The Adjunction Inequality for Weyl-Harmonic Maps

In this paper we study an analog of minimal surfaces called Weyl-minimal surfaces in conformal manifolds with a Weyl connection $(M^4,c,D)$. We show that there is an Eells-Salamon type correspondence between nonvertical $\mathcal{J}$-holomorphic curves in the weightless twistor space and branched Weyl-minimal surfaces. When $(M,c,J)$ is conformally almost-Hermitian, there is a canonical Weyl connection. We show that for the canonical Weyl connection, branched Weyl-minimal surfaces satisfy the adjunction inequality \begin{equation}\label{adj} \chi(T_f\Sigma)+\chi(N_f\Sigma) \le \pm c_1(f^*T^{(1,0)}M). \end{equation} The $\pm J$-holomorphic curves are automatically Weyl-minimal and satisfy the corresponding equality.


Introduction
This paper describes an extension of the notion of minimal surfaces to the setting of conformal manifolds with a Weyl connection. Particular attention is given to almost-Hermitian 4-manifolds endowed with their canonical Weyl connection. We first review the relevant standard theory.
According to Eells and Salamon [5], branched minimal surfaces in an oriented Riemannian 4-manifold M have a one-to-one correspondence with nonvertical J -holomorphic curves in the twistor space Z, where J is the canonical non-integrable almost-complex structure on Z. Applying twistor techniques, they further show that if M is almost-Kähler with almostcomplex structure J, the ±J-holomorphic curves are minimal. When M is Kähler they prove the adjunction inequality, where T f Σ is the tangent bundle to Σ ramified at the branch points of f , and N f Σ is its normal bundle in f * T M. Concurrently, Webster [15] obtained his formulas (1), (2) for a minimal surface in a Kähler 4-manifold, which imply the adjunction inequality. The adjunction inequality was extended to minimal surfaces in almost-Kähler 4-manifolds by Chen-Tian [3], Ville [14], and Ma [10]. This leads to the following picture for almost-Kähler manifolds: The adjunction inequality holds for minimal surfaces; every ±J-holomorphic curve is minimal, and equality holds in (3) with the corresponding sign.
For an almost-Hermitian manifold, in general, the ±J-holomorphic curves are not minimal, and in [3] they remark that the (3) will not hold for minimal surfaces. In this paper we show that the above scenario for almost-Kähler manifolds can be extended to almost-Hermitian manifolds when considering a conformally invariant condition on surfaces related to the minimal condition. We now briefly describe this condition and list our main theorems.
Let M be a manifold with conformal metric c and Weyl connection ∇ D , to be described in detail later. For i : Σ → M an immersed submanifold, the Weyl second fundamental form B is defined in [13] as follows. Taking g ∈ c, there is a one-form α g such that ∇ D g = −2α g ⊗ g.
Let A g be the usual second fundamental form, then We say the submanifold is Weyl-minimal if tr i * g B = 0. Branched Weyl-minimal then has the obvious meaning.
We extend the Eells-Salamon twistor correspondence as follows. There are complex structures J ± on the weightless twistor spaces Z ± which give a 1-to-1 correspondence between non-vertical J ± -holomorphic curvesf ± : Σ → Z ± and non-constant weakly conformal branched Weyl-minimal immersions f : Σ → M.
If the Weyl derivative is exact, then the Weyl-minimal surfaces are minimal for a preferred metric in c, and this is just the usual correspondence for that metric. This and the previous theorem imply the following corollary. Finally we prove that Webster's formulas hold for branched Weyl-minimal immersions.
The adjunction inequality follows from P and Q being positive.
The corresponding equality holds for ±J-holomorphic curves.

Weyl Geometry
By definition, the density bundle on an n-dimensional manifold M is L := |Λ n T M| 1 n . The tensor bundles L w ⊗ T M j ⊗ T * M k are said to have weight w + j − k. A Weyl derivative D is a connection on the density bundle. A conformal metric c is a metric on the weightless tangent bundle L −1 T M satisfying the normalizing condition |det c| = 1. This can also be considered as a metric on T M with values in L 2 . The bundle L is trivial, and a nowhere zero section of L, µ, is called a length scale. This defines a metric in the conformal class c by g µ = µ −2 c. The section µ gives a trivialization of L which has a corresponding trivializing connection D µ . This defines a one form There is a unique torsion free connection ∇ D on T M making c parallel, where ∇ gµ is the Levi-Civita connection for the metric g µ .

Weightless Twistor Space
When M is oriented, c defines a section ν c of the orientation bundle L n Λ n T * M. This can be used to define the conformal Hodge star For n = 4 and m = 0, ⋆ : Λ 2 T * M → Λ 2 T * M is an involution with ±1 eigenspaces Λ 2 ± T * M. The weightless twistor spaces [2] can be constructed as the sphere bundles We now review the construction of an almost-complex structure J ± on Z ± . This can be seen by working at a point q ± ∈ Z ± which projects to p ∈ M. For U a neighborhood of p, and a local section s ± : U → Z ±|U satisfying s ± (p) = q ± , there is a weightless Kähler form σ ± given by this section and a corresponding almost-complex structure J ± on T p M given by As the fiber of Z ± at p is a sphere in L 2 Λ 2 ± T * p M, the vertical tangent space at q ± is the space perpendicular to σ ± in L 2 Λ 2 ± T * p M. This is the space of weightless J ± -anti-invariant 2-forms [4].
There is an induced almost-complex structure acting on To see that this is an almost-complex strucure, first note that J ± β is a two form as Second, Finally, it is easily seen that J 2 ± β = −β. Extending β to be complex bilinear gives . This isomorphism and the connection induce an isomorphism, X → X h , from T p M to the horizontal tangent space The almost-complex structure on Z ± is now given by linearly extending In [5] Eells and Salamon used a similar complex structure to study weakly conformal harmonic maps. Their complex structure is the same as the complex structure of Penrose, studied by Atiyah, Hitchen and Singer in [1], except that it reverses the orientation of the fibers. The complex structure defined in (5) and (6) differs from that of Eells and Salamon only in the use of a Weyl connection to define the horizontal space rather than the Levi-Civita connection.

Submanifold Geometry
Let (M, c, D) be a Weyl manifold, and i : Σ → M an immersed submanifold. Then Σ inherits a conformal structurec and Weyl derivativeD. One way to see this is to choose a length scale, µ ∈ Γ(L). Then the metric g µ and the one form α µ can be pulled back to Σ as is a section of the density bundle of Σ. The inherited conformal metric isc =μ 2ḡ µ and the inherited Weyl derivative is The connection∇D on Σ is defined so thatc is parallel, The Weyl second fundamental form [13] is given by The Weyl mean curvature is where H gµ is the usual mean curvature of Σ with respect to the metric g µ . The harmonic map equation can also be generalized to this setting. In [8] the second fundamental form of a map f : Σ → M is defined for manifolds Σ and M with torsion-free connections ∇ Σ and ∇ M . If ∇ is the induced connection on T * Σ ⊗ f * T M, then the second fundamental form is just ∇df . If η is a metric on Σ then the tension of the map can be defined as A map is psuedo-harmonic if the tension field is zero. We study the case where the domain (Σ, η) is a Riemannian manifold with its Levi-Civita connection ∇ η and the target manifold (M, c, D) is a Weyl manifold. This is opposite of the case studied in [8], where the domain is Weyl and the target is Riemannian.
From this point we only consider the case where Σ has dimension two. Using local isothermal coordinates on Σ so that η = e 2λ (dx 2 + dy 2 ), the tension field is where f x = df (∂ x ). In terms of the complex coordinate z = x + iy this is just This can also be written more explicitly as Thus, in this case, the definition of Weyl-harmonic depends only on the conformal class of η. We are also interested in the case where f is weakly conformal. If z = x + iy is a complex coordinate on Σ so that η = e 2λ (dx 2 + dy 2 ), then the equations are conformally invariant. If f is weakly conformal then this implies that Extending the conformal inner product to be complex bilinear, these are equivalent to the equation where f z = 1 2 (f x − if y ). Any point where df is not full rank is called a singular point. A branch point p is a singular point where in some neighborhood of p, f z = z k Z and Z p = 0. for some non-zero homogeneous degree m polynomial h. The zeros of dh are isolated, so the zeros of df must be as well. In fact, h must also be weakly conformal and harmonic, thus the only zeros of dh are branch point singularities. Details and further analysis of the structure of these branch points is contained in [12].
If f : Σ → M is an immersion with only branch point singularities, then it is called a branched immersion. When f is a branched immersion the conformal class f * c on Σ can be defined across the branch points and f is weakly conformal when f * c = [η]. A branched immersion which is Weyl-minimal away from the branch points is called a branched Weylminimal immersion. Proof. In this case, away from the branch points, Comparing this with equation (7)   Then the Lee form is θ µ = −J ⋆ dω µ = J(a 1 e 1 + a 2 e 2 + a 3 e 3 + a 4 e 4 ) = a 1 e 2 − a 2 e 1 + a 3 e 4 − a 4 e 3 , furthermore θ µ ∧ ω µ = dω µ .

Conformally Almost-Hermitian Manifolds
Then for Weyl derivative D = d + α µ , The canonical Weyl derivative of (M, c, J) is then determined by setting α µ = − 1 2 θ µ . The induced connection on T M is given by The Nijenhuis Tensor of J is given by This is J antilinear in both slots, that is N(JX, Y ) = −JN(X, Y ) = N(X, JY ). For any vector field, X, ∇ D X J is also J antilinear. Proposition 2.3. For any almost-Hermitian manifold

Twistor Correspondence
Following [5] we now show there is a correspondence between weakly conformal Weylharmonic maps and non-vertical J ± -holomorphic maps into the weightless twistor space with complex structure given by (5) and (6). The twistor lifts of a weakly conformal map f : Σ → M are given bỹ The natural isomorphism ♭ c : L −1 T M → LT * M preserves weights, but interchanges the holomorphic and anti-holomorphic spaces. Proof. The twistor liftsf ± are J ± -holomorphic provided df ± (∂ z ) ∈ T (1,0) Z ± . We have It follows thatf ± is pseudo-holomorphic map if and only if (∇ D ∂z fz) (1,0) ± = kf z , for some function k. Taking the conformal inner-product with fz gives Since fz ∈ T For f weakly conformal, this shows that k = 0. Thereforef ± is J -holomorphic if and only if (∇ D ∂z fz) (1,0) ± = 0, but since ∇ D ∂z fz is real, this can only be true when it is zero.
There is a one-to-one correspondence between weakly conformal Weyl-harmonic maps to (M, c, D) and non-vertical J ± -holomorphic maps to the twistor space.
Proof. It only remains to show that for a non-vertical J -holomorphic curve, φ : Σ → Z ± the projectionφ : Σ → M is weakly conformal and Weyl-harmonic. It is clearly weakly conformal asφ z is holomorphic with respect to the complex structure defined by φ, which implies c(φ z ,φ z ) = 0. It is Weyl-harmonic as φ is its twistor lift and is J -holomorphic. Proof. The composition with the section s : M → Z + determined by J is a J + -holomorphic curve of Z + .

Adjunction Inequality
In this section we prove theorem 1.3 Proof. Fix a metric , in the conformal class. For a holomorphic normal coordinate z on Σ, split f z into its holomorphic and antiholomorphic parts we have α, α = 0 = β, β .
Thus α and β are Hermitian orthogonal and away from their zeros span the holomorphic tangent bundle f * T (1,0) M. The Weyl-harmonic map equation in coordinates is Since ∇ D does not preserve the almost-complex structure, we write the equation using the connection ∇ D,J , given by This connection preserves the complex structure, and thus preserves the holomorphic and anti-holomorphic tangent spaces. In terms of this connection the Weyl-harmonic map equation is Since ∇ D ∂z J is J anti-linear, it maps from T (1,0) M to T (0,1) M and from T (0,1) M to T (1,0) M. The Weyl-harmonic map equation can then be written in terms of α andβ as Using proposition 2.4, a weakly conformal Weyl-harmonic map must satisfy where the last line follows fromᾱ,β ∈ f * T (0,1) M, which implies N(ᾱ,β) ∈ f * T (1,0) M. This implies that away from the zeros of α and β, Similarly This gives By the Koszul-Malgrange theorem [9], there are holomorphic structures on f * T (1,0) M and f * T (0,1) M so that∂ X = ∇ D,J ∂z X ⊗ dz. Then for a Weyl-harmonic map∂ The Bers-Vekua similarity principle (see [6]) implies that near any point p ∈ Σ we have α = γ p e σp ,β = δ p e τp , for some local holomorphic sections γ p of f * T (1,0) M, δ p of f * T (0,1) M, and some bounded functions σ p , τ p . This can be used to define the indices These are the total ramification index R, the number of anti-complex points Q, and the number of complex points P . Following [5], these determine the degrees of the line bundles spanned by the vector valued one forms f z dz, αdz andβdz respectively.
We also have f * T (1,0) M = [α] ⊕ [β], and since α andβ span a negatively oriented, maximal isotropic subspace of f * T M ⊗ C which contains f z , it must be that f * T Since Since P and Q are both non-negative, this implies the adjunction inequality (3) of corollary 1.2 4 Examples

Hopf Surfaces
The primary Hopf surface M = S 1 × S 3 is fibered over S 2 with fiber T 2 . The bundle projection is just the projection to the S 3 component followed by the Hopf map. There is a Hermitian structure on M induced by the standard Hermitian structures on the base and fiber. The Lee form is just dφ, where φ is the angle along S 1 . Every fiber is J-holomorphic and is therefore Weyl-minimal. It is also minimal as θ ♯ is tangent to the fiber. In addition, there is a Lagrangian Weyl-minimal surface for every great circle γ in the base S 2 . To see this consider the Clifford torus in S 3 which maps to γ under the Hopf map. This torus contains two great circles on S 3 , one tangent to the fiber and one perpendicular to the fiber. The great circle perpendicular to the fiber times the product S 1 gives a Lagrangian totally geodesic T 2 to which θ ♯ is tangent, and is therefore Weyl-minimal.
Since θ = dφ is closed we can look at the universal coverM = R × S 3 . Using φ as the coordinate on R the metric is just gM = dφ 2 + g S 3 .
Therefore the Weyl-minimal surfaces will lift to minimal surfaces of the conformal metric e 2φ gM = e 2φ dφ 2 + e 2φ g S 3 = (de φ ) 2 + e 2φ g S 3 .
Using the new coordinate r = e φ this is just the (incomplete) flat metric on R 4 \ 0 ∼ =M . dr 2 + r 2 g S 3 .
Any surface lifted from M will be invariant under deck transformation φ → φ + 2π or r → e 2π r. The Weyl-minimal surfaces described above correspond to the planes through the origin in R 4 .
Therefore the Lee form is θ = F 1 β 2 − F 2 β 1 . For a constant curvature connection, this will be closed. The Hopf surface is a special case for this example where Σ = S 2 with the round metric, and M has associated bundle M × C 2 /U(1) × U(1) = C ⊕ K. As in that case, the fiber is always a J-holomorphic curve and therefore Weyl-minimal. If a closed geodesic on γ : S 1 → Σ has a closed horizontal liftγ and the connection has constant curvature theñ γ(s) · (e −iF 2 t , e iF 1 t ) parametrizes a Lagrangian minimal torus on M to which θ ♯ is tangent, and thus Weyl-minimal.