Lobachevskii Journal of Mathematics Vol.3, 1999, 73–95 \begin{matrix}c \\ ◯ \end{matrix}A. Fujioka and J. Inoguchi

A. Fujioka* and J. Inoguchi

ON SOME GENERALISATIONS OF CONSTANT MEAN CURVATURE SURFACES

(submitted by B.N.Shapukov)

Abstract. In this paper we shall study relations between three generalisations of CMC surfaces– Bonnet surfaces, H-surfaces and HIMC surfaces. We introduce the notion of HIMC surfaces in space forms and show that HIMC surfaces have similar properties to CMC surfaces. Furthermore we shall introduce a special one-parameter family of framings for H- surfaces in the space forms. Through these framings we reveal relationship between the associated family of H-surfaces and the extended solutions for harmonic maps into SO(3).

Introduction

In recent fifteen years geometry of surfaces in 3-dimensional space forms have been paid much attentions of differential geometers. Because surface geometry in space forms might be a treasure box of integrable systems. In fact, we can find many integrable equations which appeared in surface geometry as Gauss-Codazzi equations of special surfaces. In particular surfaces with constant mean curvature (CMC surfaces) in 3-dimensional space forms have been studied extensively. (See [1] and references therein.)

It seems to be interesting to generalise CMC surfaces from various viewpoints. In this paper we shall consider this problem from the three viewpoints mentioned below.

At first we shall consider variational theoretic generalisations of CMC surfaces. In the previous paper [15], the first named author introduced the equations whose solutions are considered as generalisations of simply connected minimal surfaces or constant mean curvature surfaces in the 3-dimensional space forms. We call these equations H-surface equations.

1991 Mathematical Subject Classification. Primary 53A10; Secondary 53A04, 58B20.

*Partially supported by Grant-in-Aid for Scientific Research number 10740028, the Ministry of Education, Science and Culture, Japan

Secondly we shall generalise CMC surfaces in terms of integrability theory. On each CMC surface free of umbilics we can take an isothermic coordinate system such that its Gauss-Codazzi equations take the form of elliptic Sinh-Gordon equation. Thus we can apply methods in the theory of integrable systems to the study of CMC surfaces. The starting point of such methods is a zero curvature representation (ZCR), more precisely, a representation of the Gauss-Codazzi equations in a form of compatibility condition for the Lax equations with spectral parameter.

From the view point of integrability theory, it is natural to consider the following problem:

What kind of surfaces can be considered as integrable generalisations of CMC surfaces?

In 1994, A. I. Bobenko [1] has introduced the notion of surfaces with harmonic inverse mean curvature (HIMC surfaces) in Euclidean 3-space E^{3}. A surface in E^{3} is said to be a surface with harmonic inverse mean curvature if the reciprocal of the mean curvature is a harmonic function. He showed that HIMC surfaces are natural generalisation of CMC surfaces in terms of integrability theory. In fact, he showed HIMC surfaces admit ZCR with variable spectral parameter (cf. [7].) Further, A. I. Bobenko, U. Eitner and A. Kitaev [3] have developed detailed study on HIMC surfaces via the theory of Painlevé transcendents P_{V} and P_{V I}.

A. S. Fokas and I. M. Gelfand [13] found another characterisation of HIMC surfaces. They showed that the class of HIMC surfaces is the only class of surfaces which admit a Lie-point group of symmetries. In [17], the first named author generalised the notion of HIMC surfaces in Euclidean 3-space to those in 3-dimensional space forms.

The third generalisations of CMC surfaces are purely geometric ones– Bonnet surfaces. In 1867, O. Bonnet proved that every umbilic free CMC surface in E^{3} admits nontrivial isometric and principal curvature preserving deformations. Traditionary surfaces in space forms which admit such deformations are called Bonnet surfaces. The study of Bonnet surfaces is a subject of wide interest. There are much progress on this subject. In particular Bobenko and Eitner have obtained many remarkable results on Bonnet surfaces in E^{3} via the theory of Painlevé transcendents [2].

In this paper we shall study relations between these three generalisations of CMC surfaces in 3-dimensional space forms.

More precisely, in section 4 we shall obtain a special one-parameter family of solutions for the H-surface equations. As a result, we can reveal relationship between the associated family of H-surfaces and the well-known one-parameter family

– so called extended solutions– for harmonic maps, into special orthogonal group SO(3). (See [33].)

In section 5 we shall introduce the notion of HIMC surfaces in space forms. Our definition coincides with that of [1] if the ambient space is E^{3}. We shall show that HIMC surfaces in space forms admit natural one-parameter deformations. Further we shall establish the Lawson correspondences between HIMC surfaces in section 6.

In sections 6 and 7 we shall give a survey on our recent work [18] and brief survey on this area.

1. H-surfaces in Euclidean 3-space

Let E^{3} be a Euclidean 3-space with metric ⟨·, ·⟩. The metric ⟨·, ·⟩ is expressed as ⟨·, ·⟩ = dξ_{1}^{2} + dξ_{2}^{2} + dξ_{3}^{2} in terms of natural coordinate system.

Minimal surfaces in E^{3} are defined as conformal immersions from Riemann surfaces into E^{3} which extremise the area functional. On the other hand minimal surfaces can be characterised as conformal harmonic maps. Namely a minimal surface is a conformal smooth map f : M → E^{3} from a Riemann surface M which extrimises the Dirichelet energy

E(f) = ∫_{M} \frac{1}{2} |df|^{2} dV.  (1.1)

The Euler-Lagrange equation of the energy E is given by

f_{xx} + f_{yy} = 0  (1.2)

with respect to an isothermal coordinate system (x, y).

Surfaces with nonzero constant mean curvature in E^{3} have a variational characterisation. A smooth conformal immersion f : M → E^{3} of a Riemann surface is a surface of constant mean curvature H ≠ 0 if and only if f extrimises the area functional under the volume preserving variations. Such surfaces have another variational aspect. In fact, constant mean curvature surfaces are characterised as conformal immersions which are critical points of the following functional:

E_{H}(f) := E(f) + ∫_{M} \frac{2H}{3} ⟨f, f_{x} × f_{y}⟩dV, H ∈ R.  (1.3)

Here (x, y) is an isothermal coordinate system. Critical points (which are not necessarily conformal) of the functional E_{H} are called H-surfaces.

The Euler-Lagrange equation of the functional E_{H} is given by

f_{xx} + f_{yy} = 2H(f_{x} × f_{y})  (1.4)

with respect to an isothermal coordinates (x, y). Here × denotes the vector product operation of E^{3}. We refer to M. Struwe [30] for analytical aspects of H-surfaces.

2. Gauge theoretic equations for H-surfaces

In this section we shall introduce the notion of H-surfaces in 3-dimensional space forms in gauge theoretic formulation [15].

We start with Riemannian symmetric space expressions of 3-dimensional space forms. We shall denote 𝔐^{3}(c) by the simply connected 3-dimensional space form of constant curvature c.

We denote G_{c} the identity component of the isometry group of 𝔐^{3}(c). More explicitly G_{c} is given by

G_{0} = SO(3) ⋉ R^{3} = \{( \begin{matrix}T & s \\ 0 & 1 \end{matrix} ) ∈ GL_{4}R ; T ∈ SO(3), s ∈ R^{3}\} ,
G_{c} = SO(4)

for c > 0 and

G_{c} = SO^{+}(3, 1)
= {X = (x_{ij}) ∈ GL_{4}R ; ^{t}XJX = J, det X = 1, x_{44} > 0}

for c < 0, where J is the signature matrix of Minkowski 4-space E_{1}^{4} defined by J =diag(1, 1, 1, −1). Set

K_{c} = \{( \begin{matrix}T & 0 \\ 0 & 1 \end{matrix} ) ∈ GL_{4}R ; T ∈ SO(3)\} .

An involutive automorphism σ_{c} of G_{c} is defined by

σ_{c} (X) = JXJ, X ∈ G_{c}.

The Lie algebra 𝔤_{c} and 𝔨_{c} of G_{c} and K_{c} are

𝔤_{c} = \{( \begin{matrix}A & b \\ −sign(c)^{t}b & 0 \end{matrix} ) ∈ M_{4}R ; A ∈ 𝔰𝔬(3), b ∈ R^{3}\} ,

𝔨_{c} = \{( \begin{matrix}A & 0 \\ 0 & 0 \end{matrix} ) ∈ M_{4}R ; A ∈ 𝔰𝔬(3)\} .

The tangent space 𝔪_{c} of 𝔐^{3}(c) at the origin o = K_{c} is

𝔪_{c} = \{( \begin{matrix}0 & b \\ −sign(c)^{t}b & 0 \end{matrix} ) ∈ M_{4}R ; b ∈ R^{3}\} ,

where

sign(c) = \{ \begin{matrix}1 & if c > 0, \\ 0 & if c = 0, \\ −1 & if c < 0. \end{matrix}

An Ad_{G_{c}}K_{c}-invariant metric on 𝔪_{c} is defined by

g_{c} (( \begin{matrix}0 & b \\ −sign(c)^{t}b & 0 \end{matrix} ) , ( \begin{matrix}0 & b′ \\ −sign(c)^{t}b′ & 0 \end{matrix} )) = L^{2}(c)⟨b, b′⟩,

where b, b′ ∈ R^{3} and

L(c) = \{ \begin{matrix}1 & if c = 0, \\ \frac{1}{\sqrt{|c|}} & if c ≠ 0. \end{matrix}

The quintet (G_{c}, K_{c}, σ_{c}, g_{c}) is the Riemannian symmetric data of 𝔐^{3}(c). We denote by π_{c} the natural projection π_{c} : G_{c} → 𝔐^{3}(c) = G_{c}/K_{c}.

Let M be a simply connected Riemann surface. For any smooth map f : M → 𝔐^{3}(c), there always exists a map F : M ⟶ G_{c} such that π_{c} ∘ F = f. Such a map F is called a framing of f. Then we have a decomposition F^{−1}dF =: α = α_{𝔨_{c}} + α_{𝔪_{c}}. Since M is a Riemann surface, we have a type decomposition α_{𝔪_{c}} = α𝔪_{c}^{′} +α𝔪_{c}^{′′}, where α𝔪_{c}^{′} is an 𝔪_{c}^{C}-valued (1, 0)-form with complex conjugate α𝔪_{c}^{′′}. We write the decomposition α = α_{𝔨_{c}} + α𝔪_{c}^{′} + α𝔪_{c}^{′′} simply as α = α_{𝔨} + α𝔪^{′} + α𝔪^{′′}. When we write α_{𝔪} as

α_{𝔪} = ( \begin{matrix}0 & b \\ −sign(c)^{t}b & 0 \end{matrix} )

for b ∈ Ω^{1}(M; R^{3}), we shall define an 𝔪_{c}-valued 2-form α_{𝔪} × α_{𝔪} by

α_{𝔪} × α_{𝔪} = ( \begin{matrix}0 & b × b \\ −sign(c)^{t}(b × b) & 0 \end{matrix} ) .

Then the vector field

n_{f} = \frac{L(c)}{2}dπ (AdF ∗ (α_{𝔪} × α_{𝔪}))

is independent of the choice of framing F. Here ∗ is the Hodge star operator on M. We may regard n_{f} as a unit normal vector field along the H-surface f even if f is not a conformal immersion.

We now fix a base point p_{0} ∈ M. For H ∈ C^{∞}(M), c ∈ R, we set

H_{H,c} = \{f : M ⟶ 𝔐^{3}(c) ; \frac{1}{2}τ(f) = Hn_{f}, f(p_{0}) = o\} ,

C_{H,c} = {f ∈ H_{H,c} ; f is conformal} .

Here τ(f) is the tension field of f defined by τ(f) := trace(∇df). Then C_{H,c} is the set of based conformal immersion with mean curvature function H. Thus H-surfaces f ∈ H_{H,c} may be considered as generalisations of conformally immersed surface with prescribed mean curvature function H [25].

Let f ∈ H_{H,c} and F be a framing of f. Then direct computation shows that \frac{1}{2}τ(f) = Hn_{f} is equivalent to

(dα′_{𝔪} + [α_{𝔨} ∧ α′_{𝔪}]) − (dα′′_{𝔪} + [α_{𝔨} ∧ α′′_{𝔪}]) = 2\sqrt{−1}HL(c)α′_{𝔪} × α′′_{𝔪}. (2.1)

Taking the 𝔪_{c}- and 𝔨_{c}-parts of the Maurer-Cartan equations for α, we have

(dα′_{𝔪} + [α_{𝔨} ∧ α′_{𝔪}]) + (dα′′_{𝔪} + [α_{𝔨} ∧ α′′_{𝔪}]) = 0  (2.2)

and

dα_{𝔨} + \frac{1}{2}[α_{𝔨} ∧ α_{𝔨}] + [α′_{𝔪} ∧ α′′_{𝔪}] = 0.  (2.3)

Equations (2.1) and (2.2) are equivalent to

dα′_{𝔪} + [α_{𝔨} ∧ α′_{𝔪}] = \sqrt{−1}HL(c)α′_{𝔪} × α′′_{𝔪}.  (2.4)

Conversely, let α = α_{𝔨} + α𝔪^{′} + α𝔪^{′′} be a 𝔤_{c}-valued 1-form which satisfies (2.3) and (2.4). Then from the standard fact, there exists a map F : M → G_{c}, unique up to left translation by a constant element of K_{c}, satisfying F^{−1}dF = α and π ∘ F ∈ H_{H,c}. We shall call equations (2.3) and (2.4) H-surface equations.

We set

A_{H,c} = {F ∈ C^{∞}(M, G_{c}) ; α = F^{−1}dF satisfies (2.3) and (2.4)}

and

K_{c} = C^{∞}(M, K_{c}).

The group K_{c} acts on A_{H,c} from the right. The Lie group K_{c} acts on A_{H,c} by left translations. Thus we get the following isomorphism:

H_{H,c} ≅ K_{c}\A_{H,c}/K_{c}.

See [15] for more details on the actions of these groups.

Given f ∈ H_{H,c} and F a framing of f. Since α′_{𝔪} can be written as

α′_{𝔪} = ( \begin{matrix}0 & b \\ −sign(c)^{t}b & 0 \end{matrix} ) for b = ( \begin{matrix}b^{1} \\ b^{2} \\ b^{3} \end{matrix} ) ∈ Ω^{1,0}(M, C^{3}),

we shall define ι(α′_{𝔪}) ∈ Ω^{1,0}(M; 𝔰𝔬(3)^{C}) by

ι(α′_{𝔪}) = ( \begin{matrix}0 & −b^{3} & b^{2} \\ b^{3} & 0 & −b^{1} \\ −b^{2} & b^{1} & 0 \end{matrix} ) .

We regard ι(α′_{𝔪}) naturally as a 𝔨_{c}^{C}-valued (1,0)-form and define ι(α′′_{𝔪})

∈ Ω^{0,1}(M; 𝔰𝔬(3)^{C}) in a similar way. Then the H-surface equations are equivalent to

dα_{𝔨} + \frac{1}{2}[α_{𝔨} ∧ α_{𝔨}] + sign(c)[ι(α′_{𝔪}) ∧ ι(α′′_{𝔪})] = 0,  (2.5)

dι(α′_{𝔪}) + [α_{𝔨} ∧ ι(α′_{𝔪})] = \sqrt{−1}HL(c) [ι(α′_{𝔪}) ∧ ι(α′′_{𝔪})] .  (2.6)

These equations (2.5)–(2.6) can be interpreted gauge theoretically as follows.

Let P be the trivial principal bundle P = M×SO(3) over a Riemann surface M with structure group SO(3). Then A := α_{𝔨} defines a connection d_{A} = d + A on P. Define an 𝔰𝔬(3)^{C}-valued one-form ψ by

ψ = ψ′ + ψ′′, ψ′ = ι(α′_{𝔪}), , ψ′′ = ι(α′′_{𝔪}).

Since P is trivial ψ is naturally regarded as an adjoint bundle Ad(P)-valued one-form.

Then the H-surface equations (2.5)–(2.6) become

F (A) + \frac{1}{2}sign(c)[ψ ∧ ψ] = 0,  (∗_{H,c})

d′′_{A}ψ′ = \sqrt{−1}L(c)H[ψ′ ∧ ψ′′]

We shall call the equations (∗_{H,c}) the gauge theoretic equations for Hsurfaces

Remark. The H-surface equations (∗_{H,c}) may be considerd as generalisations of gauge theoretic equations for harmonic map or self duality equations over M.

(1) (∗_{0,0}) is the harmonic map equations of M into SO(3) (cf. [6].)

(2) (∗_{0,c}) for c < 0 is the self duality equations over M studied by N. Hitchin [24].

(3) (∗_{0,c}) for c > 0 is the gauge theoretic equations for harmonic maps into SO(3).

We refer to Hitchin [23] for the equations (∗_{0,c}) with c > 0. M. Mukai [26] and G. Valli [35] studied the equatios (∗_{0,c}) with c > 0 over general principal bundles over a Riemann surface. Mukai and Ohnita [24] studied gauge theoretic equations for harmonic maps into Riemannian symmetric spaces.

3. A reduction of H-surface equations.

In this section we shall consider the following reduction condition:

α_{𝔨} = pι(α′_{𝔪}) + p̄ι(α′′_{𝔪}), p ∈ C.  (⋆_{p})

for the H-surface equations with constant H. First we shall determine the necessary condition for p. Substituting (⋆_{p}) into (2.5), we have

d (pι(α′_{𝔪}) + p̄ι(α′′_{𝔪})) + \frac{1}{2} [(pι(α′_{𝔪}) + p̄ι(α′′_{𝔪})) ∧ (pι(α′_{𝔪}) + p̄ι(α′′_{𝔪}))]
+ sign(c) [ι(α′_{𝔪}) ∧ ι(α′′_{𝔪})] = 0.

Using (2.6), we shall obtain

p (− [α_{𝔨} ∧ ι(α′_{𝔪})] + \sqrt{−1}HL(c) [ι(α′_{𝔪}) ∧ ι(α′′_{𝔪})])
+ p̄(− [α_{𝔨} ∧ ι(α′′_{𝔪})] − \sqrt{−1}HL(c) [ι(α′_{𝔪}) ∧ ι(α′′_{𝔪})])
+ |p|^{2} [ι(α′_{𝔪}) ∧ ι(α′′_{𝔪})] + sign(c) [ι(α′_{𝔪}) ∧ ι(α′′_{𝔪})] = 0.

Using (⋆_{p}) again, we have

\{−|p|^{2} + \sqrt{−1}HL(c)p − \sqrt{−1}HL(c)p̄ + sign(c)\} [ι(α′_{𝔪}) ∧ ι(α′′_{𝔪})] = 0.

Hence we have

\{|p + \sqrt{−1}HL(c)|^{2} − (H^{2}L^{2}(c) + sign(c))\} [ι(α′_{𝔪}) ∧ ι(α′′_{𝔪})] = 0. (3.1)

We observe

H^{2}L^{2}(c) + sign(c) = \{ \begin{matrix}\frac{H^{2}+c}{c} & c > 0, \\ H^{2} & c = 0, \\ \frac{H^{2}+c}{−c} & c < 0. \end{matrix}

Thus if we assume H^{2} + c ≥ 0, we can obtain some p ∈ C such that (3.1) holds.

Lemma 3.1. Let c = 0, H ∈ R and p ∈ C satisfy |p + \sqrt{−1}H|^{2} = H^{2}. Then for any f ∈ H_{H,0}, there exists a framing of f such that condition (⋆_{p}) is satisfied.

Proof. Since c = 0, 𝔐^{3}(c) = E^{3}. Let z be a local complex coordinate on M. Since f ∈ H_{H,0}, we have

\frac{∂^{2}f}{∂z∂z̄} = −\sqrt{−1}H \frac{∂f}{∂z} × \frac{∂f}{∂z̄}.  (3.2)

Now we can see that a framing F of f satisfies the condition (⋆_{p}) if and only if F satisfies

T^{−1}dT = pι(α′_{𝔪}) + p̄ι(α′′_{𝔪})  (3.3)

and

T^{−1}df = α′_{𝔪} + α′′_{𝔪}.  (3.4)

From (3.3) and (3.4), direct computation shows that

dT · T^{−1} = pι (\frac{∂f}{∂z} ) dz + p̄ι (\frac{∂f}{∂z̄}) dz̄.

Put B = pι (\frac{∂f}{∂z} ) dz + p̄ι (\frac{∂f}{∂z̄}) dz̄. Then

dB − \frac{1}{2}[B ∧ B]

= −pι (\frac{∂^{2}f}{∂z∂z̄}) dz ∧ dz̄ + p̄ι (\frac{∂^{2}f}{∂z∂z̄}) dz ∧ dz̄

− |p|^{2} [ι (\frac{∂f}{∂z̄}) dz ∧ ι (\frac{∂f}{∂z̄}) dz̄]

= (−|p|^{2} + \sqrt{−1}Hp − \sqrt{−1}Hp̄) ι (\frac{∂f}{∂z} × \frac{∂f}{∂z̄}) dz ∧ dz̄ by (3.4)

= 0.

Thus we have T : M → SO(3) such that dT · T^{−1} = B. Obviously the

framing F = ( \begin{matrix}T & f \\ 0 & 1 \end{matrix} ) satisfies (3.3) and (3.4). □

Theorem 3.2. Let H, c ∈ R such that H^{2} + c ≥ 0 and p ∈ C satisfy |p + \sqrt{−1}HL(c)|^{2} = H^{2}L^{2}(c) + sign(c). Then for any f ∈ H_{H,c}, there exists a framing of f satisfyning the condition (⋆_{p}) .

Proof. It is proved in [15] that if sign(H^{2} + c) =sign(H′^{2} + c′) for H, H′, c, c′ ∈ R, there exists a bijective map

ϕ : A_{H,c} → A_{H′,c′}

such that ϕ gives a bijective correspondence ϕ̄: H_{H,c} → H_{H′,c′}.

Since H^{2} + c ≥ 0, we can choose H′ = \sqrt{H^{2} + c} and c′ = 0. In this case, ϕ(F) ∈ A_{\sqrt{H^{2}+c},0} (F ∈ A_{H,c}) is determined by

ϕ(F)^{−1}dϕ(F) = α_{𝔨} + L(c)α′_{𝔪} + L(c)α′′_{𝔪}
+ \sqrt{−1} (H − \sqrt{H^{2} + c}) L(c)ι(α′_{𝔪})
− \sqrt{−1} (H − \sqrt{H^{2} + c}) L(c)ι(α′′_{𝔪}),

where F^{−1}dF = α_{𝔨} + α′_{𝔪} + α′′_{𝔪}.

By Lemma 3.1, we can obtain a framing F′ of ϕ̄(f) such that condition (⋆_{p′}) is satisfied for p′ = \frac{1}{L(c)}p + \sqrt{−1} (H − \sqrt{H^{2} + c}) .

Note that p′ satisfies |p′ + \sqrt{−1}\sqrt{H^{2} + c}|^{2} = H^{2} + c. Then it is easy to see that ϕ^{−1}(F′) is a framing of f such that condition (⋆_{p}) is satisfied. □

4. The associated family of harmonic maps into SO(3)

In this section we shall concentrate our attention to the case c = 0 and H ∈ R^{∗}. Take any μ_{0} ∈ S^{1} = \{w ∈ C ; |w| = 1\} and put p = −\sqrt{−1}H + \sqrt{−1}Hμ_{0}^{−1}. Then p satisfies the condition in Lemma 3.1. Let f ∈ H_{H,0} and F a framing of f as in Lemma 3.1. Then direct computation shows that (2.4) is equivalent to

dα′_{𝔪} = \sqrt{−1}Hμ_{0}α′_{𝔪} × α′′_{𝔪}.  (4.1)

Proposition 4.1. If μ_{0} = ±1, T is a harmonic map into SO(3).

Proof. First we observe

T^{−1}dT = \sqrt{−1}H(μ_{0}^{−1} − 1)ι(α′_{𝔪}) − \sqrt{−1}H(μ_{0} − 1)ι(α′′_{𝔪}).  (4.2)

Thus we have

d ∗ (T^{−1}dT)
= d \{\sqrt{−1}H(μ_{0}^{−1} − 1)ι(−\sqrt{−1}α′_{𝔪}) − \sqrt{−1}H(μ_{0} − 1)ι(\sqrt{−1}α′′_{𝔪})\}
= H(μ_{0}^{−1} − 1)ι(dα′_{𝔪}) + H(μ_{0} − 1)ι(dα′′_{𝔪})
= H(μ_{0}^{−1} − 1)ι(\sqrt{−1}Hμ_{0}α′_{𝔪} × α′′_{𝔪})
+ H(μ_{0} − 1)ι(−\sqrt{−1}Hμ_{0}^{−1}α′_{𝔪} × α′′_{𝔪}) by (4.1)
= \sqrt{−1}H^{2}(−μ_{0} + μ_{0}^{−1})ι(α′_{𝔪} × α′′_{𝔪})
= 0.

Hence T^{−1}dT satisfies the harmonic map equations. □

Remark. From (4.2), if μ_{0} = 1, T is a constant map. If μ_{0} = −1, the covering harmonic map T̃ : M → { \atop SO(3)} = SU(2) of T is also obtained in [15].

In the following we fix the framing for μ_{0} = −1. We shall study harmonic map T : M → SO(3) as above. In [33] K. Uhlenbeck defined a spectral family for each harmonic map into Lie groups (so-called extended solutions.) Then the spectral family for our T is

B_{μ} = \sqrt{−1}H(μ^{−1} − 1)ι(α′_{𝔪}) − \sqrt{−1}H(μ − 1)ι(α′′_{𝔪}),

where μ ∈ S^{1}. For any μ ∈ S^{1}, we have a map T_{μ} : M → SO(3) such that T_{μ}^{−1}dT_{μ} = B_{μ}. Then we have

Theorem 4.2. For any μ ∈ S^{1}, there exists f_{μ} : M → E^{3} such that

T_{μ}^{−1}df_{μ} = −μ^{−1}α′_{𝔪} − μα′′_{𝔪}.

Then f_{μ} ∈ H_{H,0} and {f_{μ}}_{μ∈S^{1}} is the associated family of f = f_{−1}.

Proof. We put α̃_{𝔨} = B_{μ}, α̃′_{𝔪} = −μ^{−1}α′_{𝔪} and α̃′′_{𝔪} = −μα′′_{𝔪}. Since T_{−1} is a harmonic map, we have

dα̃_{𝔨} + \frac{1}{2} [α̃_{𝔨} ∧ α̃_{𝔨}] + [α̃′_{𝔪} ∧ α̃′′_{𝔪}] = 0.

Furthermore we have

dα̃′_{𝔪} + [α̃_{𝔨} ∧ α̃′_{𝔪}]
= d(−μ^{−1}α′_{𝔪})
+ [(−\sqrt{−1}H(μ^{−1} − 1)ι(α′_{𝔪}) − \sqrt{−1}H(μ − 1)ι(α′′_{𝔪})) ∧ (−μ^{−1}α′_{𝔪})]
= −μ^{−1}dα′_{𝔪} + \sqrt{−1}H(1 − μ^{−1})α′_{𝔪} × α′′_{𝔪}
= \sqrt{−1}Hμ^{−1}α′_{𝔪} × α′′_{𝔪} + \sqrt{−1}H(1 − μ^{−1})α′_{𝔪} × α′′_{𝔪} by (4.1)
= \sqrt{−1}Hα̃′_{𝔪} × α̃′′_{𝔪}.

Since α̃ = α̃_{𝔨} + α̃′_{𝔪} + α̃′′_{𝔪} satisfies H-surface equations (2.3) and (2.4) for c = 0, we have f_{μ} ∈ H_{H,0}.

The associated family of f is determined by

α_{λ} = α_{𝔨} + λ^{−1}α′_{𝔪} + λα′′_{𝔪}
− \sqrt{−1}H(λ^{−1} − 1)ι(α′_{𝔪}) + \sqrt{−1}H(λ − 1)ι(α′′_{𝔪}),

where α = α_{𝔨} + α′_{𝔪} + α′′_{𝔪} = F^{−1}dF for any framing F of f and λ ∈ S^{1}. In our case, since

α_{𝔨} = −2\sqrt{−1}Hι(α′_{𝔪}) + 2\sqrt{−1}Hι(α′′_{𝔪}),

we have

α_{λ} = λ^{−1}α′_{𝔪} + λα′′_{𝔪}
+ \sqrt{−1}H(−λ^{−1} − 1)ι(α′_{𝔪}) − \sqrt{−1}H(−λ − 1)ι(α′′_{𝔪}).

If we put λ = −μ, we have α_{λ} = α̃. □

5. HIMC surfaces in space forms

In this section we shall consider the following reduction:

α_{𝔨} = hι(α′_{𝔪}) + h̄ι(α′′_{𝔪}).  (⋆_{h})

for the H-surface equations with H ∈ C^{∞}(M). Here h is a holomorphic function on M. This ansatz is a generalisation of (⋆_{p}) discussed in the previous section. We shall determine the necessary condition of the reduction (⋆_{h}).

Substituting (⋆_{h}) into (2.5) we get

hdι(α)′ + h̄dι(α)′′ + |h|^{2} [ι(α)′ ∧ ι(α)′′] + c[ι(α)′ ∧ ι(α)′′] = 0.

Using (2.6) and (⋆_{h}) we have

\{\sqrt{−1}HL(c)(h − h̄) − (|h|^{2} − sign(c) )\} [ι(α)′ ∧ ι(α)′′] = 0.  (5.1)

Thus if we assume that

H = \frac{|h|^{2} − sign(c)}{\sqrt{−1}L(c)(h − h̄)}  (5.2)

then (5.1) is always satisfied.

Now we shall study fundamental properties of H-surfaces with mean curvature (5.2). Direct computation gives the following:

Proposition 5.1. Let h be a holomorphic function on a Riemann surface M such that Im h ≠ 0 and |h|^{2} − sign(c) ≠ 0. Then the function φ on M defined by

φ = \frac{\sqrt{−1}(h − h̄)}{|h|^{2} − sign(c)}

satisfies the following partial differential equation:

\frac{∂^{2}φ}{∂z∂z̄} − \frac{2cφ}{1 + cφ^{2}} |\frac{∂φ}{∂z}|^{2} = 0.  (5.3)

Here z is a local complex coordinate of M.

To understand the differential geometric meaning of the form (5.2) of H, we shall introduce the following 1-dimensional Riemannian manifold I(c);

I(c) = \{ \begin{matrix}(R(t), g(c) ) & c ≥ 0, \\ (R(t) \ \{\frac{±1}{\sqrt{−c}}\}, g(c) ) & c < 0, \end{matrix}

g(c) = \frac{dt^{2}}{1 + ct^{2}} .

Note that these Riemannian manifolds may be considered as 1-dimensional space forms.

We can get easily the following proposition.

Proposition 5.2. Let φ : (M, z) → I(c) be a smooth map. Then φ is a harmonic map if and only if φ is a solution to (5.3).

Note that if c = 0 then the harmonicity of φ coincides with usual harmonicity of functions. The harmonic map equation (5.3) can be solved explicitly as follows.

Proposition 5.3. All the solutions of (5.3) are given by

φ = \{ \begin{matrix}\frac{\sqrt{−1}(h−h̄)}{|h|^{2}−sign(c)} & c = 0, \\ ^{\frac{\sqrt{−1}(h−h̄)}{|h|^{2}−sign(c)}} or \frac{|h|^{2}−sign(c)}{\sqrt{−1}(h−h̄)} & c ≠ 0 \end{matrix}  (5.4)

for some holomorphic function h on M.

These observations lead us to the following definition.

Definition 5.4. ([17]) Let f : M → 𝔐^{3}(c) be a conformal H-surface. Then (M, f) is said to be a surface with harmonic inverse mean curvature if its reciprocal mean curvature is a harmonic map into I(c).

It is easy to see that this definition coincides with that of A. I. Bobenko [1] if c = 0. In Bobenko [1], it is proved that each surface with harmonic inverse mean curvature (HIMC surface) in Euclidean 3-space has natural associated family (conformal deformations.) We can prove that every HIMC surfaces in space forms admit natural deformations.

Under the deformation

H ⟼ H + 2t, t ∈ R,

the mean curvature function H is transformed into

H_{t} = \frac{|h + 2t|^{2} − sign(c)}{\sqrt{−1}L(c)(h − h̄)}.

We shall define one parameter family {β_{t}}_{t∈R} of 𝔤_{c}-valued one forms on M by

β_{t} = β_{𝔨} + β′_{𝔪} + β′′_{𝔪},
β_{𝔨} = α_{𝔨} + p_{t}ι(α′_{𝔪}) + p̄_{t}ι(α′′_{𝔪}) ∈ Ω^{1}(M; 𝔨_{c}),
β′_{𝔪} = r_{t}α′_{𝔪} ∈ Ω^{1,0}(M; 𝔪_{c}^{C}), β′′_{𝔪} = r̄_{t}α′′_{𝔪} ∈ Ω^{0,1}(M; 𝔪_{c}^{C}).

Here the functions p_{t} and r_{t} are defined by

p_{t} = −\frac{h(h + 2t) + sign(c)}{(h + 2t)^{2} − sign(c)}t, r_{t} = \frac{|h|^{2} − sign(c)}{(h + 2t)^{2} − sign(c)}.

Then we have

Proposition 5.5. The one form β_{t} satisfies

dβ_{𝔨} + \frac{1}{2}[β_{𝔨} ∧ β_{𝔨}] + [β′_{𝔪} ∧ β′′_{𝔪}] = 0  (5.5)

and

dβ′_{𝔪} + [β_{𝔨} ∧ β′_{𝔪}] = \sqrt{−1}H_{t}L(c)β′_{𝔪} × β′′_{𝔪}.  (5.6)

Comparing (5.5)–(5.6) and (2.3)–(2.4), we get the following.

Theorem 5.6. Let f ∈ H_{H,c} (respectively C_{H,c} ) with mean curvature function H = (|h|^{2} − sign(c)) /\{\sqrt{−1}(h − h̄)\}. Then f has a natural one parameter deformation {f_{t}}_{t∈R} through f = f_{0} such that f_{t} ∈ H_{H_{t},c} (respectively C_{H_{t},c}.) In particular if f ∈ C_{H,c} then each f_{t} is a surface with harmonic inverse mean curvature in 𝔐^{3}(c).

Proof. We define f_{t} by integrating (5.5)–(5.6). Note that f is conformal if and only if α′_{𝔪} ⊗ α′_{𝔪} = 0 and α′_{𝔪} ⊗ α′′_{𝔪} ≠ 0 [15]. Hence we have f_{t} ∈ C_{H_{t},c} if f ∈ C_{H,c}.

Remark. If f is a HIMC surface in E^{3} then the family {f_{t}}_{t∈R} is just the one obtained by Bobenko [1]. One can easily check this fact by computing the first and second fundamental forms of each f_{t}.

In the next section we shall study the familty {f_{t}} by 2 by 2 matrix formulation.

6. Integrable generalisations of CMC surfaces

In this section we shall consider integrable generalisations of CMC surfaces. We shall start with the following list of characteristic properties of CMC tori.

Every CMC torus

(0) is umbilic free; (I) is covered by isothermic coordinates; (II) has associated family (locally); (III) admits zero curvature representation; (IV) has harmonic Gauss map and (V) has local correspondents in other space forms.

Here isothermic coordinates means an isothermal-curvature line coordinates. (See [5].) With respect to isothermic coordinates, the Gauss-Codazzi equations are written in the form of elliptic Sinh-Gordon equation.

Note that for each CMC torus the property (0) implies the existence of associated family (II). In fact each umbilic free CMC surface is isometrically deformable preserving its mean curvature. (O. Bonnet [4]. see also K. Voss [36].)

We shall pay attention to the properties (II) and (III). The generalisations of CMC surfaces from the view point of (II) are Bonnet surfaces. Namely surfaces which admit one parameter family of nontrivial isometric deformation preserving the mean curvature. Here a nontrivial isometric deformation means an isometric deformation which can not be obtained by one paramerer subgroup of isometry group of the ambient space.

The following theorem due to M. Umehara [34] is fundamental in the global study of Bonnet surfaces. (cf. [32].)

Theorem 6.1. Let M be a compact surface of positive genus immersed in 3-dimensional space form 𝔐^{3}(c). If M is a Bonnet surface then the mean curvature is constant.

Now we shall recall fundamental equations of surfaces in 𝔐^{3}(c). Let M be a simply connected Riemann surface and f : M → 𝔐^{3}(c) a conformal immersion with a unit normal vector field n_{f}. The first fundamental form I of M can be expressed as

I = e^{u}dzdz̄  (6.1)

in terms of local complex coordinate z = x + \sqrt{−1}y. The Hopf differential Q^{#} of M is defined by Q^{#} = Qdz^{2}, Q = g_{c}(f_{zz}, n_{f}). The Gauss-Codazzi equatios of M are given by

u_{zz̄} + \frac{1}{2}(H^{2} + c)e^{u} − 2|Q|^{2}e^{−u} = 0, H_{z} = 2e^{−u}Qz̄.  (GC_{c})

For later use we shall recall the following results on isothermic surfaces.

Lemma 6.2. A surface (M, f) in 𝔐^{3}(c) is isothermic if and only if there exists a local isothermic coordinate (x, y) such that the coefficient Q of the Hopf differential with respect to z = x + \sqrt{−1}y is real around any point of M. Namely there exists a smooth real-valued function q such that Q = q/2.

Proposition 6.3. Let (M, f) be an isothermic surface in E^{3} and 𝔇 a simply connected isothermic coordinate region of M. Then there exists an immersion f^{∗} : 𝔇 → E^{3} such that

(1) (𝔇, f^{∗}) is isothermic;

(2) (𝔇, f^{∗}) is anti conformal to (M, f) and

(3) the fundamental quantities of f^{∗} are given by e^{u^{∗}} = e^{−u}, H^{∗} = q, Q^{∗} = \frac{H}{2} .

The immersion (𝔇, f^{∗}) is called the Christoffel transform of M or dual surface of M.

The following characterisation of Bonnet surfaces is due to L. Raffy [28] and W. C. Graustein [22].

Proposition 6.4. A surface (M, f) free of umbilics in 𝔐^{3}(c) is a Bonnet surface if and only if

(1) (M, f) is isothermic and

(2) Δ(1/Q) = 0 with respect to isothermic coordinates.

Remark. Graustein proved that each Bonnet surface in E^{3} is a Weingarten surface. It is easy to check that every Bonnet surface in 𝔐^{3}(c) of positive curvature c is Weingarten. However Bonnet surfaces in 𝔐^{3}(c) of negative curvature c are not necessarily Weingarten. See Voss [36] and Chen and Li [9]. There exist Bonnet surfaces in 𝔐^{3}(c) with c < 0 which are not Weingarten but Willmore. U. Eitner provides us computer graphics of Bonnet-Willmore surfaces in the homepage of TU-Berlin (http://www-

sfb288.math.tu-berlin.de/).

Next, we shall consider generalisations of CMC surfaces in E^{3} from the view point of (III).

Now we shall use 2 by 2 matrix formalism [1]. Let us denote the skew field of quaternions by H and its natural basis by {1, i, j, k}. The skew field H is naturally identified with Euclidean 4-space and the group Sp(1) of unit quaternions are identified with 3-sphere S^{3}. The Lie algebra 𝔰𝔭(1) of Sp(1) is Im H and hence identified with Euclidean 3-space E^{3}.

Since the field H = C ⊕ kC is a right complex linear space of dimension 2, H has a matricial expression in the linear space M_{2}C of all complex 2 by 2 matrices:

ξ_{0}1 + ξ_{1}i + ξ_{2}j + ξ_{3}k ⟷ ( \begin{matrix}ξ_{0} + \sqrt{−1}ξ_{1} & −ξ_{3} + \sqrt{−1}ξ_{2} \\ ξ_{3} + \sqrt{−1}ξ_{2} & ξ_{0} − \sqrt{−1}ξ_{1} \end{matrix} ) .  (6.2)

Under this identification the Lie group Sp(1) is identified with special linear group SU(2). Hence Euclidean 3-space E^{3} is identified with the Lie algebra 𝔰𝔲(2) of SU(2). We shall take an SU(2)-framing Φ of f defined by

Ad(Φ)(i, j, k) = (n_{f}, e^{− \frac{u}{2}} f_{x}, e^{− \frac{u}{2}} f_{y}).  (6.3)

We denote ω the pulled-back one form of the Maurer-Cartan form of SU(2) by Φ. Namely ω = Φ^{−1}dΦ. We shall write ω explicitly as follows:

ω = Udz + V dz̄,  (6.4)
U = ( \begin{matrix}\frac{u_{z}}{4} & \frac{H}{2} e^{\frac{u}{2}} \\ −Qe^{− \frac{u}{2}} & −\frac{u_{z}}{4} \end{matrix} ) , V = ( \begin{matrix}−\frac{uz̄}{4} & Q̄e^{− \frac{u}{2}} \\ −\frac{H}{2} e^{\frac{u}{2}} & \frac{uz̄}{4} \end{matrix} ) .  (6.5)

We shall insert a variable spectral parameter λ, i.e., an additional complex parameter depending on z and z̄, in ω. There are three possibilities of natural way of inserting λ:

ω_{λ}^{[i]} = U_{λ}^{[i]}dz + V_{λ}^{[i]}dz̄, i = 1, 2, 3;  (6.6)

(1) U_{λ}^{[1]}, V_{λ}^{[1]}: Deform Q ↦ Q_{λ} = λQ in ω;

(2) U_{λ}^{[2]}, V_{λ}^{[2]}:

U_{λ}^{[2]} = ( \begin{matrix}\frac{u_{z}}{4} & λ\frac{H}{2} e^{\frac{u}{2}} \\ −Qe^{− \frac{u}{2}} & −\frac{u_{z}}{4} \end{matrix} ) , V_{λ}^{[2]} = ( \begin{matrix}−\frac{uz̄}{4} & Q̄e^{− \frac{u}{2}} \\ −λ^{−1} \frac{H}{2} e^{\frac{u}{2}} & \frac{uz̄}{4} \end{matrix} );  (6.7)

(3) U_{λ}^{[3]}, V_{λ}^{[3]}:

U_{λ}^{[3]} = ( \begin{matrix}\frac{u_{z}}{4} & λ^{−1} \frac{H}{2} e^{\frac{u}{2}} \\ −λ^{−1}Qe^{− \frac{u}{2}} & −\frac{u_{z}}{4} \end{matrix} ) , V_{λ}^{[3]} = ( \begin{matrix}−\frac{uz̄}{4} & λQ̄e^{− \frac{u}{2}} \\ −λ\frac{H}{2} e^{\frac{u}{2}} & \frac{uz̄}{4} \end{matrix} ) . (6.8)

By computing the curvature F (ω_{λ}^{[i]}) of ω^{[i]} we get the following.

Proposition 6.5. Assume that Q ≠ 0 and H ≠ 0. Then

(1) F (ω_{λ}^{[1]}) = 0 if and only if M is a Bonnet surface. With respect to a suitable isothermic coordinate z = x + \sqrt{−1}y, the spectral parameter λ and Q are written in the following form:

λ = \frac{1 − 2\sqrt{−1}tḡ}{1 + 2\sqrt{−1}tg}, \frac{1}{Q} = g + ḡ.  (6.9)

Here g is a holomorphic function satisfying

g_{zz}(g + ḡ) − g_{z}^{2} = ḡ_{z̄z̄}(g + ḡ) − ḡ2 ; z̄.  (Bo)

The Gauss-Codazzi equations become

(\frac{H_{zz̄}}{H_{z}} )z̄ − \frac{Hz̄}{g_{z}} = |Q|^{2} (2gz̄ − \frac{H^{2}}{H_{z}} ) g_{z},  (6.10)

g_{z}H_{z} = gz̄Hz̄, e^{u} = −\frac{2g_{z}}{(g + ḡ)^{2}H_{z}} .  (6.11)

(2) F (ω_{λ}^{[2]}) = 0 if and only if M is a HIMC surface. The spectral parameter λ and H are given by

λ = \frac{ḡ}{g}, \frac{1}{H} = g + ḡ  (6.12)

for some holomorphic function g.

(3) F (ω_{λ}^{[3]}) = 0 if and only if M is a CMC surface. The spectral parameter λ is independent of z and z̄.

We can solve the Maurer-Cartan equation for ω_{λ}^{[i]}:

(Φ_{λ}^{[i]})^{−1}dΦ_{λ}^{[i]} = ω_{λ}^{[i]}  (6.13)

over a simply connected Riemann surface M. In particular Φ_{λ}^{[3]} is an extended framing of a harmonic (Gauss) map of M into a unit 2-sphere S^{2}= Ad(SU(2))i in E^{3}. These three framings are gauge-equivalent in the following way:

Φ_{λ}^{[2]} = Φ_{λ}^{[1]} ( \begin{matrix}\frac{1}{\sqrt{λ}} & 0 \\ 0 & \sqrt{λ} \end{matrix} ) , Φ_{λ^{−1}}^{[3]} = Φ_{λ^{2}}^{[1]} ( \begin{matrix}\frac{1}{\sqrt{λ}} & 0 \\ 0 & \sqrt{λ} \end{matrix} ) .  (6.14)

We saw two generalisations of CMC surfaces in this section, i.e., Bonnet surfaces and HIMC surfaces. Bonnet surfaces and HIMC surfaces are related by Christoffel transformations.

Corollary 6.6. The Christoffel transform of a Bonnet surface in E^{3} is a HIMC surface in E^{3}.

Since every Bonnet surface is isothermic, each Bonnet surface has a HIMC Chirstoffel transform. However HIMC surfaces not always have Bonnet-Chistoffel transforms since HIMC surfaces are not necessarily isothermic.

Remark. By the complex coordinate change

w = ∫ \frac{dz}{g_{z}}  (6.15)

in Proposition 6.5–(1), we can see that |Q| and u depend only on w + w̄. Gauss equation (6.10) becomes

(\frac{H′′}{H′} )′ − H′ = |Q|^{2} (2 − \frac{H^{2}}{H′} ) .  (Ha)

Here the prime means the derivation with respect to w + w̄. The ordinary equation (Ha) is called the Hazzidakis equation. (cf. Cartan [8].) In [2] Bobenko and Eitner solved (Ha) by using Painlevé transcendents. In this paper we prefer to use isothermic coordinate description of (Ha). We refer to Chern [10] and Colares and Kenmotsu [12] for the reader who prefers differential form fashion.

Hazzidakis equations for Bonnet surfaces in 𝔐^{3}(c) are described in [36]. (See also [9].) Recently U. Eitner studied Hazzidakis equations for Bonnet surfaces in 𝔐^{3}(c) by Painlevé techniques in his Ph.D. thesis.

Theorem 6.7. ([17]) Let M be a simply connected Riemann surface and h a holomorphic function on M such that |h|^{2} − sign(c) ≠ 0 and Im h ≠ 0. Define a function H(c) by the right hand side of (5.2), i.e.,

H(c) = \frac{|h|^{2} − sign(c)}{\sqrt{−1}L(c)(h − h̄)}.

Then for any real numbers c and c′ the moduli spaces C_{H(c),c} and C_{H(c′),c′} are isomorphic.

This theorem may be considered as a generalisation of the Lawson correspondences for CMC surfaces. This theorem tells us that HIMC surfaces are generalisations of CMC surfaces from the view point of (V). We can observe these correspondences via the immersion formulae for HIMC surfaces.

Hereafter we assume c = 0, ±1 for simplicity.

Let f : M → E^{3} be a HIMC surface and Φ[t] a one parameter family of framings for (M, f) defined by Φ[t] = Φ_{λ}^{[2]}, λ = \frac{1+2t/h̄}{1+2t/h}, t ∈ R. (cf. (7.4) in [1].) Since the zero curvature equation (6.13) is completelty integrable, the solution Φ[t] can be extended for all t ∈ C. We denote the extended solution of Φ[t] by the same letter and call it complexified framing.

Theorem 6.8. Let Φ[t] : M × C → SL_{2}C be a complexified framing. Then (c = 0) For every t ∈ R

f_{t} = \frac{∂}{∂t}Φ[t] · Φ[t]^{−1}

is a HIMC surface in E^{3} described in Theorem 5.6.

(c = 1) For any t_{1}, t_{2} ∈ R; t_{1} ≠ t_{2}

f_{S}(t_{1}, t_{2}) := Φ[t_{1}] · Φ[t_{2}]^{−1}

is a HIMC surface in S^{3} = SU(2).

(c = −1) For any t ∈ R

f_{H}(t) := Φ[\sqrt{−1}t] · Φ[\sqrt{−1}t]^{∗},

is a HIMC surface in H^{3} with mean curvature H^{2} > 1. Here we identified Minkowski 4-space E_{1}^{4} with the linear space of all Hermitian 2-matrices:

Her(2) = \{Ξ = ( \begin{matrix}ξ_{4} + ξ_{1} & ξ_{3} − \sqrt{−1}ξ_{2} \\ ξ_{3} + \sqrt{−1}ξ_{2} & ξ_{4} − ξ_{1} \end{matrix} ) ; ξ_{1}, ξ_{2}, ξ_{3}, ξ_{4} ∈ R\} .

Under this identification the hyperbolic 3-space H^{3} is represented by

H^{3} = { Ξ ∈ Her(2) ; det Ξ = −1}.

In particular the Lawson correspondents of f = f_{0} in S^{3} and H^{3} are f_{S}(1/2, −1/2) and f_{H}(1/2).

Remark. Surfaces with harmonic inverse mean curvature 0 < H^{2} < 1 in hyperbolic 3-space have no Euclidean correspondents. However we can obtain Lax representations and immersion formulae for such surfaces [17].

7. Bonnet surfaces with constant curvature

In this section we assume that surfaces are free of umbilics. Bonnet surfaces in 𝔐^{3}(c) with constant Gaussian curvature are investigated more precisely than general case. In fact I. M. Roussos [29] showed that if a Bonnet surface E^{3} is flat then it is a piece of some deformations of the logarithmic spiral cylinder or a generalised cone. A. G. Colares and K. Kenmotsu [12] proved that if a Bonnet surface in E^{3} is of constant curvature K then K = 0 and the same results as above. H. Takeuchi [31] showed that if a Bonnet surface in 𝔐^{3}(c) is of constant curvature K then K = c or 0. Later W. Chen and H. Li [9] obtained simpler proof of this fact.

In this section we shall apply our results on Lawson correspondences to the classification of Bonnet surfaces with constant curvature. We shall only describe the statements of our classification. For more details we refer to [18].

By the theorem due to Takeuchi and Chen–Li as above, in order to classify Bonnet surfaces with constant curvature in 𝔐^{3}(c), we have only to consider the following cases

(1) Bonnet surface with flat extrinsic curvature (K = c, c = ±1),

(2) flat Bonnet surfaces.

Let f : M → 𝔐^{3}(c) be a Bonnet surface. As saw in the preceeding section every Bonnet surface is isothermic. We shall take an isothermic coordinate system z = x + \sqrt{−1}y such that Q = q/2. Here q is a real-valued function.

(i) K = c = ±1.

With respect to the isothermic coordinate z the Gauss-Codazzi equations become

q^{2} = e^{2u}H^{2}, qz̄ = e^{u}H_{z}.  (7.1)

In particular we get

q = ±e^{u}H  (7.2)

We can solve explicitly (7.1)–(7.2) and hence we get the following.

Theorem 7.1. ([18]) Assume that c = ±1. Then the first fundamental form, Hopf differential and the mean curvature of a Bonnet surface in 𝔐^{3}(c) with constant curvature c are given as follows:

(u, q, H) = (u(η), \frac{e^{\frac{u}{2}}}{s(ξ)}, \frac{ϵe^{\frac{u}{2}}}{s(ξ)}) ,

where (η, ξ, ϵ) = (x, y, −1) or (y, x, 1).

u(η) = \{ \begin{matrix}log \frac{α^{2}}{cosh^{2}(αη+β)} & c = 1, \\ log \frac{α^{2}}{sinh^{2}(αη+β)}, log \frac{α^{2}}{cosh^{2}(αη+β)} or log \frac{1}{(η+β)^{2}} & c = −1, \end{matrix}

s(ξ) = \{ \begin{matrix}C_{1} cos αξ + C_{2} sin αξ & c = 1, \\ C_{1} cos αξ + C_{2} sin αξ, C_{1}e^{αξ} + C_{2}e^{−αξ} or C_{1}ξ + C_{2} & c = −1, \end{matrix}

α > 0, β ∈ R, (C_{1}, C_{2}) ∈ R^{2} \ {0}.

In the case c = −1, this result is obtained by Takeuchi [31] by differential form computations. We can explicitly solve Gauss–Weingarten formulae for Bonnet surfaces in Theorem 7.1. See our paper [18].

(ii) K = 0.

In this case the Gauss-Codazzi equations become

q^{2} = e^{2u}(H^{2} + c), qz̄ = e^{z}H_{z}.  (7.3)

Since M is a flat Bonnet surface, the functions 1/q, u are harmonic with respect to the isothermic coordinate z. Hence we get

\frac{|H_{z}|^{2}}{H^{2} + c} = (\frac{HH_{z}}{H^{2} + c})z̄  (7.4)

The equation (7.4) is equivalent to the harmonic map equation of φ = 1/H into I(c). Namely M is a HIMC surface in 𝔐^{3}(c). For HIMC surfaces in 𝔐^{3}(c), flatness and isothermic property are preseved by Lawson correspondence. Hence we get the following

Theorem 7.2. ([18]) All flat Bonnet surfaces in S^{3} and H^{3} are obtained by Lawson correspondence from flat Bonnet surfaces in E^{3}.

Remark.

(1) We can obtain similar results on classification of spacelike or timelike Bonnet surfaces with constant curvature in Lorentzian space forms. [19], [20], [21].

(2) We can formulate dressing actions of loop groups on the moduli spaces of H-surfaces [16]. It seems to be interesting to introduce the dressing actions on the moduli space of HIMC surfaces.

References

1. A. I. Bobenko, Surfaces in terms of 2 by 2 matrices– Old and new integrable cases, Harmonic Maps and Integrable Systems (A. P. Fordy and J. C. Wood, eds.), Aspects of Math., vol. E 23, Viewig, Braunschweig, 1994, pp. 83–127.

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Graduate school of Natural Science and Technology, Kanazawa University, Kakuma-machi, Kanazawa, Ishikawa, 920–1192, Japan

E-mail address: fujioka@kappa.s.kanazawa-u.ac.jp

Department of Applied Mathematics, Fukuoka University, Nanakuma, Jonanku, Fukuoka, 814-0180, Japan E-mail address: inoguchi@bach.sm.fukuoka-u.ac.jp

Received July 27, 1999