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Timelike surfaces with harmonic inverse mean curvature

In this paper we introduce the notion of timelike surface with harmonic inverse mean curvature in 3-dimensional Lorentzian space forms, and study their fundamental properties.


Introduction
In this paper we introduce the notion of timelike surface with harmonic inverse mean curvature in 3-dimensional Lorentzian space forms, and study their fundamental properties.
In classical differential geometry, surfaces of constant mean curvature (CMC surfaces) have been studied extensively [1]. As a generalization of CMC surfaces, Bobenko [2] introduced the notion of surface with harmonic inverse mean curvature (HIMC surface). He showed that HIMC surfaces admit Lax representation with variable spectral parameter. In [5], Bobenko, Eitner and Kitaev showed that the Gauss equations of θ-isothermic HIMC surfaces reduce to the ordinary differential equation: , q ′ (t) < 0, with c = θ 2 > 0. Here the coefficient function S(t) is 1/ sin 2 (2t), 1/ sinh 2 (2t) or 1/t 2 . This ordinary differential equation is called the generalized Hazzidakis equation. Bobenko, Eitner and Kitaev [5] solved ( * ) in terms of Painlevé transcendents P V and P VI . For c < 0, solutions to ( * ) do not describe surfaces in Euclidean 3-space. It seems to be interesting to find "corresponding surfaces" to such solutions.
The first author extended the notion of HIMC surface in Euclidean 3-space to that of Riemannian 3-space forms [7]. Moreover he generalized a theorem due to Lawson (Lawson correspondence) to HIMC surfaces. By using the Lawson correspondence for HIMC surfaces, we have classified Bonnet surfaces with constant curvature in Riemannian 3-space forms [8].
On the contrary very little is known about (integrable) timelike surfaces of nonconstant mean curvature in Lorentzian 3-space forms. Timelike Bonnet surfaces are investigated by present authors very recently [11].
In this paper we introduce the notion of timelike surface with harmonic inverse mean curvature (THIMC surface) in Lorentzian 3-space forms. We shall show that every solution to the generalized Hazzidakis equation with c < 0 describes a THIMC surface in Minkowski 3-space. This is one of the motivations to study THIMC surfaces.
Because of the indefiniteness of metric, timelike surface geometry has many aspects different from Euclidean surface geometry. For instance, there exist timelike (HIMC) surfaces with imaginary principal curvatures. Moreover there exist non totally umbilical timelike surfaces with real repeated principal curvatures. Both of such surfaces have no counterparts in Euclidean surface geometry and spacelike surface geometry. Thus the geometry of THIMC surfaces has its own interest.
The second motivation of the present study is to give new examples of Lax equations with variable spectral parameter, namely, Lax equations whose spectral parameters depend on the variables. Burtsev, Zakharov and Mikhailov [6] exhibited some examples of Lax equations with variable spectral parameter appeared in theoretical physics. In differential geometry, HIMC surfaces and Bianchi surfaces are known examples. (See [2], [15] and [16]) . We shall show that THIMC surfaces Lorentzian 3-space forms admit Lax representation with variable spectral parameter. Moreover we shall show that in de Sitter 3-space or anti de Sitter 3-space, THIMC surfaces admit Lax representation with two independent variable spectral parameters. This paper is organized as follows. After recalling fundamental facts on Lorentzian geometry, we introduce the notion of THIMC surface in Minkowski 3-space in Section 3. We give a Lax representation and an immersion formula (Sym-formula) for THIMC surfaces.
Some elementary examples will be given in Section 3. In the next Section 4, we introduce the notion of ± isothermic timelike surface. We shall give a duality between timelike Bonnet surfaces and ± isothermic THIMC surfaces.
In Section 5, we shall investigate the normal forms of the Gauss equations of THIMC surfaces. More precisely we show that (θ-isothermic or anti θ-isothermic) THIMC surfaces in Minkowski 3-space are derived from solutions to the generalized Hazzidakis equation with c = −θ 2 < 0.
In Section 6, we shall generalize the notion of THIMC surface to Lorentzian 3-space forms and establish a Lawson-type correspondence for THIMC surfaces.
The authors would like to express their gratitude to the referee for careful reading of the manuscript.

Lorentzian space forms
1.1. First of all, we shall describe Lorentzian 3-space forms, i.e., complete and connected Lorentzian 3-manifolds M 3 1 (c) of constant curvature c explicitly. Without loss of generality, we may assume that c = 0 or ±1. On a Cartesian 4-space R 4 , we equip the following scalar product ·, · = ·, · c : The resulting semi-Euclidean 4-space (R 4 , ·, · ) is of index 1 for c = 1 and of index 2 for c = 0 or −1 respectively. The Lorentzian 3-space forms M 3 1 (c) are embedded in the semi-Euclidean space (R 4 , ·, · c ) as For more details on semi-Riemannian geometry, we refer to O'Neill [18].
1.2. Next we recall 2 by 2 matrix models of M 3 1 (c) for later use. First the semi-Euclidean 4-space E 4 2 = (R 4 , ·, · −1 ) is identified with the linear space M 2 R of all 2 by 2 real matrices via the isomorphism: The semi-Euclidean metric of E 4 2 corresponds to the following scalar product on M 2 R.
Under the identification (1.1), the Minkowski 3-space E 3 1 (p 1 , p 2 , p 3 ) is identified with the Lie algebra g = sl 2 R: for all X ∈ M 2 R, the anti de Sitter 3-space H 3 1 ⊂ E 4 2 corresponds to the real special linear group: Since the Lorentzian metric of G is bi-invariant, the product group G×G acts transitively and isometrically on H 3 1 as follows: Moreover G acts isometrically on E 3 1 via the Ad-action: 1.4. Finally we recall a 2 by 2 matrix model of S 3 1 . The Minkowski 4-space E 4 1 = (R 4 , ·, · 1 ) is identified with the space H of all Hermitian 2-matrices via the following isomorphism: Under the identification (1.4), the scalar product ·, · 1 of E 4 1 corresponds to the following scalar product on H: In particular det X = − X, X 1 under (1.4). Thus the de Sitter 3-space S 3 1 is represented by The complex special linear group SL 2 C acts transitively and isometrically on S 3 1 by Here g * denotes the transposed complex conjugate of g. The isotropy subgroup of SL 2 C at i ′ is SL 2 R. Hence the de Sitter 3-space S 3 1 is represented by S 3 1 = G C /G as a Lorentzian symmetric space. The natural projection p S : G C → G C /G is given explicitly by p S (g) = µ(g) i ′ = g i ′ g * , g ∈ G C .

Timelike surfaces in Lorentzian space forms.
We start with some preliminaries on the geometry of timelike surfaces in Lorentzian space forms M 3 1 (c).

2.1.
Let M be a connected 2-manifold and F : M → M 3 1 (c) an immersion. The immersion F is said to be timelike if the induced metric I of M is Lorentzian. Hereafter we may assume that M is an orientable timelike surface in M 3 1 (c) immersed by F . The induced Lorentzian metric I of M determines a Lorentzian conformal structure on M . We treat M as a Lorentz surface with respect to this conformal structure and F as a conformal immersion. Our general reference on "Lorentz surfaces" is Weinstein [21].
On a timelike surface M , there exists a local coordinate system (x, y) such that Such a local coordinate system (x, y) is called a Lorentz isothermal coordinate system. Let (u, v) be the local null coordinate system of M derived from (x, y). Namely (u, v) is defined by u = x + y, v = −x + y. Then the induced metric I can be written as Now, let N be a unit normal vector field to M . The second fundamental form II of (M, F ) derived from N is defined by The shape operator S of (M, F ) relative to N is defined by The (complex) eigenvalues of S are called principal curvatures of (M, F ). The mean curvature H of (M, F ) is defined by H = tr S/2. The Gaussian curvature K of (M, I) is computed by the formula: K = det S. The Gauss-Codazzi equations of (M, F ) have the following form: Here the functions Q = F uu , N and R = F vv , N define global null 2-differentials Q # = Qdu 2 and R # = Rdv 2 on M . These two differentials are called the Hopf differentials of M . The Gauss equation implies The first and second fundamental forms are related by the formula: This formula implies that the common zero of Q and R coincides with the umbilic point of (M, F ). Even if S has real and same eigenvalues, (M, F ) is not necessarily totally umbilic. In fact, there exist timelike surfaces with QR = 0 but II − HI = 0. See Example 3.3.

2.2.
In the study of timelike surfaces, we also use the following local coordinate system: be a timelike surface. Then there exists a local coordinate system (ǔ,v) such that With respect to this coordinate system, the Gauss-Codazzi equations are written as We call the local coordinate system (ǔ,v) an anti isothermal coordinate system. Anti isothermal coordinate systems will be used for introducing the notion of the Christoffel transformation of an anti isothermic surface. See Proposition 4.14.

Timelike HIMC surfaces in Minkowski 3-space
In this section we shall consider a generalization of timelike CMC surfaces in Minkowski 3-space in terms of integrability theory.

3.1.
We start with recalling the Lax representation for timelike surfaces in E 3 1 . Hereater we assume H = 0.
Let F : M → E 3 1 be a timelike surface. Let us take an SL 2 R-valued framing Φ defined by Thus we get the following Lax representation of Gauss-Codazzi equations: Now we shall insert a variable spectral parameter λ , i.e., an additional real parameter λ depends on the coordinate system (u, v) into the Lax pair (3.2) in the following way: Then the compatibility condition These equations (3.6) can be easily solved as follows: where f (u) and g(v) are smooth functions. It is easy to see that the mean curvature H is invariant under the one parametric deformation Under this deformation, the spectral parameter λ is transformed as Note that λ(u, v; 0) ≡ 1. Example 3.1. (THIMC cylinders.) Let a(y) = (a 2 (y), a 3 (y)) be a curve in Euclidean plane E 2 (ξ 2 , ξ 3 ) parametrized by the arclength parameter y ∈ I. Here I is an interval. A timelike cylinder over the curve a is a flat timelike surface in E 3 1 defined by the immersion ). It is straightforward to see that the mean curvature of the cylinder is H = κ(y)/2. Here κ is the curvature of a. Thus the cylinder F is a THIMC surface if and only if the base curve has the curvature 1 C 1 y+C 2 , C 1 , C 2 ∈ R. It is well known that curves with curvature 1 C 1 y+C 2 are logarithmic spirals or circles. Hence all the THIMC cylinders over a Euclidean curve are cylinders over a logarithmic spiral or a circular cylinder.
Example 3.2. (THIMC cylinders over timelike curves.) Let a(x) = (a 1 (x), a 2 (x)) be a timelike curve in Minkowski plane E 2 1 (ξ 1 , ξ 2 ) parametrized by the proper time parameter x defined on an interval I. A timelike cylinder over the timelike curve a is a flat timelike surface in E 3 1 defined by the immersion F : We can see that timelike curves with curvature 1 C 1 x+C 2 are logarithmic pseudo-spirals or timelike hyperbolas. (cf. Appendix of [10].) Hence all the THIMC cylinders are cylinders over a logarithmic pseudo-spiral or a timelike hyperbola.
The functions κ and τ are called the curvature and torsion of γ respectively. The ruled surface F (s, t) = γ(s) + tB(s) is called the B-scroll of γ. (See Graves [12] and McNertney [17]). The mean curvature of F is the torsion τ (s). It is straightforward to check that for any null Frenet curve with τ = 0, its B-scroll is a THIMC surface.
Remark. The Gaussian curvature of the B-scroll is τ 2 . Thus every B-scroll satisfies QR = 0 but is not totally umbilical (II − HI = 0). The property QR = 0 implies that every B-scroll is a timelike Bonnet surface. Here timelike Bonnet surfaces are timelike surfaces which admit nontrivial isometric deformation preserving mean curvature [11]. Conversely we proved that every timelike Bonnet surface with QR = 0 are B-scrolls [11].

3.3.
In [14], we have obtained a one-parameter "isometric" deformation of timelike surfaces with constant mean curvature (TCMC surfaces). For THIMC surfaces in E 3 1 , we get the following one-parameter family of "conformal" deformation.
1 be a timelike surface with harmonic inverse mean curvature. Express the mean curvature H as in terms of null coordinate system (u, v). Here f (u) and g(v) are smooth functions. Then F admits the following Lax representation with variable spectral parameter λ(u, v; τ ) = (1 − 2τ g(v))/(1 + 2τ f (u)), τ ∈ R: Then The fundamental associated quantities of F λ are given as follows: (3.14) The formula (3.14) implies that the members of the one parameter family F λ have the same ratio of the principal curvatures.

± isothermic timelike surfaces
4.1. In the study of HIMC surfaces in Riemannian space forms, isothermic surfaces play a fundamental role. In this section we shall consider such surfaces in timelike surface geometry. Here an isothermal-curvature line coordinate system is a local Lorentz-isothermal coordinate system such that both of parameter curves are curvature lines. It should be remarked that isothermic property implies the positivity of the descriminant D for the characteristic equation for the shape operator S. Equivalently, every isothermic timelike surface has real distinct principal curvatures.
The isothermic property for timelike surfaces in M 3 1 (c) can be reformulated in terms of associated null coordinate system as follows.
Here q is a real smooth function and ̺ and σ are positive Lorentz holomorphic and anti holomorphic functions respectively.
Remark. On a Lorentz surface M with null coordinate system (u, v), a smooth function f on M depends only on u [resp. v] is called a Lorentz holomorphic function [resp. Lorentz anti holomorphic function].
Hereafter we shall call a null coordinate system derived from an isothermic coordinate system simply an isothermic coordinate system.  cosh(ax), c(y) , a ∈ R * so that

Remark. Isothermic timelike surfaces in
With respect to this isothermic coordinate system, the mean curvature is given by With respect to this isothermic coordinate system, the mean curvature is given by With respect to this isothermic coordinate system, the mean curvature is given by Here a linear null frame means a basis of E 3 1 such that With respect to this isothermic parametrization, the mean curvature of F is given by Proposition 4.7. For any THIMC surface of revolution with nonconstant mean curvature, there exists an isothermic coordinate system (x, y) such that H(x) = 1/x or H(y) = 1/y. Furthermore φ is a solution to the third Painlevé equation of trigonometric form: (4.2) y {φ ′′ (y) − 2 sin(2φ(y))} + φ ′ (y) + 2 sin φ(y) = 0.
Furthermore φ is a solution to the third Painlevé equation of hyperbolic form: Furthermore φ is a solution to the ordinary differential equation: Remark 4.11. The ordinary differential equations (4.2) and (4.3) are related to the third Painlevé equation. More precisely let w = w(x) be a solution to the third Painlevé equation: with unit modulus, i.e., w(x) = e √ −1ψ(x) for some real valued function ψ(x). Then (P III ) is equivalent to the following ordinary differential equation: x {ψ ′′ (x) + 2γ sin(2ψ(x))} + ψ ′ (x) + 2α sin ψ(x) = 0.
Timelike HIMC surfaces of revolution with null axis can be classified as follows: Proposition 4.12. Let F : M −→ E 3 1 be a timelike surface of revolution with null axis parametrized as in Example 4.6 with harmonic inverse mean curvature 1/H = 4x. Then the function a(x) is a solution to the following ordinary differential equation: (4.5) x a ′′ (x)a(x) + a ′ (x) 2 = a 2 (x)a ′ (x).
This ordinary differential equation is explicitly solved by quadratures. In fact the solution a(x) is given as follows.
4.3. Next, to study timelike surfaces with imaginary principal curvatures we shall introduce the notion of anti isothermic surface.
Definition 4.13. Let F : M → M 3 ν (c) be a timelike surface. A null coordinate system (u, v) is said to be anti isothermic if its Hopf differentials take the following form: In addition (M, F ) is said to be anti isothermic if there exists an anti isothermic coordinate system around any point of M .
Note that anti isothermic property implies that M has imaginary principal curvatures. In M 3 1 (c), c ≥ 0, anti isothermic surfaces have non negative Gaussian curvature. (See (2.4).) The following result plays a fundamental role in the study of isothermic timelike surfaces and anti isothermic timelike surfaces in E 3 1 . We write these alternatives together as ± isothermic.
Proposition 4.14. Let (M, F ) be a ± isothermic timelike surface in E 3 1 and (D; u, v) a simply connected ± isothermic coordinate region so that Then the formulas: The conformal structure of D induced by F * is anti conformal to the original conformal structure determined by I. The fundamental quantities of F * are given as follows: (4.9) The new immersion F * is called the Christoffel transform of F or dual of F .
In particular for ± isothermic THIMC surfaces, we have the following.
Corollary 4.15. Every ± isothermic THIMC surface in E 3 1 is dual to a ± timelike Bonnet surface in E 3 1 and vice versa.

The Hazzidakis equation
In this section we shall investigate normal forms of Gauss equation for THIMC surfaces.

Timelike surfaces with ± holomorphic inverse mean curvature.
Let F : M → E 3 1 be a ± isothermic timelike surface with ± holomorphic inverse mean curvature.
These formulas show that timelike surfaces with ± holomorphic inverse mean curvature are flat Bonnet surfaces with ±holomorphic mean curvature described in [11, Theorem 3.1]. In particular the case α = 0 corresponds to timelike CMC cylinders.
Proposition 5.1. Let M be a timelike surface in E 3 1 with ± holomorphic inverse mean curvature. If M is ± isothermic then M is a flat isothermic timelike Bonnet surface.
The notion of ± isothermic surfaceual can be generalized to the notion of "(ε, ϑ)-isothermic surface" in the following way: Definition 5.2. A timelike surface (M, F ) is said to be (ε, ϑ)-isothermic if there exists a local null coordinate system (u, v) around any point of M such that the Hopf differentials Q and R have the following form: Here q is a real smooth function, ̺ and σ are ± Lorentz-holomorphic functions and ϑ is a real constant. If ε = + [resp. ε = −], then we call M a ϑ-isothermic surface [resp. an anti ϑ-isothermic surface].

Then
(1) if ε = +, there exists a timelike immersion (2) if ε = −, there exists a timelike immersion The timelike immersion F * is called a dual surface of F . In particular if F is a THIMC surface then F * is a timelike Bonnet surface and vice versa.
Remark. In section 6, we shall prove a Lawson correspondence between THIMC surfaces in Lorentzian space forms. Combining the duality in the preceding proposition and Lawson correspondence, we get a duality between THIMC surfaces and timelike Bonnet surfaces in H 3 1 . 5.2. Timelike surfaces with non ± holomorphic inverse mean curvature.
To adapt our computations to [5] and [10], and avoid a plethora of unnecessary 1/2's in the description, we shall use the following convention: And we call (ξ, η) simply an (ε, θ)-isothermic coordinate system.
Now we introduce a new null coordinate system (u, v) by Then the formula (5.11) implies that q depends only on t := εu + v. We should separate our consideration to the following two cases: (1) 2 − QR/R ξ = 0, (2) 2 − QR/R ξ = 0.
(2) If θ = 0 then the dual surface F * in E 3 1 is defined by the data: The dual surface F * is an anti isothermic timelike Bonnet surface in E 3 1 . We call a THIMC surface (M, F ) generic if (M, F ) does not correspond to a solution of 2R ξ − QR = 0.

Case 2: 2R
In this case, inserting into (5.8) and by the assumption 2R ξ − QR = 0, we can define the following function The following theorem is proved by much the same way in [5] and [10]. .
Here q(t) is a solution to the generalized Hazzidakis equation: Here the coefficient function S(t) in the generalized Hazzidakis equation is given by
The harmonic map equation (6.1) may be considered as a nonlinear generalization of the classical linear wave equation ϕ uv = 0. As is well known classical linear wave equation can be solved by the d'Alembert formula. The following is regarded as a nonlinear d'Alembert formula for (6.1).
Proposition 6.2. The harmonic map equation (6.1) can be solved as follows: The following definition is a generalization of that in Section 3.   1 (c), c = ±1. Before describing immersion formulas for THIMC surfaces, we point out the following invariance of (6.1): Let ϕ be a solution to (6.1) of the form: Then the replacements: f −→ 2τ f, g −→ 2τ g, τ ∈ R * produce a new solution to (6.1). More precisely, the function ϕ[τ ] defined by is still a solution to (6.1). Let Φ λ be a solution of the zero curvature equations (3.8) with variable spectral parameter λ. To describe immersion formulas we shall use the following notational convention.
Since the zero curvature equation (3.8) is completely integrable, (3.8) has also solutions for all τ ∈ C.
describes a THIMC surface in E 3 1 given in Proposition 3.4. (c = −1) For any τ ∈ R * , is a THIMC surface in H 3 1 with unit normal vector field , τ ∈ R be a complexified solution to (3.8).
In particular, for c = ±1, F (±1) (1/2) is the Lawson correspondent of F = F (0) . The conformal deformations of THIMC surfaces in M 3 1 (c) preserve K/(H 2 + c). Remark. The conformal deformation of HIMC surfaces in Riemannian space forms [resp. spacelike HIMC surfaces in Lorentzian space forms] preserves K/(H 2 +c) [resp. K/(H 2 −c)] Note that in case c = 0, the constancy of K/(H 2 − c) is equivalent to the constancy of the ratio of principal curvatures.
Computing the Gaussian curvature or K/(H 2 + c), we have the following theorem. As an application of Lawson correspondence as above, one can classify ± isothermic flat timelike Bonnet surfaces in Lorentzian space forms. In fact, since the Lawson correspondence preserves ± isothermic property or flatness, we have obtained the following ([11, Theorem 6.2]). Theorem 6.8. Flat simply connected ± isothermic timelike Bonnet surfaces in one Lorentzian 3-space form correspond to those in another Lorentzian 3-space form.
Moreover in [11], timelike Bonnet surfaces in M 3 1 (c) with constant Gaussian curvature are classified.
6.4. Finally we consider THIMC surfaces in H 3 1 with mean curvature H 2 < 1. To investigate such surfaces, we use the following invariance of (6.1) with c = ±1. Let H be a solution of (6.1) of the form: .
Then for any τ ∈ R * , the replacements: f −→ τ f, g −→ τ −1 g produce a new solution of (6.1). Namely the function H[τ ] defined by is also a solution of (6.1). Based on this deformation, we define two auxiliary functions (variable spectral parameters): Then we have the following.
Since the Lax equation (6.2) with two variable spectral parameters λ and ν is completely integrable, (6.2) has solutions for all τ ∈ C. Such complexified solutions Ψ[τ ] to (6.2) describe another kind of surfaces in S 3 1 . Theorem 6.10. Let Ψ[τ ] : M × C → G C be a complexified solution to (6.2). Then for any τ ∈ R * , is a timelike surface in S 3 1 with unit normal vector field and mean curvature τ f − τ −1 g 1 − cf g .
The inverse mean curvature of F (1) [τ ] in Theorem 6.10 is a harmonic map into I[−1].