1. Introduction. Surfaces of Voss and Guichard

Let us first recall two equivalent definitions of these surfaces: a geometric one and an analytic one. Our notation follows the review of Gambier [4].

Geometrically, the surfaces of Voss [11] and Guichard [6] are by definition those which admit a conjugate net made of geodesics. For instance, every minimal surface is such a surface.

Analytically, they can be characterized by their three fundamental quadratic forms dF.dF, −dF.dN, dN. dN in which F(u, v) is the current point of the surface and N(u, v) a unit vector normal to the tangent plane. Choosing the coordinates (u, v) defined by the geodesic conjugate net, these are [4, p. 362]

with the notation X_u = ∂X(u,v) / ∂u, . . . , they depend on three functions ω, X, Y of two variables, and 2ω is the angle between the two conjugate geodesics. It is remarkable that, among the three Gauss-Codazzi equations [4, p. 362] the first one characterizes the surfaces with a constant total (Gauss) curvature.

2. Their Lax pair

Gambier succeeded in introducing a deformation parameter λ, thus upgrading the moving frame equations to a Lax pair, but he did not write this Lax pair explicitly, so let us do it here.

The moving frame equations (Gauss-Weingarten equations) only depend on the coefficients of the first and second fundamental forms, and the spectral parameter is introduced, as in the case of surfaces with a constant mean curvature, by noticing the invariance of the Gauss-Codazzi equations (1.2) under the scaling transformation (u, v)→( λu, λ⁻¹v). The traceless Lax pair is

with the zero-curvature condition, denoting E_j, j = 1; 2; 3 the lhs of (1.2), and E, F, G the coefficients of the first fundamental form,

3. Surfaces applicable on a surface of revolution

Gambier [3, p. 99] investigated surfaces whose first fundamental form I, Eq. (1.1), has coefficients X_u, Y_v, ω only depending on the single variable x = u + v. Denoting for shortness X_u = ξ, Y_v = η, he first obtains

with C₁, C₂ two integration constants. After a possible conformal transformation, this defines two reductions of the Gauss-Codazzi equations, either or

The first reduction (3.2) integrates with elliptic functions and is handled in full detail by Gambier [3, pp. 100–105].

As to the second reduction (3.3), Gambier unexpectedly fails to integrate it. This ordinary differential equation (ODE) belongs to the class of second order first degree ODEs

whose property is to be form-invariant under the group of point transformations Roger Liouville [7] enumerated equivalence classes of (3.4) modulo the group (3.5) but, as later pointed out by Babich and Bordag [1], he forgot the important class, to which the ODE (3.3) belongs, when the invariants which he denotes v₅ and w₁ both vanish.

When v₅ and w₁ both vanish, the coefficients A₃, A₂, A₁ in the class (3.4) can be set to zero by a transformation (3.5), thus defining the five remarkable four-parameter nonautonomous ODEs

in which the summation in the first equation runs over the four half-periods ω_j of the Weierstrass elliptic function ℘.

The third one is precisely, up to rescaling, the ODE (3.3) isolated by Gambier, and the main result of Ref. [1] is the existence of a point transformation mapping these five four-parameter ODEs to the representation of the Painlevé equations chosen by Garnier [5],[2] (i.e. five equations with four parameters, the last one unifying P_II and P_I),

The point transformations which realize this mapping are, respectively,

Therefore the mapping between the ODE (3.3) for ω(x) and the third Painlevé equation (3.6) for u(ξ) is either

or equivalently

As is well known, the third Painlevé equation has three kinds of solutions:

1. (i)

   two-parameter transcendental solutions, which is the generic case, and one cannot proceed beyond the description of Gambier [3, pp. 105–106];

2. (ii)

   one-parameter Riccati-type solutions, but for our case γδ ≠ 0 this does not happen;

3. (iii)

   zero-parameter rational^a solutions, the only ones being, with the choice (3.7),

   u=(−βα)1/2ξ1/2, γ=0, δ=0,(3.9)

   or equivalently with the choice (3.8),

   u=(−δγ)1/4ξ1/2, α=0, β=0.(3.10)

However, these rational solutions correspond to sin (2ω) = 0, forbidden because the second fundamental form would vanish. Consequently, all solutions of (3.3) are transcendental.

4. Future developments

The equation (1.2)₁ for constant total curvature surfaces (sine-Gordon equation) possesses many closed form solutions which obey neither (3.2) nor (3.3), for instance the factorized solution [10]

in which J₁ and J₂ are Jacobi elliptic functions, a degeneracy of which is or the N -soliton solution [9], which depends on 2N arbitrary constants. The difficulty to build Voss-Guichard surfaces from such solutions is the integration of the linear system (1.2)_2,3 for X(u, v) and Y(u, v).

Another useful development would be to find a Darboux transformation for the system (1.2).

Acknowledgments

This work was partially funded by the National Natural Science Foundation of China grant 11471182, and the Hong Kong GRF grant HKU 703313P and GRF grant 17301115. The second author also thanks the Institute of Mathematical Research (IMR), HKU for the financial support of his visit to IMR in November, 2017.