Surfaces: from generalised Weierstrass representation to Cartan moving frame
Laboratoire de Physique Fondamentale, FRANCE
19 April 2020
Konopelchenko’s generalised Weierstraß representation of a surface immersed in the three dimensional Euclidean space is derived by the way of the conserved current, 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 from a prescribed Gauß map is investigated, and this is then applied to inverse scattering transform for solving some nonlinear differential equations.
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Introduction
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] used 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 current, 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 fixed 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. This is done through a lift to the moving frame under the condition of remaining in the given formalism. The geometrical representation of integrable nonlinear differential equations follows directly, and in section §4 it is shown how this can be applied to solving the nonlinear 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: $$\DeclareMathOperator{\Tr}{Tr}\DeclareMathOperator{\d}{d\!}\DeclareMathOperator{\e}{e} \sigma^1=\pmatrix{0& \phantom-1\\1& \phantom-0},\quad \sigma^2=\pmatrix{0& -i\\i& \phantom-0},\quad \sigma^3=\pmatrix{1& \phantom-0\\0& -1}\tag{1.1}$$ as follow (the choice of the Dirac matrices is not a whim…): $$\{-i\sigma^2\partial_1+i\sigma^1\partial_2-m\}\psi=0.\tag{1.2}$$ Here, the mass parameter $m$ (more usually called the potential) is generalised to a real function of $\xi^1$ and $\xi^2,$ the Cartesian coordinates, 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 $$\partial=\tfrac12(\partial_1-i\partial_2),\quad\bar\partial=\tfrac12(\partial_1+i\partial_2),\tag{1.3}$$ 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: $$\left\{\eqalign{ \bar\partial\varphi& =\phantom-\tfrac12m\chi\\ \partial\chi& =-\tfrac12m\varphi. }\right.\tag{1.7a,b}$$
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 $\Phi\Phi^\dagger=\rho{\rm I}.$ Let us remark that if we make the substitution $$\matrix{ \varphi& \to& -\bar\chi\\ \chi& \to& \phantom-\bar\varphi, }\tag{1.10}$$ 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 Dirac equation has the symmetry group $GL(2\Bbb C).$ The substitution above (1.10) belongs to this group, which makes charge conjugaison a continuous transformation (i.e. continuously connected to the identity.)
We then consider the associated (right-invariant) Maurer-Cartan $1$-form: $$\d\Phi\Phi^{-1}=\rho^{-1}\d\Phi\Phi^\dagger.\tag{1.11}$$ Its entries can be calculated from the Dirac equation, with the result $$\rho^{-1}\d\Phi\Phi^\dagger= \frac12\pmatrix{2\partial u& Q\\-m& 0}\d\zeta+ \frac12\pmatrix{0& m\\-\bar Q& 2\bar\partial u}\d\bar\zeta,\tag{1.12}$$ where the following defining relation has been used: $$\bar\chi\partial\varphi-\varphi\partial\bar\chi=:\tfrac12\rho Q,\quad \tfrac12\rho\bar Q:=\chi\bar\partial\bar\varphi-\bar\varphi\bar\partial\chi.\tag{1.13}$$ In addition, the following differential constraints have been found: $$\varphi\partial\bar\varphi+\bar\chi\partial\chi=0=\bar\varphi\bar\partial\varphi+\chi\bar\partial\bar\chi,\tag{1.14a}$$ $$\chi\partial\bar\varphi-\bar\varphi\partial\chi=\tfrac12\rho m=\bar\chi\bar\partial\varphi-\varphi\bar\partial\bar\chi,\tag{1.14b}$$ $$\bar\varphi\partial\varphi+\chi\partial\bar\chi=\rho\partial u,\quad \rho\bar\partial u=\varphi\bar\partial\bar\varphi+\bar\chi\bar\partial\chi.\tag{1.14c}$$ They prove very useful in the sequel. For completeness, let us list two further constraints: $$\bar\varphi\partial\bar\varphi+\bar\chi\bar\partial\bar\chi=0 =\varphi\bar\partial\varphi+\chi\partial\chi,\tag{1.15a}$$ $$\bar\varphi\partial\chi-\varphi\bar\partial\bar\chi=0 =\chi\partial\bar\varphi-\bar\chi\bar\partial\varphi.\tag{1.15b}$$ 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 (1.13) and using the Dirac equation (1.7), we directly get $$\rho^{-1}\bar\partial(\rho Q)=\rho\partial(\rho^{-1}m),\tag{1.16}$$ that was firstly proven by Hopf [2 chap. VI, sec. 1.2]. We later use the quantity $$J:=\rho Q,\tag{1.17}$$ which is called the Hopf function. The value of $Q$ could be called the mirror mass, and by calculating $\partial\Phi$ and $\bar\partial\Phi$ with (1.12), we have the mirror Dirac equation: $$\left\{\eqalign{ \partial(\rho^{-1}\varphi)& =\phantom-\tfrac12Q(\rho^{-1}\chi)\\ \bar\partial(\rho^{-1}\chi)& =-\tfrac12\bar Q(\rho^{-1}\varphi). }\right.\tag{1.18a,b}$$ It is actually the one satisfied by $\rho^{-1}\sigma^1\Phi\sigma^1$ or $\rho^{-1}\sigma^2\Phi\sigma^2,$ the mirror spinor.
Now we use the well known vector representation. A matrix such as $\Phi$ is associated to a rotation times a scaling in $\Bbb R^3$ as follow: $$\boldsymbol V=v_i\sigma^i\to\boldsymbol V'=\Phi\boldsymbol V\Phi^\dagger.\tag{1.19}$$ 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 $\boldsymbol V$: $$\langle\boldsymbol V_1,~\boldsymbol V_2\rangle=\frac12\Tr[\boldsymbol V_1\boldsymbol V_2],\tag{1.20}$$ then $$e^j_i=\tfrac12\Tr[\sigma_i\Phi\sigma^j\Phi^\dagger],\tag{1.21}$$ and after some straightforward calculations: $$e={\small\pmatrix{\boldsymbol e_+:={\sqrt2\over2}(\boldsymbol e_1-i\boldsymbol e_2)\\ \boldsymbol e_-:={\sqrt2\over2}(\boldsymbol e_1+i\boldsymbol e_2)\\ \boldsymbol e_3}} ={\small\pmatrix{{\sqrt2\over2}(\varphi\varphi-\bar\chi\bar\chi)& -i{\sqrt2\over2}(\varphi\varphi+\bar\chi\bar\chi)& \sqrt2~\varphi\bar\chi\\ {\sqrt2\over2}(\bar\varphi\bar\varphi-\chi\chi)& \phantom-i{\sqrt2\over2}(\bar\varphi\bar\varphi+\chi\chi)& \sqrt2~\bar\varphi\chi\\ -(\varphi\chi+\bar\varphi\bar\chi)& i(\varphi\chi-\bar\varphi\bar\chi)& \varphi\bar\varphi-\chi\bar\chi}.\tag{1.22}}$$ 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 $$\boldsymbol E_i:=e^j_i\sigma_j=\Phi^\dagger\sigma_i\Phi,\tag{1.23}$$ with which we calculate $$\d\boldsymbol e_i\cdot\boldsymbol e_j=\langle\d\boldsymbol E_i,~\boldsymbol E_j\rangle =\frac12\Tr[\d\,(\Phi^\dagger\sigma_i\Phi)\Phi^\dagger\sigma_j\Phi].\tag{1.24}$$ Differentiating, this gives $$\d\boldsymbol e_i\cdot\boldsymbol e_j= \frac12\rho\Tr[\d\Phi\Phi^\dagger\sigma_j\sigma_i+(\d\Phi\Phi^\dagger\sigma_j\sigma_i)^\dagger]= \rho~{\frak Re}\{\Tr[\d\Phi\Phi^\dagger\sigma_j\sigma_i]\},\tag{1.25}$$ from which the vector representation of the Maurer-Cartan form is deduced: $$\rho^{-2}\d ee^\dagger={\small \pmatrix{2\partial u& 0& \phantom-{\sqrt2\over2}Q\\0& 0& -{\sqrt2\over2}m\\ {\sqrt2\over2}m& -{\sqrt2\over2}Q& \phantom-\partial u}}\d\zeta+{\small \pmatrix{0& 0& -{\sqrt2\over2}m\\0& 2\bar\partial u& \phantom-{\sqrt2\over2}\bar Q\\ -{\sqrt2\over2}\bar Q& {\sqrt2\over2}m& \phantom-\bar\partial u}}\d\bar\zeta.\tag{1.26}$$
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 $$\Sigma_1=\frac{\sqrt2}2{\small\pmatrix{\phantom-0& \phantom-0& -i\\\phantom-0& \phantom-0& \phantom-i\\-i& -i& \phantom-0}},\quad \Sigma_2=\frac{\sqrt2}2{\small\pmatrix{0& \phantom-0& -1\\0& \phantom-0& -1\\1& \phantom-1& \phantom-0}},\quad \Sigma_3={\small\pmatrix{i& \phantom-0& \phantom-0\\0& -i& \phantom-0\\0& \phantom-0& \phantom-0}.\tag{1.27}}$$ 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 $$\e^{-2uS_0}\d RR^\dagger=\d u~S_0 -i(\partial u\d\zeta-\bar\partial u\d\bar\zeta)S_3+\\ +\tfrac i2[(m+Q)\d\zeta-(m+\bar Q)\d\bar\zeta]S_1 +\tfrac12[(m-Q)\d\zeta+(m-\bar Q)\d\bar\zeta]S_2,\tag{1.28}$$ 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.
In the usual way, from the Dirac equation we get a conserved current by multiplying it on the left by $\Phi^\dagger,$ writing its hermitian conjugate $$\eqalignno{ \Phi^\dagger\{-i\sigma^2\partial_1+i\sigma^1\partial_2-m\}\Phi& =0,& {\rm(1.29a)}\\ \Phi^\dagger\{-i\overset{\small\leftarrow}{\partial_1}\sigma^2 +i\overset{\small\leftarrow}{\partial_2}\sigma ^1+m\}\Phi& =0& {\rm(1.29b)} }$$ and adding them. The result is $$\partial_1(\Phi^\dagger\sigma^2\Phi)-\partial_2(\Phi^\dagger\sigma^1\Phi)=0.\tag{1.30}$$ In the vector representation (1.21) and with complex variables, this equation becomes the conservation law $$(-\bar\partial,~\partial,~0)e=\partial\boldsymbol e_--\bar\partial\boldsymbol e_+=0.\tag{1.31}$$ From (1.22) we get the explicit components of the three conserved currents: $$\pmatrix{\phantom-\frac i2(\varphi\varphi-\bar\chi\bar\chi)\\-\frac i2(\bar\varphi\bar\varphi-\chi\chi)},\quad \pmatrix{\frac12(\varphi\varphi+\bar\chi\bar\chi)\\\frac12(\bar\varphi\bar\varphi+\chi\chi)},\quad \pmatrix{\phantom-i\vphantom{\frac12}\varphi\bar\chi\\-i\vphantom{\frac12}\bar\varphi\chi}.\tag{1.32}$$ According to Noether’s theorem, they are associated to symmetries of the equation of motion, which are indeed the right multiplication of $\Phi$ by a $SU(2)$ matrix. The currents (1.32) correspond to the generators $\frac i2\sigma^i$ of $SU(2).$ The third one is just the Dirac electromagnetic current, as expected since the transformation of the $U(1)$ electromagnetic gauge is associated to $\frac i2\sigma^3.$
From this conservation law, we deduce that the $\Bbb R^3$-valued differential form $$\d\boldsymbol r=\tfrac{\sqrt2}2\boldsymbol e_+\d\zeta+\tfrac{\sqrt2}2\boldsymbol e_-\d\bar\zeta\tag{1.33}$$ 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: $$\eqalignno{ x+iy& =\int_\Gamma(\varphi^2\d\zeta'-\chi^2\d\bar\zeta')+x_0+iy_0& {\rm(1.34a)}\\ x-iy& =\int_\Gamma(-\bar\chi^2\d\zeta'+\bar\varphi^2\d\bar\zeta')+x_0-iy_0& {\rm(1.34b)}\\ z& =\int_\Gamma(\varphi\bar\chi\d\zeta'+\bar\varphi\chi\d\bar\zeta')+z_0& {\rm(1.34c)} }$$ for the point $\boldsymbol r$ of the surface with coordinates $(x,y,z)$ in $\Bbb R^3.$ That is the same as in [1 equ. (2)] up to some cosmetic rearrangements. The surface is then normal to $\boldsymbol e_3.$ This method is not suited for generalisations, it is just a coincidence working only for surfaces in $\Bbb R^3.$
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 $${\rm I}:=\d\boldsymbol r\cdot\d\boldsymbol r=\rho^2\d\zeta~\d\bar\zeta\tag{1.35}$$ (it is therefore a conformal immersion with isothermal coordinates $\xi^1$ and $\xi^2.$) The second fundamental form can be directly calculated too, using $\boldsymbol e_3\cdot\d\boldsymbol r=0$: $$\eqalign{ {\rm II}:=& \d\,(\rho^{-1}\boldsymbol e_3)\cdot\d\boldsymbol r =\rho^{-1}\d\boldsymbol e_3\cdot\d\boldsymbol r\\ =& -\tfrac12\rho(Q\d\zeta\d\zeta-2m\d\zeta\d\bar\zeta+\bar Q\d\bar\zeta\d\bar\zeta).\tag{1.36} }$$ $${\rm III}:=\d\,(\rho^{-1}\boldsymbol e_3)\cdot\d\,(\rho^{-1}\boldsymbol e_3) =-mQ\d\zeta\d\zeta+(m^2+Q\bar Q)\d\zeta\d\bar\zeta-m\bar Q\d\bar\zeta\d\bar\zeta\\ =(Q\d\zeta-m\d\bar\zeta)(\overline{Q\d\zeta-m\d\bar\zeta}).\tag{1.36'}$$ Finally the usual formulæ gives the mean curvature $H$ and the Gauß curvature $K$: $$H=\tfrac12\Tr[{\rm II}\cdot{\rm I}^{-1}]=\rho^{-1}m,\tag{1.37}$$ $$K=\det({\rm II}\cdot{\rm I}^{-1})=\rho^{-2}(m^2-Q\bar Q)=H^2-\rho^{-2}Q\bar Q.\tag{1.38}$$ Konopelchenko [1 equ. (4)] gives a different expression: $$K=-4\rho^{-2}\partial\bar\partial u,\tag{1.39}$$ which is known as the Gauß-Riemann or intrinsic curvature. But Ferapontov and Grundland [3 equ. (2.6)] proved that they are equivalent, in accordance with Gauß’ Theorema Egregium. Then the modulus of $Q$ is a function of $\rho$ and $m$: $$Q\bar Q=m^2+4\partial\bar\partial u.\tag{1.40}$$
It is interesting to note that since the difference between the principal curvatures is given by $$(\Delta\kappa)^2=4(H^2-K),\tag{1.41}$$ it holds $$|\Delta\kappa|=2\rho^{-1}|Q|,\tag{1.42}$$ 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 $$\kappa=\rho^{-1}(m\pm|Q|).\tag{1.43}$$ Thence the quantity $\rho^{-1}|Q|$ is called the skew curvature.
§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. Expressing the different quantities in this fixed frame was deemed awkward by Cartan, so he developed the powerful method of the moving frame [4] from the Darboux frame. 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 (2.6a) is the equivalent, expressed in the moving frame language, of the requirement that the form $\d\boldsymbol r$ be exact.
From the expression of $\d\boldsymbol r$ (1.33) we immediately get $$\omega^+=\tfrac{\sqrt2}2d\zeta,\quad\omega^-=\tfrac{\sqrt2}2\d\bar\zeta,\quad\omega^3=0,\tag{2.7}$$ 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 (1.26), 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 $${\cal Z}={\small\pmatrix{ 2\partial u& 0& \phantom-{\sqrt2\over2}Q\\ 0& 0& -{\sqrt2\over2}m\\ {\sqrt2\over2}m& -{\sqrt2\over2}Q& \phantom-\partial u }},\quad\tilde{\cal Z}={\small\pmatrix{ 0& 0& -{\sqrt2\over2}m\\ 0& 2\bar\partial u& \phantom-{\sqrt2\over2}\bar Q\\ -{\sqrt2\over2}\bar Q& {\sqrt2\over2}m& \phantom-\bar\partial u }},\tag{2.15}$$ 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: $$Q\bar Q=m^2+4\partial\bar\partial u.\tag{2.G}$$ $$\rho^{-1}\bar\partial(\rho Q)=\rho\partial(\rho^{-1}m).\tag{2.CM}$$ Many integrable nonlinear partial differential equations are expressed within this formalism. The last equation (2.17) is then called the zero curvature condition, and is the nonlinear equation represented as a compatibility condition of a linear system. The double $(Z,\tilde Z)$ is called the Lax pair.
In the spinor representation, the corresponding matrices are readily obtained from the Maurer-Cartan form (1.12), they are $$Z=\frac12\pmatrix{ 2\partial u& Q\\ -m& 0},\quad \tilde Z=\frac12\pmatrix{ 0& m\\ -\bar Q& 2\bar\partial u },\tag{2.18}$$ 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 (2.6). 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. The connection has no torsion, i.e. extrinsic part of $\d\omega^i,$ 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 (2.G) and (2.CM) 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 $$\eqalignno{ \omega^3_+& =h_{++}\omega^++h_{+-}\omega^-& {\rm(2.24a)}\\ \omega^3_-& =h_{-+}\omega^++h_{--}\omega^-,& {\rm(2.24b)} }$$ 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: $${\rm I}=2\rho^2\omega^+\omega^-,\tag{2.27}$$ $$\eqalign{ {\rm II}& =\rho(\omega^+_3\omega^-+\omega^-_3\omega^+)\\ & =-\rho(h_{++}\omega^+\omega^++(h_{+-}+h_{-+})\omega^+\omega^-+h_{--}\omega^-\omega^-).\tag{2.28} }$$ 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 (1.35-38), 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 meromorphic, as is known for a long time. Actually, $\bar w$ is precisely the meromorphic function of the Enneper-Weierstraß representation, while the holomorphic $1$-form is $\varphi^2\d\zeta.$ That generalises for any non minimal surface with $w$ being any function. On the other hand, if $Q=0,$ the definition of $Q$ (1.13) 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.
Now we tackle the problem of reconstructing the surface from the Gauß map. In our setting, it is easily answered through the spinor, and more precisely the unit row spinor $$V=\e^{-u/2}\pmatrix{\varphi,-\bar\chi},\quad VV^\dagger=1\tag{3.4a}$$ and its orthogonal unit spinor $$\tilde V=\e^{-u/2}\pmatrix{\chi,\bar\varphi},\quad\tilde V\tilde V^\dagger=1,\tag{3.4b}$$ satisfying $$\tilde VV^\dagger=V\tilde V^\dagger=0.\tag{3.4c}$$ Together they form the unitary matrix $\rho^{-1/2}\Phi.$ Taking into account that $w=-\chi/\bar\varphi,$ a parametrisation of them is $$V={1\over\sqrt{1+w\bar w}}(1,\bar w)\Big({\varphi\over\bar\varphi}\Big)^{1/2},\quad \tilde V={1\over\sqrt{1+w\bar w}}(-w,1)\Big({\varphi\over\bar\varphi}\Big)^{-1/2}.\tag{3.5a,b}$$ Next, using the differential constraints (1.14b) we calculate $$\partial w=\frac12{\rho m\over\bar\varphi^2},\tag{3.6}$$ and dividing the latter equation by it complex conjugate gives $$\left({\varphi\over\bar\varphi}\right)^2={\partial w\over\bar\partial\bar w}.\tag{3.7}$$ Then we have $V$ and $\tilde V$ as functions of $w$ only: $$\left\{\eqalign{ V& ={1\over\sqrt{1+w\bar w}}(\phantom-1,\bar w)\left({\partial w\over\bar\partial\bar w}\right)^{1/4}\e^{in\pi/2},\\ \tilde V& ={1\over\sqrt{1+w\bar w}}(-w,1)\left({\partial w\over\bar\partial\bar w}\right)^{-1/4}\e^{-in\pi/2} }\right.,\quad n\in\Bbb N.\tag{3.8a,b}$$ We have achieved the lift (or framing) of the spherical map to $SU(2),$ but to get the surface completely, $u$ is still lacking. Yet, from the differential constraints (1.14b) and (1.13), with (3.4a) and (3.4b) we have $$m=-2\partial\tilde VV^\dagger =2\bar\partial V\tilde V^\dagger,\\ Q=2\partial V\tilde V^\dagger,\quad \bar Q=-2\bar\partial\tilde VV^\dagger\tag{3.9a,b}$$ then with (3.8), cancellations because of (3.4c) lead to $$m=\pm{2\partial w\over1+w\bar w}\left({\partial w\over\bar\partial\bar w}\right)^{-1/2} =\pm{2(\partial w~\bar\partial\bar w)^{1/2}\over1+w\bar w},\\ Q=\pm{2\partial\bar w\over1+w\bar w}\left({\partial w\over\bar\partial\bar w}\right)^{1/2},\quad \bar Q=\pm{2\bar\partial w\over1+w\bar w}\left({\partial w\over\bar\partial\bar w}\right)^{-1/2}\tag{3.10a,b}$$ yielding the simpler formulæ $$\eqalignno{ m^2& ={4\partial w~\bar\partial\bar w\over(1+w\bar w)^2},& {\rm(3.11a)}\\ mQ& ={4\partial w~\partial\bar w\over(1+w\bar w)^2},& {\rm(3.11b)}\\ Q\bar Q& ={4\bar\partial w~\partial\bar w\over(1+w\bar w)^2}.& {\rm(3.11c)} }$$ So, $m$ and $Q$ can be written as a function of $w$ only. Similarly, using once more the differential constraint (1.14c), that is $$\tfrac12\partial u=\partial VV^\dagger =-\partial\tilde V\tilde V^\dagger,\\ \tfrac12\bar\partial u=\bar\partial\tilde V\tilde V^\dagger =-\bar\partial VV^\dagger,\tag{3.12a,b}$$ we get the remaining components of the Maurer-Cartan form $$\left\{\eqalign{ \frac12\partial u & =\phantom-\frac14\partial\log\left({\partial w\over\bar\partial\bar w}\right) +\frac12{w\partial\bar w-\bar w\partial w\over1+w\bar w},\\ \frac12\bar\partial u & =-\frac14\bar\partial\log\left({\partial w\over\bar\partial\bar w}\right) +\frac12{\bar w\bar\partial w-w\bar\partial\bar w\over1+w\bar w}. }\right.\tag{3.13a,b}$$ Under the condition that this system be compatible, $u$ is also determined by integration up to a constant multiplicative factor, which corresponds to a mere change of scale of the surface, and it is obvious that the Gauß map is invariant under such a transformation. However 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. Indeed, a gauge transformation is a rotation of the axes $\boldsymbol e_+$ and $\boldsymbol e_-,$ then as $w$ is given as a function of $\zeta$ and $\bar\zeta,$ and the latter variables are defined by this axis system, the gauge is fixed.
Now we are in position to 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, by inserting it in (3.14), it 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 (3.11) it is easily seen that one of the integrability conditions (2.G) is fulfilled. Then a tedious but straightforward calculation shows that the other one (2.CM) is as well fulfilled. Therefore it is a sufficient condition too.
Because of the system (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: $$t=\tfrac14\rho^2\tau.\tag{3.21}$$
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 $V,$ $\tilde V$ (3.8) 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 (3.8). Its values of same parity give the same surface because the induction formula (1.34) 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. 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. 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. It 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 $1,$ while $Q\to pQ.$ 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 putting $\varphi$ and $\chi$ in the induction formula (1.34). Then as already discussed, recalling that the holomorphic $1$-form is $\varphi^2\d\zeta$ we recognise the Weierstraß representation, where the argument of $p$ is the Bonnet angle.
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:
- The function $F$ in (3.22) depends on $H$ alone, viz. $$\partial F=H^{-1}\partial H,\quad F=\log|H|,\tag{3.27}$$ and we fall back to the result of Kenmotsu [5].
- For a constant mean curvature surface, $\partial H=0,$ and thus as $t=0$ we also have $$\bar\partial J=\partial\bar J=0,\tag{3.28}$$ that is, $J$ (see equ. 1.17) is a holomorphic function.
- As $$(mQ)^{-1}\bar\partial(mQ)=(HJ)^{-1}\bar\partial(HJ)={t\over\partial w}+{\bar t\over\partial\bar w},\tag{3.29}$$ the function $$f(\zeta,\bar\zeta)={4\partial w\partial\bar w\over(1+w\bar w)^2}=mQ\tag{3.30}$$ is a first integral of the harmonic equation (3.20), that is $\bar\partial f=0$ if $w$ is a solution.
To conclude this section, let us look back at the road traveled. We got at an integrability equation of the Gauß map for a constant mean curvature surface, while we started from another equation, but that is not restricted to particular surfaces, and that contains a prescribed function. We shall then investigate how it can be modified so that it describes a constant mean curvature surface too. One way is to directly replace $m$ by $(\varphi\bar\varphi+\chi\bar\chi)H$ in the equation, that becomes nonlinear. Konopelchenko has shown that it is integrable [1]. But there is another way by keeping the prescribed function $m.$ We can use the mirror Dirac equation (1.18) if $Q$ can be calculated, and it turns out it can. The equation then remains linear.
The case of $m$ identically zero is already known, there are many minimal surfaces. The case of $H$ negative is deduced from the one of the opposite mean curvature, as we have seen the surface is spatially inverted. So we consider the case of $H$ positive. Then $m$ must be a strictly positive function. To begin with, the definition (1.37) yields $$u=\log(m)-\log(H).\tag{3.31}$$ Then we make use of the integrability equations in turn. From (2.G) along with the condition that $H$ be constant we first get $$Q\bar Q=m^2+4\partial\bar\partial\log(m).\tag{3.32}$$ As $Q\bar Q\geqslant0,$ there is a second constraint on $m$: $$\partial\bar\partial\log(m)\geqslant-m^2/4.\tag{3.33}$$ The equation (2.CM) reduces to $$\bar\partial Q+Q\bar\partial u=0,\tag{3.34}$$ and using the parametrisation $$Q=q\e^{i\theta},\quad \theta\in\Bbb R, \quad q=\sqrt{m^2+4\partial\bar\partial\log(m)}\in\Bbb R\tag{3.35}$$ in it, we finally obtain the result $$iq\bar\partial\theta+\bar\partial q+q\bar\partial\log(m)=0,\\ \bar\partial\theta=\tfrac i2\bar\partial\log([m^2+4\partial\bar\partial\log(m)]m^2).\tag{3.36}$$ This equation has a unique solution up to an additive constant, since it is linear and of first order, and this determines $Q$ entirely. As a global change of the argument of $Q$ gives a different surface, there is a one-parameter family of surfaces, which is the associated family. Remark that the function $\theta$ is harmonic, which is a mere consequence of the holomophicity of $J.$ So, $Q$ is determined by $m$, the linear Dirac type equation for a mean curvature surface is a system of four equations, and it has a unique solution if $\Phi$ is given at a point, thus all the solutions represent the same surface up to a rigid motion.
We have then arrived at: provided that the function $m(\zeta,\bar\zeta)$ be strictly positive and satisfies (3.33), among the surfaces represented by the solutions of the Dirac equation (1.2) there is an associate family of constant mean curvature surfaces, and all those deduced from them by a rigid motion. There is an apparent contradiction with the transformation (3.23), with which we said that $m$ wouldn’t remain real. But here, $m$ is fixed, and consequently we no longer have this transformation, so that $w$ isn’t invariant.
§4. Application
This application illustrates the use of the equations (2.16, 2.17) or (2.19, 2.20) to express a nonlinear partial differential equation and its representation by the compatibility condition of a linear system. As particular case we shall take a spherical surface, then $Q=0.$ Equation (1.38) simplifies to $$K=\rho^{-2}m^2,\tag{4.1}$$ where $K$ is now a constant, and equation (2.G), 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 nonlinear Liouville equation as well known. The other equation (2.CM) is trivially satisfied.
Because of the spinor representation of the Maurer-Cartan form (1.12) 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 nonlinear equation (2.20) still holds. Following Lund and Regge [7, see also 3] we choose the matrix $$G=\pmatrix{\bar k^{1/2}& 0\\0& k^{1/2}}\pmatrix{1& 0\\0& \e^{-u}}.\tag{4.6}$$ The purpose of each factor, from right to left, is to eliminate the terms in $\bar\partial u$ and to introduce the complex spectral parameter $k$ that verifies $$k\bar k=1\tag{4.7}$$ The result is the linear system $$\partial\Phi_k'=\frac12\pmatrix{2\partial u& 0\\-k& -2\partial u}\Phi_k',\quad \bar\partial\Phi_k'=\frac{\bar k}2\pmatrix{0& \e^{2u}\\0& 0}\Phi_k'\tag{4.8}$$ where $$\Phi_k'=\pmatrix{\bar k^{1/2}\varphi& -\bar k^{1/2}\bar\chi\\ k^{1/2}\e^{-u}\chi& k^{1/2}\e^{-u}\bar\varphi},\quad \det \Phi_k'=1.\tag{4.9}$$ In order that it could be used to solve the nonlinear equation by the inverse scattering transform, it must be reformulated in the $\xi$ coordinates, giving $$\partial_1\Phi_k'=\frac12\pmatrix{2\partial u& \bar k\e^{2u}\\-k& -2\partial u}\Phi_k',\quad \partial_2\Phi_k'=\frac i2\pmatrix{2\partial u& -\bar k\e^{2u}\\-k& -2\partial u}\Phi_k'.\tag{4.10}$$ Using the further factor $$\frac{\sqrt2}2\pmatrix{1& -1\\1& \phantom-1},\tag{4.11}$$ this system has the same form as the one used for the sine-Gordon equation in laboratory coordinates [8].
Our aim is not to work out the full inverse scattering transform. Actually, we already have the general solution from (4.3, 3.11a, & 3.11c). As $Q=0,$ we have $\bar\partial w=0,$ so that $$\e^u=2{\sqrt{\partial w\bar\partial\bar w}\over1+w\bar w}.\tag{4.12}$$ where $w$ is any holomorphic function. The same formula can be obtained by integrating the linear system (3.13) using the fact that $w$ is holomorphic. In addition, $\Phi$ can be determined from (3.8).
As an example, the choice $$w(\zeta)=\zeta\tag{4.13}$$ yields the complete solution $$m=e^u={2\over1+\zeta\bar\zeta},\quad \Phi=\pm{\sqrt2\over 1+\zeta\bar\zeta}\pmatrix{1& \bar\zeta\\-\zeta& 1},\\ \partial u=-{\bar\zeta\over1+\zeta\bar\zeta},\quad \bar\partial u=-{\zeta\over1+\zeta\bar\zeta}.\tag{4.14}$$ It is an easy exercice to verify that it is actually the case. This $\Phi$ stands for all the solutions of the spectral problem because there is only one spherical surface of curvature $1.$
What is shown here is that this linear system is derived from the Dirac equation used to represent the surface. That’s the explanation of the link between nonlinear 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 Lax pair 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 $$\pmatrix{a& -\bar b\\b& \phantom-\bar a},\quad a\bar a+b\bar b=1,\tag{4.15}$$ we get the transformation $$\bar\varphi\mapsto\bar a\bar\varphi-\bar b\chi,\quad \chi\mapsto a\chi+b\bar\varphi,\tag{4.16}$$ and accordingly, dividing numerator and denominator by $\bar\varphi,$ the so-called nonlinear realisation of $SU(2)$ $$-{\chi\over\bar\varphi}=w\mapsto{aw-b\over \bar bw+\bar a},\tag{4.17}$$ 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 nonlinear equation is now $$4\partial\bar\partial u-\e^{-2u}=0.\tag{4.18}$$ The antiholomorphic functions are replaced by holomorphic ones, and the surface is minimal. Since by substituting $u=-v$ the nonlinear equation is the same as in the previous case, the general solution is $$\e^{-u}=2{\sqrt{\bar\partial w\partial\bar w}\over1+w\bar w}.\tag{4.19}$$ However, $\partial w/\bar\partial\bar w$ is no longer defined. But its value is partially determined by the fact that when $m=0,$ the functions $\varphi,$ $\bar\chi,$ and $\bar w$ are holomorphic, which gives $${\bar\partial w\over\partial\bar w}\e^{2i\theta},\tag{4.19'}$$ where $\theta$ is a constant of integration.
With $w=\bar\zeta$ our example becomes $$\e^{-i\theta}Q=\e^{-u}={2\over 1+\zeta\bar\zeta},\quad \Phi_\theta=\pm\frac{\sqrt2}2\pmatrix{\e^{i\theta/2}& 0\\0& \e^{-i\theta/2}} \pmatrix{1& \zeta\\-\bar\zeta& 1},\\ \partial u={\bar\zeta\over1+\zeta\bar\zeta},\quad \bar\partial u={\zeta\over1+\zeta\bar\zeta},\tag{4.20}$$ This family of minimal surfaces is the associate family we have seen in §3, for $\theta=0$ it is known as the minimal Enneper surface. 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 nonlinear equations arises from this setting. We have seen that when the mean curvature is constant, the function $J$ (1.17) is holomorphic, so the (2.CM) equation holds. Then we write the (2.G) equation as follows: $$\rho^{-2}J\bar J=\rho^2H^2+4\partial\bar\partial u.\tag{4.21}$$ Replacing $\rho$ by $\e^u$ and rearranging, we get $$4\partial\bar\partial u+H^2\e^{2u}-J\bar J\e^{-2u}=0.\tag{4.22}$$ Our first case corresponds to $J=0$ and $H=1,$ the second one to $|J|=1$ and $H=0,$ and it is obvious in this equation that these two cases are deduced from one another through $u\to-u.$ There is also the elliptic sinh-Poisson (or elliptic sine-Gordon) equation for $|J|=H.$ Together with the $SU(2)$ group, they are associated to the isothermal coordinates.
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 (3.11a) since $m^2=\e^{2u}H^2$ (1.37). Conversely, from a solution $u$ of the nonlinear equation (4.22) 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 $$4\partial\bar\partial u+2\sinh(2u)=0\tag{4.23}$$ are determined by two constraints. First, $w$ must be a general harmonic function. Then, from $H^2=1$ and (3.11a) we get the solution given by (4.12), and from $J\bar J=1$ and (3.11c) we get another expression of the same solution given by (4.19). Both expressions must agree and this is a condition on the eligible harmonic functions. But this way doesn’t seem promising, so we only exhibit the simplest example of a constant mean curvature surface: the round cylinder. All the normal vectors are in the same plane, so that for a cylinder along the $z$ axis we have $$w\bar w=1.\tag{4.24}$$ Actually, it can be written as $$w=\e^{i(p\zeta+\bar p\bar\zeta)},\tag{4.25}$$ which is clearly harmonic. The condition that (4.12) and (4.19) should agree imposes $$p\bar p=1.\tag{4.26}$$ Then the solution is $$\e^u=1,\quad u=0.\tag{4.27}$$ It is the trivial solution of (4.23). The reader would as easily get the function $\Phi$ and see that it is a plane wave. The other directions of the cylinder are deduced from a rigid rotation with (4.17).
The Lax pair of the sinh-Poisson equation is derived as follows. The conditions $J\bar J=1$ and $H=1$ give $$Q=\e^{-u}\e^{i\theta},\quad m=\e^u.\tag{4.28}$$ Then from the equation (2.CM) we have $$\bar\partial\e^{i\theta}=0,\tag{4.29}$$ thus $\theta$ is constant since it is real. We take $k=\e^{i\theta/2}$ as spectral parameter, the Maurer-Cartan matrices are $$Z=\frac12\pmatrix{2\partial u& -k^2\e^{-u}\\-\e^u& 0},\quad \tilde Z=\frac12\pmatrix{0& \e^u\\\bar k^2\e^{-u}& 2\bar\partial u}\tag{4.30}$$ so that by the gauge transformation $$G=\pmatrix{1& 0\\0& k\e^{-u}}\tag{4.31}$$ we get the linear system $$\partial\Phi_k'=\frac12\pmatrix{2\partial u& -k\\-k& -2\partial u}\Phi_k',\quad \bar\partial\Phi_k'=\frac{\bar k}2\pmatrix{0& \e^{2u}\\\e^{-2u}& 0}\Phi_k',\tag{4.32}$$ which in the laboratory coordinates becomes $$\partial_1\Phi_k' =\frac12\pmatrix{2\partial u& -k+\bar k\e^{2u}\\-k+\bar k\e^{-2u}& -2\partial u}\Phi_k',\\ \partial_2\Phi_k' =\frac i2\pmatrix{2\partial u& -k-\bar k\e^{2u}\\-k-\bar k\e^{-2u}& -2\partial u}\Phi_k'.\tag{4.33}$$ With the further gauge transformation (4.11) it can be directly compared with [8]. This Lax pair was also derived in the vector representation by Hesheng [10].