1
− | | | |
0
0 descriptive of CMC surfaceswithin the framework ofthe generalizedWeierstrass rep-
/ resentation, decouples into a direct sum of the elliptic Sh-Gordon and Laplace equa-
h
t tions. Connections of this system with the sigma model equations are established.
a
It is pointed out, that the instanton solutions correspond to different Weierstrass
m
parametrizations of the standard sphere S2 E3.
: ⊂
v
i
X 1 Introduction
r
a
We investigate the system
2 2
∂ψ1 = (ψ1 + ψ2 )ψ2
| | | | (1.1)
∂¯ψ2 = (ψ1 2+ ψ2 2)ψ1
− | | | |
which has been derived in [1] and governs constant mean curvature (CMC) surfaces in the
conformal parametrization z,z¯ (∂ = ∂z,∂¯= ∂z¯). This system was subsequently discussed
in [2,4,6]. In this paper, we demonstrate that system (1.1) can be decoupled into a direct
Copyright c 2000 by E.V. Ferapontov and A.M. Grundland
(cid:13)
Different Analytic Descriptions of Mean Curvature Surfaces 15
sum of the elliptic Sh-Gordon and Laplace equations. Firstly, we change from ψ1, ψ2 to
the new dependent variables Q,R
Q = 2(ψ2∂ψ¯1 ψ¯1∂ψ2), R = 2(ψ1 2+ ψ2 2)2/Q .
− | | | | | |
Secondly, we introduce new independent variables η, η¯ according to
dη = Qdz, dη¯= Q¯dz¯
q
p
(theseformulaearecorrectsinceQisholomorphic). InthenewvariablesQ,R,η,η¯,system
(1.1) assumes the decoupled form
1
(lnR)ηη¯ = R,
R −
Qη¯ = Q¯η = 0,
whichisadirectsumoftheellipticsh-GordonandLaplaceequations. Thistransformation
is an immediate corollary of the known properties of CMC surfaces. Connection of system
(1.1) with the sigma-model equations
2ρ¯
∂∂¯ρ ∂ρ∂¯ρ= 0,
− 1+ ρ2
| |
(ρ = iψ¯1/ψ2) is also discussed. In terms of ρ, our transformation adopts the form:
2∂ρ∂ρ¯ ∂ρ¯
Q = , R = .
(1+ ρ2)2 (cid:12)∂ρ(cid:12)
| | (cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
2 Generalized Weierstrass representation of surfaces in R3
Following the results of [1], with any solution ψ1, ψ2 of the Dirac equations
∂ψ1 = pψ2, ∂¯ψ2 = pψ1, (2.1)
−
(p(z,z¯) is a real potential), we associate a surface M2 E3 with the radius-vector r(z,z¯)
⊂
defined by the formulae
∂r = i(ψ22+ψ¯12), ψ¯12 ψ22, 2ψ2ψ¯1 ,
− −
∂¯r= (cid:0) i(ψ¯22+ψ12), ψ12 ψ¯22, 2ψ1(cid:1)ψ¯2 .
− − −
(cid:0) (cid:1)
The latter are compatible by virtue of (2.1). The unit normal n of the surface M2 can be
calculated according to the formula n= 1∂r ∂¯r/∂r ∂¯r and is represented as follows:
i × | × |
1
n = i(ψ¯1ψ¯2 ψ1ψ2), ψ¯1ψ¯2+ψ1ψ2, ψ1ψ¯1 ψ2ψ¯2 , (2.2)
q − −
(cid:0) (cid:1)
2 2
q = ψ1 + ψ2 .
| | | |
16 E.V. Ferapontov and A.M. Grundland
Onecanverifydirectlythatthescalar productsinE3 are(n,n) = 1,(n,∂r) = (n,∂¯r) = 0.
Equations of motion of the complex frame ∂r, ∂¯r, n are of the form
∂q
∂r  2 q 0 Q  ∂r
∂ ∂¯r = 0 0 2Hq2  ∂¯r 
n  Q  n
   −H −2q2 0  
 
(2.3)
2
0 0 2Hq
∂r  ∂¯q  ∂r
∂¯ ∂¯r = 0 2 q Q¯  ∂¯r 
 
n  Q¯  n
   H 0  
 −2q2 − 
 
where Q = 2(ψ2∂ψ¯1 ψ¯1∂ψ2) and H = p/q is the mean curvature. Formulae (2.3) are
−
compatible with the scalar products
(n,n) = 1, (∂r,∂¯r) = 2q2
(all other scalar products being equal to zero). Using (2.3), one can derive the following
useful equation for the unit normal n:
∂∂¯n+(∂n,∂¯n)n+∂¯H∂r+∂H∂¯r= 0. (2.4)
ThefirstfundamentalformI = (dr,dr) andthesecondfundamentalformII = (d2r,n)
2
of the surface M are given by
2
I = 4q dzdz¯,
(2.5)
II = Qdz2+4Hq2dzdz¯+Q¯dz¯2.
2
The quantity Qdz is called the Hopf differential. The real potential p and the spinors
ψ1,ψ2 satisfying (1.1) can be viewed as the ”generalized Weierstrass data” of the surface
2
M . The corresponding Gauss-Codazzi equations which are the compatibility conditions
for (2.3), are of the form
1QQ¯
∂∂¯(lnq2) = 2H2q2,
2 q2 −
∂¯Q = 2q2∂H, (2.6)
∂Q¯ = 2q2∂¯H.
In fact, equations (2.6) are a direct differential consequence of (2.1), as can be checked
by a straightforward calculation. Gauss-Codazzi equations of surfaces in conformal para-
metrization z,z¯have been discussed in [5]. We recall also that the Gaussian curvature K
2
of the surface M can be calculated as follows:
1
K = ∂∂¯(lnq). (2.7)
−q2
Different Analytic Descriptions of Mean Curvature Surfaces 17
The unit normal n = (n1,n2,n3), given by (2.2), maps a surface M2 onto the unit
2
sphere S . Combining this map with the stereographic projection, we obtain a map ρ of
2
the surface M onto the complex plane, called the complex Gauss map. In our notation,
ρ assumes the form
n1+in2 ψ¯1
ρ = = i . (2.8)
1 n3 ψ2
−
According to the results of [3], Gauss map ρ satisfies the nonlinear equation
2ρ¯
(∂∂¯ρ ∂ρ∂¯ρ)H = ∂H∂¯ρ, (2.9)
− 1+ ρ2
| |
which formally can be viewed as a differential consequence of (2.1). In terms of ρ and H,
the initial data ψ1, ψ2 and p assume the form
ρ¯ i∂¯ρ 1 √i∂ρ¯ ∂ρ¯
ψ1 = √H 1p+ ρ2, ψ2 = √H 1+ ρ2, p = 1+| ρ| 2
| | | | | |
while the expressions for q and Q take the form
1 ∂¯ρ∂ρ¯ 2 ∂ρ∂ρ¯
q = , Q = .
H 1+ ρ2 H (1+ ρ2)2
| | | |
Remark 1. Linear problem (2.3) can be rewritten in terms of ψ1, ψ2 as follows. First of
all, we point out that
∂q = ψ1∂ψ¯1+ψ¯2∂ψ2.
Combining this equation with the definition of Q:
1
Q = ψ2∂ψ¯1 ψ¯1∂ψ2,
2 −
and solving these two equations for ∂ψ¯1, ∂ψ2, we can “close” system (2.1) as follows:
0 qH
ψ1 ψ1
∂ = Q ∂q  ,
(cid:18) ψ2 (cid:19) (cid:18) ψ2 (cid:19)
−2q q
 
(2.10)
∂¯q Q¯
∂¯ ψ1 = q 2q  ψ1 .
(cid:18) ψ2 (cid:19) (cid:18) ψ2 (cid:19)
qH 0
 − 
The compatibility conditions for system (2.10) coincide with (2.6). We point out that the
3
2 2 matrix approach to surfaces in E has been extensively developed in [5]. From the
×
point of view of the theory of integrable systems, linear system (2.3) can be regarded as
the squared eigenfunction equations corresponding to (2.10) (indeed, ∂r, ∂¯r and n are
quadratic expressions in ψ1 and ψ2).
18 E.V. Ferapontov and A.M. Grundland
3 CMC-1 surfaces
The class of CMC-1 surfaces is characterized by the constraint H = 1 or, equivalently,
p = q. Introducing this ansatz in (2.1), we arrive at the nonlinear system (1.1)
2 2
∂ψ1 = (ψ1 + ψ2 )ψ2,
| | | |
∂¯ψ2 = (ψ1 2+ ψ2 2)ψ1,
− | | | |
which is the main subject of our study. According to the previous section, system (1.1) is
equivalent to
1QQ¯
∂∂¯lnq2 = 2q2,
2 q2 − (3.1)
∂¯Q = ∂Q¯ = 0,
where q = ψ1 2 + ψ2 2, Q = 2(ψ2∂ψ¯1 ψ¯1∂ψ2). Thus, for CMC surfaces the Hopf
| |2 | | −
differential Qdz is holomorphic. Applying to system (3.1) the reciprocal transformation
dη = Qdz, dη¯= Q¯dz¯
q
p
(that is, changing from z,z¯ to the new independent variables η,η¯ which are correctly
defined in view of the holomorphicity of Q) and introducing
2
2q
R = ,
Q
| |
we transform system (3.1) into the decoupled form
1
(lnR)ηη¯ = R,
R − (3.2)
Qη¯ = Q¯η = 0.
This result provides the rationale for the change of variables which we introduce in Sec-
tion 1.
Remark 2. System (1.1) is invariant under the SU(2)-symmetry
ζ1 = αψ1+βψ¯2, ζ2 = βψ¯1+αψ2, (3.3)
−
where α,β are complex constants subject to the constraint αα¯ +ββ¯ = 1. One can check
directly, thatthequantities QandR areinvariantundertransformations (3.3), sothatthe
surfaces, correspondingto (ψ1,ψ2) and (ζ1,ζ2), have coincident fundamental forms. Thus,
3
they are identical up to a rigid motion in E . The passage from ψ1,ψ2 to Q,R can thus
be viewed as a passage to the differential invariants of the point symmetry group (3.3).
Different Analytic Descriptions of Mean Curvature Surfaces 19
Remark 3. For CMC-1 surfaces, system (2.10) allows the introduction of a spectral pa-
rameter
0 q
ψ1 ψ1
∂ = Q ∂q 
(cid:18) ψ2 (cid:19) λ (cid:18) ψ2 (cid:19)
− 2q q
 
(3.4)
∂¯q 1 Q¯
∂¯ ψ1 = q λ2q  ψ1
(cid:18) ψ2 (cid:19) (cid:18) ψ2 (cid:19)
q 0
 − 
where λ is a unitary constant, λ = 1. The gauge transformation
| |
ψ˜1 = ψ1, ψ˜2 = q−1ψ2
reduces linear spectral problem (3.4) to the SL(2) form
2
0 q
ψ˜1   ψ˜1
∂ =
(cid:18) ψ˜2 (cid:19) Q (cid:18) ψ˜2 (cid:19)
 λ 0 
 − 2q2 
 
∂¯q Q¯ (3.5)
∂¯ ψ˜1 = q 2λ  ψ˜1 .
(cid:18) ψ˜2 (cid:19)  ∂¯q (cid:18) ψ˜2 (cid:19)
 1 
 − − q 
 
The compatibility conditions for both systems (3.4) and (3.5) coincide with (3.1). From
the linear system (3.5) the radius-vector r can berecovered via the so-called Sym formula:
we refer to [5] and [9] for the further discussion of the Sym approach.
Linear system (3.5) can be readily rewritten in terms of ψ1, ψ2. Indeed, observing that
(3.5) implies
∂lnq = ∂ln(ψ1 2+ ψ2 2), ∂¯lnq = ∂¯ln(ψ1 2+ ψ2 2),
| | | | | | | |
we can take q = c(ψ1 2+ ψ2 2), c C, which, uponsubstitution in (3.5), producessystem
| | | | ∈
(1.1). Thus transformation from (3.1) to (1.1) consists of rewriting (3.5) in terms of the
ψ. Representations in terms of ψ are called eigenfunction equations, and are fundamental
in soliton theory – see eg [7]. The Lax pair for system (1.1) is of the form [6]
Q Q
φ1 2  −ψ¯1ψ2+ 2q2ψ1ψ¯2 −ψ¯12− 2q2ψ¯22  φ1
∂ =
(cid:18) φ2 (cid:19) µ+1  Q Q (cid:18) φ2 (cid:19)
 ψ22+ 2q2ψ12 ψ¯1ψ2− 2q2ψ1ψ¯2 
 
Q¯ Q¯
 −ψ1ψ¯2+ 2q2ψ¯1ψ2 ψ¯22 + 2q2ψ¯12 
∂¯ φ1 = 2 φ1 .
(cid:18) φ2 (cid:19) µ−1  −ψ12− 2Qq¯2ψ22 ψ1ψ¯2− 2Qq¯2ψ¯1ψ2 (cid:18) φ2 (cid:19)
 
20 E.V. Ferapontov and A.M. Grundland
It is interesting to note that the compatibility conditions for this linear system, which is
of the first order in the derivatives of ψ, give us exactly system (1.1). The latter is also of
the first order in ψ.
4 CMC surfaces and sigma model equations
For CMC surfaces, equations (2.9) imply the nonlinear sigma model
2ρ¯
∂∂¯ρ ∂ρ∂¯ρ= 0, (4.1)
− 1+ ρ2
| |
descriptions of the stationary two-dimensional SU(2) magnet. The transformation of the
system (4.1) into the decoupled form (3.2) now assumes the form
∂ρ∂ρ¯ ∂ρ¯
Q = 2 , R = ,
(1+ ρ2)2 |∂ρ|
| | (4.2)
dη = Qdz, dη¯= Q¯dz¯.
q
p
In terms of the unit normal vector n, equation (2.4) adopts the form of the SO(3) sigma
model
∂¯∂n+(∂n,∂¯n)n= 0, (n,n) = 1. (4.3)
Formula (2.8) establishes a link between sigma models (4.1) and (4.3). The topological
charge
1
(n,[∂n ∂¯n])dz dz¯
4π Z Z × ∧
can be written as
1
∂∂¯ln qdz dz¯
2πi Z Z ∧
2
which, in view of (2.7), is the topologically invariant integral curvature of the surface M .
Instanton solutions of system (4.3) are specified by the ansatz
∂n = in ∂n, ∂¯n= in ∂¯n,
± × ∓ ×
which, after a simple calculation, implies Q = 0. Solutions of system (1.1) specified by a
constraint Q = 0, can be represented in the form
ρ ∂¯ρ¯ √∂ρ ∂ρ
ψ1 = 1+pρ2, ψ2 = 1+ ρ2, p = 1+| ρ| 2, (4.4)
| | | | | |
where ρ(z) is an arbitrary holomorphic function. In the case when the energy
∂ρ∂¯ρ
E = dz dz¯
Z Z 1+ ρ2 ∧
| |
Different Analytic Descriptions of Mean Curvature Surfaces 21
is finite, the function ρ(z) is rational in z [8].
Geometrically, instanton solutions (4.4) parametrize the standardly embedded sphere
2 3
S E . This can be readily seen from formulae (2.5), which, in case Q = 0, imply
⊂
the proportionality of fundamental forms I and II. This example shows that different
Weierstrass data (ψ1,ψ2,p) can correspond to different parametrizations of one and the
2 3
same surface M E .
⊂
Introducing the two-component complex vector
N = (ψ1, ψ¯2), q = ψ1 2+ ψ2 2,
√q √q | | | |
one can check that N satisfies the equations of the CP1 sigma model
(N,N¯)= 1,
(4.5)
∂∂¯N = (N¯,∂¯N)∂N+(N¯,∂N)∂¯N kN,
−
where
1 1
k = 2(N¯,∂N)(N,∂¯N¯)+ (∂N,∂¯N¯)+ (∂¯N,∂N¯).
− 2 2
Equations (4.5) are associated with the Lagrangian
L = (∂¯N,∂N¯)+(∂N,∂¯N¯)+2(N¯,∂N)(N¯,∂¯N) 2k[(N,N¯) 1] dzdz¯,
Z Z { − − }
where k is the Lagrange multiplier.
Acknowledgments
One of the authors (E.V.F.) would like to thank Centre de Recherches Math´ematiques,
Universit´e de Montr´eal for generous hospitality and financial support during the period
when this investigation was performed. This work was supported by research grant from
NSERC of Canada and the Fonds FCAR du Gouvernment du Qu´ebec. We would like to
thank Professor B. G. Konopelchenko for drawing our attention to this subject and for
helpful discussions. We would like to thank the referee for useful comments.
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