Minimal surfaces and holomorphic curves

1. Minimal surfaces in \( \mathbb{R}^3 \) and Gauss map.
2. Twistor spaces and Gauss lift.
3. Twistor correspondence.

This post is my reading note for Eells.Salamon85_TwistorialConstructionHarmonic writen using a more standard notation. The goal of their paper is, for a Riemannian four-manifold \( M \), to contruct an almost complex six-manifold \( S \) such that minimal surfaces in \( M \) correspond to holomorphic curves in \( S \). The space \( S \), called twistor space of \( M \) is a \( \mathbb{S}_2 \)-bundle over \( M \) with fibre at point \( p\in M \) being the set of almost complex structures at \( p \) that are compatible with the Riemannian metric.

1 Minimal surfaces in \( \mathbb{R}^3 \) and Gauss map.

We will start with the following result of S. S. Chern chern_minimal_1965.

Let \( f:\Sigma^2 \hookrightarrow \mathbb{R}^3 \) be a compact oriented surface. The association to each point \( p\in \Sigma \) its normal vector at \( p \) gives a map \[ \tilde f: \Sigma \longrightarrow \mathbb{S}^2 \] Then \( \Sigma \) is minimal surface if and only if \( \tilde f: (\Sigma,i) \longrightarrow (\mathbb{S}^2,J) \) is antiholomorphic, where

\( i \) is the complex structure given by the conformal class of the induced metric of Euclidean metric from \( \mathbb{R}^3 \),
\( J \) is the complex structure on \( \mathbb{S}^2 \) given by the diffeomorphism
\begin{align*} \mathbb{S}^2 &\longrightarrow Q_1 = \{z_0^2 + z_1^2 + z_2^2 = 0\}\subset \mathbb{CP}^2\\ r & \longmapsto [(u_1 + iv_1, u_2+iv_2, u_3+ iv_3)] \end{align*}

where \( (u,v,r) \) form an oriented orthonormal basis of \( \mathbb{R}^3 \).

This result can also be generalised for surface \( \Sigma^2\subset \mathbb{R}^n \) by associating to each point \( p\in \Sigma \) its tangent plane \( T_p\Sigma\in \widetilde {\rm Gr}_2(\mathbb{R}^n) \) the space of oriented 2-planes in \(\mathbb{R}^n \). We can then equip \( \widetilde {\rm Gr}_2(\mathbb{R}^n) \) with a complex structure given by the diffeomorphism

\begin{align*} \widetilde {\rm Gr}_2(\mathbb{R}^n) &\longrightarrow Q_{n-2}=\{z_0^2 +\dots + z_{n-1}^2\}\subset \mathbb{CP}^{n-1}\\ u\wedge v &\longmapsto [u+iv] = [(u_1+iv_1,\dots, u_n+iv_n)] \end{align*}

where \( (u,v) \) forms an oriented orthonormal basis of \( T_p\Sigma \).

The above association \( \tilde f \) is called the Gauss map of the surface \( \Sigma \).

The above definition of \( Q_{n-2} \) suggests that we should complexify an inner product \( \langle \cdot , \cdot \rangle \) on a real vector space \( V \) to an inner product \( \langle \cdot,\cdot \rangle_{\mathbb{C}} \) on \( V\otimes \mathbb{C} \) so that the latter is symetric, i.e. \[ \langle iu, v \rangle_{\mathbb{C}} = \langle u, iv \rangle_{\mathbb{C}} = i\langle u, v \rangle_{\mathbb{C}} \] and not \( i \)-antilinear in the second parameter. This way, one has \[ \left| u+iv \right|^2 = \sum_i (u_i + iv_i)^2 = |u|^2 - |v|^2 + 2i \langle u,v \rangle = 0 \text{ if \( u,v \) form an orthogonal basis}. \]

The correspondence that we will define between minimal surface in a Riemannian four-manifold and a holomorphic curve in an almost complex six-manifold is a generalised version of the Gauss map, called Gauss lift.

2 Twistor spaces and Gauss lift.

2.1 2-forms in 4 dimensional geometry

Let \( M \) be an oriented four-manifold and \( \vol \) is a volume form on \( M \). Then there is a symmetric bilinear form on \( \Lambda^2 T^*M \) given by the wedge product

\begin{align*} \Lambda^2 T^*M &\longrightarrow \mathbb{R}\\ (u,v) &\longmapsto \frac{u\wedge v}{\vol} \end{align*}

which is of signature (3,3), i.e. the (non-unique) maximal vector subspace on which \( \wedge \) is positive (resp. negative) are of dimension 3.

Fixing a metric \( g \) on \( M \), one can dualise \( \wedge \), i.e. define the Hodge star operator

\begin{align*} \ast: \Lambda^2T^*M & \longrightarrow \Lambda^2 T^*M \\ v & \longmapsto \ast v \text{ such that } u\wedge \ast v= \langle u, v \rangle \vol_g \end{align*}

The inner product \( \langle \cdot , \cdot \rangle \) here is rescaled from the induced metric on \( \Lambda^2 T^*M \) such that \( dx^i\wedge dx^j \) is of norm 1 (and not 2) for all orthonormal \( dx^i,dx^j \).
The Hodge star (on 2-forms in 4D) is conformal invariant. In fact, if \( g_\theta = e^{2\theta} g\) then \( e^{-4\theta}\langle u, \ast_\theta v \rangle = \langle u, \ast_\theta v \rangle_\theta = e^{-4\theta} \frac{u\wedge v}{\vol_g} = e^{-4\theta} \langle u, \ast v \rangle\). On the other hand, knowing \( \ast \), one can recover the conformal class of \( g \).

Since \( \ast \) is auto-adjoint, i.e. \( \langle u,\ast v \rangle = \frac{u\wedge v}{\vol} = \langle \ast u, v \rangle \), and \( (\ast)^2 = 1 \), its Eigenvalues are \( \pm 1 \) with Eigenspaces \( \Lambda_\pm \) of dimension 3.

Similarly, one can define the bilinear form \( \wedge \) and the Hodge star \( \ast \) for 2-vectors in \( \Lambda^2 TM \).

The above constructions are point-wise, so let us consider a 4 dimensional vector space \( V \) that will be in our case the tangent space \( T_xM \) at a point \( x\in M \).

The null-cone of 2-vectors (resp. 2-vectors) \( u \) such that \( u\wedge u =0 \) is exactly the set of simple 2-vectors (resp. 2-forms). This is because of the standard form of 2-vectors (resp. 2-forms), which in 4D can only be \( e_1\wedge f_1 \) or \( e_1\wedge f_1 + e_2\wedge f_2 \) where \( \{e_i,f_i,i=1,2\} \) form a basis of \( V \). These simple 2-forms represents an oriented 2 dimensional subspace of \( V \).

Now let \( g \) be a metric on \( V \), as before we can talk about (anti)-self-dual 2-vectors (resp. 2-forms) on \( V \).

The set of complex structures \( J \) compatible with \( g \) is in one-to-one correspondence with the set of self-dual 2-vectors (if \( J \) preserves orientation) and anti-self-dual 2-vectors (if \( J \) reverses orientation). In other words, the musical \( J^\sharp \) is of form \[ J^\sharp = e_1\wedge e_2 + e_3\wedge e_4 \] for an orthonormal basis \( \{e_i\} \) of \( g \). The orientation of \( J \) is given by the orientation of the basis \( (e_1,e_2,e_3,e_4) \).

The surjectivity part of the following result may be useful when one needs a good basis of \( V \).

The group \( SO(V) \) acts separately and isometrically on \( \Lambda_+ \) and \( \Lambda_- \) and the covering \[ SO(V) \longrightarrow SO(\Lambda_+) \oplus SO(\Lambda_-) \] is two-to-one.

The separate action is because \( \ast \) (and hence \( \Lambda_\pm \)) only depends on \( g \).
Surjectivity follows from 2-to-1, connectedness, compactness of the \(SO \) groups and dimension: The image of \( SO(V) \) has to be a connected, compact subgroup of \( SO(\Lambda_+)\oplus SO(\Lambda_-) \) having the same dimension.
2-to-1: This is the only computational part of the proof. Let \( \alpha\) be an element of \( SO(V) \) sending an orthonormal base \( e_i \) to another orthonormal base \( f_i \) such that \( \alpha \) acts trivially on \( \Lambda_\pm \), then clearly \( \alpha \) is trivial on \( \Lambda^2 \). One has \( e_i\wedge e_j = \alpha(e_i)\wedge \alpha(e_j) = \alpha^h_i e_h \wedge \alpha^k_j e_k \), hence \[ (\delta_i^h\delta_j^k -\delta_i^k\delta_j^h)e_h\wedge e_k = (\alpha_i^h\alpha_j^k -\alpha_i^k\alpha_j^h)e_h\wedge e_k \quad\forall h,k,i,j \] Hence \( (\alpha_i^h)^2=(\delta_i^h)^2 \), meaning that \( \alpha = \pm \text{Id} \).

An application of Theorem thm:SOV-cover is the immediate (without computation) proof of the following result. Denote by \( S_\pm \) the unit sphere of \( \Lambda_\pm \).

For any \( \omega_+\in S_+ \) and \( \omega_-\in S_- \), the sum \( \omega_+ + \omega_- \) is in the null-cone

By surjectivity in Theorem thm:SOV-cover, there exists \( \varphi\in SO(V) \) that maps \(\omega_\pm\) to \( e_1\wedge e_2 \pm e_3\wedge e_4 \), hence it maps \( \omega_+ +\omega_- \) to \( 2e_1\wedge e_2 \) which is in the null-cone. It remains to evoke that the null-cone is preserved by \( SO(V) \).

2.2 Twistor spaces and their natural complex structures

Given a Riemannian four-manifold \( (M,g) \), then \( \Lambda^2 TM \) splits as \( \Lambda^2 TM = \Lambda_+ \oplus \Lambda_- \) and one obtains 2 \( \mathbb{S}^2 \)-bundle over \( M \) \( S_\pm = S(\Lambda_\pm)\) whose fibres are unit spheres in \( \Lambda_{\pm} \). Again, then metric on \( \Lambda_\pm \) is renormalised so that \( e_1\wedge e_2 \pm e_3\wedge e_4 \) are of norm 1. In particular, if \( (e_1,e_2, e_3,e_4) \) is an oriented basis then an orthonormal basis of \( S_- \) is \( \{e_1\wedge e_2 -e_3\wedge e_4, e_1\wedge e_3 +e_2\wedge e_4, e_1\wedge e_4 -e_2\wedge e_3\} \). The six-dimensional total spaces of these bundles are called Twistor spaces over \( M \).

The distinction between \( S_+ \) and \( S_- \) only depends on the orientation of \( M \), i.e. if one reverses the orientation, one becomes the other.

We will show that there are two natural (almost) complex structure on twistor spaces \( S_\pm \). By Remark rem:orient-twistor, let us define these complex structure on \( S_- \) since the same construction applies for \( S_+ \).

Take \( \omega\in S_- \) an anti-self-dual 2-vector over a point \( p\in M \), then by Remark rem:complex-str-2-vect the musical \( \omega_\flat \) gives rise to an orientation-reversing complex structure \( J \) on \( T_pM \). Under \( J \), \( \Lambda^2TM \) splits as \[ \mathbb{C}\otimes_{\mathbb{R}}\Lambda^2 T_p M = \Lambda^{2,0}_p \oplus \Lambda^{1,1}_p \oplus\Lambda^{0,2}_p \] where the factors are of complex dimension 1, 4 and 1 respectively and \( \omega\in \Lambda^{1,1}_p \). In particular, if \( \omega \) is given by \(\omega = e_1\wedge e_2 -e_3\wedge e_4 \), which we denote briefly as \( 12-34 \) then

\( \Lambda^{1,1}_p \) is \( \mathbb{C} \)-generated by \( 12,34,14+23. 13-24 \), i.e. \( \{\omega, 12+34, 14+23, 13-24\} \) is an orthonormal basis of \( \Lambda^{1,1}_p \).
\( \Lambda^{2,0} \) is \( \mathbb{C} \)-generated by \( (14-23)-i(13+24) \) and \( \Lambda^{0,2} \) is \( \mathbb{C} \)-generated by \( (14-23)+i(13+24) \).

Writing \[ \mathbb{C}\otimes \Lambda^2T_pM = (\mathbb{C}\otimes\Lambda_{+.p}) \oplus (\mathbb{C}\otimes\Lambda_{-,p}) = \Lambda^{2,0}_p \oplus \Lambda^{1,1}_p \oplus\Lambda^{0,2}_p, \] one see

from the first point shows that \( \Lambda_p^{1,1} = \mathbb{C}\omega\oplus^\perp (\mathbb{C}\otimes \Lambda_{+,p}) \)
from the second point shows that \( \mathbb{C}\otimes \Lambda_{-,p} = \left( \Lambda^{2,0}_p \oplus \Lambda^{0,2}_p \right) \oplus^\perp \mathbb{C}\omega \)

Therefore the complexified vertical tangent of \( S_- \) at \( \omega \) is \( \Lambda^{2,0}_p\oplus \Lambda^{0,2}_p \) which is the complexification of \( \mathbb{R}(14-23)\oplus \mathbb{R}(13 + 24) \).

The following result allows us to define Koszul-Malgrange almost complex structure on \( S_\pm \).

Let \( (M^4,g) \) be a Riemannian four-manifold and \( S_\pm \) be its twistor spaces. Then the Levi–Civita connection \( \nabla \) reduces to \( S_\pm \), meaning that the horizontal 4-planes at \( \omega\in S_+\subset \Lambda^2T^*M \) (resp. \( \omega\in S_-\subset\Lambda^2 T^*M \)) are in \( T_\omega S_+\subset T_\omega \Lambda^2 T^*M\) (resp. \( T_\omega S_-\subset T_\omega \Lambda^2 T^*M\)).

The two natural complex structure \( J_1, J_2 \) of \( S_- \) at \( \omega \) is given by setting

\begin{equation*} T^{1,0}(\omega) = \begin{cases} (T^{1,0})^h \oplus \Lambda^{2,0} \text{ for } J_1\\ (T^{1,0})^h \oplus \Lambda^{0,2} \text{ for } J_2 \end{cases} \end{equation*}

where \( T^{1,0} \) is the holomorphic tangent at the point \( p \), the base point of \( \omega\in S_- \) on \( M \), under the complex structure given by \( \omega \), parallelly lifted to the horizontal in plane at \( \omega \) using the Levi–Civita connection of \( (M,g) \).

Another way to define these 2 complex structures is to say that

One has a natural complex structure on the fibers of \( S_- \), under which, if one supposes \( \omega = 12 - 34 \), \( J_1^v(14-23) = 13+24 \) and \( J_2^v = -J_1^v \).
The complex structures on \( T_\omega S_- = H\oplus V \) is given by the sum of \( J_1^v \) (resp. \( J_2^v \)) on the vertical component \( V \) and the complex structure (given by \( \omega \) through horizontal lift) on the horizontal component \( H \).

We will know prove that the almost complex structure \( J_2 \) is never integrable. The idea is to say that if this was true than there are very few holomorphic sections, which contradicts the fact that under Koszul–Malgrange Koszul.Malgrange58_CertainesStructuresFibrees complex structure (component-wise complex structure) a section is holomorphic if and only if its image is a complex sub manifold and there are plenty complex sub manifold in an integrable complex manifold.

We will start by explaining two ways to complexify a connection.

2.3 Two different complexifications of a linear map.

The majority of computational details in the proof of holomorphic curve-minimal surface correspondence will be very intuitive if one is able to go back and forth between the following two ways to complexify a linear map \( f: (V,J_V) \longrightarrow (W,J_W) \) between complex vector spaces.

	First way (A)	Second way (B)
Definition	\begin{align} F_1: V⊗ \mathbb{C} &\longrightarrow W\\iX &\longmapsto J_W f(X) \end{align}	\begin{align} F_2: V⊗ \mathbb{C} &\longrightarrow W⊗ \mathbb{C}\\ iX &\longmapsto i f(X)\end{align}
\( f \) is holomorphic iff	\( F_1 (X + i J_V X) = 0 \), i.e. \(F_1( \frac{\partial }{\partial \bar z})=0\)	\(F_2(V^{1,0}) \subset W^{1,0}\)
Advantage	Do not need to complexify \( W \)	\( F_2 \) preserves type

While the advantage of (B) is clear, the fact that in (A), we do not have to complexify \( W \) is quite convenient in certain cases, for example when \( f \) is a connection on a complex vector bundle \( F \) (with complex structure on each fibre) over an almost complex manifold \( M \). Imagine we have to complexify the fibre and then the defining horizontal planes of certain connection \( \nabla \) in order to talk about \( \nabla_{\frac{\partial}{\partial z}} \). A section \( s \) is then called holomorphic if and only if \( \nabla_{\frac{\partial }{\partial \bar z}}s=0 \). This means that \( s \) is holomorphic as a map from \( M \) to the total space \( F \) equipped with the Koszul-Malgrange complex structure.

Now let us first use the complexification (B) for the Levi–Civita connection \( \nabla \) on \( S^- \). First note that if \( s:U\subset M \longrightarrow S_- \) is a section then \( U \) can be equipped with a natural complex structure \( J \) which is \( s(p) \) at every point \( p\in U\subset M \). The following result is straight-forward.

Given a section \( s: (U,J) \longrightarrow (S_-, J_\alpha), \alpha=1,2 \), then

One only has to check the vertical component to prove \( s \) is holomorphic.
To prove that \( s \) is holomorphic, take any \( \xi\in T^{1,0}_pM \) and check if \(v=\nabla_\xi s \) is in \(\Lambda^{2,0}\) (for \( J_1 \)) or in \( \Lambda^{0,2} \) (for \( J_2 \)).
\( s \) is holomorphic if and only if \( s_* T_pM \) is a complex subspace of \( T_{s(p)}S_- \).

For the 'if' part of the last statement, note that \( s_*(J\xi) \) and \( Js_*\xi \) live in the same four-plane \( s_*T_pM \) with the same horizontal projection, therefore they coincide.

Using complexification (B), we can see \( \nabla \) as a section of \( (\mathbb{C}\otimes T^*M)\otimes (\Lambda^{2,0}\oplus \Lambda^{1,1}\oplus \Lambda^{0,2}) \). Denote by \( D_1,D_2 \) the projection of \( \nabla \) on \[ \mathfrak{S}_1:=((T^{0,1})^*\otimes \Lambda^{2,0})\oplus ((T^{1,0})^*\otimes \Lambda^{0,2}), \] and \[ \mathfrak{S}_2:=((T^{0,1})^*\otimes \Lambda^{0,2})\oplus ((T^{1,0})^*\otimes \Lambda^{2,0}). \] then one has

For any section \( s \) over \( U \) and J the complex structure coming from \( s \), there is no \( \Lambda^{1,1} \) component in \( \nabla_X s \) for any vector \( X\in T_pM \). In other words, \[ \nabla s= D_1 s + D_2 s. \]

Let \( (e_1,e_2,e_3,e_4) \) be an orthonormal frame such that \( Je_1 = e_2, Je_3 = e_4 \) and \( s = e_1\wedge e_2 + e_3\wedge e_4 \) and denote \( \alpha_1 = \frac{1}{2}(e_1-ie_2) \) and \( \alpha_2 = \frac{1}{2}(e_3 - ie_4) \). One has \( \alpha_i\in T^{1,0} \) and \[ 2is = \alpha_1\wedge \bar \alpha_1 +\alpha_2\wedge \bar \alpha_2 \] hence the \( \Lambda^{1,1} \) component of \( 2i\nabla_X s = \nabla_x \alpha_i\wedge \bar \alpha_i + \alpha_i\wedge \nabla_X\bar\alpha_i \) is \( ( 2i\nabla_X s)^{1,1}=\lambda_i^j \alpha_j\wedge\bar\alpha_i + \tilde \lambda_i^j\alpha_i\wedge\bar\alpha_j\) where

\begin{align*} \nabla_X\alpha_i &= \lambda_i^j \alpha_j + \dots \text{ terms in } \bar\alpha_j\\ \nabla_X \bar\alpha_i &= \tilde\lambda_i^j \bar\alpha_j + \dots \text{ terms in } \alpha_j \end{align*}

since \( \lambda_i^j = \langle \nabla_X \alpha_i,\bar \alpha_j \rangle = - \langle \alpha_i,\nabla_X \bar\alpha_j \rangle = -\tilde\lambda_j^i \), one has \[ ( 2i\nabla_X s)^{1,1}=\lambda_i^j \alpha_j\wedge\bar\alpha_i - \lambda_j^i\alpha_i\wedge\bar\alpha_j =0. \]

2.4 Non-integrability of \( (S_-, J_2) \).

It follows from Proposition prop:holo-sec-S and Proposition prop:no-1-1 that

A section \( s: (U,J) \longrightarrow (S_-, J_\alpha), \alpha=1,2 \) is holomorphic if and only if \( D_\alpha s = 0 \).

The equation \( D_1 s = 0 \) is equivalent to the Nijenhuis tensor of \( J \) being 0, therefore one has:

Let \( s \) be a section of \( S_- \) on \( U \) and \( J \) be the corresponding complex structure on \( U \). Then \( D_1 s = 0 \) if and only if \( J \) is integrable.

Now let us derive non-integrability of \( (S_-,J_2) \) using the previous results. If \( (S_-,J_2) \) was integrable, then take any 2 (complex) dimensional sub manifold of \( S_- \) that is graph of a non-parallel section \( s \). Then by Proposition prop:holo-sec-S, \( s \) would be holomorphic meaning that \( D_2s = 0 \) by Proposition prop:holo-and-D. Also, integrability of \( J_2 \) and holomorphicity of \( s \) would imply integrability of the complex structure \( J \) coming from \( s \) on \( U \), which means, by Theorem thm:D1-integ, \( D_1 s = 0\). By Proposition prop:no-1-1, \( \nabla s =D_1 s + D_2 s= 0\) and \( s \) would be therefore parallel, which is a contradiction.

\( (S_-, J_-) \) is never integrable.

2.5 Gauss lift.

We finalise this section by defining the generalised Gauss map (i.e. Gauss lift) and giving the exact statement of Twistorial correspondence.

Let \( f: (\Sigma,i) \longrightarrow (M^4,g) \) be an immersion of a Riemann surface \( \Sigma \) to a Riemannian four-manifold \( M \). Define \( \tilde f_- \) to be the anti-self-dual projection of the 2-vector associated to \( f_*T_p\Sigma \) (Remark rem:null-cone-simple), i.e.

\begin{align*} \tilde f_-:\Sigma &\longrightarrow S_- \\ p &\longmapsto (1-\ast)(f_*T_pM) \end{align*}

Another way to define \( \tilde f_- \) is to say that it is the unique anti-self-dual 2-vector such that \( f \) is holomorphic under the corresponding complex structure (Remark rem:complex-str-2-vect).

It is clear that the Gauss lift can be continuously defined in case where \( f \) is a branched immersion Gulliver.etal73_TheoryBranchedImmersions. We can now state the correspondence in Eells.Salamon85_TwistorialConstructionHarmonic.

There is a one-to-one correspondence between non-vertical \( J_2 \)-holomorphic curves in \( S_- \) and conformal harmonic map from \( (\Sigma,i) \) to \( (M,g) \) given by the Gauss lift.

3 Twistor correspondence.

The correspondence is between conformal harmonic branched immersions in \( M \) and \( J_2 \)-holomorphic curves in \( S_- \) via the Gauss lift. Therefore a branch point of \( f:\Sigma \longrightarrow M \) comes from

either a critical point of the holomorphic curve \( \tilde f_-: \Sigma \longrightarrow S_- \)
or a point \( p\in\Sigma \) where \( \tilde f_- \) is tangent to the twistor lines \( (S_-)_{f(p)} \).

We will show that in both cases, these points are isolated, hence of finite number by compactness of \( \Sigma \).

3.1 Review: A few local properties of (pseudo)holomorphic curves.

Let \( u:\Sigma \longrightarrow S \) be a holomorphic curve between a compact connected Riemann surface \( \Sigma \) and an almost complex manifold \( S \). Then \( u \) is uniquely determined by its \( \infty \)-jet at one point \( p\in\Sigma \), i.e. if there is another holomorphic curve \( v:\Sigma \longrightarrow M \) such that the \( \infty \)-jets of \( v \) and \( u \) coincide then the two maps \( u,v \) coincide.

For a proof of this Theorem thm:infty-jet, see McDuff.Salamon00_JholomorphicCurvesQuantum . The idea is to write down the generalised Cauchy-Riemann equation for \( u \) and \( v \) then apply a PDE estimate. We are interested in how this theorem can be applied to prove that critical point of \( u \) is isolated.

Given a non-constant holomorphic curve \( u; (\Sigma,i) \longrightarrow S \) from a Riemann surface \( \Sigma \) to an almost complex manifold \( S \). Then critical points \( p\in\Sigma \) where \( du(p)=0 \) are isolated in \( \Sigma. \)

This result is immediate when \( S \) has a holomorphic function \( \xi_1 \) locally defined at \( p \), for example when the complex structure on \( S \) is integrable. In that case, it suffices to notice that \( p \) has to be a critical point of \( \xi_1\circ u \) which, in a local chart of \( \Sigma \), is a zero of the holomorphic function \( \frac{(\partial \xi_1\circ u)}{\partial z} \).

We can suppose that \( u(0)=0\in \mathbb{C}^n, du(0)=0 \) and the complex structure on \( M \) is \(J(\xi)\in GL(\mathbb{R}^{2n})\) with \( J^2 = -Id \) and \( J(0)=J_0 \) the standard complex structure of \( \mathbb{C}^n \). Let \( s,t \) be the isothermal coordinates of \( \Sigma \), the generalised Cauchy–Riemann equation reads

\begin{equation} \label{eq:gen-cauchy-rie} \frac{\partial u}{\partial s} + J(u)\frac{\partial u}{\partial t}=0 \end{equation}

Since \( u \) is non-constant, using Theorem thm:infty-jet, one can find an \( l\geq 1 \) such that \[ \left| u(z) \right| = O(|z|^l) \ne O(|z|^{l+1}) \] therefore \( J(u(z)) = J_0 + O(|z|^l) \). Take the \( |z|^{l-1} \) term in \eqref{eq:gen-cauchy-rie}, one has \[ \frac{\partial T_l(u)}{\partial s} + J_0 \frac{\partial T_l(u)}{\partial t}=0 \] where \( T_l(u) \) is the expansion of \( u \) at \( 0 \) upto order \( l \). This means \( T_l \) is holomorphic w.r.t \( J_0 \), i.e. \( T_l(u) = (a_1 z^l, \dots, a_n z^l) \) for a non-zero vector \( a\in \mathbb{C}^n \). Now that \[ u(z) = az^l + O(|z|^{l+1}),\quad du(z) = alz^{l-1} + O(|z|^l), \] one sees that for \( |z| <\epsilon \) sufficiently small \( du(z)\ne 0 \).

In case where \( S \longrightarrow M \) is a twistor space with the twistor lines \( S_q \) over \( q\in M \) being a complex submanifold, using the same technique, one can prove that

Let \( u:\Sigma \longrightarrow S\) be a non-vertical holomorphic curve from a compact Riemann surface \( \Sigma \) to a twistor space \( S \), then the points \( p\in \Sigma \) where the tangent \( u_* T_p\Sigma \) is vertical at \( u(p)\in S \) are isolated and therefore of finite number.

The statement being local, we can suppose \( p=0\in \mathcal{U} \subset \Sigma= \mathbb{C} \). Let \( v: \mathcal{V}=\mathcal{O}p(0) \longrightarrow S=\mathbb{C}^n \) be the vertical twistor line \( z \longmapsto z\times \{0\} \) touching \( u \) at \( u(0)\equiv v(0)=0 \). This means we also suppose that the restriction of complex structure on \( S=\mathbb{C}^n\) onto \( \mathbb{C}\times\{0\} \) is the standard one (we can always do this!). Choose isothermal coordinates \( \frac{\partial}{\partial s} \) and \( \frac{\partial}{\partial t} \) of \( \mathcal{U}, \mathcal{V} \) such that \( u_* \frac{\partial}{\partial s} = v_* \frac{\partial}{\partial_s} \) and \( u_* \frac{\partial}{\partial t} = v_* \frac{\partial}{\partial_t} \) at \( 0 \). In other words, we suppose that \( u(0) = v(0) = 0\in \mathbb{C}^n \) and \( du(0) = dv(0) \).

Subtracting the generalised Cauchy–Riemann equations for \( u \) and \( v \), one has

\begin{equation} \label{eq:u-v-vertical} \partial_s (u-v) + (J(u)-J(v)) \partial_t u + J(v) \partial_t (u-v) = 0 \end{equation}

Since \( u \) is not vertical, by Theorem thm:infty-jet, there exists \( l\geq 2 \) such that \( u-v = O(|z|^l)\ne O(|z|^{l+1}) \). Therefore \( J(u)-J(v) = O(|z|^l) \). Take the \( O(|z|^{l-1}) \) part in \eqref{eq:u-v-vertical}, one sees that \( T_l(u-v) \) is holomorphic in the usual sense, hence there exists \( 0\ne a\in \mathbb{C}^n \) such that \[ (u-v)(z) = (a_1 z^l,\dots, a_n z^l) \]

The proof is finished if there exists an \( a_i\ne 0, i\in \overline{2,n} \). If not, we can replace \( v \) by \( \tilde v: z\longmapsto (z+a_1 z^l)\times \{0\} \) which is still a holomorphic parametrisation of the twistor line \( \mathbb{C}\times\{0\} \), but \( |u(z)- v(z)| = O(|z|^{l+1}) \). By repeating this argument finitely many times, one reaches a moment when there exists a non-zero \( a_i \) for \( i\geq 2 \). The conclusion then follows.

3.2 Review: Harmonic maps from a Riemann surface.

First of all this term makes sense because energy of a map \( f:\Sigma \longrightarrow M \) is a conformal invariant when \( \dim \Sigma = 2 \). We can therefore try to write the equation of harmonicity \( \tau(f)=0 \) using the complex structure of \( \Sigma \).

Using a complex coordinate \( z \), one has \( df = \nabla_{\partial_z}f dz + \nabla_{\partial_{\bar z}}f\ d\bar z \) where \( \nabla_{\partial_z}f:=f_* \partial_z \), \( \nabla_{\partial_{\bar z}}f:= f_* \partial_{\bar z} \). The vector bundle \( f^*M \) over \( \Sigma \) is equiped with a fiberwise metric and a metric connection inherited from \( M \). Combining this with the Levi–Civita connection on \( (\Sigma,h) \) where \( [h]=i \) the complex structure on \( \Sigma \), one obtains a connection on all \( T^*M^{\otimes k}\otimes TM^{\otimes l}\otimes f^* TM\) that is compatible with tensor product and contraction. We now use complexification (B) when we talk about covariant derivative \( \nabla_{\partial_z}s \) of a section \( s \).

\(\nabla_{\partial_z}\) and \( \nabla_{\partial_{\bar z}} \) commute on \( f \). In fact \[ \nabla_{\partial_z}\nabla_{\partial_{\bar z}}f - \nabla_{\partial_{\bar z}}\nabla_{\partial_{z}}f = \nabla^M_{f_* \partial_z}f_* \partial_{\bar z} - \nabla^M_{f_* \partial_{\bar z}}f_* \partial_{z} = [f_* \partial_z, f_* \partial_{\bar z}] = f_*[\partial_z,\partial_{\bar z}]=0 \] This means that \( \nabla_{\partial_z}\nabla_{\partial_{\bar z}}f \) is always a real vector field along \(f \).

We will write down the equation of harmonicity \( \tr_h \nabla df=0 \) explicitely using \( \partial_z, \partial_{\bar z} \) and the fact that \( \tr_h (dz\otimes dz) = \tr_h(d\bar z\otimes d\bar z)=0 \):

\begin{align*} \nabla df &= \nabla_{\partial_z}(\nabla_{\partial_z}f dz + \nabla_{\partial_{\bar z}}f\ d\bar z)\otimes dz + \nabla_{\partial_{\bar z}}(\nabla_{\partial_z}f dz + \nabla_{\partial_{\bar z}}f\ d\bar z)\otimes d\bar z \\ &= (\nabla_{\partial_z}\nabla_{\partial_{\bar z}}f) (dz\otimes d\bar z + d\bar z\otimes dz) + \text{ terms in } dz\otimes dz \text{ and } d\bar z\otimes d\bar z \end{align*}

in which we claim that the sum \[ \mathcal{A} = \left(\nabla_{\partial_z}f \nabla_{\partial_z}dz + \nabla_{\partial_{\bar z}}f\nabla_{\partial_z}d\bar z\right)\otimes dz + \left(\nabla_{\partial_{z}}f\nabla_{\partial_{\bar z}}dz + \nabla_{\partial_{\bar z}}f \nabla_{\partial_{\bar z}}d\bar z\right)\otimes dz \] is in fact a "term in \(dz\otimes dz\) and \(d\bar z\otimes d\bar z\)". This can be seen by rewriting the first sum in \( \mathcal{A} \) as \[ -\Gamma_{z\bar z}^{ z} f_* \partial_z - \Gamma_{z\bar z}^{\bar z}f_* \partial_{\bar z} + \text{ terms in } dz\otimes dz \text{ and } d\bar z\otimes d\bar z \] where \( \Gamma \) is the (B-complexified) Christoffel symbols of \( (\Sigma,h) \), and noticing that \( \Gamma_{z\bar z}^{z} f_* \partial_z + \Gamma_{z\bar z}^{\bar z}f_* \partial_{\bar z} = f_* \nabla^\Sigma_{\partial_z} \partial_{\bar z}\). But \[ \nabla^\Sigma_{\partial_z}\partial_{\bar z} = \nabla_{\partial_x} \partial_x + \nabla_{J \partial_x}J \partial_x = 0 \] since Riemann surfaces are Kahler manifolds (\( SO(2) = U(1) \)).

An application of the last formula is that the tension field is \[ \tau(f) = -\tr_h \nabla df = - 4 \nabla_{\partial_z}\nabla_{\partial_{\bar z}}f, \] hence \[ \frac{\partial}{\partial \bar z} \left\langle \frac{\partial f}{\partial z}, \frac{\partial f}{\partial z} \right\rangle_g = -\frac{1}{2} \left \langle \tau(f), f_* \frac{\partial}{\partial z} \right\rangle_g \] Therefore the Hopf's quadratic differential \( q(f,[h],h):= \left \langle \frac{\partial f}{\partial z}, \frac{\partial f}{\partial z} \right \rangle_g dz\otimes dz \) is holomorphic if and only if the tension field \( \tau(f) \) is orthogonal to the tangent \( f_* T\Sigma \) of the surface. In particular if \( f \) is harmonic then its quadratic differential is holomorphic.

A consequence is that all harmonic map from the 2-sphere \( \mathbb{S}^2 \) is quadratic. This is because \( \mathbb{S}^2 \) carries no non-zero holomorphic quadratic differential.

3.3 Proof of Theorem thm:twistor-cor.

We have now acquired enough technology to prove Theorem thm:twistor-cor of Eells-Salamon. Keeping the same notation as before, with \( f:(\Sigma,i) \longrightarrow (M,g) \) the immersion of the surface \( \Sigma \) and \( \tilde f_- \) its Gauss lift, we can suppose that \( f \) has no branch point and its Gauss lift does not touch the vertical twistor lines. In other words, we only need to prove a "smooth version" of Theorem thm:twistor-cor. This is because the Gauss lift can be continuously defined at branch points, which are of finite number, so there is no problem of associating a holomorphic curve on \( S_- \) to a harmonic conformal \( f:\Sigma \longrightarrow M \). On the other hand, given a holomorphic curve \( u: (\Sigma,i) \longrightarrow (S_-, J_2) \) and denote \( f:\Sigma \longrightarrow M \) its projection to \( M \), by Proposition prop:crit-isol and Proposition prop:vert-tang, the singular points of \( f \) (where \( f \) fails to be an immersion) are of finite number. Holomorphicity of \( u \) implies that, at regular points of \( f \), \( f \) is holomorphic with respect to the complex structure given by \( u \), but this is the defining property of Gauss lift \( \tilde f_- \), we have \( u\equiv \tilde f_- \) at non-singular points of \( f \). The continuity argument finishes the proof.

It remains now to prove the smooth version of Theorem thm:twistor-cor.

Given an immersion \( f:(\Sigma^2,J) \longrightarrow (M^4,g) \) of a Riemann surface to a Riemannian four-manifold. Then \( f \) is conformal and harmonic if and only if \( \tilde f_-:(\Sigma,J) \longrightarrow (S_-,J_2) \) is holomorphic.

We will denote by \( \sigma \) the 2-form (section of \( f^* S_- \)) corresponding to \( \tilde f_- \), i.e. \( (\tilde f_-)_*X = \nabla_X\sigma \oplus f_*X \) where the former is vertical and the latter is the horizontal lift of \( f_* X \). Then \( \tilde f_- \) is holomorphic if and only if, using complexification (A), \( (\tilde f_-)_*(X+iJX) = 0 \), which means, for horizontal and vertical components:

\begin{align} f_*(X+iJX) &= 0, \text{ which means } f_* X + (\tilde f_-) f_*JX=0 \text{ where \( (\tilde f_-) \) is the corresponding complex structure} \label{eq:smooth-twistor-1} \\ \nabla_{X+iJX}\sigma &= 0, \text{ which means } \nabla_X\sigma + J_2\nabla_{JX}\sigma = 0 \label{eq:smooth-twistor-2} \end{align}

This is because the induced complex structure of \( f^*S_-\subset S_- \) coincides with the Koszul–Malgrange complex structure of \( f^*S_- \) as a vector bundle on \( \Sigma \) (the last statement is because \( f: (\Sigma,i) \longrightarrow (f(\Sigma), \tilde f_-) \) is holomorphic).

One sees that \eqref{eq:smooth-twistor-1} is equivalent to \( f \) being conformal because \( f_*X \) and \( f_* JX \) have the same length and are orthogonal in \( M \).

We will prove that under \eqref{eq:smooth-twistor-1}, \eqref{eq:smooth-twistor-2} is equivalent to \( f \) being harmonic. To see that, we will translate \eqref{eq:smooth-twistor-2} to an equivalent condition using complexification (B): \[ \nabla_X\sigma + J_2\nabla_{JX}\sigma = 0 \forall X\in T\Sigma \iff \nabla_{\frac{\partial}{\partial z}}\sigma=\nabla_{X-iJX}\sigma:=\nabla_X\sigma -i\nabla_{JX}\sigma=\nabla_X\sigma -iJ_2\nabla_X\sigma\in T^{1,0}S_- \] with respect to the complex structure \( J_2 \). But \( \nabla_{\frac{\partial}{\partial z}}\sigma \) is vertical and \( T^{1,0}S_- = (T^{1,0})^h\oplus (\Gamma^{0,2})^v \), one has \eqref{eq:smooth-twistor-2} is equivalent to \( \nabla_{\frac{\partial}{\partial z}}\sigma\in \Lambda^{0,2} \). Since we knew from Proposition prop:no-1-1 that \( \nabla_{\frac{\partial}{\partial z}} \sigma \) only has a \( \Lambda^{2,0} \) component and a \( \Lambda^{0,2} \) component (under \( \tilde f_- \)), \eqref{eq:smooth-twistor-2} means the former vanishes.

We will denote by \( \delta \) and \( \bar\delta \) the covariant derivatives \( \nabla_{\frac{\partial}{\partial z}} \) and \( \nabla_{\frac{\partial}{\partial \bar z}} \) (complexification (B)) respectively. Since \( f \) is conformal its Gauss lift \( \sigma \) is a multiple of \( i(1-\ast) (\delta f\wedge \bar\delta f) \), let suppose that \( \sigma = -ic(1-\ast)(\delta f\wedge\bar\delta f) \) where the scalar \( c \) may varies from point to point. One has \[ \delta \sigma = (c^{-1}\delta c)\sigma -ic(1-\ast)(\delta^2 f\wedge \bar\delta f+\delta f\wedge \delta\bar\delta f). \]

At the point of \( M \) in question and under the complex structure \( \tilde f_- \), one has

\( \mathbb{C}\otimes\Lambda^2_- = \Lambda^{0,2}\oplus \Lambda^{2,0}\oplus \mathbb{C} \sigma \),
\( \bar\delta f \) is of type (0,1) (\( f \) is holomorphic under \( \tilde f_- \))
\( \delta f \) is of type (1,0).

Therefore the \( \Lambda^{2,0} \) component of \( \delta\sigma \) is \[ -ic \delta f\wedge (\delta\bar\delta f)^{1,0} =: A \] The only way for \( A \) to be 0 without \( (\delta\bar\delta f)^{1,0} \) vanishing is that \( (\delta\bar\delta f)^{1,0} \) is a non-zero multiple of \( \delta f \), which is impossible because one always has \[ g(\bar\delta f, (\delta\bar\delta f)^{1,0}) = g(\bar\delta f, \delta\bar\delta f) = -g(\delta \bar\delta f,\bar\delta f)=0 \]

So \( A=0 \iff (\delta\bar\delta f)^{1,0}=0 \iff \delta\bar\delta f=0 \) because \( \delta\bar\delta f \) is real according to Remark rem:ddbarf.

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