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Surfaces: from generalised Weierstrass representation to Cartan moving frame

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Claude Pierre Massé

Laboratoire de Physique Fondamentale, FRANCE

21 April 2018

Konopelchenko’s generalised Weierstraß representation of a surface immersed in the three dimensional Euclidean space is derived by the way of the conserved currents, and then expressed in the framework of the moving frame. The data of classical surface theory are given each time. The problem of determining the surfaces having a given Gauß map is investigated, and this is then applied to solving some non-linear equations by the inverse scattering transform.

Latest version: https://phy.clmasse.com/surfaces-weierstrass-cartan-v3.html


The classical Enneper-Weierstraß representation allows to represent a minimal surface immersed in $\Bbb R^3$ by a meromorphic function and a holomorphic $1$-form. Recently, Konopelchenko [1] proposed a similar representation with two functions given by a solution of a two-dimensional Dirac equation. In this article, that formalism is presented in a transparent and systematic way, so that the Cartan’s moving frame can be introduced naturally. In section §1, we first derive the generalised Weierstraß representation by the way of the conserved currents, taking advantage of the vector representation of the spinor. Up to there, everything is expressed in an arbitrary, fixed reference frame. Then in section §2, the representation is recast along the lines of the Cartan’s moving frame, so that the former frame becomes unnecessary. The mathematical expressions of the traditional data of a surface are obtained each time. Singling out one vector of the moving frame and relating it to the Gauß map, in section §3 is investigated whether there exist surfaces having this Gauß map, and if yes, how they are determined by it. The geometrical representation of integrable non-linear differential equations follows directly, and in section §4 it is shown how this can be applied to solving the non-linear Liouville equation and its generalisations through the inverse scattering transform.

§1. The generalised Weierstraß representation

In the Euclidean two-dimensional space, the Dirac equation can be written using the Pauli matrices: $$\DeclareMathOperator{\Tr}{Tr}\DeclareMathOperator{\d}{d\!}\DeclareMathOperator{\e}{e}\def\Dslash{\rlap{\,/}D} \sigma_1=\pmatrix{0& 1\\1& 0},\quad\sigma_2=\pmatrix{0& -i\\i& 0},\quad\sigma_3=\pmatrix{1& 0\\0& -1}\tag{1.1}$$ as follow: $$\Dslash\psi:=\{i\sigma_1\partial_1+i\sigma_2\partial_2+m\}\psi=0.\tag{1.2}$$ Here, the mass parameter $m$ is generalised to a real function of $\xi^1$ and $\xi^2,$ the Cartesian coordinates of the space, with an obvious notation for the partial differential operators. We shall work with complex numbers for compactness, and denote the complex conjugate by an overbar. We first define the Wirtinger differential operators $$\partial=\frac12(\partial_1-i\partial_2),\quad\bar\partial=\frac12(\partial_1+i\partial_2),\tag{1.3}$$ or using the complex variable: $$\zeta=\xi^1+i\xi^2,\tag{1.4}$$ $$\partial=\partial_\zeta,\quad\bar\partial=\partial_{\bar\zeta}.\tag{1.5}$$ Writing the wave function as: $$\psi=\binom\varphi\chi,\tag{1.6}$$ the Dirac equation (1.2) is now decomposed into a system of two simple and handy equations: $$\left\{\eqalign{ \bar\partial\varphi& =\frac i2m\chi\\ \partial\chi& =\frac i2m\varphi. }\right.\tag{1.7a,b}$$

We first focus on the spinor. It is uniquely extended as follow: $$\Phi=\pmatrix{\varphi& -\bar\chi\\\chi& \phantom-\bar\varphi}.\tag{1.8}$$ It is a $SU(2)$ matrix multiplied by the factor $\sqrt\rho,$ with the definitions: $$\rho=\e^u:=\det\Phi=\varphi\bar\varphi+\chi\bar\chi,\tag{1.9}$$ so that $\Phi\Phi^\dagger=\rho{\rm I}.$ We then consider the associated (right-invariant) Maurer-Cartan $1$-form: $$\d\Phi\Phi^{-1}=\rho^{-1}\d\Phi\Phi^\dagger.\tag{1.10}$$ Its entries can be calculated from the Dirac equation, with the result $$\rho^{-1}\d\Phi\Phi^\dagger= \frac12\pmatrix{2\partial u& iQ\\im& 0}\d\zeta+ \frac12\pmatrix{0& im\\i\bar Q& 2\bar\partial u}\d\bar\zeta,\tag{1.11}$$ where the following defining relation has been used: $$\bar\chi\partial\varphi-\varphi\partial\bar\chi=:\frac i2\rho Q,\quad \frac i2\rho\bar Q:=\bar\varphi\bar\partial\chi-\chi\bar\partial\bar\varphi.\tag{1.12}$$ In addition, the following differential constraints have been found: $$\varphi\partial\bar\varphi+\bar\chi\partial\chi=0=\bar\varphi\bar\partial\varphi+\chi\bar\partial\bar\chi,\tag{1.13a}$$ $$\bar\varphi\partial\chi-\chi\partial\bar\varphi=\frac i2\rho m=\bar\chi\bar\partial\varphi-\varphi\bar\partial\bar\chi,\tag{1.13b}$$ $$\bar\varphi\partial\varphi+\chi\partial\bar\chi=\rho\partial u,\quad \rho\bar\partial u=\varphi\bar\partial\bar\varphi+\bar\chi\bar\partial\chi.\tag{1.13c}$$ They prove very useful in the sequel. For completeness, let us list two further constraints: $$\bar\varphi\partial\bar\varphi-\bar\chi\bar\partial\bar\chi=0=\chi\partial\chi-\varphi\bar\partial\varphi,\tag{1.14a}$$ $$\bar\varphi\partial\chi+\varphi\bar\partial\bar\chi=0=\chi\partial\bar\varphi+\bar\chi\bar\partial\varphi.\tag{1.14b}$$ They are in the form of a conservation laws, a more elaborate version of which will be derived below.

Taking the derivative of the defining relation (1.12) and using the Dirac equation (1.7), we directly get $$\rho^{-1}\bar\partial(\rho Q)=\rho\partial(\rho^{-1}m).\tag{1.15}$$ The value of $Q$ could be called the mirror mass, and by solving (1.12) and (1.13c) we have the mirror Dirac equation: $$\left\{\eqalign{ \partial(\rho^{-1}\varphi)& =\frac i2Q(\rho^{-1}\chi)\\ \bar\partial(\rho^{-1}\chi)& =\frac i2\bar Q(\rho^{-1}\varphi). }\right.\tag{1.16a,b}$$ It is actually the one satisfied by $\rho^{-1}\sigma_1\Phi$ or $\rho^{-1}\sigma_2\Phi,$ the mirror spinor. The quantity $J$ used by Bracken [3] (eq. (3.7)) is $$J:=\rho Q.\tag{1.17}$$

Now we use the well known vector representation. A matrix such as $\Phi$ represents a rotation times a scaling in $\Bbb R^3$ as follow: $$\boldsymbol V=v^i\sigma_i\to\boldsymbol V'=\Phi\boldsymbol V\Phi^\dagger.\tag{1.18}$$ The matrix $\Phi$ can thus be represented by a multiple of an orthogonal $3\times3$ real matrix, whose elements are easily found by using the inner product in the space of $\boldsymbol V$: $$\langle\boldsymbol V_1,~\boldsymbol V_2\rangle=\frac12\Tr[\boldsymbol V_1\boldsymbol V_2],\tag{1.19}$$ then $$e^j_i=\frac12\Tr[\sigma_i\Phi\sigma_j\Phi^\dagger],\tag{1.20}$$ and after some straightforward calculations: $$e={\small\pmatrix{\boldsymbol e_+:={\sqrt2\over2}(\boldsymbol e_1-i\boldsymbol e_2)\\ \boldsymbol e_-:={\sqrt2\over2}(\boldsymbol e_1+i\boldsymbol e_2)\\ \boldsymbol e_3}} ={\small\pmatrix{{\sqrt2\over2}(\varphi\varphi-\bar\chi\bar\chi)& -i{\sqrt2\over2}(\varphi\varphi+\bar\chi\bar\chi)& \sqrt2~\varphi\bar\chi\\ {\sqrt2\over2}(\bar\varphi\bar\varphi-\chi\chi)& \phantom-i{\sqrt2\over2}(\bar\varphi\bar\varphi+\chi\chi)& \sqrt2~\bar\varphi\chi\\ -(\varphi\chi+\bar\varphi\bar\chi)& i(\varphi\chi-\bar\varphi\bar\chi)& \varphi\bar\varphi-\chi\bar\chi}.\tag{1.21}}$$ It is an $SO(3)$ matrix multiplied by a factor $\rho,$ so that $ee^\dagger=\rho^2{\rm I}.$

According to the properties of the inner product, we can write $$\boldsymbol E_i:=e^j_i\sigma_j=\Phi^\dagger\sigma_i\Phi,\tag{1.22}$$ with which we calculate $$\d\boldsymbol e_i\cdot\boldsymbol e_j=\langle\d\boldsymbol E_i,~\boldsymbol E_j\rangle =\frac12\Tr[\d\,(\Phi^\dagger\sigma_i\Phi)\Phi^\dagger\sigma_j\Phi].\tag{1.23}$$ Differentiating, this gives $$\d\boldsymbol e_i\cdot\boldsymbol e_j= \frac12\rho\Tr[\d\Phi\Phi^\dagger\sigma_j\sigma_i+(\d\Phi\Phi^\dagger\sigma_j\sigma_i)^\dagger]= \rho~{\frak Re}\{\Tr[\d\Phi\Phi^\dagger\sigma_j\sigma_i]\},\tag{1.24}$$ from which the vector representation of the Maurer-Cartan form is deduced: $$\rho^{-2}\d ee^\dagger={\small \pmatrix{2\partial u& 0& -i{\sqrt2\over2}Q\\0& 0& i{\sqrt2\over2}m\\ -i{\sqrt2\over2}m& i{\sqrt2\over2}Q& \partial u}}\d\zeta+{\small \pmatrix{0& 0& -i{\sqrt2\over2}m\\0& 2\bar\partial u& i{\sqrt2\over2}\bar Q\\ -i{\sqrt2\over2}\bar Q& i{\sqrt2\over2}m& \bar\partial u}}\d\bar\zeta.\tag{1.25}$$

Remark: There is a two-ways fast lane for switching between both representations. Given that the Lie algebras of $SU(2)$ and $SO(3)$ are locally the same, if the generators of $SU(2)$ associated with $\Phi$ are $\frac i2\sigma_i,$ the corresponding generators of $SO(3)$ associated with $e$ are $$\Sigma_1=\frac{\sqrt2}2{\small\pmatrix{0& 0& -i\\0& 0& i\\-i& i& 0}},~ \Sigma_2=\frac{\sqrt2}2{\small\pmatrix{0& 0& -1\\0& 0& -1\\1& 1& 0}},~ \Sigma_3={\small\pmatrix{i& 0& 0\\0& -i& 0\\0& 0& 0}.\tag{1.26}}$$ Similarly, the generator of the scaling is $\frac12{\rm I}$ for he spinor representation, and ${\rm I}$ for the vector representation, where ${\rm I}$ is the unit matrix of the appropriate size. The Maurer-Cartan form belongs to the respective Lie algebra. One passes from one to the other just by replacing the basis of the Lie algebra representation. Denoting this abstract basis by $S_i,$ $i$ running from $0$ to $3,$ the form reads $$\e^{-2uS_0}\d RR^\dagger=\d u~S_0 -i(\partial u\d\zeta-\bar\partial u\d\bar\zeta)S_3+\\ +\frac12[(m+Q)\d\zeta+(m+\bar Q)\d\bar\zeta]S_1 -\frac i2[(m-Q)\d\zeta-(m-\bar Q)\d\bar\zeta]S_2,\tag{1.27}$$ where $R$ stands either for $\Phi$ or $e.$ The same works as well for $\Phi$ and $e$ using the exponential map of the Lie algebra, from which the Maurer-Cartan form can be derived. It is still simpler using the step generators $S_\pm=S_1\pm iS_2$ and $\tilde S_\pm=S_0\pm iS_3,$ but we shall not spoil the fun of the reader, who may want to prove the claim too.

We are looking for a closed differential $1$-form with values in $\Bbb R^3,$ so that we get a surface by quadrature. It can be constructed from a conserved current, whose components are divergence-free. Such a current is given from the global symmetries of the Lagrangian through Noether’s theorem. For investigating them, let us first remark that if we make the substitution $$\matrix{ \varphi& \to& -\bar\chi\\ \chi& \to& \phantom-\bar\varphi, }\tag{1.28}$$ which is in fact the charge conjugaison, the Dirac equation (1.7) still holds. That is, we get the same solutions if we put $\Phi$ instead of $\psi$ in equation (1.2). Then, $\Phi$ multiplied on the right by any constant non singular matrix is a solution of the equation if $\Phi$ is so. Thus, the full symmetry group is $GL(2\Bbb C).$ The substitution above belongs to this group, which makes charge conjugaison a continuous transformation (i.e. continuously connected to identity.)

We write the Lagrangian of the Dirac equation (1.2) as $${\scr L}=\frac12\Tr[\Phi^\dagger\Dslash\Phi] =i\bar\varphi\partial\chi-i\chi\partial\bar\varphi+i\bar\chi\bar\partial\varphi -i\varphi\bar\partial\bar\chi+m(\varphi\bar\varphi+\chi\bar\chi).\tag{1.29}$$ It is manifestly invariant under the multiplication of $\Phi$ on the right by a constant $SU(2)$ matrix. The currents corresponding to the generators $\frac i2\sigma_i$ of $SU(2)$ are then $$\pmatrix{-\frac12(\varphi\varphi-\bar\chi\bar\chi)\\-\frac12(\bar\varphi\bar\varphi-\chi\chi)},\quad \pmatrix{\phantom-\frac i2(\varphi\varphi+\bar\chi\bar\chi)\\-\frac i2(\bar\varphi\bar\varphi+\chi\chi)},\quad \pmatrix{-\vphantom{\frac12}\varphi\bar\chi\\-\vphantom{\frac12}\bar\varphi\chi}.\tag{1.30}$$ In (1.21) we see that they are proportional to the components of $\boldsymbol e_+$ and $\boldsymbol e_-,$ thus the conservation law is $$(\bar\partial,~\partial,~0)e=\partial\boldsymbol e_-+\bar\partial\boldsymbol e_+=0,\tag{1.31}$$ as can be checked with (1.25). The third component is just the Dirac electromagnetic current, as expected since the transformation associated to $\frac i2\sigma_3$ is the one of the $U(1)$ electromagnetic gauge.

This method is not suited for generalisations, it is just a coincidence working only for surfaces in $\Bbb R^3.$ Alternatively, write the Dirac equation (1.2) and its Hermitian conjugate as $$\Phi^\dagger\{i\sigma_1\partial_1+i\sigma_2\partial_2+m\}\Phi=0\\ \Phi^\dagger\{i\overset{\small\leftarrow}{\partial_1}\sigma_1+i\overset{\small\leftarrow}{\partial_2}\sigma_2-m\}\Phi=0\tag{1.32}$$ and add them. The result is $$\partial_1(\Phi^\dagger\sigma_1\Phi)+\partial_2(\Phi^\dagger\sigma_2\Phi)=0.\tag{1.33}$$ From (1.20), it is clear that it is the same equation as derived above (1.31).

From this conservation law, we deduce that the $\Bbb R^3$-valued differential form $$\d\boldsymbol r=i\frac{\sqrt2}2\boldsymbol e_+\d\zeta-i\frac{\sqrt2}2\boldsymbol e_-\d\bar\zeta\tag{1.34}$$ is closed. In addition it is real because $\boldsymbol e_-=\bar{\boldsymbol e}_+.$ Moreover, it is exact in virtue of the Poincaré’s lemma since every loop in $\Bbb C$ can be shrunk to a point. Therefore, integrating it along a path $\Gamma$ in the $(\zeta,~\bar\zeta)$ plane from a fixed point $\zeta_0,$ we arrive at the wanted formula of the surface: $$\eqalignno{ x+iy& =i\int_\Gamma(\varphi^2\d\zeta'+\chi^2\d\bar\zeta')+x_0+iy_0&(\text{1.35a})\\ x-iy& =i\int_\Gamma(-\bar\chi^2\d\zeta'-\bar\varphi^2\d\bar\zeta')+x_0-iy_0&(\text{1.35b})\\ z& =i\int_\Gamma(\varphi\bar\chi\d\zeta'-\bar\varphi\chi\d\bar\zeta')+z_0&(\text{1.35c}) }$$ for the point $\boldsymbol r$ of the surface with coordinates $(x,y,z)$ in $\Bbb R^3.$ That is the same as in [1] (equation 2) up to some cosmetic rearrangements. The surface is then normal to $\boldsymbol e_3.$

We now have all what is necessary to get the traditional data of a surface. Since $\boldsymbol e_\pm^2=0$ and $\boldsymbol e_+\cdot\boldsymbol e_-=\rho^2,$ the first fundamental form is $${\rm I}:=\d\boldsymbol r\cdot\d\boldsymbol r=\rho^2\d\zeta~\d\bar\zeta\tag{1.36}$$ (it is therefore a conformal immersion with isothermal coordinates $\xi^1$ and $\xi^2.$) The second fundamental form can be directly calculated too, using $\boldsymbol e_3\cdot\d\boldsymbol r=0$: $$\eqalign{ {\rm II}:=& -\d\,(\rho^{-1}\boldsymbol e_3)\cdot\d\boldsymbol r =-\rho^{-1}\d\boldsymbol e_3\cdot\d\boldsymbol r\\ =& -\frac12\rho(Q\d\zeta\d\zeta+2m\d\zeta\d\bar\zeta+\bar Q\d\bar\zeta\d\bar\zeta).\tag{1.37} }$$ $${\rm III}:=\d\,(\rho^{-1}\boldsymbol e_3)\cdot\d\,(\rho^{-1}\boldsymbol e_3) =mQ\d\zeta\d\zeta+(m^2+Q\bar Q)\d\zeta\d\bar\zeta+m\bar Q\d\bar\zeta\d\bar\zeta\\ =(Q\d\zeta+m\d\bar\zeta)(\overline{Q\d\zeta+m\d\bar\zeta}).\tag{1.37'}$$ Finally the usual formulæ gives the mean curvature $H$ and the gauß curvature $K$: $$H=\frac12\Tr[{\rm II}\cdot{\rm I}^{-1}]=-\rho^{-1}m,\tag{1.38}$$ $$K=\det({\rm II}\cdot{\rm I}^{-1})=\rho^{-2}(m^2-Q\bar Q)=H^2-\rho^{-2}Q\bar Q.\tag{1.39}$$ Konopelchenko [1] (equation 4) gives a different expression: $$K=-4\rho^{-2}\partial\bar\partial u,\tag{1.40}$$ which is known as the Gauß-Riemann curvature. But Ferapontov and Grundland [2] (equation 2.6) proved that they are equivalent, in accordance with Gauß’ Theorema Egregium. Then the modulus of $Q$ is a function of $\rho$ and $m$: $$Q\bar Q=m^2+4\partial\bar\partial u.\tag{1.41}$$

It is interesting to note that since the difference between the principal curvatures is given by $$(\Delta\kappa)^2=4(H^2-K),\tag{1.42}$$ it holds $$|\Delta\kappa|=2\rho^{-1}|Q|,\tag{1.43}$$ that is, the modulus of $Q$ is a measure of the local deformation from a spherical surface, like $m$ is a measure of the local deformation from a minimal surface. Indeed $$\kappa=-\rho^{-1}(m\pm|Q|).\tag{1.44}$$

§2. The Cartan moving frame

So far, we implicitly used a fixed referential frame in $\Bbb R^3.$ Indeed, by varying this frame, with the same solution of the Dirac equation we get a whole set of surfaces that are deduced from each other by a rigid motion. This way has been deemed awkward by Cartan, so he developed the powerful method of the moving frame [4]. To an infinitesimal displacement in the $(\zeta,~\bar\zeta)$ plane is now associated an infinitesimal change of the origin and of the basis vectors, but this time expressed with respect to the moving frame itself. By using differential $1$-forms (also called Pfaffian forms,) this is written down as $$\left\{\eqalign{ \d\boldsymbol r& =\omega^j\boldsymbol e_j\\ \d\boldsymbol e_i& =\omega^j_i\boldsymbol e_j }\right.,\quad i,j=+,-,3,\tag{2.1a,b}$$ which is called the first system of structure equations by Cartan. The vectors $\boldsymbol e_i$ satisfy the orthonormality conditions $$\boldsymbol e_\pm^2=0,\quad\boldsymbol e_+\cdot\boldsymbol e_-=\rho^2, \quad\boldsymbol e_\pm\cdot\boldsymbol e_3=0,\quad\boldsymbol e_3^2=\rho^2.\tag{2.2}$$ Differentiating them, and using the structure equations (2.1), we get relations among the differential forms, e.g.: $$\d\boldsymbol e_+^2=0\Rightarrow 2\boldsymbol e_+\cdot\d\boldsymbol e_+=0\Rightarrow \omega^-_+=0.\tag{2.3}$$ By a similar method, we collect the following relations: $$\omega^+_-=\omega^-_+=0,\tag{2.4a}$$ $$\omega^-_3+~\omega^3_+=\omega^+_3+~\omega^3_-=0.\tag{2.4b}$$ $$\omega^-_-+~\omega^+_+=2\d u,\tag{2.4c}$$ $$\omega^3_3=\d u,\tag{2.4d}$$ As $\bar{\boldsymbol e}_+=\boldsymbol e_-$ and $\bar{\boldsymbol e}_3=\boldsymbol e_3,$ with the complex conjugates of the same equations we moreover find $$\eqalignno{ \bar\omega^3_+& =\omega^3_-,&\text{(2.5a)}\\ \bar\omega^+_+& =\omega^-_-.&\text{(2.5b)} }$$ Then assuming the structure equations (2.1) are integrable, by differentiating them and substituting $\d\boldsymbol e_i$ (2.1b) wherever possible, we get compatibility equations, called the second system of structure equations: $$\left\{\eqalign{ \d\omega^i& =\omega^k\wedge\omega^i_k\\ \d\omega^i_j& =\omega^k_j\wedge\omega^i_k. }\right.\tag{2.6a,b}$$ The second equality (2.6b) is always true as long as the frames are given, and the first one (2.6b) is the equivalent, expressed in the moving frame language, of the requirement that the form $\d\boldsymbol r$ be exact.

From the expression of $\d\boldsymbol r$ (1.34) we immediately get $$\omega^+=i\frac{\sqrt2}2d\zeta,\quad\omega^-=-i\frac{\sqrt2}2\d\bar\zeta,\quad\omega^3=0,\tag{2.7}$$ and $$\bar\omega^+=\omega^-.\tag{2.8}$$

The expression of the remaining $1$-forms are easily obtained by writing the structure equation (2.1b) as a matrix: $$\d e=\Omega e,\tag{2.9}$$ where $$\Omega=\pmatrix{ \omega^+_+& \omega^-_+& \omega^3_+\\ \omega^+_-& \omega^-_-& \omega^3_-\\ \omega^+_3& \omega^-_3& \omega^3_3\\ }.\tag{2.10}$$ Multiplying (2.9) by $e^\dagger$ on the right, there results $$\Omega=\rho^{-2}\d ee^\dagger\tag{2.11}$$ which is already known, it is the Maurer-Cartan form (1.25), the preceding relations (2.4), (2.5) are incidentally satisfied. Further writing $$\Theta=(\omega^+,~\omega^-,~0),\quad \d\Theta=0,\tag{2.12}$$ the compatibility equations (2.6) become $$\left\{\eqalign{ \d\Theta& =\Theta\wedge\Omega=0\\ \d\Omega& =\Omega\wedge\Omega. }\right.\tag{2.13a,b}$$ If we decompose $\Omega$ as follow: $$\Omega={\cal Z}\d\zeta+\tilde{\cal Z}\d\bar\zeta,\tag{2.14}$$ with $${\cal Z}={\small\pmatrix{ 2\partial u& 0& -i{\sqrt2\over2}Q\\ 0& 0& i{\sqrt2\over2}m\\ -i{\sqrt2\over2}m& i{\sqrt2\over2}Q& \partial u }},\quad\tilde{\cal Z}={\small\pmatrix{ 0& 0& -i{\sqrt2\over2}m\\ 0& 2\bar\partial u& i{\sqrt2\over2}\bar Q\\ -i{\sqrt2\over2}\bar Q& i{\sqrt2\over2}m& \bar\partial u }},\tag{2.15}$$ the structure equation (2.1b) is then $$\partial e={\cal Z}e,\quad\bar\partial e=\tilde{\cal Z}e,\tag{2.16}$$ which is known in the theory of surfaces as the Gauß-Weingarten equations. The second compatibility equation (2.6b) reads $$\bar\partial{\cal Z}-\partial\tilde{\cal Z}+[{\cal Z},~\tilde{\cal Z}]=0,\tag{2.17}$$ also known as the Gauß-Codazzi-Mainardi equations. We already met them, here they are gathered: $$Q\bar Q=m^2+4\partial\bar\partial u.\tag{Gauß}$$ $$\rho^{-1}\bar\partial(\rho Q)=\rho\partial(\rho^{-1}m).\tag{Codazzi-Mainardi}$$ Many integrable non-linear partial differential equations are expressed within this formalism. The last equation (2.17) is then called the zero curvature condition, and is the non-linear equation represented as a compatibility condition of a linear system.

In the spinor representation, the corresponding matrices are readily obtained from the Maurer-Cartan form (1.11), they are $$Z=\frac12\pmatrix{ 2\partial u& iQ\\ im& 0},\quad \tilde Z=\frac12\pmatrix{ 0& im\\ i\bar Q& 2\bar\partial u },\tag{2.18}$$ and the equations (2.16), (2.17) become $$\partial\Phi=Z\Phi,\quad\bar\partial\Phi=\tilde Z\Phi,\tag{2.19}$$ $$\bar\partial Z-\partial\tilde Z+[Z,\tilde Z]=0.\tag{2.20}$$

Remark: The differential form $\Omega$ (2.11) is a connection, and actually an affine connection of $\Bbb R^3$ that is flat on the surface, which is the meaning of the second system of structure equations. We have thus the stunning result that the (two-dimensional) Dirac equation is but a way of writing an affine connection. Its restriction to the surface is $$\Omega_{/2}=2\pmatrix{ \partial u\d\zeta& 0\\ 0& \bar\partial u\d\bar\zeta }\tag{2.21}$$ and its intrinsic scalar curvature is precisely the Riemann-Gauß curvature of the surface. The restriction of the structure equations doesn’t necessarily hold, and indeed we have: $$\d\omega_+^+ =\omega_+^+\wedge\omega_+^++\omega_+^-\wedge\omega_-^++\omega_+^3\wedge\omega_3^+ =\omega_+^3\wedge\omega_3^+.\tag{2.22}$$ But $\omega_+^3$ isn’t defined on the surface, thus the term $\omega_+^3\wedge\omega_3^+=\omega^3_-\wedge\omega^3_+$ stands for the affine curvature form, whose explicit expression taken in three dimensions, that is extrinsically, is the Gauß curvature. That’s still another way of viewing the Theorema Egregium. There is no torsion from the sheer fact that $\omega^3=0,$ and thus it is indeed the Levi-Civita connection.

Spelling out the compatibility equations (2.6), we easily derive the already known equations (Gauß) and (Codazzi-Mainardi) again. In addition, as $\omega^3=0,$ from the compatibility equation for $\d\omega^3$ we have $$0=\omega^+\wedge\omega^3_++\omega^-\wedge\omega^3_-.\tag{2.23}$$ According to the Cartan’s lemma, $\omega^3_+$ and $\omega^3_-$ are then equal to a linear combination of $\omega^+$ and $\omega^-,$ like $$\eqalign{ \omega^3_+& =h_{++}\omega^+& +h_{+-}\omega^-\\ \omega^3_-& =h_{-+}\omega^+& +h_{--}\omega^-, }\tag{2.24a,b}$$ and $$h_{-+}=h_{+-},\tag{2.25}$$ as can be seen by substituting $\omega^3_+$ and $\omega^3_-$ in equation (2.23). Finally by identification we get $$h_{++}=-Q,\quad h_{--}=-\bar Q,\\ h_{+-}=h_{-+}=m.\tag{2.26}$$

Now using these coefficients together with the structure equations (2.1), we are able to express the data of the surface in term of the Pfaffian forms: $${\rm I}=2\rho^2\omega^+\omega^-,\tag{2.27}$$ $$\eqalign{ {\rm II}& =-\rho(\omega^+\omega^3_++\omega^-\omega^3_-)\\ & =-\rho(h_{++}\omega^+\omega^++(h_{+-}+h_{-+})\omega^+\omega^-+h_{--}\omega^-\omega^-).\tag{2.28} }$$ The surface element is $$\d S=i\rho^2\omega^+\wedge\omega^-,\tag{2.29}$$ and the corresponding surface element on the Gauß map (see next section) is $$\d\sigma=i\omega^3_-\wedge\omega^3_+=i(h_{+-}h_{-+}-h_{++}h_{--})\omega^+\wedge\omega^-.\tag{2.30}$$ The total curvature is the ratio of these two surfaces: $$K={\d\sigma\over\d S}=\rho^{-2}(h_{+-}h_{-+}-h_{++}h_{--}),\tag{2.31}$$ and the mean curvature is $$\eqalign{ H& =\frac i2\rho{\omega^-\wedge\omega^3_--\omega^+\wedge\omega^3_+\over\d S}\\ & =-\frac i2\rho{(h_{+-}+h_{-+})\omega^+\wedge\omega^-\over \d S}=-\frac12\rho^{-1}(h_{+-}+h_{-+}). }\tag{2.32}$$ The results are the sames as in the end of the previous section (1.36-39), as expected.

§3. The Gauß map

Given a moving frame, and under the condition that it is integrable, the surface is defined up to a translation. Conversely, among the three vectors of this frame, only one is determined uniquely by the surface, the normal vector. It is called the Gauß map (or spherical map,) it is a map from the parameter plane to the two-dimensional sphere of radius $1.$ The question addressed in this section is then, to which extend is the surface determined by the Gauß map alone?

From our derivation, it is obvious that the Gauß map is directly given by: $$\boldsymbol\phi=\boldsymbol e_3/\rho.\tag{3.1}$$ The north pole corresponds to $\chi=0,$ and the south pole to $\varphi=0.$ To go further, let us remark that in the vector representation, we use row vectors while the spinor is a column. If we consider the first column of $\Phi^\dagger$ and the associated fundamental field: $$w=-\chi/\bar\varphi,\tag{3.2}$$ then, dividing the numerator and the denominator by $\bar\varphi\varphi$ in the expression of $\boldsymbol\phi$ (3.1), we obtain $$\boldsymbol\phi=(w+\bar w,~i(\bar w-w),~1-w\bar w)/(1+w\bar w),\tag{3.3}$$ that is a function of $w$ alone. In other words, $w$ is the stereographic projection of the Gauß map from the south pole. Moreover, for a minimal surface ($m=0,$) it is readily shown that $\bar w$ is an analytic function of $\zeta,$ as is known for a long time. Actually, $\bar w$ is precisely the holomorphic function of the Enneper-Weierstraß representation. That generalises for any non-minimal surface with $w$ being any function. On the other hand, if $Q=0,$ the definition of $Q$ (1.12) implies $\partial\bar w=0,$ that is, $\bar w$ is an antiholomorphic function. In the following, we shall then use $w$ throughout, and call it the Gauß map.

Using the differential constraints (1.12) and (1.13b) we calculate $$\partial w=-\frac i2{\rho m\over\bar\varphi^2},\quad \partial\bar w=\frac i2{\rho Q\over\varphi^2},\tag{3.4a,b}$$ which yields $$\eqalign{ m^2& =4{\partial w~\bar\partial\bar w\over(1+w\bar w)^2},\\ mQ& =4{\partial w~\partial\bar w\over(1+w\bar w)^2},\\ Q\bar Q& =4{\bar\partial w~\partial\bar w\over(1+w\bar w)^2}.\tag{3.5a,b,c} }$$

So, $m$ and $Q$ can be written as a function of $w$ only. The question arises whether the last component of the Maurer-Cartan form, $\partial u,$ can too. The answer is yes. In order to derive it, we use the differential constraint (1.13c): $$\partial u=\rho^{-1}(\bar\varphi\partial\varphi+\chi\partial\bar\chi)\\ =\e^{-u/2}\partial(\varphi,-\bar\chi)\e^{-u/2}(\varphi,-\bar\chi)^\dagger.\tag{3.6}$$ The first factor is entered into the differentiation calculating $$\partial[\e^{-u/2}(\varphi,-\bar\chi)]\e^{-u/2}(\varphi,-\bar\chi)^\dagger =-\frac12\partial u+\e^{-u/2}\partial(\varphi,-\bar\chi)\e^{-u/2}(\varphi,-\bar\chi)^\dagger,\tag{3.7}$$ since $(\varphi,-\bar\chi)(\varphi,-\bar\chi)^\dagger=\e^u.$ Then the result is $$\frac12\partial u=\partial\big[\e^{-u/2}(\varphi,-\bar\chi)\big]\big[\e^{-u/2}(\varphi,-\bar\chi)\big]^\dagger =\partial VV^\dagger.\tag{3.8}$$ The row spinor $V=e^{-u/2}(\varphi,-\bar\chi)$ satisfies $VV^\dagger=1.$

The remaining task is to express $V$ as a function of $w.$ Taking into account that $w=-\chi/\bar\varphi,$ a parametrisation is $$V={1\over\sqrt{1+w\bar w}}(1,\bar w)\Big({\varphi\over\bar\varphi}\Big)^{1/2}.\tag{3.9}$$ Next, dividing equation (3.4a) by its complex conjugate gives $$\left({\varphi\over\bar\varphi}\right)^2=-{\partial w\over\bar\partial\bar w}.\tag{3.11}$$ Then we have $V$ as a function of $w$ only: $$V={1\over\sqrt{1+w\bar w}}(1,\bar w)\left({\partial w\over\bar\partial\bar w}\right)^{1/4}\e^{i(2n+1)\pi/4},\tag{3.12}$$ where $n$ is any integer, and inserting $V$ into the expression of $\partial u$ (3.8) provides us with the searched result: $$\left\{\eqalign{ \frac12\partial u & =\phantom-\frac14\partial\log\left({\partial w\over\bar\partial\bar w}\right) +\frac12{w\partial\bar w-\bar w\partial w\over1+w\bar w},\\ \frac12\bar\partial u & =-\frac14\bar\partial\log\left({\partial w\over\bar\partial\bar w}\right) +\frac12{\bar w\bar\partial w-w\bar\partial\bar w\over1+w\bar w}. }\right.\tag{3.13a,b}$$ In general, there is no simple integral of the second term. We see that $\partial u/2$ has the form of a potential with a fixed gauge. It is fixed because $w$ is given as a fonction of $\zeta$ and $\bar\zeta,$ then the directions of the axes $\boldsymbol e_+$ and $\boldsymbol e_-$ are fixed, while a gauge transformation is a rotation of them.

Now we investigate under which condition a given complex function $w(\zeta,\bar\zeta)$ is the Gauß map of some surface. We get a necessary condition by cross differentiating the linear system (3.13), which amounts to the same as requiring that $\partial\bar\partial u$ be real. This gives $$\partial\bar\partial u= -\frac12\partial\bar\partial\log\left({\partial w\over\bar\partial\bar w}\right)+\\ +{\partial\bar w\bar\partial w-\bar\partial\bar w\partial w -(1+w\bar w)(w\partial\bar\partial\bar w-\bar w\partial\bar\partial w) -\bar w\bar w\partial w\bar\partial w+ww\partial\bar w\bar\partial\bar w\over(1+w\bar w)^2},\tag{3.14a}$$ $$\bar\partial\partial u= \phantom-\frac12\partial\bar\partial\log\left({\partial w\over\bar\partial\bar w}\right)+\\ +{\partial\bar w\bar\partial w-\bar\partial\bar w\partial w +(1+w\bar w)(w\partial\bar\partial\bar w-\bar w\partial\bar\partial w) +\bar w\bar w\partial w\bar\partial w-ww\partial\bar w\bar\partial\bar w\over(1+w\bar w)^2}.\tag{3.14b}$$ The necessary condition then reads $$\partial\bar\partial\log\left({\partial w\over\bar\partial\bar w}\right) +2{(1+w\bar w)(w\partial\bar\partial\bar w-\bar w\partial\bar\partial w) +\bar w\bar w\partial w\bar\partial w-ww\partial\bar w\bar\partial\bar w\over(1+w\bar w)^2}=0.\tag{3.15}$$ It is a single real condition, since the $\log$ term is purely imaginary, and the fraction is the difference of two expressions that are the complex conjugate of each other. If it is satisfied, that means that $$\partial\bar\partial u ={\bar\partial w\partial\bar w-\partial w\bar\partial\bar w\over(1+w\bar w)^2},\tag{3.16}$$ and therefore from (3.5) it is easily seen that one of the integrability conditions (Gauß) is fulfilled. Then a tedious but straightforward calculation shows that the other one (Codazzi-Mainardi) is as well fulfilled. Therefore it is a sufficient condition too.

Because of the potential form of $\partial u/2$ in (3.13), an alternative expression can still be given: $$\partial\bar\partial u =\partial\Big(\frac12{\bar w\bar\partial w-w\bar\partial\bar w\over1+w\bar w}\Big) +\bar\partial\Big(\frac12{w\partial\bar w-\bar w\partial w\over1+w\bar w}\Big)=\frac14\rho^2K,\tag{3.17}$$ where the curvature nature of $K$ is manifest. Due to cancellations, in addition we have the short hand expressions $$\partial\bar\partial u =\partial\left({\bar w\bar\partial w\over1+w\bar w}\right) +\bar\partial\left({-\bar w\partial w\over1+w\bar w}\right) =\partial\left({-w\bar\partial\bar w\over1+w\bar w}\right) +\bar\partial\left({w\partial\bar w\over1+w\bar w}\right).\tag{3.18}$$ Similarly, the integrability condition (3.15) can be written as a zero-curvature condition: $$\partial\Big({\partial\bar\partial w\over\partial w} -2{\bar w\bar\partial w\over1+w\bar w}\Big) -\bar\partial\Big({\partial\bar\partial\bar w\over\bar\partial\bar w} -2{w\partial\bar w\over1+w\bar w}\Big)=0.\tag{3.19}$$ It then comes as obvious that if $$t(w,\bar w):=\partial\bar\partial w-{2\bar w\over1+w\bar w}\partial w\bar\partial w\equiv0,\tag{3.20}$$ the condition is automatically satisfied. But that is just the equation satisfied by the Gauß map of a constant mean curvature surface [5], which is harmonic. Thus $t$ is proportional to the tension field $\tau,$ actually: $$t=\frac14\rho^2\tau.\tag{3.21}$$

Up to now, we have derived the results of Hoffman and Osserman [6] for a surface in $\Bbb R^3,$ arguably in an more intelligible way. To proceed, notice that the zero-curvature condition (3.19) means that $t/\partial w$ is the gradient of some real fonction $F,$ or $$t=\partial w\bar\partial F(\zeta,\bar\zeta).\tag{3.22}$$ Beside constant mean curvature surfaces with harmonic Gauß map, other families of surfaces can be constructed by prescribing $F$ for which solutions of the above equation are known.

Let us remark that, having obtained $V$ (3.12) and $\partial u,\bar\partial u$ (3.13), we have also found $\Phi$ as a function of $w,$ up to a constant real factor, therefore the surface is determined up to a scaling factor by $w$ only. But it is not unique as it depends on the chosen fourth root, given by $n$ in (3.12). Its values of same parity give the same surface because the induction formula (1.35) is quadratic in $\Phi,$ thus there are two different surfaces, which are deduced from each other by a space inversion with $m\to-m$ (or by a negative scaling factor if one want.) Yet, $\partial u$ is the same since the constant exponential factor cancels out.

We have tacitly assumed that $m\ne0,$ for otherwise $\partial w=0,$ and $\partial u$ can’t be determined. We shall not consider isolated zeros, but the case where $m\equiv0$ in an open set, which corresponds to a portion of a minimal surface. Actually, the Dirac equation (1.7) reduces to $\bar\partial\varphi=0$ and $\partial\chi=0,$ so that a solution is given by an arbitrary holomorphic function $\varphi,$ and we have $\chi=-\bar\varphi w$ which is automatically antiholomorphic. But all these solutions don’t necessarily represent different surfaces. Thus we shall take $\partial u=0$ and $\rho=1,$ and we will see this is sufficient to represent all the minimal surfaces. Now it is well known that for a given Gauß map $w$ such that $\partial w=0,$ there is a one-parameter family of surfaces called the associate family, that is obtained through the transformation $$\varphi\to p^{1/2}\varphi,\quad\chi\to\bar p^{1/2}\chi,\tag{3.23}$$ that keep both $w$ and $\rho$ invariant if $p$ is a complex constant of modulus $1.$ This is not possible if $m\ne0$ because $m$ wouldn’t remain real, hence in the latter case the only allowed values of $p$ are $\e^{in\pi}.$ The surface can now be constructed by taking $V$ and replacing the phase factor by $p^{1/2}.$ As $\rho\equiv1,$ we directly get $$\varphi={p^{1/2}\over\sqrt{1+w\bar w}},\quad \chi=-{\bar p^{1/2}w\over\sqrt{1+w\bar w}}.\tag{3.24}$$ When this is put in the induction formula (1.35), we recognise the Weierstraß representation for $p=1.$

For the parameters of the surface that are proportional to a power of $\rho,$ one can still compute the logarithmic derivatives: $$H^{-1}\partial H ={\partial\bar\partial\bar w\over\bar\partial\bar w}-2{w\partial\bar w\over1+w\bar w} ={\bar t\over\bar\partial\bar w},\quad H^{-1}\bar\partial H ={t\over\partial w},\tag{3.25}$$ $$J^{-1}\bar\partial J ={\partial\bar\partial\bar w\over\partial\bar w}-2{w\bar\partial\bar w\over1+w\bar w} ={\bar t\over\partial\bar w},\quad \bar J^{-1}\partial\bar J ={t\over\bar\partial w},\tag{3.26}$$ Three conclusions can be drawn:

§4. Application

This application illustrates the use of the equations (2.16), (2.17) or (2.19), (2.20) to express a non-linear partial differential equation and its representation by the compatibility condition of a linear problem. As particular case we shall take a spherical surface, then $Q=0.$ Equation (1.39) simplifies to $$K=\rho^{-2}m^2,\tag{4.1}$$ where $K$ is now a constant, and equation (Gauß), which is deduced from (2.17) or (2.20), becomes $$4\partial\bar\partial u+m^2=0.\tag{4.2}$$ For the special choice $K=1,$ as $m=\rho,$ substituting $$m=\e^u,\tag{4.3}$$ it gives $$4\partial\bar\partial u+\e^{2u}=0,\tag{4.4}$$ and the only remaining variable is $u.$ It is the non-linear Liouville equation as well known. The other equation (Codazzi-Mainardi) is trivially satisfied.

Because of the spinor representation of the Maurer-Cartan form (1.11) from which $Z$ and $\tilde Z$ are deduced (2.18), for any non singular complex matrix function $G$ there is the gauge transformation $$\eqalign{ \Phi& \to G\Phi\\ Z& \to GZG^{-1}+\partial GG^{-1}\\ \tilde Z &\to G\tilde ZG^{-1}+\bar\partial GG^{-1}, \tag{4.5}}$$ for which the non-linear equation (2.20) still holds. Following Lund and Regge [7], (see also [2]) we choose the matrix $$G=\pmatrix{\bar k^{1/2}& 0\\0& k^{1/2}}\pmatrix{\e^{u/2}& 0\\0& \e^{-u/2}}\e^{-u/2}.\tag{4.6}$$ The purpose of each factor, from right to left, is to:

The result is a linear system that can be used to solve the non-linear equation by the inverse scattering transform: $$\partial\Phi_k'=\frac12\pmatrix{2\partial u& 0\\ik& -2\partial u}\Phi_k',\quad \bar\partial\Phi_k'=i\frac{\bar k}2\pmatrix{0& \e^{2u}\\0& 0}\Phi_k'.\tag{4.8}$$ where $$\Phi_k'=\pmatrix{\bar k^{1/2}\varphi& -\bar k^{1/2}\bar\chi\\k^{1/2}\e^{-u}\chi& k^{1/2}\e^{-u}\bar\varphi},\quad \det \Phi_k'=1.\tag{4.9}$$ A system similar to the AKNS one [8] is obtained by using a further factor $$\frac{\sqrt2}2\pmatrix{1& -1\\1& \phantom-1},\tag{4.10}$$ that is $$\partial\Phi_k''=\frac14\pmatrix{-ik& 4\partial u-ik\\ 4\partial u+ik& ik}\Phi_k'',\quad \bar\partial\Phi_k''=i\frac{\bar k}4\pmatrix{-\e^{2u}& \e^{2u}\\-\e^{2u}& \e^{2u}}\Phi_k''.\tag{4.11}$$

Our aim is not to work out the full inverse scattering transform. Instead, let us make some observations that directly lead to a general solution, using our previous study of the Gauß map. As $Q=0,$ from the formula (3.5c) we have $\bar\partial w=0,$ and in this case moreover $$\partial\log(\bar\partial\bar w)=\bar\partial\log(\partial w)=0,\tag{4.12}$$ $${w\partial\bar w-\bar w\partial w\over1+w\bar w} ={-\bar w\partial w\over1+w\bar w}=-\partial\log(1+w\bar w),\tag{4.13}$$ so that an integral of the linear system (3.13) is $$\e^u=2{\sqrt{\partial w\bar\partial\bar w}\over1+w\bar w}.\tag{4.14}$$ The immaterial multiplicative constant of integration is chosen according to (4.3). The general solution is then given by (4.14), where $w$ is any holomorphic function. In addition, $\varphi$ can be determined from (3.4a) and (4.3), giving $$\varphi=\pm\frac{1+i}2\e^u/\sqrt{\bar\partial\bar w}.\tag{4.15}$$

As an example, the choice $$w(\zeta)=i\zeta\tag{4.16}$$ yields the complete solution $$m=e^u={2\over 1+\zeta\bar\zeta},\quad \Phi={\sqrt2\over 1+\zeta\bar\zeta}\pmatrix{i& \bar\zeta\\-\zeta& -i}.\tag{4.17}$$ It is an easy exercice to verify that it is actually the case. This $\Phi$ solution stands for all the solutions of the spectral problem because there is only one spherical surface of curvature $1.$

What is shown is that this linear system is derived from the Dirac equation used to represent the surface. That’s the explanation of the link between non-linear evolution equations and geometry. In fact, it is not the usual representation with a surface of constant negative gaußian curvature and isometry group $SL(2\Bbb R),$ but the AKNS system can be derived the same way.

We have seen that if $\Phi$ is multiplied on the right by a constant $SU(2)$ matrix, it is still a solution of the Dirac equation, and in addition its determinant remains identical to $\e^u.$ Writing this matrix as $$\pmatrix{a& -\bar b\\b& \phantom-\bar a},\quad a\bar a+b\bar b=1,\tag{4.18}$$ we get the transformation $$\bar\varphi\to\bar a\bar\varphi-\bar b\chi,\quad \chi\to a\chi+b\bar\varphi,\tag{4.19}$$ and accordingly, dividing numerator and denominator by $\bar\varphi,$ the non-linear transformation $$-{\chi\over\bar\varphi}=w\to{aw-b\over \bar bw+\bar a},\tag{4.20}$$ which is a (proper) isometry of the Gauß sphere, and then of the spherical surface. That is to say, the isometry group is $SU(2)$ instead of $SL(2\Bbb R).$

Because of the mirror symmetry, it is natural to wonder whether the rôles of $m$ and $Q$ can be swapped. And that is indeed true, it is a recently discovered duality [9] that is more transparent in this framework. So, setting $m=0,$ for the choice $K=-\e^{-4u}$ the non-linear equation is now $$4\partial\bar\partial u-\e^{-2u}=0.\tag{4.21}$$ The antiholomorphic functions are replaced by holomorphic ones, and the surface is minimal. Since by substituting $u=-v$ the non-linear equation is the same as in the previous case, the general solution is $$\e^{-u}=2{\sqrt{\bar\partial w\partial\bar w}\over1+w\bar w},\tag{4.22}$$ while our example becomes $$\e^{-i\theta}Q=\e^{-u}={2\over 1+\zeta\bar\zeta},\quad \Phi_\theta=\frac{\sqrt2}2\pmatrix{\e^{i\theta/2}& 0\\0& \e^{-i\theta/2}}\pmatrix{1& -i\zeta\\-i\bar\zeta& 1},\tag{4.23}$$ where $\theta$ is a real constant. This family of minimal surfaces is the associate family we have seen in §3. No further details are given since the reasoning is strictly parallel. Let us only remark that these two surface families have the same isometry group, the Gauß map being merely inverted. The improper isometries are generated by $w(\zeta)\to w(\bar\zeta),$ which transform a surface into its mirror image (the mirror is not plane, as it were.)

It happens we have used two constant mean curvature surfaces. This is no coincidence. Actually, a full family of non-linear equations arises from this setting. We have seen that when the mean curvature is constant, the function $J$ (1.17) is holomorphic, so the (Codazzi-Mainardi) equation holds. Then we write the (Gauß) equation as follows: $$\rho^{-2}J\bar J=\rho^2H^2+4\partial\bar\partial u.\tag{4.24}$$ Replacing $\rho$ by $\e^u$ and rearranging, we get $$4\partial\bar\partial u+H^2\e^{2u}-J\bar J\e^{-2u}=0.\tag{4.25}$$ Our first case corresponds to $J=0$ and $H=1,$ the second one to $|J|=1$ and $H=0.$ There is also the sinh-Poisson (or elliptic sinh-Gordon) equation for $|J|=H,$ and many others more complicated, and indeed uninteresting since they explicitly involve $\zeta.$

But whatever the equation, it is a reduction of a single one: the harmonic equation (3.20). The above two special cases correspond to the finite action solutions satisfying $\partial w=0$ or $\bar\partial w=0.$ For the general case, from a harmonic function $w,$ and choosing a value of $H$ that is but a scale factor, the solution is given by (3.5a) since $m^2=\e^{2u}H^2$ (1.38). Conversely, from a solution $u$ of the non-linear equation (4.25) where $H$ and the holomorphic function $J$ are given, we can calculate $\partial u,$ $m=-\e^uH,$ and $Q=\e^{-u}J.$ The Gauß map is then deduced from the integration of the Weingarten equations (2.16) or (2.19), starting from any initial value of $e$ or $\Phi$ at $\zeta=0,$ which fixes the Gauß map at $\zeta=0,$ and thus selects a single element in the set of surfaces differing only by a rotation.

For instance, the solutions of the sinh-Poisson equation $$4\partial\bar\partial u+2\sinh(u)=0\tag{4.26}$$ can be found by simply imposing $J\bar J=1,$ so that the general solution is given by (4.22), where now $w$ is neither holomorphic nor antiholomorphic, but harmonic. The mean curvature is then constant, and is brought to $1$ by a simple rescaling of $\zeta.$ Using known harmonic functions, we get known constant mean curvature surfaces [10].

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