Formal Killing fields for minimal Lagrangian surfaces in complex space forms

Journal of Geometry and Physics 114 (2017) 291–328

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Journal of Geometry and Physics journal homepage: www.elsevier.com/locate/jgp

Formal Killing fields for minimal Lagrangian surfaces in complex space forms Joe S. Wang Seoul, South Korea

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Article history: Received 18 November 2013 Received in revised form 15 April 2016 Accepted 6 December 2016 Available online 23 December 2016 MSC: 53C43 35A27 Keywords: Minimal Lagrangian surface Infinite prolongation Formal Killing field Recursion Jacobi field Conservation law

abstract The differential system for minimal Lagrangian surfaces in a 2C -dimensional, non-flat, complex space form is an elliptic integrable system defined on the Grassmann bundle of oriented Lagrangian 2-planes. This is a 6-symmetric space associated with the Lie group SL(3, C), and the minimal Lagrangian surfaces arise as the primitive maps. Utilizing this property, we derive the inductive differential algebraic formulas for a pair of the formal loop algebra sl(3, C)[[λ]]-valued canonical formal Killing fields. For applications, (a) we give a complete classification of the (pseudo) Jacobi fields for the minimal Lagrangian system, (b) we obtain an infinite sequence of conservation laws from the components of the canonical formal Killing fields. © 2016 Elsevier B.V. All rights reserved.

1. Introduction 1.1. Minimal Lagrangian surfaces In relation to the developments in string theory, special Lagrangian submanifolds in Calabi–Yau manifolds have received much attention recently, [1–11]. When the ambient Calabi–Yau manifold is Cm+1 (flat case), the link of a special Lagrangian cone in the unit sphere S2m+1 ⊂ Cm+1 is called a special Legendrian submanifold. Under the Hopf map S2m+1 → CPm , a special Legendrian submanifold corresponds to a minimal Lagrangian submanifold in CPm . In the 2-dimensional case, Schoen and Wolfson gave a variational analysis of the area minimizing (Hamiltonian stationary) Lagrangian surfaces in a Kähler surface, [12]. They proved the existence of an area minimizer in a given Lagrangian homology class, possibly with the certain admissible conical singularities. In the absence of the singularities, the area minimizer is minimal Lagrangian. Haskins and Kapouleas gave a gluing construction of the compact high-genus special Legendrian surfaces in the 5-sphere, [8]. For the integrable system aspects of the theory on the minimal Lagrangian tori in CP2 , we refer to [13] and the references therein. In case of the hyperbolic complex space form CH2 , Loftin and McIntosh gave an analysis of the minimal Lagrangian surfaces in relation to the surface group representations in the Lorentzian special unitary group SU(1, 2), [14].

E-mail address: [email protected]. http://dx.doi.org/10.1016/j.geomphys.2016.12.002 0393-0440/© 2016 Elsevier B.V. All rights reserved.

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1.2. Elliptic Tzitzeica equation The partial differential equation which locally describes a minimal Lagrangian surface (away from the zero divisor of Hopf differential, Section 2.3) in a 2C -dimensional1 complex space form is the elliptic Tzitzeica equation (4.38). It is a rank-1 Toda field equation, which is a well known example of elliptic integrable equation, and the infinite sequence of Jacobi fields and conservation laws were completely determined in [15,16]. The elliptic Tzitzeica equation admits a sl(3, C)-valued Lax representation, i.e., a loop algebra sl(3, C)[λ−1 , λ]-valued2 1-form ψλ which satisfies the Maurer–Cartan equation dψλ + ψλ ∧ ψλ = 0. The main idea of construction in [16] is to consider the associated Killing field equation with respect to ψλ , dXλ + [ψλ , Xλ ] = 0,

(1.1)

where the Killing field Xλ takes values in sl(3, C)[[λ , λ]]. When Eq. (1.1) is expanded as a formal series in the spectral parameter λ, it exhibits a pair of 6-step recursion relations embedded in the infinite jet space of the elliptic Tzitzeica equation. Via an analysis of the conservation laws as characteristic cohomology, a repeated application of the recursion generates the infinite sequence of Jacobi fields and conservation laws. −1

1.3. Purpose In the previous work [17] on constant mean curvature (CMC) surfaces, we gave an interpretation of the classical work [18] by Pinkall and Sterling via the associated loop algebras and showed that a CMC surface in a 3-dimensional Riemannian space form admits a loop algebra sl(2, C)[[λ]]-valued canonical formal Killing field. As a consequence, the infinite sequence of higher-order Jacobi fields and conservation laws were determined from the components of the formal Killing field. We claim that the results of [17] are true for the integrable matrix Lax equations in general; the existence of canonical formal Killing fields is likely a universal property. In the present paper, we verify this claim for the differential system for minimal Lagrangian surfaces in a 2C -dimensional, non-flat, complex space form. We show that such a minimal Lagrangian surface admits a pair of the loop algebra sl(3, C)[[λ]]-valued canonical formal Killing fields, Theorems 5.19 and 5.21. This can be considered as an extension and refinement of the results for the elliptic Tzitzeica equation in [15,16]. The main idea of construction is similar to [16,17]. We apply the recursion relations from the formal Killing field equation to obtain the inductive differential algebraic formulas for the canonical formal Killing fields. 1.3.1. sl(2, C) vs. sl(3, C) The underlying Lie algebra for the analysis of the CMC surfaces in [17] is sl(2, C), which has rank 1. For the minimal Lagrangian surfaces, it is sl(3, C), which has rank 2. In hindsight, this accounts for the appearance of two canonical Killing fields. More generally, consider for example the two dimensional harmonic map equation into the compact special unitary group SUN +1 , which is also a well known integrable matrix Lax equation. In this case, it is known that there exists an N = rank(SUN +1 )-dimensional space of canonical formal Killing fields. 1.4. Main results See Notation 5.1 for the relevant notations for loop algebras. 1.4.1. A pair of canonical formal Killing fields We give a construction of a pair of sl(3, C)[[λ]]-valued canonical formal Killing fields by explicit inductive differential algebraic formulas, Theorems 5.19 and 5.21. The construction relies on a 6-step recursion process, which involves two steps at which one needs to solve certain ∂ξ equations, see Section 5.1.4. The idea is to bypass this problem by imposing the compatible algebraic constraints on the characteristic polynomials of the formal Killing fields. As a corollary, for example, it implies that a minimal Lagrangian torus in CP2 admits a pair of canonical polynomial Killing fields, and hence a pair of spectral curves.

1 Here ‘‘2 -dimensional’’ means ‘‘2 complex dimensional’’. C 2 See Notation 5.1.

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1.4.2. Classification of (pseudo) Jacobi fields The structure equations for the Killing fields show that there exist two kinds of ‘Jacobi fields’ for the minimal Lagrangian system; the Jacobi fields which generate the symmetries of the minimal Lagrangian system, and the pseudo-Jacobi fields which generate the symmetries of the underlying elliptic Tzitzeica equation, Section 4.4. We give a complete classification of the (pseudo) Jacobi fields, Theorem 6.1. Modulo the classical (pseudo) Jacobi fields, the space of (pseudo) Jacobi fields is generated by the infinite sequence of higher-order (pseudo) Jacobi fields from the components of the canonical formal Killing fields and their complex conjugates, Corollaries 5.20 and 5.22. 1.4.3. Conservation laws The conservation laws are defined as the elements in the 1st characteristic cohomology of the infinite prolongation of the minimal Lagrangian system. We obtain an infinite sequence of conservation laws from the components of the canonical formal Killing fields, Theorem 7.6. Assuming they are nontrivial, an analysis indicates that Noether’s theorem holds and there exists a canonical isomorphism between the Jacobi fields and the conservation laws. 1.5. Contents The paper can be divided into the following three parts. Sections 2 and 3. Classical Killing fields, Sections 4 and 5. Formal Killing fields, Sections 6 and 7. Applications. In Sections 2 and 3, the differential system for minimal Lagrangian surfaces in a 2C -dimensional complex space form is defined, and we give an analysis of the associated classical Killing fields. We obtain the classical (pseudo) Jacobi fields and the classical conservation laws from the components of the classical Killing fields. In Sections 4 and 5, the infinite prolongation of the minimal Lagrangian system is defined, and we give an inductive construction for the pair of canonical formal Killing fields. In Sections 6 and 7, we give applications of the canonical formal Killing fields to the (pseudo) Jacobi fields and conservation laws. We shall work within the smooth (C ∞ )-category. 2. Minimal lagrangian surfaces in a complex space form After a brief summary of the structure equations for a 2C -dimensional complex space form, the differential equation for minimal Lagrangian surfaces is defined as a homogeneous exterior differential system on the Grassmann bundle of oriented Lagrangian 2-planes. The structure equations for the minimal Lagrangian system established in Section 2.2, together with their infinite prolongation in Section 4, will be the basis of our analysis for the rest of the paper. 2.1. Complex space form We will summarize the basic formulas of Kähler geometry for a 2C -dimensional complex space form. We refer to [19] for the details. In order to avoid repetitions, we agree on the following range for the indices: 1 ≤ A, B, C ≤ 2. The Einstein summation convention will be used for repeated indices. 2.1.1. 2C -dimensional complex space form Let M be the 2C -dimensional simply connected complex space form of constant holomorphic sectional curvature 4γ 2 .3 In the case γ 2 = 0 and M = C2 , it turns out that the minimal Lagrangian surfaces in M are equivalent to the holomorphic curves in C2 under a different covariant constant complex structure. We restrict to the case

γ 2 ̸= 0, where the differential equation for minimal Lagrangian surfaces is genuinely nonlinear. The squared expression γ 2 is introduced for the sake of convenience, for the quantity γ appears frequently in the analysis. We adopt the following convention for γ .

 γ = Here i =

 + γ 2 +i −γ 2

if

γ 2 > 0, γ 2 < 0.

√ −1 denotes the unit imaginary number.

3 See (2.1). The numerical factor 4 is ornamental.

(2.1)

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2.1.2. Unitary coframe bundle Let U(2) be the group of 2-by-2 unitary matrices. Let

/F

U(2)

π



M be the principal U(2)-bundle of unitary coframes. An element u ∈ F is by definition a Hermitian isometry

u : Tπ(u) M −→ C2 , where C2 is equipped with the standard Hermitian vector space structure. The structure group U(2) acts on F on the right by

u → g −1 ◦ u ,

for g ∈ U(2).

Let (ζ , ζ ) be the C2 -valued tautological 1-form on F . The corresponding action of the structure group U(2) on (ζ , ζ 2 )t is 1

2 t

1

 1  1 ζ −1 ζ →g , ζ2 ζ2

for g ∈ U(2).

By definition, the Kähler structure on M is given by the pair

ζA ◦ ζ

g :=

i

ϖ :=

A

ζ ∧ζ A

2

(Riemannian metric), A

(2.2)

(symplectic form).

Let u(2) be the space of 2-by-2 skew-Hermitian matrices, which is the Lie algebra of U(2). There exists a unique u(2)valued connection 1-form (ζBA ) on F such that the following structure equations hold. dζ A = −ζBA ∧ ζ B ,

(2.3)

B A

ζ = −ζ . A B

The u(2)-valued curvature 2-form (ΩBA ) is in turn defined by

ΩBA := dζBA + ζCA ∧ ζBC .

(2.4)

For the case at hand of the complex space form, the curvature 2-form is given by

 ΩBA

=γ

2

B

ζ ∧ ζ +δ A

A B

2 

 ζ ∧ζ C

C

.

(2.5)

C =1

Here δBA is the Kronecker delta. 2.1.3. Real structure equations For the analysis of the minimal Lagrangian surfaces, we decompose the structure equations (2.3)–(2.5) into the real, and imaginary parts as follows.4 Set



ζ11 ζ12

ζ A = ωA + iµA ,     1 ζ21 · ρ β − 3γ 2 θ0 = +i 2 −ρ · ζ2 β2

 β2 , −β 1 − 3γ 2 θ0

4 Note the isomorphism U(2) = SO(4) ∩ Sp(2, R). Here the Lie groups U(2), SO(4)(special orthogonal group), and Sp(2, R) (symplectic group) are considered as subgroups of SL(4, R). Accordingly, M can be considered as a 4R -dimensional manifold equipped with the pair (g, ϖ ) in (2.2).

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for the set of real 1-forms {ωA , µA , ρ, β A , θ0 }. Here ‘ · ’ denotes ‘0’. In terms of these 1-forms, the structure equations (2.3)– (2.5) are written as follows. dω1 = −ρ ∧ ω2 + (β 1 ∧ µ1 + β 2 ∧ µ2 ) − 3γ 2 θ0 ∧ µ1 ,

(2.6)

dω = +ρ ∧ ω + (β ∧ µ − β ∧ µ ) − 3γ θ0 ∧ µ , 2

1

2

1

1

2

2

2

dµ1 = −ρ ∧ µ2 − (β 1 ∧ ω1 + β 2 ∧ ω2 ) + 3γ 2 θ0 ∧ ω1 , dµ2 = +ρ ∧ µ1 − (β 2 ∧ ω1 − β 1 ∧ ω2 ) + 3γ 2 θ0 ∧ ω2 , dρ = γ 2 (ω1 ∧ ω2 + µ1 ∧ µ2 ) + 2β 1 ∧ β 2 , dθ0 = −(µ1 ∧ ω1 + µ2 ∧ ω2 ), dβ 1 = −2ρ ∧ β 2 + γ 2 (µ1 ∧ ω1 − µ2 ∧ ω2 ), dβ 2 = +2ρ ∧ β 1 + γ 2 (µ2 ∧ ω1 + µ1 ∧ ω2 ). 2.2. Exterior differential system With this preparation, we proceed to define the differential system for minimal Lagrangian surfaces. 2.2.1. Grassmann bundle of oriented Lagrangian 2-planes Let C2 be the standard 2C -dimensional Hermitian vector space. Let (g0 , ϖ0 ) be the U(2)-invariant compatible pair of a Riemannian metric g0 and a symplectic form ϖ0 on C2 which define the Hermitian structure. A 2R -dimensional subspace of C2 is Lagrangian if the restriction of the symplectic form ϖ0 vanishes. Let Lag(C2 ) = {Lagrangian 2-planes in C2 } be the Grassmannian of oriented Lagrangian 2-planes. It is well known that U(2) acts transitively on Lag(C2 ), with SO(2) = U(2)∩ SL(2, R) as the stabilizer subgroup. It follows that Lag(C2 ) is the homogeneous space, Lag(C2 ) = U(2)/ SO(2). Let X −→ M be the Grassmann bundle of oriented Lagrangian 2-planes in TM. By the analysis above, X = F / SO(2), and we have the following commutative diagram:

F

~ ~~ ~ ~ ~ ~ U(2) X @ @@ @@ U(2)/SO(2) @@   SO(2)

M

2.2.2. Differential system for Lagrangian surfaces An immersed surface x : Σ ↩→ M is Lagrangian if its tangent space at each point is Lagrangian, i.e., if the restriction of the symplectic form vanishes on Σ , x∗ ϖ = 0. By definition of X , an immersed oriented Lagrangian surface in M admits a unique tangential lift to X . We first define the differential equation for such tangential lifts as the canonical contact differential system on X . Consider the real structure equation (2.6). By a standard moving frame analysis, set the contact ideal I0 on X generated by

I0 := ⟨ µ1 , µ2 , dµ1 , dµ2 ⟩.

(2.7)

Note that I0 , which is originally defined on F , is invariant under the induced action by the structure group SO(2), and I0 descends to X . By construction, an immersed oriented integral surface of I0 in X corresponds to a (possibly singular) oriented Lagrangian surface in M, Remark 4.5.

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2.2.3. Differential system for minimal Lagrangian surfaces The condition that an immersed Lagrangian surface is minimal (i.e., its mean curvature vector vanishes) is a second order constraint. The differential system for minimal Lagrangian surfaces is obtained by supplementing the canonical contact ideal I0 to encode this additional minimality condition. From Eqs. (2.6), we claim that this is equivalent to,

θ0 = 0. To see this, consider the real structure equations (2.6) restricted to the tangential lift of an immersed Lagrangian surface Σ ↩→ M. With the contact 1-forms {µ1 , µ2 } being set to 0, the induced Riemannian metric on Σ is given by I := (ω1 )2 + (ω2 )2 . An elementary moving frame computation shows that the second fundamental form of Σ can be identified with the symmetric cubic differential

I := ωA ◦ Im(ζBA ) ◦ ωB . Hence the mean curvature vector vanishes whenever the corresponding trace vanishes, trI (I) = 0. This is equivalent to the vanishing of θ0 , which is up to a constant scale the trace of Im(ζBA ). Proposition 2.8. Let M be a 2C -dimensional complex space form. Let X → M be the Grassmann bundle of oriented Lagrangian 2-planes. Let I0 be the canonical contact differential system on X , (2.7). (a) The differential system for minimal Lagrangian surfaces is given by

I := ⟨ µ1 , µ2 , θ0 , φ + , φ − ⟩,

(2.9)

where

φ + = β 1 ∧ ω1 + β 2 ∧ ω2 , φ − = β 2 ∧ ω1 − β 1 ∧ ω2 . The structure equations (2.6) show that the differential ideal I, which is originally defined on F , is invariant under the induced action by the structure group SO(2). The differential ideal I is well defined on X . (b) An immersed oriented minimal Lagrangian surface in M admits a unique tangential lift to X as an integral surface of I. Conversely, an immersed integral surface of I in X projects to a (possibly singular) minimal Lagrangian surface in M. Note from Eqs. (2.6) that dθ0 , dµA ≡ 0

mod I,

and I is differentially closed. Remark 2.10. In terms of Cartan’s theory of exterior differential systems, the differential system (X , I) is involutive and the local moduli space of solutions depends on two arbitrary real functions of 1 real variable, [20, Chapter III]. 2.2.4. Complexified structure equations The differential system for minimal Lagrangian surfaces is an integrable extension over the elliptic Tzitzeica equation, [15]. The characteristic directions are complex and they induce a complex structure on a minimal Lagrangian surface. In order to utilize this property, we introduce a set of complex differential forms on F adapted to I. Set

ξ := ω1 + i ω2 ,

(2.11)

θ1 := µ − iµ , 1

2

η2 := β 1 − iβ 2 . Note

φ + − iφ − = η2 ∧ ξ , and the differential ideal (2.9) can be re-written as

I = ⟨θ0 , θ1 , η2 ∧ ξ ⟩.

(2.12)

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In terms of these complex 1-forms, the structure equations (2.6) simplify to the following equations. dξ = iρ ∧ ξ − 3γ 2 θ0 ∧ θ 1 − θ1 ∧ η2 ,

(2.13)

 1 dθ0 = − θ1 ∧ ξ + θ 1 ∧ ξ , 2

dθ1 = −iρ ∧ θ1 − η2 ∧ ξ + 3γ 2 θ0 ∧ ξ , dη2 = −2iρ ∧ η2 + γ 2 θ1 ∧ ξ , dρ =

i  2

 γ 2 ξ ∧ ξ − 2η2 ∧ η2 − γ 2 θ1 ∧ θ 1 .

From now on, the differential analysis for minimal Lagrangian surfaces will be carried out based on this structure equation. Let us mention here a relevant notation which will be frequently used. For a scalar function f : F → C, the covariant derivatives are written in the upper-index notation, ¯ ¯ df ≡ f ξ ξ + f ξ ξ + f 0 θ0 + f 1 θ1 + f 1 θ 1 + f 2 η2 + f 2 η2

mod ρ.

(2.14)

2.3. Hopf differential The induced local geometric structures on a minimal Lagrangian surface consist of a triple of data called admissible triple, Definition 2.20. In particular, it contains a holomorphic cubic differential called Hopf differential which arises as a complexified version of the second fundamental form. 2.3.1. Hopf differential Let x : Σ ↩→ X be an immersed integral surface of the differential system for minimal Lagrangian surfaces. On the induced SO(2)-bundle x∗ F → Σ ,

θ0 , θ1 = 0, η2 ∧ ξ = 0. The structure equation (2.13) restricted to x∗ F then become, dξ = i ρ ∧ ξ ,

(2.15)

0 = η2 ∧ ξ , dη2 = −2iρ ∧ η2 , dρ =

i  2

 γ 2 ξ ∧ ξ − 2η2 ∧ η2 .

The first equation shows that Σ inherits a complex structure for which any section of ξ is a (1, 0)-form. The second, and third equations show that the cubic differential of type (3, 0)

η2 ◦ ξ 2 is invariant under the action by the structure group SO(2), and it is a well defined holomorphic cubic differential on Σ with respect to the induced complex structure. Let K → Σ denote the canonical line bundle. Definition 2.16. Let x : Σ ↩→ X be an immersed integral surface of the differential system for minimal Lagrangian surfaces. Consider the induced structure equation (2.15). The Hopf differential of x is the holomorphic cubic differential

I = η2 ◦ ξ 2 ∈ H 0 (Σ , K 3 ).

(2.17)

The umbilic divisor U = (I)0 is the zero divisor of I. When Σ is compact, Riemann–Rochh theorem implies that deg(U) = 6 genus(Σ ) − 6. 2.3.2. Admissible triple Suppose that an integral surface Σ ↩→ X of the minimal Lagrangian system I on X is the tangential lift of an immersed oriented minimal Lagrangian surface in M. By definition of the tautological forms on F , it satisfies the independence condition

ξ ∧ ξ ̸= 0,

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and the induced (non-degenerate) Riemannian metric on Σ is given by I = ξ ◦ ξ. This implies that the SO(2)-bundle x∗ F → Σ can be identified with the principal SO(2)-bundle of the Riemannian metric I on Σ . Under this identification, ξ is the tautological unitary (1, 0)-form, and ρ is the Levi-Civita connection form. From the second equation of (2.15), there exists in this case a coefficient h3 defined on x∗ F such that

η2 = h3 ξ .

(2.18)

The Hopf differential can now be written as

I = h3 ξ 3 . From the fourth equation of (2.15), the Gauß equation, the curvature R of the induced metric I satisfies the compatibility equation R = γ 2 − 2h3 h¯ 3 .

(2.19)

Summarizing this, we give a definition of the compatible Bonnet data for minimal Lagrangian surfaces. Definition 2.20. Let M be the 2C -dimensional complex space form of constant holomorphic sectional curvature 4γ 2 . An admissible triple for a minimal Lagrangian surface in M consists of a Riemann surface, a conformal metric, and a holomorphic cubic differential which satisfy the compatibility equation (2.19). An analogue of the classical Bonnet theorem can be stated as follows. It can be verified by a standard ODE argument, and we omit the proof. Theorem 2.21. Let Σ be a Riemann surface. Let (I, I) be a pair of a conformal metric and a holomorphic cubic differential on Σ  → Σ be such that (Σ , I, I) form an admissible triple for a minimal Lagrangian surface in the complex space form M. Let π : Σ  ↩→ M which realizes the simply connected universal cover. Then there exists a conformal minimal Lagrangian immersion x : Σ (π ∗ I, π ∗ I) as the induced Riemannian metric and the Hopf differential. Such an immersion x is unique up to motion by the Kähler isometries of M.

2.3.3. Cube root of Hopf differential The analysis of the canonical formal Killing fields in Section 5 will inevitably involve the object

 3

h3 ,

or, equivalently, the cube root of Hopf differential

√ 3

I.

It is generally a multi-valued holomorphic 1-form on a minimal Lagrangian surface. In order to better accommodate this, we introduce the triple cover of a minimal Lagrangian surface defined by Hopf differential. Definition 2.22. Let x : Σ ↩→ X be an immersed integral surface of the differential system for minimal Lagrangian surfaces. Let I ∈ H 0 (Σ , K 3 ) be the Hopf differential (2.17). The triple cover

ˆ →Σ ν:Σ associated with I is the Riemann surface of the complex curve

  Σ ′ = κ ∈ K |κ 3 = I ⊂ K . √ 3

ˆ obtained by The cube root ω = I of the Hopf differential is the well defined (single-valued) holomorphic 1-form on Σ the pull-back of restriction of the tautological (1, 0)-form on K to Σ ′ . By construction, the projection ν is a triple covering branched over the umbilics of degree 1 or 2 mod 3.

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3. Classical Killing fields Let G = SU(3),

or

SU(1, 2)

(special unitary group) 5

be the group of Kähler isometries of the complex space form M, depending on the sign of the holomorphic sectional curvature 4γ 2 . The induced action of G on X is transitive, and by construction the differential system for minimal Lagrangian surfaces (X , I) is G-invariant, i.e., the group G acts as a symmetry of the differential system (X , I). In this section, we examine the classical Killing fields generated by the corresponding infinitesimal action of the Lie algebra of G,

g = su(3),

or su(1, 2).

The structure equation for the classical Killing fields reveals its recursive structure when it is written in terms of an adapted basis of g, Eq. (3.8). From this, we extract two kinds of Jacobi fields called classical Jacobi fields, and their variants classical pseudo-Jacobi fields, as well as the recursion relations between them, Section 3.2, Section 3.2.3. The structure equation also shows that there exists an eight dimensional space of classical conservation laws, Section 3.3. The analysis of the classical Killing fields in this section will serve as a model for the construction of the formal Killing fields in Section 5. At the end of the section, we comment on the corresponding higher-order extension via the associated loop algebras. 3.1. Structure equations 3.1.1. Maurer–Cartan form Recall the complexified differential forms (2.11). Consider the following g-valued 1-form on F ,

ψ = ψ+ + ψ0 + ψ− ,

(3.1)

where

 · −γ (ξ − iθ1 ) iγ (ξ + iθ1 ) γ (ξ + iθ1 ) iη2 −η2 ψ− = , 2 −iγ (ξ − iθ ) −η2 −iη2 1  2  2iγ θ0 · · ψ0 =  · −iγ 2 θ0 ρ , · −ρ −iγ 2 θ0   · −γ (ξ − iθ 1 ) −iγ (ξ + iθ 1 ) 1 . ψ+ =  γ (ξ + iθ 1 ) iη 2 η2 2 iγ (ξ − iθ 1 ) η2 −i η 2 1



A computation shows that Eq. (2.13) implies (is equivalent to) the Maurer–Cartan structure equation for ψ , dψ + ψ ∧ ψ = 0 . 3.1.2. Classical Killing fields The Killing field equation, (3.3), characterizes the classical Killing fields generated by the infinitesimal action of the Lie algebra g on X . Let

gC := g ⊗ C = sl(3, C) be the complexification of g. Definition 3.2. Let X = F / SO(2) → M be the Grassmann bundle of oriented Lagrangian 2-planes. A classical Killing field (complexified classical Killing field respectively) is an SO(2)-equivariant, g-valued (gC -valued) function X on F which satisfies the Killing field equation dX + [ψ, X] = 0.

(3.3)

By definition, the space of classical Killing fields (complexified Killing fields) is isomorphic to g (gC ), and we identify these spaces from now on.

5 To be precise, one needs to take a quotient of G by the diagonal subgroup Z of order 3. This is a minor detail, and will not affect our analysis. 3

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Fig. 3.1. Decomposition of g.

Definition 3.4. Let (X , I) be the differential system for minimal Lagrangian surfaces. A vector field V ∈ H 0 (TX ) is a classical symmetry if it preserves the differential ideal I under the Lie derivative,

LV I ⊂ I. The R-Lie algebra of classical symmetries is denoted by S(0) . It is clear that a classical Killing field generates a classical symmetry. The converse is also true, and a classical symmetry is necessarily generated by a classical Killing field. Proposition 3.5.

S(0) ≃ g. The proof follows by a direct computation. We omit the proof. 3.1.3. Recursive structure equations It will be convenient to write the Killing field equation (3.3) in components. Consider the following decomposition of g, Fig. 3.1. Here {p, b, c, f, a, g, s, t} are the scalar variables which satisfy the following reality conditions:

 g=

su(3) then su(1, 2)

a = a, p = p, t = −c, s = −b, g = −f, a = a, p = p, t = −c, s = +b, g = +f.

(3.6)

Remark 3.7. Without the reality conditions (3.6), Fig. 3.1 represents a decomposition of gC . For simplicity, and reference, we record here the Killing field Eq. (3.3) in terms of these variables as restricted to a minimal Lagrangian surface. dp ≡ (iγ b + 2ih3 c)ξ + (iγ s + 2ih¯ 3 t)ξ , db + ibρ ≡ ih3 fξ +

i 2

(3.8)

γ pξ ,

dc − 2icρ ≡ iγ fξ + ih¯ 3 pξ , df − ifρ ≡

3i 2

γ aξ + (iγ c + ih¯ 3 b)ξ ,

da ≡ iγ gξ + iγ fξ , dg + igρ ≡ (−iγ t − ih3 s)ξ + ds − isρ ≡

i 2

3i 2

γ aξ ,

γ pξ − ih¯ 3 gξ ,

dt + 2itρ ≡ ih3 pξ − iγ gξ ,

mod I.

Note here we applied the substitution (2.18),

η2 ≡ h3 ξ ,

η2 ≡ h¯ 3 ξ

(‘‘mod I′′ ).6

3.1.4. Total derivatives Let us introduce here the relevant notations for total derivatives. For a scalar function u on F , we define the total derivatives ∂ξ u, ∂ξ u by

∂ξ u = uξ := ξ -coefficient of du, ∂ξ u = uξ := ξ -coefficient of du,

(3.9) ‘‘mod I . ′′

6 Here ‘‘mod I’’ means modulo the infinite prolongation I(∞) , Section 4. See the argument below Proposition 3.13.

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As the notations suggest, they represent the covariant derivatives on a minimal Lagrangian surface with respect to ξ , ξ . For example, from Eq. (3.8) we have a ξ = iγ g ,

aξ = iγ f.

3.2. Classical Jacobi fields and classical pseudo-Jacobi fields From the apparent symmetry of Eq. (3.8), we extract two kinds of Jacobi fields for the minimal Lagrangian system; classical Jacobi fields, and classical pseudo-Jacobi fields. 3.2.1. Classical Jacobi fields Consider the component a in Eqs. (3.8). One finds aξ = iγ g, 3

∂ξ (aξ ) = aξ ,ξ = iγ gξ = − γ 2 a. 2

Definition 3.10. A scalar function A : X → C is a classical Jacobi field if it satisfies

E (A) := Aξ ,ξ +

3 2

γ 2 A = 0.

(3.11)

The C-vector space of classical Jacobi fields is denoted by J(0) = J0 . By definition, we have the associated map a : gC −→ J(0) .

(3.12)

An analysis of the Killing field equation (3.3) shows that the a-component generates a complexified classical Killing field X in the sense that if the a-component of X vanishes identically, then X vanishes. This implies that the associated map is injective, a : gC ↩→ J(0) . In fact, this is an isomorphism. Proposition 3.13.

gC ≃ J(0) .

(3.14)

The claim follows by a direct computation. Let us give a sketch of the relevant ideas, and omit the details of proof. Let A be a classical Jacobi field. Denote the covariant derivatives of A by dA = Aξ ξ + Aξ ξ + A0 θ0 + A1 θ1 + A1 θ 1 + A2 η2 + A2 η2 . We adopt the similar notation for the successive derivatives of A; Aξ , Aξ ,ξ , . . . , etc. Applying the total derivative operators (3.9), one gets

∂ξ A = Aξ = Aξ + A2 h3 , ∂ξ (Aξ ) = Aξ ,ξ = Aξ ,ξ + Aξ ,2 h¯ 3 + (A2,ξ + A2,2 h¯ 3 )h3 . The Jacobi equation (3.11) becomes



E (A) = Aξ ,ξ +

3 2

 γ 2 A + Aξ ,2 h¯ 3 + A2,ξ h3 + A2,2 h¯ 3 h3 = 0.

(3.15)

Since the Jacobi field A is defined on X , while h3 , h¯ 3 are the prolongation variables7 , each coefficient of {h3 , h¯ 3 , h3 h¯ 3 } in (3.15) must vanish separately. As a result, a classical Jacobi field must satisfy the set of equations, Aξ ,ξ +

3

γ 2 A = 0, Aξ ,2 = A2,ξ = A2,2 = 0. 2 From this, a standard over-determined PDE analysis shows that A is the a-component of a complexified classical Killing field. 7 See Section 4.

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3.2.2. Classical pseudo-Jacobi fields Consider next the coefficient p in Eqs. (3.8). One finds pξ = iγ b + 2ih3 c, 1 pξ ,ξ = iγ bξ + 2ih3 cξ = − (γ 2 + 4h3 h¯ 3 )p. 2 Definition 3.16. A scalar function P : X → C is a classical pseudo-Jacobi field if it satisfies

E ′ (P ) := Pξ ,ξ +

1 2

(γ 2 + 4h3 h¯ 3 )P = 0.

(3.17)

The C-vector space of classical pseudo-Jacobi fields is denoted by J′(0) = J′0 . By definition, we have the associated map p : gC −→ J′(0) .

(3.18)

The similar argument as above shows that this is also an isomorphism, Proposition 3.19.

gC ≃ J′(0) .

(3.20)

Corollary 3.21.

J(0) ≃ J′(0) (≃ gC ). 3.2.3. Recursion relations The structure equations (3.8) contain a pair of 3-step recursion relations between the classical pseudo-Jacobi field p and the classical Jacobi field a. We record and emphasize these structures here, for they are the main technical ingredients in the construction of the formal Killing fields later on. [From p to a] Let the classical pseudo-Jacobi field p be given. From Eqs. (3.8), suppose (either of) the equations

∂ξ b =

i 2

γ p,

∂ξ c = ih¯ 3 p

were solved. Differentiate b, c,

∂ξ b = ih3 f,

∂ξ c = iγ f,

and one gets f. Differentiate f,

∂ξ f =

3i 2

γ a,

and one gets the classical Jacobi field a. The process can be summarized by the following diagram. ∂ −1

∂ξ

ξ

∂ξ

p −−−−→ b, c −−−−→ f −−−−→ a. [From a to p] In a similar way, starting from the classical Jacobi field a, one may reach p by the following process. ∂ξ

∂ −1 ξ

∂ξ

a −−−−→ g −−−−→ s, t −−−−→ p. Note that by combining the pair of processes, one gets a 6-step recursion relation for the classical (pseudo) Jacobi fields, [16]. 3.3. Classical conservation laws Another important invariants of the minimal Lagrangian system are conservation laws. In this section, we show that they can also be read off Eqs. (3.8).

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3.3.1. Definition Let (Ω ∗ (X ), d) be the de-Rham complex of C-valued differential forms on X . Since I is a differential ideal, closed under the exterior derivative, the quotient complex

( Ω ∗ , d) is well defined, where Ω ∗ = Ω ∗ (X )/I, and d = d mod I. The characteristic cohomology of the differential system (X , I) is by definition the cohomology of the quotient complex (Ω ∗ , d). Let H q ( Ω ∗ , d) denote the cohomology at Ω q , q = 0, 1, 2. Definition 3.22. Let (X , I) be the differential system for minimal Lagrangian surfaces. A classical conservation law is an element in the 1st characteristic cohomology H 1 (Ω ∗ , d) of (X , I). The C-vector space of classical conservation laws is denoted by

C (0) = C 0 := H 1 (Ω ∗ , d).

3.3.2. Classical conservation laws from complexified classical Killing fields From Eqs. (3.8), consider the 1-form

ϕa = bξ + sξ .

(3.23)

One finds that dϕa ≡ 0

mod I,

and the 1-form ϕa represents a classical conservation law. 3.3.3. Noether’s theorem for classical conservation laws Let [ϕa ] ∈ C (0) denote the conservation law represented by the 1-form ϕa (which is globally defined on X ). We claim that the associated map

gC ≃ J(0) −→ C (0) given by a −→ [ϕa ] is an isomorphism. Theorem 3.24 (Noether’s Theorem). Let M be the simply connected 2C -dimensional complex space form of constant holomorphic sectional curvature 4γ 2 . Let G = SU(3), or SU(1, 2) be the group of Kähler isometries of M. Let g be the Lie algebra of G, and let gC = sl(3, C) be its complexification. Let J(0) , and J′(0) be the space of classical Jacobi fields, and classical pseudo-Jacobi fields, Definitions 3.10 and 3.16. There exist the isomorphisms,

gC ≃ J(0) ≃ J′(0) ≃ C (0) . The claim can be verified by introducing the space of classical differentiated conservation laws H (0) , and then establishing the isomorphisms between C (0) and H (0) , and gC and H (0) separately. We do not give a proof of this claim, and refer the reader to [17, Part 1] for the related details for the case of the constant mean curvature surfaces. Remark 3.25 (Moment Conditions). Consider a closed integral curve Γ ⊂ X of I. Such Γ corresponds to a pair (c, Π ) of a closed curve c ⊂ M and a field of oriented Lagrangian 2-planes Π tangent to c such that the associated certain (2, 0)-vector field is covariant constant along c. By definition, the integrals of the classical conservation laws over Γ yield the moment conditions for (c, Π ) to bound a (sufficiently smooth) minimal Lagrangian surface, with the given boundary c and the prescribed tangent planes Π along c. Compare this with [21].

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In the remainder of the paper, we will extend the classical objects treated in this section, Killing fields, (pseudo) Jacobi fields, conservation laws, to their higher-order analogues in the framework of the infinite prolongation of the differential system (X , I), Section 4. The extension process relies on the extension of the Maurer–Cartan form

ψ −→ ψλ = λψ+ + ψ0 + λ−1 ψ− , obtained by inserting the spectral parameter λ. This leads to a generalization of the g-valued classical Killing fields X to the formal loop algebra gC [[λ]]-valued formal Killing fields Xλ , X −→ Xλ . In practice, this amounts to extending the scalar components {p, b, c, f, a, g, s, t} of X to the appropriate formal series in λ, Section 5. As a first step toward the higher-order analysis, we begin by introducing the infinite prolongation of the differential system (X , I), and record its basic structure equations. 4. Infinite prolongation The minimal Lagrangian surfaces under consideration are locally described by the elliptic Tzitzeica equation (4.38), which is a well known integrable equation, [22,23]. In particular, it possesses an infinite sequence of higher-order symmetries and conservation laws, [16]. In order to access the corresponding higher-order structures for the minimal Lagrangian system, it is necessary to introduce the infinite prolongation. In Section 4.1, we give a construction of the infinite prolongation of the minimal Lagrangian system, and determine the basic structure equations. In Section 4.2, we define a (branched) triple cover of the infinite prolongation. It turns out that the canonical formal Killing fields to be constructed later on are defined on this triple cover, rather than on the original infinite prolongation space. In Section 4.3, a sequence of adapted scalar functions called balanced coordinates are introduced, and we record their basic structure equations. We will find that the coefficients of the canonical formal Killing fields are weighted homogeneous polynomials in the balanced coordinates (up to scaling). In Section 4.4, we give a definition for the (pseudo) Jacobi fields, which naturally extends the classical (pseudo) Jacobi fields. 4.1. Infinite prolongation 4.1.1. Infinite sequence of prolongations Let

(X (0) , I(0) ) := (X , I) be the original differential system (2.12). Inductively define

(X (k+1) , I(k+1) ),

k ≥ 0, as the first prolongation of (X (k) , I(k) ). By definition, X (k+1) ⊂ Gr+ (2, TX (k) )8 is the bundle of oriented integral 2-planes of (X (k) , I(k) ). Let

πk+1,k : X (k+1) → X (k) ,

k ≥ 0,

be the associated projection. The Pfaffian system I(k+1) is generated by the pull-back πk∗+1,k I(k) and the restriction of the canonical contact system on Gr+ (2, TX (k) ) to X (k+1) . An elementary analysis shows that for each k ≥ 0, the prolongation space X (k+1) is a smooth manifold, and that the projection πk+1,k is a smooth submersion with two dimensional fibers isomorphic to CP1 , see Section 4.2. Definition 4.1. The infinite prolongation (X (∞) , I(∞) ) of the differential system (X , I) for minimal Lagrangian surfaces is defined as the projective limit X (∞) = lim X (k) , ←−

I(∞) = ∪k≥0 I(k) . Let

π∞,k : X (∞) → X (k) ,

k ≥ 0,

∗ (k) be the associated sequence of projections. Note that we identify I(k) with its image π∞, ⊂ Ω ∗ (X (∞) ). By construction, kI

the sequence of Pfaffian systems I(k) satisfy the inductive closure conditions dI(k) ≡ 0

mod I(k+1) , k ≥ 1.

8 The Grassmann bundle of oriented 2-planes in the tangent bundle of X (k) .

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305

The defining properties of the sequence of prolongations can be summarized as follows. Proposition 4.2. Let x : Σ ↩→ X be an immersed integral surface of the differential system for minimal Lagrangian surfaces. For each k = 1, 2, . . ., there exists a unique k-th prolongation x(k) : Σ ↩→ X (k) , such that (a) x(k) is integral to I(k) , (b) x(k−1) = πk,k−1 ◦ x(k) .

The infinite prolongation x(∞) of x is defined as the associated limit x(∞) = lim x(k) . ←−

By construction, x(∞) is integral to I(∞) . 4.1.2. Associated sequence of principal SO(2)-bundles The actual analysis will be carried out on the following associated principal SO(2)-bundles. Recall the original principal SO(2)-bundle,

Π : F → X = F / SO(2). Let F = F , and set F (k) := Π ∗ X (k) , k ≥ 1. (0)

Define

F (∞) := lim F (k) , ←−

and it is a principal SO(2)-bundle,

F (∞) → X (∞) .

We continue to use I(k) , I(∞) to denote the corresponding Pfaffian systems on F (k) , F (∞) . One of the advantage of introducing F (∞) is that the infinitely prolonged Pfaffian system I(∞) possesses a well defined set of generators on F (∞) . The higher-order differential analysis for the minimal Lagrangian system will be carried out on F (∞) in an SO(2)-equivariant way so that it has a well defined meaning on X (∞) . 4.1.3. Zariski open sets (1)

⊂ X (1) be the open subset defined by the independence condition

(1)

:= {E ∈ X (1) |(ξ ∧ ξ )|E ̸= 0}.9

Let X0 X0

(4.3)

Inductively define the corresponding sequence of open subsets, (k+1)

X0

:= πk−+11,k (X0(k) ),

k ≥ 1.

Set (∞)

X0

:= lim X0(k) ⊂ X (∞) . ←−

(k)

(∞)

Let F0 ⊂ F (k) , F0 ⊂ F (∞) denote the corresponding open subsets. Dually, define the complementary open subset (1) X∞ := {E ∈ X (1) |(η2 ∧ η2 )|E ̸= 0}.

(4.4)

(k) (∞) (k) (∞) Define the associated open subsets X∞ , X∞ , F∞ , F∞ similarly as above. (∞) (∞) Note that the pair of open subsets {X0 , X∞ } form an open covering of (∞)

(∞)

X (∞) (and similarly the pair {F0

, F∞(∞) } of

).

F

(∞)

(∞)

Remark 4.5. The analysis can be carried out on X∞ , F∞ as follows. Given the 2-form η2 ∧ξ in the ideal I, recall the prolongation variable h3 defined by (2.18). Introduce the new prolongation variable p3 such that

ξ = p3 η2 (1)

(1)

(1)

1 on the integral 2-planes in F∞ (and hence p3 = h− on F0 ∩ F∞ ). By switching from h3 to p3 , it is clear that one 3 (∞) may proceed to the infinite prolongation on F∞ in such a way that the associated formulas agree on the intersection

(∞)

F0

∩ F∞(∞) .

9 Note that the tangential lift of an immersed minimal Lagrangian surface in M takes values in X (1) . 0

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4.1.4. Infinitely prolonged structure equations Recall the induced structure Eq. (2.15) on an immersed minimal Lagrangian surface in M. By the general theory of prolongations, [20, Chapter VI], set

η2 = h3 ξ for the prolongation variable h3 . From the third equation of (2.15), mod ξ , I(∞) .

dh3 + 3ih3 ρ ≡ 0

Inductively define the higher-order derivatives of h3 by mod I(∞) , j = 3, 4, . . . ,

dhj + ijhj ρ ≡ hj+1 ξ + Tj ξ

(4.6)

where T3 = 0,

(4.7)

j−3

Tj+1 =



aj,s hj−s ∂ξs R,

for j ≥ 3,

s =0

aj , s =

(j + 2s + 3) 2(j − 1)



j−1



s+2

,

∂ξs R = δ0s γ 2 − 2h3+s h¯ 3 . ∞ Here the sequence { hj }∞ j=3 are the prolongation variables. The sequence of ξ -coefficients { Tj+1 }j=4 are uniquely determined

by requiring that d(dhj ) ≡ 0 mod I(∞) for j = 3, 4, . . . . This implies the recursive relation Tj+1 = ∂ξ Tj +

j

T3 = 0 , T4 =

3

R hj . 2 For example, the first few terms are, 2

(4.8)

h3 (γ 2 − 2h3 h¯ 3 ), T5 =

7 2

γ 2 h4 − 10h3 h¯ 3 h4 .

Remark 4.9. For a scalar function f on F (∞) , the notation ∂ξ f , or fξ means the total derivative with respect to ξ , i.e., the

ξ -coefficient of df modulo I(∞) , etc. See (3.9), (4.13).

Based on Eqs. (4.6), (4.7), we define the following set of differential forms and vector fields on the open subset ⊂ F0(∞) .10

(∞)

F∗∗

(a) Differential forms: Set

ηj θj

= dhj = ηj

+ijhj ρ

(k)

−Tj ξ ,

−h j + 1 ξ ,

for j ≥ 3, for j ≥ 2.

(4.10)

(k)

On the open subset F∗∗ ⊂ F0 , the set of 1-forms

{ρ, ξ , ξ , θ0 , θ1 , θ 1 , . . . , θk+1 , θ k+1 , ηk+2 , ηk+2 } (∞)

form a coframe. On the open subset F∗∗

⊂ F0(∞) , the set of 1-forms

{ρ, ξ , ξ , θ0 , θ1 , θ 1 , θ2 , θ 2 , . . .}

(4.11)

form a coframe. By construction, I(∞) = ⟨θ0 , θ1 , θ 1 , θ2 , θ 2 , . . .⟩. (∞)

(b) Vector fields: On the open subset F∗∗

⊂ F0(∞) , let

{Eρ , ∂ξ = Eξ , ∂ξ = Eξ , E0 , E1 , E1¯ , E2 , E2¯ , . . .}

(4.12)

be the dual frame of (4.11). By definition,

∂ξ := total derivative with respect to ξ , ∂ξ := total derivative with respect to ξ 10 See Section 4.3.1.

(4.13) mod I

(∞)

.

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307

In terms of these 1-forms, for example, the covariant derivatives of Tj are written as ¯

dTj + i (j − 1)Tj ρ = (∂ξ Tj )ξ + (∂ξ Tj )ξ + Tj3 θ 3 +

j−1 

Tjs θs ,

j ≥ 3,

s =3

where Tjs = Es (Tj ), Tjs¯ = Es¯ (Tj ). With this preparation, we proceed to determine the structure equations satisfied by the 1-forms { θj }. Recall from (2.13), dξ = iρ ∧ ξ − 3γ 2 θ0 ∧ θ 1 − θ1 ∧ θ 2 − h¯ 3 θ1 ∧ ξ , dθ0 = −

1 2

(4.14)

θ1 ∧ ξ + θ 1 ∧ ξ , 

dθ1 + iρ ∧ θ1 = −θ2 ∧ ξ + 3γ 2 θ0 ∧ ξ , dθ2 + 2iρ ∧ θ2 = −θ3 ∧ ξ + 3h3 γ 2 θ0 ∧ θ 1 + h3 θ1 ∧ θ 2 + (γ 2 + h3 h¯ 3 )θ1 ∧ ξ , dρ =

 i  Rξ ∧ ξ − 2θ2 ∧ θ 2 − γ 2 θ1 ∧ θ 1 − 2h¯ 3 θ2 ∧ ξ + 2h3 θ 2 ∧ ξ .

2

Lemma 4.15. For j ≥ 3, dθj + ijρ ∧ θj = −θj+1 ∧ ξ + 3γ 2 θ0 ∧ (Tj θ1 + hj+1 θ 1 ) +

jhj  2

γ 2 θ1 ∧ θ 1 + 2θ2 ∧ θ 2

+ Tj θ 1 ∧ θ2 + hj+1 θ1 ∧ θ 2 + τj′ ∧ ξ + τj′′ ∧ ξ ,



(4.16)

where

τj = h3 Tj θ 1 − jh3 hj θ 2 , ′

 τj′′ = h¯ 3 hj+1 θ1 + jh¯ 3 hj θ2 −

3¯

Tj θ 3 +

j −1 

 Tjs θs

.

s=3

Proof. Direct calculation.



Corollary 4.17. The dual frame (4.12) satisfies the following commutation relations.

[Eℓ , Eξ ] =



Tjℓ Ej ,

ℓ ≥ 3.

(4.18)

j≥ℓ+1

Proof. Let θ be a differential 1-form, and let Ea , Eb be vector fields. The corollary follows from Cartan’s formula dθ (Ea , Eb ) = Ea (θ (Eb )) − Eb (θ (Ea )) − θ ([Ea , Eb ]).  Notation 4.19. We introduce the following notations in order to keep track of orders (k ≥ 2):

• O (k): functions on X (∞) , F (∞) which do not depend on hj for j > k. • O (−k): functions on X (∞) , F (∞) which do not depend on h¯ j for j > k. Note for example that Tj ∈ O (j − 1) ∩ O (−3). 4.2. Triple cover 4.2.1. Motivation ˆ → Σ of an immersed integral surface of (X , I) defined in Definition 2.22 prompts to define a global The triple cover Σ triple cover Xˆ (∞) → X (∞) such that we have the following commutative diagram:

ˆ Σ

/ Xˆ (∞)



 / X (∞)

Σ 4.2.2. Definition Let KF → F ,

KηF2 → F ,

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be the (trivial) complex line bundles generated by the 1-forms ξ , η2 respectively. Recall the principle SO(2)-bundle Π : F → X , and the structure equation (2.13). Let K → X,

Kη 2 → X ,

be the corresponding induced line bundles such that

Π ∗ Kη2 = KηF2 .

Π ∗K = K F ,

Note from (2.13) that Kη2 = K −2 . Consider F (1) = Π ∗ (X (1) ). Since an element in F (1) is defined by the equation

η2 − h3 ξ = 0, we have

F (1) ∼ = P(K F ⊕ KηF2 ) ∼ = F × CP1 . It follows that X (1) ∼ = P(K p+1 ⊕ K p−2 ) for any integer p. For our purpose, we choose X (1) ∼ = P(K 3 ⊕ C). Definition 4.20. The triple cover Xˆ (1) of the first prolongation X (1) is defined by Xˆ (1) := P(K ⊕ C). Let

ν : Xˆ (1) → X denote the triple covering map (projection). Define similarly the triple cover

Fˆ (1) := P(K F ⊕ C). The corresponding triple covering map is also denoted by ν . Note that, when restricted to the CP1 -fibers of prolongation, ν becomes the usual triple covering map,

ν : CP1 → CP1 , → [x3 , y3 ], branched at the two points 0 = [0, 1] and ∞ = [1, 0]. Let ˆI(1) = ν ∗ I(1) be the pulled-back Pfaffian system, and set the new differential system

(Xˆ (1) , ˆI(1) ). ˆ → Σ admits a unique lift to As claimed in Section 4.2.1, it is clear that the triple cover of a minimal Lagrangian surface Σ Xˆ (1) as an integral surface of ˆI(1) . Definition 4.21. The sequence of triple covers of the prolongations are defined by Xˆ (k) := ν ∗ (X (k) ),

Fˆ (k) := ν ∗ (F (k) ),

for 2 ≤ k ≤ ∞.

By definition, we have the commutative diagram: Xˆ (k)



νk

πˆ k,1

Xˆ (1)

ν

/ X (k) 

πk,1

/ X (1)

(and similarly for Fˆ (k) , Fˆ (1) , etc.). Set the ideals

ˆI(k) := νk∗ (I(k) ).

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Proposition 4.22. Let x : Σ ↩→ X be an immersed integral surface of the differential system for minimal Lagrangian surfaces. Let x(k) : Σ ↩→ X (k) ,

1 ≤ k ≤ ∞,

ˆ → Σ be the triple cover defined by the Hopf differential of x, Definition 2.22. be the sequence of prolongations of x. Let ν : Σ (1) ( 1) ˆ ˆ There exists a lift xˆ : Σ ↩→ X and the associated sequence of prolongations xˆ

(k)

ˆ ↩→ Xˆ (k) , :Σ

2 ≤ k ≤ ∞,

such that (k)

(a) each xˆ (b) x

(k)

is integral to ˆI(k) ,

◦ ν = νk ◦ xˆ (k) .

The lift xˆ

(1)

and its prolongation sequence { xˆ

(k)

} are uniquely determined by these properties.

4.3. Balanced coordinates As is often the case with integrable differential equations, the infinite prolongation space Xˆ (∞) supports a preferred set of functions called balanced coordinates. 4.3.1. Set up (k)

(k)

Recall the open subsets X0 , X∞ , 1 ≤ k ≤ ∞, from (4.3) and below. Let (1)

X∗(1) := X0

(1) ∩ X∞ = { E ∈ X (1) | h3 |E ̸= 0, ∞ }. (k+1)

Inductively define the sequence of open subsets X∗

(k) X∗∗ := X∗(k) ∩ { hj ̸= ∞, 4 ≤ j ≤ k + 2 } ⊂ X∗(k) ,

:= πk−+11,k (X∗(k) ) ⊂ X (k+1) . Let k ≥ 2.

Denote the corresponding open subsets in the triple cover by (k)

(k) ˆ (k) ˆ (k) Xˆ 0 , Xˆ ∞ , X∗ , X∗∗ , etc. (∞)

(∞)

The balanced coordinates to be constructed will be smooth scalar functions on Xˆ ∗∗ . Note that Xˆ ∗∗ is connected. (k) (k) (k) (k) (k) (k) (k) (k) Let {F0 , F∞ , F∗ , F∗∗ , Fˆ0 , Fˆ∞ , Fˆ∗ , Fˆ∗∗ } denote the corresponding open subsets of F (k) , Fˆ (k) . 4.3.2. Definition (∞)

Definition 4.23. The balanced coordinates zj : Xˆ ∗∗

→ C, j ≥ 4, are defined by

−j

zj := h3 3 hj . The balanced coordinate ring R is the polynomial ring generated by the balanced coordinates,

R := C[z4 , z5 , . . . ]. (∞)

Originally defined on Fˆ∗∗ , these functions are invariant under the action by the structure group SO(2). They are well (∞) defined smooth functions on Xˆ ∗∗ . Note that the balanced coordinates {z4 , z5 , . . . } form a complex coordinate system when (∞) (1) restricted to a fiber of the projection Xˆ ∗∗ → Xˆ ∗ . 4.3.3. Structure equations Set

  r      ω

1 := |h3 | = (h3 h¯ 3 ) 2 ,

ζj      ˆ  T j

:= h3 3 θj ,

1

:= h33 ξ , −j

− j−31

:= h3

Tj ,

j ≥ 0, j ≥ 3.

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(∞)

They are all invariant under the action by the structure group SO(2) and descends to Xˆ ∗∗ . For example, this enables to (∞) express the ideal ˆI(∞) on Xˆ ∗∗ directly by

ˆI(∞) = ⟨θ0 , ζj , ζ j ⟩. Note the following structure equations. r

(z4 ω + z 4 ω),  j 2 dzj ≡ zj+1 − z4 zj ω + Tˆj r− 3 ω dr ≡

(4.24)

2



3

Tˆj+1 = ∂ω Tˆj +

j−1 3

z4 Tˆj +

j 2

mod ˆI(∞) ,

(γ 2 − 2r2 )zj ,

for j ≥ 4,

for j ≥ 3.

Here we set z3 = 1 for convenience, and denote by ∂ω the scaled total derivative operator −1

∂ω := h3 3 ∂ξ . 4.3.4. Spectral weight From these formulas, we note an important property of Tˆj . Definition 4.25. The spectral weights of the balanced coordinates are weight(zj ) = j − 3,

weight(z j ) = −(j − 3).

(4.26)

Set weight(r) = 0. Assign the weights for the 1-forms weight(ω) = −1,

weight(ω) = +1.

Lemma 4.27. For j ≥ 3, Tˆj ∈ R ⊕ r 2 R. Tˆj is weighted homogeneous of spectral weight j − 4. Proof. From the identities j

∂ω zj = zj+1 − z4 zj , 3

∂ω (r2 ) = r2 z4 , it follows that the operator ∂ω increases the spectral weight by +1 when acting on C[r2 , z4 , z5 , z6 , . . .]. Note the initial term Tˆ4 = 32 (γ 2 − 2r2 ) is of spectral weight 0. The rest follows from the inductive formula for Tˆj , (4.24).  4.3.5. Spectral weight vs. order The prolongation variable h3 represents the second fundamental form of a minimal Lagrangian surface, and hence it is a second order object. Consequently, the jet order of the balanced coordinate zj is j − 1. But, in order to match the order with the index notations for convenience, we re-define the order of the functions zj , z j as follows: zj zj

order j j

spectral weight j−3 −(j − 3)

(4.28)

4.4. Jacobi fields and pseudo-Jacobi fields The Jacobi operator E , and the pseudo-Jacobi operator E ′ , (3.11), (3.17), were originally defined for the scalar functions on X . From the analysis in Section 4.1.4 and the definition of the total derivative operators ∂ξ , ∂ξ in (4.13), it is natural to (∞)

extend them for the scalar functions on Xˆ ∗∗ . The Jacobi fields, and pseudo-Jacobi fields are in turn defined as the elements in the kernels of these operators. In Section 4.4.3, we show that the Jacobi fields are the generating functions of symmetries of the infinitely prolonged minimal Lagrangian system. In Section 4.4.4, we show that the pseudo-Jacobi fields are the generating functions of symmetries of the elliptic Tzitzeica equation underlying the minimal Lagrangian system.

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4.4.1. Definition (∞) Definition 4.29. A scalar function A : Xˆ ∗∗ → C is a Jacobi field if it satisfies the Jacobi equation

E (A) := ∂ξ ∂ξ A +

3 2

γ 2 A = 0.

(4.30)

The C-vector space of Jacobi fields is denoted by J(∞) . (∞) A scalar function P : Xˆ ∗∗ → C is a pseudo-Jacobi field if it satisfies the pseudo-Jacobi equation

E ′ (P ) := ∂ξ ∂ξ P +

1 2

(γ 2 + 4h3 h¯ 3 )P = 0.

(4.31)

The C-vector space of pseudo-Jacobi fields is denoted by J′(∞) . A higher-order Jacobi field is a Jacobi field in the ring R ⊕ R. The subspace of higher-order Jacobi fields is denoted (∞) by Jh . (k)

Let J(k) ⊂ J(∞) be the subspace of Jacobi fields of order ≤k + 2 defined on Xˆ ∗∗ . The space of Jacobi fields of order k + 2 is defined as the quotient space

Jk = J(k) /J(k−1) ,

k ≥ 1,

J0 = J(0) = {classical Jacobi fields}. ′(∞)

The subspaces of pseudo-Jacobi fields Jh

, J′(k) , J′ k are similarly defined.

4.4.2. Examples Example 4.32. A direct computation shows that z4 is a pseudo-Jacobi field (of order 4 and weight 1), and z5 −

5 3

z42

is a Jacobi field (of order 5 and weight 2). These examples provide a hint for the existence of higher-order (pseudo) Jacobi fields. 4.4.3. Interpretation of Jacobi fields From the geometric theory of differential equations, [24], a generating function of symmetry of a differential equation is characterized as an element in the kernel of its linearization. For the minimal Lagrangian system, it turns out that the generating functions of symmetries are Jacobi fields. (∞) Recall the coframe of Fˆ∗∗ ,

{ρ, ξ , ξ , θ0 , θ1 , θ 1 , θ2 , θ 2 , . . .}, and its dual frame

{Eρ , Eξ , Eξ , E0 , E1 , E1¯ , E2 , E2¯ , . . .}. (∞)

(∞)

By a vector field on Xˆ ∗∗ , we mean a vector field (derivation) on Fˆ∗∗ V = Vξ Eξ + Vξ Eξ + V0 E0 +

∞ 

of the form

(Vj Ej + V¯j E¯j )

j =1

(with complex coefficients Vξ , Vξ , V0 , . . . ) which is invariant under the action by the structure group SO(2) of the principal (∞)

bundle Fˆ∗∗

(∞) (∞) (∞) → Xˆ ∗∗ . We denote the set of vector fields on Xˆ ∗∗ by H 0 (T Xˆ ∗∗ ⊗ C). (∞)

Definition 4.33. A vector field V ∈ H 0 (T Xˆ ∗∗ derivative LV preserves the ideal ˆI(∞) ,

(∞) ˆ(∞) ⊗ C) is a symmetry of the differential system (Xˆ ∗∗ , I ) if the formal Lie

LV ˆI(∞) ⊂ ˆI(∞) . The C-Lie algebra of symmetry vector fields is denoted by S. A symmetry V ∈ S is vertical when Vξ = Vξ = 0 and it has no (horizontal) Eξ , Eξ components. The subspace of vertical symmetries is denoted by Sv .

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J.S. Wang / Journal of Geometry and Physics 114 (2017) 291–328

(∞)

Note by definition that a vertical symmetry V ∈ H 0 (T Xˆ ∗∗ V = V0 E 0 +

∞ 

⊗ C) is written as

(Vj Ej + V¯j E¯j ) ∈ H 0 (T Xˆ (∞) ).

(4.34)

j =1

Proposition 4.35. For a vertical symmetry V ∈ Sv of the minimal Lagrangian system, the E0 coefficient V0 is a Jacobi field. Conversely, a Jacobi field A uniquely determines a vertical symmetry V of the form (4.34) with the generating function V0 = A. As a consequence, there exists a canonical isomorphism

J(∞) ≃ Sv . Proof. The claim follows from the defining relations,

LV θ0 , LV θj , LV θ j ≡ 0 mod ˆI(∞) , for j = 1, 2, . . . . Let us check the first three 1-forms {θ0 , θ1 , θ 1 }. The equation LV θ0 ≡ 0 mod ˆI(∞) shows that

 dV0 − V y

1 2



1

(θ1 ∧ ξ + θ 1 ∧ ξ ) ≡ dV0 − (V1 ξ + V1¯ ξ ) 2

≡ 0 mod ˆI(∞) . One gets 1

dV0 ≡

2

(V1 ξ + V1¯ ξ ) mod ˆI(∞) .

(4.36)

By a similar computation, the equations LV θ1 , LV θ 1 ≡ 0 mod ˆI(∞) show that dV1 + iV1 ρ ≡ V2 ξ

− 3γ 2 V0 ξ ,

dV1¯ − iV1¯ ρ ≡ V2¯ ξ

− 3γ 2 V0 ξ ,

(4.37)

mod ˆI(∞) .

Eqs. (4.36), (4.37) imply that V0 is a Jacobi field. The sequence of equations LV θj , LV θ j ≡ 0 mod ˆI(∞) , for j = 2, 3, . . . , show that the rest of the coefficients Vj , V¯j , for j = 2, 3, . . . , are determined inductively by the successive derivatives of the generating function V0 . The compatibility of these inductive defining equations for {V0 , V1 , V1¯ , . . .} can be checked from the formula for Tj and its differential consequences. This is a straightforward computation.  4.4.4. Interpretation of pseudo-Jacobi fields We show that the pseudo-Jacobi fields correspond to the symmetries of the elliptic Tzitzeica equation which underlies the minimal Lagrangian system. Away from the umbilic divisor on a minimal Lagrangian surface, take a local holomorphic coordinate z such that 1

dz = h33 ξ , and the Hopf differential is normalized to I = (dz )3 . Set u

3 h3 = e− 2 u

ξ = e 2 dz ,

for a real scalar function u. The connection 1-form ρ is given by ρ = R = −2e

−u

i 2

(uz dz − uz dz ), and the curvature R is,

uzz .

The compatibility equation (2.19) becomes the elliptic Tzitzeica equation uzz +

1 2

 γ 2 eu − 2e−2u = 0.

(4.38) 1

−1

On the other hand, consider the pseudo-Jacobi operator (4.31). From dz = h33 ξ , one gets ∂z = h3 3 ∂ξ . Hence

∂ξ ∂ξ = e−u ∂z ∂z , and the pseudo-Jacobi operator becomes ′

−u

E =e



1

∂z ∂z + (γ e + 4e 2

2 u

−2u

 ) .

Up to scaling, this is the linearization of (4.38).

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313

5. Formal Killing fields The Grassmann bundle of oriented Lagrangian 2-planes X = Gr+ (2, TM ) = F / SO(2), on which the original minimal Lagrangian system is defined, is a 6-symmetric space associated with the Lie group SL(3, C) (complexification of G), and the minimal Lagrangian surfaces arise as the primitive maps, [16]. From the well known theory of integrable systems, the g-valued 1-form ψ restricted to a minimal Lagrangian surface, (3.1) mod ˆI(∞) , admits an extension to a loop algebra gC [λ−1 , λ]-valued11 1-form ψλ , (5.3), by inserting the auxiliary spectral parameter λ. The structure equations for the minimal Lagrangian system imply that the extended 1-form ψλ continues to satisfy the Maurer–Cartan equation, dψλ + ψλ ∧ ψλ ≡ 0

mod ˆI(∞) .

In order to make use of this extra structure of the minimal Lagrangian system, we give a construction of two12 formal loop algebra gC [[λ]]-valued canonical formal Killing fields with respect to ψλ . The recursive structure equations for the formal Killing fields reveal that; (a) an infinite sequence of higher-order (pseudo) Jacobi fields arise as the coefficients of the formal Killing fields, (b) there exists a pair of 3-step recursion relations between these Jacobi fields and pseudo-Jacobi fields, which generalize the recursion relations from the classical Killing fields, Section 3.2.3. For two particular sets of initial data drawn from Example 4.32, we give the explicit inductive differential algebraic formulas for the formal Killing fields, Theorems 5.19 and 5.21. Consequently, we obtain an infinite sequence of higher-order (pseudo) Jacobi fields, Corollaries 5.20 and 5.22. In Section 5.1, we record the recursive structure equations for the formal Killing fields with respect to the extended Maurer–Cartan form. In Section 5.2, we determine the first few terms of the formal Killing fields for the two initial ansätze from Example 4.32. In Section 5.3, we give the inductive differential algebraic formulas for the formal Killing fields. Notation 5.1. Let λ denote the auxiliary spectral parameter. Recall the following list of associated rings.

C[λ] : ring of polynomials, −1

C[λ

, λ] : ring of Laurent polynomials,

C[[λ]] : ring of formal series, C((λ)) : field of formal Laurent series. The tensor products of the Lie algebra gC with these rings will be denoted by,

gC [λ], gC [λ−1 , λ], gC [[λ]], gC ((λ)). 5.1. Structure equations 5.1.1. Extended Maurer–Cartan form Recall the g-valued Maurer–Cartan form ψ , (3.1). Evaluating modulo ˆI(∞) ,

ψ ≡ ψ+ + ψ0 + ψ− ,

mod ˆI(∞) ,

where

· γ ψ− := 2 −i γ  · · · ψ0 := · · −ρ  1

ψ+ :=

1 2



0

γ iγ

−γ ih3

−h 3  · ρ , · −γ ih¯ 3 h¯ 3

iγ −h 3 ξ , −ih3



 −i γ h¯ 3  ξ . −ih¯ 3

11 See Notation 5.1 for the loop algebra related notations. 12 Two = rank of g.

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J.S. Wang / Journal of Geometry and Physics 114 (2017) 291–328

Definition 5.2. The extended Maurer–Cartan form is the gC [λ−1 , λ]-valued 1-form on Fˆ (∞) given by

ψλ := λψ+ + ψ0 + λ−1 ψ− .

(5.3)

The structure equations for the minimal Lagrangian system show that it continues to satisfy the structure equation mod ˆI(∞) .

dψλ + ψλ ∧ ψλ ≡ 0

(5.4)

Note that the extended 1-form ψλ takes values in g when λ is a unit complex number. 5.1.2. Formal Killing fields Recall the decomposition of gC in Fig. 3.1. By extending each of the scalar variables {p, b, c, f, a, g, s, t} to a formal series in C[[λ]], we give an abridged definition of the formal Killing fields. Definition 5.5. Let ψλ be the extended Maurer–Cartan form (5.3). A formal Killing field is a formal loop algebra valued function (∞) Xλ : Fˆ∗∗ → gC [[λ]],

such that, (a) under the decomposition Fig. 3.1, the components of Xλ are the formal series in λ given by p=



a=



b=



a6k+7 λ6k+5 ,

g=



c=



g 6k+2 λ6k ,

c 6k+5 λ6k+3 ,

s=



s6k+3 λ6k+1 ,

f=



f 6k+6 λ6k+4 ,

t=



t 6k+3 λ6k+1 .

p6k+4 λ6k+2 , b6k+5 λ6k+3 ,

(5.6)

Here the sums are over the integer index k from 0 to ∞. (b) it satisfies the Killing field equation mod ˆI(∞) .

dXλ + [ψλ , Xλ ] ≡ 0

(5.7)

5.1.3. Recursive structure equations The Killing field equation (5.7) implies the following recursive structure equations. [n th equation] dp6n+4 = (iγ b6n+5 + 2ih3 c 6n+5 )ξ + (iγ s6n+3 + 2ih¯ 3 t 6n+3 )ξ , db6n+5 + ib6n+5 ρ = ih3 f 6n+6 ξ +

i 2

γ p6n+4 ξ ,

dc 6n+5 − 2ic 6n+5 ρ = iγ f 6n+6 ξ + ih¯ 3 p6n+4 ξ , df 6n+6 − if 6n+6 ρ =

3i 2

γ a6n+7 ξ + (iγ c 6n+5 + ih¯ 3 b6n+5 )ξ ,

da6n+7 = iγ g 6n+8 ξ + iγ f 6n+6 ξ , dg 6n+8 + ig 6n+8 ρ = (−iγ t 6n+9 − ih3 s6n+9 )ξ + ds6n+9 − is6n+9 ρ =

i 2

3i 2

γ a6n+7 ξ ,

γ p6n+10 ξ − ih¯ 3 g 6n+8 ξ ,

dt 6n+9 + 2it 6n+9 ρ = ih3 p6n+10 ξ − iγ g 6n+8 ξ ,

(mod ˆI(∞) ).

The following lemma is a formal Killing field analogue of Eqs. (3.12), (3.18). Lemma 5.9. Given a formal Killing field Xλ , (a) p6n+4 is a pseudo-Jacobi field, (b) a6n+7 is a Jacobi field. Thus, one gets an infinite sequence of (pseudo) Jacobi fields from a formal Killing field.

(5.8)

J.S. Wang / Journal of Geometry and Physics 114 (2017) 291–328

315

Fig. 5.1. Recursion diagram for formal Killing fields.

5.1.4. Recursion diagram The recursive structure equations (5.8) can be summarized by the following infinite schematic diagram of period 6, Fig. 5.1. Two nodes are connected by an arrow only if there exists a first order differential relation between them from the structure equations. From a node, one moves to the right by applying ∂ξ , and to the left by applying ∂ξ . The upper indices are designated to match (roughly) the jet orders of the coefficients. The structure equations show that the right-arrows are differential, which means that the coefficients {f ∗ , a∗ , g ∗ , p∗ } are obtained from the left-adjacent term(s) by applying ∂ξ . But, the left-arrows decrease the jet order. In addition, note that

∂ξ p6n+4 = iγ b6n+5 + 2ih3 c 6n+5 , ∂ξ g 6n+8 = −iγ t 6n+9 − ih3 s6n+9 . In order to continue the recursion process, one needs to solve for the coefficients {b∗ , c ∗ , s∗ , t ∗ }. The main idea of construction is to impose the following algebraic constraints in terms of the characteristic polynomial of Xλ ; det(µI3 + Xλ ) = µ3 + cλ3 , for a constant c ∈ C∗ . This allows one to solve for {b∗ , c ∗ , s∗ , t ∗ } differential algebraically not just by using the left-adjacent terms, but by using all of the lower-order terms (the left hand side terms in the diagram above). The relevant explicit formulas using the truncated formal Killing fields are given in Section 5.3. 5.2. Initial analyses Recall from Example 4.32 that z4 is a pseudo-Jacobi field, and z5 − 53 z42 is a Jacobi field. We start the process of solving the formal Killing field equation by determining the first few terms generated by these initial data. 5.2.1. Case p4 = z4 Set g 2 = 0. By inspection, set 3i

s3 = −

−1

γ h3 3 ,

3i

t3 =

2

h33 .

2 2 Differentiating these, one gets p4 = z4 as expected. Solving the equation ∂ξ b5 = b5 = −

i

1 3

3γ

 z5 −

h3

5 3

z42

i 2



γ z4 , one gets .

From the equation ∂ξ p4 = iγ b5 + 2ih3 c 5 , this implies i −2 c 5 = − h3 3 3

 z5 −

7 6

z42



.

Successive derivatives of b5 give f6 = − a7 = 8

g =

1

− 13

3γ

2i

h3

9γ 3

z6 −

14 3

 z7 − 7z6 z4 −

9γ 2 2



1 3

h3

 z8 −

28 3

z5 z4 + 14 3

35

z52 +

z7 z4 −

49 3

9

z43



245 9

,

z5 z42 −

z6 z5 +

455 9

455 27

z44

z6 z42



+

,

70z52 z4

−

5005 27

z5 z43

+

7280 81

z45



.

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J.S. Wang / Journal of Geometry and Physics 114 (2017) 291–328

Proceed with the similar computations as above by solving (by inspection) the associated ∂ξ -equations. This gives s9 =

2i

27γ 1976

+

27



z9 − 11z8 z4 − 22360

z53 −

27

79

689

z7 z5 +

3 108680

z52 z42 +

81

9

z7 z42 − 16z62 + 286z6 z5 z4 − 380380

z5 z44 −

2 3

729 33

758

3380 9

z6 z43



z46 ,

901 3770 h z9 − 12z8 z4 − z7 z5 + z7 z42 − z62 + z6 z5 z4 − z6 z43 27γ 4 3 3 9 2 3 9 1847 3 47255 2 2 120380 432250 6  + z5 − z5 z4 + z5 z44 − z4 , 27 54 81 729  43 112 1118 175 4979 21164 4 z10 − z9 z4 − z8 z5 + z8 z42 − z7 z6 + z7 z5 z4 − z7 z43 = 4 27γ 3 3 9 3 9 27 1066 2 4550 116324 301340 + z6 z4 + z6 z52 − z6 z5 z42 + z6 z44 3 9 27 81 165776 3 286520 2 3 3151720 9509500 7  − z5 z4 + z5 z4 − z5 z45 + z4 . 81 27 243 2187 4i

t9 =

p10

− 13

h 3 3

76



By Lemma 5.9, a7 is a Jacobi field, and p10 is a pseudo-Jacobi field. 5.2.2. Case a5 = z5 − 35 z42 For the formal Killing field generated by the Jacobi field z5 − 53 z42 , it is convenient to lower the upper indices of the formal Killing field coefficients by 2 to match the order. The resulting structure equations are, p=



a=



b=



a6k+5 λ6k+3 ,

g=



c=



g 6k+6 λ6k+4 ,

c 6k+3 λ6k+1 ,

s=



s6k+7 λ6k+5 ,

f=



f 6k+4 λ6k+2 ,

t=



t 6k+7 λ6k+5 .

p6k+2 λ6k , b6k+3 λ6k+1 ,

(5.10)

[n th equation ′ ] dp6n+2 = (iγ b6n+3 + 2ih3 c 6n+3 )ξ + (iγ s6n+1 + 2ih¯ 3 t 6n+1 )ξ , db6n+3 + ib6n+3 ρ = ih3 f 6n+4 ξ +

i 2

(5.11)

γ p6n+2 ξ ,

dc 6n+3 − 2ic 6n+3 ρ = iγ f 6n+4 ξ + ih¯ 3 p6n+2 ξ , df 6n+4 − if 6n+4 ρ =

3i 2

γ a6n+5 ξ + (iγ c 6n+3 + ih¯ 3 b6n+3 )ξ ,

da6n+5 = iγ g 6n+6 ξ + iγ f 6n+4 ξ , dg 6n+6 + ig 6n+6 ρ = (−iγ t 6n+7 − ih3 s6n+7 )ξ + ds6n+7 − is6n+7 ρ =

i 2

3i 2

γ a6n+5 ξ ,

γ p6n+8 ξ − ih¯ 3 g 6n+6 ξ , (mod ˆI(∞) ).

dt 6n+7 + 2it 6n+7 ρ = ih3 p6n+8 ξ − iγ g 6n+6 ξ , We proceed to solve for the first few terms. Set p2 = 0. By inspection, set 1 9 b3 = − γ h33 , 2

c3 =

9 4

−2

γ 2 h3 3 .

Differentiating these equations successively, one gets 4

f =

3i 2

− 31

γ h3 z4 ,

5

a = z5 −

Note that a5 is as expected.

5 3

z42

,

6

g =−

i

γ

1 3

h3

 z6 − 5z5 z4 +

40 9

z43



.

J.S. Wang / Journal of Geometry and Physics 114 (2017) 291–328

317

Similar computations as in the previous case yield, s7 = t7 =

1 3γ 2 3γ

− 31



16

2 3



3 29

z7 − 6z6 z4 −

h3

2

h3 z7 − 7z6 z4 −

z52 +

z52 +

220 9 250

z5 z42 −

z5 z42 −

385 27 935



z44 ,



z44 ,

6 9 54  2i 26 50 418 616 2 1540 6545 5  p8 = − 2 z 8 − z7 z4 − z6 z5 + z6 z42 + z5 z4 − z5 z43 + z4 , 3γ 3 3 9 9 9 81  2 1 74 748 902 3740 b9 = − 3 h33 z9 − 12z8 z4 − z7 z5 + z7 z42 − 17z62 + z6 z5 z4 − z6 z43 9γ 3 9 3 9 1760 3 23320 2 2 118745 425425 6  + z5 − z5 z4 + z5 z44 − z4 , 27 27 81 729  2 77 2 − 682 33 2 c 9 = − 2 h3 3 z9 − 11z8 z4 − z7 z5 + z7 z42 − z6 + 286z6 z5 z4 − 374z6 z43 9γ 3 9 2 1892 3 22066 2 2 752675 6  107525 z5 − z5 z4 + z5 z44 − z4 . + 27 27 81 1458 Successively differentiating b9 , c 9 , one gets, 1144 176 1672 21692 z8 z5 + z8 z42 − z7 z6 + z7 z5 z4 − z7 z43 3 9 3 3 27 118184 309485 164560 3 + 363z62 z4 + z6 z52 − z6 z5 z42 + z6 z44 − z5 z4 9 27 81 81  1075250 9784775 7 871420 2 3 z5 z4 − z5 z45 + z4 , + 81 81 2187  55 154 1760 2948 41140 4 286 z11 − z10 z4 − z9 z5 + z9 z42 − z8 z6 + z8 z5 z4 − z8 z43 = 27γ 4 3 3 9 3 3 27 14014 9482 268532 247775 176 2 z7 + z7 z6 z4 + z7 z52 − z7 z5 z42 + z7 z44 − 3 9 9 27 27 12199 2 173723 2 2 158950 5344460 10343905 + z6 z5 − z6 z4 − z6 z52 z4 + z6 z5 z43 − z6 z45 9 27 9 81 243 164560 4 11133980 3 2 283758475 8  36171410 2 4 320101925 − z5 + z5 z4 − z5 z4 + z5 z46 − z4 . 81 243 243 2187 6561

f 10 =

a11

2i

9γ

3

− 13

h3



z10 −

44

3 4466

z9 z4 −

110

By Lemma 5.9, a5 , a11 are Jacobi fields, and p8 is a pseudo-Jacobi field. 5.3. Inductive formulas Based on the initial analyses, we give the inductive differential algebraic formulas for the formal Killing fields in the following two subsections. 5.3.1. Case p4 = z4 Assume the initial data from Section 5.2.1. We wish to extend this to a formal Killing field X(p4 ) which satisfies the algebraic constraint det(µI3 + X(p4 )) = µ3 +



27 2

 γ 2 λ3 .

Here I3 denotes the 3-by-3 identity matrix. [Formulas for s6n+3 , t 6n+3 ]. Suppose all the coefficients up to g 6n+2 are known, n ≥ 1. We determine the formulas for 6n+3 {s , t 6n+3 }. Set the truncated formal Killing field

−2ia := −b + f + g + s −ib − if + ig − is 

X6n+2

b+f+g−s ic + ia − it p+c+t

ib − if + ig + is −p + c + t , −ic + ia + it



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J.S. Wang / Journal of Geometry and Physics 114 (2017) 291–328

where p

b

c

=

=

=

n 

p6k+4 λ6k+2 ,

a

=

k=0

k=0

n−1 

n 

b6k+5 λ6k+3 ,

g

=

k=0

k=0

n−1 

n 

c

λ

6k+5 6k+3

,

s

=

k=0

f

=

n−1 

n−1 

a6k+7 λ6k+5 , g 6k+2 λ6k , (5.12) s

λ

6k+3 6k+1

,

k=0

f 6k+6 λ6k+4 ,

t

=

k=0

n 

t 6k+3 λ6k+1 .

k=0

Here the unknown coefficients are {s6n+3 , t 6n+3 , p6n+4 }. The determinant is given by det(X6n+2 ) = i(4gsp − 4fga − 4b2 c − 4f2 t + 4g2 c + 4s2 t + 2a3 − 2ap2 + 8act − 4bsa + 4bfp). Expanding this as a series in λ, let us denote det(X6n+2 ) :=

3n 

6j+3

x6n+2 λ6j+3 .

j =0

From the determinant formula and by induction, one finds by checking the λ-degrees that 6j+3

x6n+2 = 0,

for 1 ≤ j ≤ n − 1.

Consider now the derivative

∂ξ (det(X6n+2 )). The structure equation shows that this term stems from the absence of b6n+5 , c 6n+5 -terms in X6n+2 . Hence only the terms that contain p6n+4 contribute to ∂ξ (det(X6n+2 )), and we get 6n+3 ∂ξ x6n +2 = 0.

By Corollary 6.5 and weighted homogeneity, this implies 6n+3 x6n +2 = 0.

Note from the determinant formula, +3 6n+3 3 3 6n+3 + (s3 )2 t 6n+3 ) + y6n x6n +2 = 4i(2s t s 6n+2 ,

(5.13)

where ∈ O (6n + 2). On the other hand, 6n+3 y6n +2

∂ξ g 6n+2 = −ih3 s6n+3 − iγ t 6n+3 .

(5.14)

Combining (5.13), (5.14), s

6n+3

=

t 6n+3 =

i 27γ

−1



h3



i 27γ 2

9γ ∂ξ g

6n+2

+

2 3

+3 h3 y6n 6n+2 2

+3 18γ ∂ξ g 6n+2 − h33 y6n 6n+2





,

(5.15)

.

[Formulas for b6n−1 , c 6n−1 ]. Suppose all the coefficients up to p6n−2 are known, n ≥ 1. We determine the formulas for

{b6n−1 , c 6n−1 }.

Given the truncated formal Killing field X6n+2 as above, let det(µI3 + X6n+2 ) = µ3 + σ2 (X6n+2 )µ + det(X6n+2 ) be the characteristic polynomial. For the case at hand, we utilize σ2 (X6n+2 ). It is given by

σ2 (X6n+2 ) = 3a2 + p2 − 4ct − 4bs − 4fg. Expanding as a series in λ, let us denote

σ2 (X6n+2 ) :=

2n  j =0

6j+4

x6n+2 λ6j+4 .

J.S. Wang / Journal of Geometry and Physics 114 (2017) 291–328

319

By the similar argument as above, 6j−2

x6n+2 = 0,

for 1 ≤ j ≤ n.

Note from the formula for σ2 (X6n+2 ), −2 −2 3 6n−1 x6n − 4t 3 c 6n−1 + y6n 6n+2 = −4s b 6n+2 ,

(5.16)

6n−2 where y6n +2 ∈ O (6n − 2). On the other hand,

∂ξ p6n−2 = iγ b6n−1 + 2ih3 c 6n−1 .

(5.17)

Combining (5.16), (5.17),

  1 −2 −3∂ξ p6n−2 + h33 y6n 6n+2 , 9γ   1 i 1 6n−2 3 6n−2 = − h− + h 6 ∂ p y ξ 3 3 6n+2 .

b6n−1 = c 6n−1

i

(5.18)

18

Theorem 5.19. Given the ansatz p4 = z4 and the initial data described in Section 5.2.1, (a) there exists a gC [[λ]]-valued canonical formal Killing field X(p4 ) which extends these data. Its components are determined by the structure equations (5.8), and the algebraic constraint det(µI 3 + X(p4 )) = µ3 +



27 2

 γ 2 λ3 .

Here I 3 denotes the 3-by-3 identity matrix. (b) each of the series coefficients of X(p4 ) is a weighted homogeneous polynomial in R, up to scaling by an appropriate power 1

of h33 . Corollary 5.20. Given the formal Killing field X(p4 ),

{p6n+4 , p6n+4 }∞ n=0 are distinct higher-order pseudo-Jacobi fields, and

{a6n+7 , a6n+7 }∞ n =0 are distinct higher-order Jacobi fields. 5.3.2. Case a5 = z5 − 53 z42 Recall that we follow (5.10), (5.11). Assume the initial data from Section 5.2.2. By the same analysis as in Section 5.3.1, we extend this to a formal Killing field X(a5 ) which satisfies the algebraic constraint det(µI3 + X(a )) = µ − 5

3



729 4

iγ

4



λ3 .

We omit the details of the arguments for this case. Theorem 5.21. Given the ansatz a5 = z5 − 53 z42 and the initial data described in Section 5.2.2, (a) there exists a gC [[λ]]-valued canonical formal Killing field X(a5 ) which extends these data. Its components are determined by the structure equation (5.11), and the algebraic constraint det(µI 3 + X(a5 )) = µ3 −



729 4

iγ 4



λ3 .

(b) each of the series coefficients of X(a5 ) is a weighted homogeneous polynomial in R, up to scaling by an appropriate power 1

of h33 .

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J.S. Wang / Journal of Geometry and Physics 114 (2017) 291–328

Corollary 5.22. Given the formal Killing field X(a5 ),

{a6n+5 , a6n+5 }∞ n =0 are distinct higher-order Jacobi fields, and

{p6n+8 , p6n+8 }∞ n =0 are distinct higher-order pseudo-Jacobi fields. Definition 5.23. The pair of formal Killing fields X(p4 ), X(a5 ) are canonical formal Killing fields for the minimal Lagrangian system. 6. Classification of (pseudo) Jacobi fields In this section, we give a complete classification of the (pseudo) Jacobi fields. (∞)

Theorem 6.1. Let Xˆ ∗∗ be the triple cover of the infinite prolongation space of the differential system for minimal Lagrangian (∞) surfaces. Let J(∞) , and J′(∞) be the C-vector spaces of Jacobi fields, and pseudo-Jacobi fields on Xˆ ∗∗ respectively, Definition 4.29. (a) There exists a unique (up to constant scale) nontrivial weighted homogeneous polynomial Jacobi field in R of degree d ≥ 2 for each d ≡ 2, 4

mod 6

(hence of order ≡ 5, 1 mod 6). They are given by the coefficients of the canonical formal Killing fields

{a6n+5 , a6n+7 }∞ n =0 .

(6.2)

The classical Jacobi fields, and these higher-order Jacobi fields and their complex conjugates generate J(∞) . (b) There exists a unique (up to constant scale) nontrivial weighted homogeneous polynomial pseudo-Jacobi field in R of degree d ≥ 1 for each d ≡ 1, 5

mod 6

(hence of order ≡ 4, 2 mod 6). They are given by the coefficients of the canonical formal Killing fields

{p6n+4 , p6n+8 }∞ n=0 .

(6.3) ′(∞)

The classical pseudo-Jacobi fields, and these higher-order pseudo-Jacobi fields and their complex conjugates generate J

.

We present the proof of the theorem in the following three subsections. In Section 6.1, we state two technical lemmas regarding the ∂ξ -equation on Xˆ (∞) . In Section 6.2, we show that a (pseudo) Jacobi field decomposes as a sum of a classical (pseudo) Jacobi field and a higher-order (pseudo) Jacobi field. Combining these with the results from [15,16], we give a complete classification of the (pseudo) Jacobi fields in Section 6.3. 6.1. Two lemmas 6.1.1. Lemma 6.4 Lemma 6.4. Let f : U ⊂ Xˆ (∞) → C be a scalar function on an open subset U ⊂ Xˆ (∞) such that − 31

∂ξ f = ch3

for a constant c. Then c = 0 necessarily, and f is a constant. Corollary 6.5. Let f : U ⊂ Xˆ (∞) → C be a scalar function on an open subset U ⊂ Xˆ (∞) such that

∂ξ f = 0.

(6.6)

Then f is a constant. Remark 6.7. The corollary states that the minimal Lagrangian differential system under consideration (γ 2 ̸= 0) is not Darboux integrable at any order, see [25] for a discussion of Darboux integrability (in the hyperbolic case). For example, this implies that no matter how many times one differentiates, the minimal Lagrangian system does not admit a Weierstraß type of holomorphic representation formula. Lemma 6.4 can be proved by an induction on k and straightforward computations. The full proof is involved, and since it is similar to the proof of the corresponding lemma for the differential system for constant mean curvature surfaces given in [17, §5], we omit the proof.

J.S. Wang / Journal of Geometry and Physics 114 (2017) 291–328

321

6.1.2. Lemma 6.8 Recall the spectral weights, Definition 4.25. Define the associated sequences of vector spaces of polynomials filtered by weight as follows.

 Pd    Pd  

Pd (k)

Qd      Qd

Qd (k)

= {weighted homogeneous polynomials of degree d ≥ 0 in R}, = ⊕di=0 Pi ⊂ R, = Pd ∩ O (k) ⊂ R ∩ O (k), = Pd ⊕ (h3 h¯ 3 )Pd , = ⊕di=0 Qi , = Qd ∩ O (k). 1

The following lemma records a rigidity property of the subspace Pd (k) under the differential operator h33 ∂ξ . (∞) (∞) Lemma 6.8. Let v : U ⊂ Xˆ ∗∗ → C be a scalar function on an open subset U ⊂ Xˆ ∗∗ such that

v ∈ O (k),

k ≥ 4.

Suppose 1

h33 ∂ξ v ∈ Qd (k). Then

v ∈ Pd+1 (k). (∞) (∞) Corollary 6.9. Let u : U ⊂ Xˆ ∗∗ → C be a scalar function on an open subset U ⊂ Xˆ ∗∗ such that

u ∈ O (k),

k ≥ 4.

Let uk = Ek (u). Suppose 1

k

h33 ∂ξ (h33 uk ) ∈ Qd (k). k

Then h33 uk ∈ Pd+1 (k), and hence u ∈ Pd+(k−2) (k) mod O (k − 1). k

Proof. Substitute v = h33 uk in Lemma 6.8.



Proof of Lemma 6.8. We apply induction on k. Consider the case k = 4. For scalar functions in O (4), Corollary 4.17 implies that we have the commutation relation 1

[E4 , ∂ξ ] = 0. By applying E4 repeatedly to h33 ∂ξ v ∈ Qd (4), we get 1

h33 ∂ξ E4m (v) = 0 for some m ≤ d + 1. By Corollary 6.5, v is a polynomial in z4 and we have

v = vm z4m + vm−1 z4m−1 + · · · + v1 z4 + v0 , where vj ∈ O (3), 0 ≤ j ≤ m, and vm is a constant. 1

1

Substitute this to the given equation h33 ∂ξ v ∈ Qd (4). Note h33 ∂ξ z4 = 1 3 ′′ h33 ∂ξ vj + (j + 1)vj+1 R = cj′ γ 2 + cj h3 h¯ 3 , 2

′′

3 R, and one gets the recursive sequence of equations 2

j = m − 1, m − 2, . . . 0,

for constants cj′ , cj . Since vm is a constant and vj ∈ O (3), Lemma 6.4 and an induction on j from m − 1 to 0 show that all the coefficients vj must be constant. Suppose now the claim is true up to k − 1. For scalar functions in O (k), Corollary 4.17 implies that we have the commutation relation [Ek , ∂ξ ] = 0. By the similar argument as above, v is a polynomial in zk and we have

v = vm zkm + vm−1 zkm−1 + · · · + v1 zk + v0 ,

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J.S. Wang / Journal of Geometry and Physics 114 (2017) 291–328 1

where vj ∈ O (k − 1), 0 ≤ j ≤ m, and vm is a constant. Substitute this to the given equation h33 ∂ξ v ∈ Qd (k), and one gets the recursive sequence of equations 1

h33 ∂ξ vj + (j + 1)vj+1 Tˆk ∈ Qd−j(k−3) (k − 1),

j = m − 1, m − 2, . . . 0.

By Lemma 4.27, Tˆk ∈ Qk−4 (k − 1). An induction on j from m − 1 to 0 shows that

vj ∈ Pd−j(k−3)+1 (k − 1),

m − 1 ≥ j ≥ 0. 

6.2. Decomposition 6.2.1. Stability lemma Recall Tˆk ∈ Qk−4 (k − 1). Lemma 6.10. Let E , E ′ be the Jacobi, and pseudo-Jacobi operators, Definition 4.29. (a) j

E (zj ) = Tˆj+1 −

3 j

E ′ (zj ) = Tˆj+1 −

3

3

(zj Tˆ4 + z4 Tˆj ) + γ 2 zj , 2 1

(zj Tˆ4 + z4 Tˆj ) + (γ 2 + 4r 2 )zj . 2

This implies,

E (R), E ′ (R) ⊂ R ⊕ (h3 h¯ 3 )R. (b) The operators E , E ′ preserve the spectral weight when acting on R. 6.2.2. Symbol lemma The highest order term of a (pseudo) Jacobi field admits the following normal form. Lemma 6.11. Let A ∈ J(k) ⊂ O (k + 2) (or A ∈ J′(k) ⊂ O (k + 2)). Suppose Ak+2 = Ek+2 (A) ̸= 0. Then A is at most linear in the highest order variable zk+2 , and (up to a constant scale) A = zk+2 + O (k + 1). Proof. For A ∈ O (k + 2), Aξ ≡ hk+3 Ak+2 Aξ ,ξ ≡ hk+3 ∂ξ (A

mod O (k + 2), k+2

) mod O (k + 2),

≡ 0 mod O (k + 2),

for E (A) = 0 (or

E ′ (A) = 0).

This forces ∂ξ (Ak+2 ) = 0. The claim follows from Corollary 6.5. 6.2.3. Decomposition Proposition 6.12. (∞)

J(∞) = Jh

⊕ J(0) .

(∞)

By definition, Jh ⊂ R ⊕ R and the space of higher-order Jacobi fields is generated by the un-mixed weighted homogeneous polynomial Jacobi fields. The space of pseudo-Jacobi fields admits the similar decomposition ′(∞)

J′(∞) = Jh

⊕ J′(0) .

Proof. Consider first the Jacobi fields case. The claim can be verified by a direct computation for the Jacobi fields in O (4) ∩ O (−4). Let A ∈ O (k) ∩ O (−k), k ≥ 5, be a Jacobi field. By Lemma 6.11, A = zk + u(1) ,

u(1) ∈ O (k − 1).

J.S. Wang / Journal of Geometry and Physics 114 (2017) 291–328

323

Applying the Jacobi operator, one finds (k−1)

1

−E (zk ) ≡ h33 ∂ξ (h3

3

uk(1−)1 )zk

mod O (k − 1).

By Lemma 6.10, E (zk ) ∈ Qk−3 (k). By Corollary 6.9, one gets u(1) = p(1) + O (k − 2), where p(1) ∈ Pk−3 (k − 1). Suppose by induction we arrive at the formula A = zk + p(1) + p(2) + · · · + p(j) + u(j+1) ,

u(j+1) ∈ O (k − (j + 1)),

(6.13)

where each p(i) ∈ Pk−3 (k − i) such that q(i) := E zk + p(1) + p(2) + · · · + p(i) ∈ O (k − i).





By Lemma 6.10, q(j) ∈ Qk−3 (k − j). Applying the Jacobi operator to the refined normal form (6.13), one gets k−(j+1) 3

1

−q(j) ≡ h33 ∂ξ (h3

k−(j+1)

u(j+1)

)zk−j mod O (k − (j + 1)).

By Corollary 6.9, one gets u(j+1) = p(j+1) + u(j+2) , p(j+1) ∈ Pk−3 (k − (j + 1)), u(j+2) ∈ O (k − (j + 2)), such that q(j+1) := E zk + p(1) + p(2) + · · · + p(j+1) ∈ O (k − (j + 1)).





Continuing this process, we arrive at the normal form A = p + u(k−3) , p = zk + p(1) + p(2) + · · · + p(k−4) ,

u(k−3) ∈ O (3),

where p(k−4) ∈ Pk−3 (4) such that q(k−4) := E (p) ∈ Qk−3 (4). Since E is a real operator, the complex conjugate of this analysis implies that the Jacobi field A decomposes into A = f + g, where f is an un-mixed pure polynomial in zi , z i , and g is a function on Xˆ (1) . By construction, we have

E (f ) = −E (g ) ∈ Qk−3 (4) ⊕ Qk−3 (4). But E (g ) ∈ O (4) ∩ O (−4) is at most linear in z4 , z 4 , and hence

E (f ) = −E (g ) = (c1 z4 + c2 z 4 + c3 ) + h3 h¯ 3 (c1′ z4 + c2′ z 4 + c3′ ) ∈ Q1 (4) ⊕ Q1 (4), for constants c1 , c2 , c3 , c1′ , c2′ , c3′ . Now, the Jacobi operator E preserves the spectral weight. Hence, up to the higher-order Jacobi fields of spectral weight ≥2, or ≤ − 2, one may assume that f is of the form f = a1 z 4 + a2 z 4 + a3 , for constants a1 , a2 , a3 . Thus the analysis is reduced to the case A ∈ O (4) ∩ O (−4). Consider next the pseudo-Jacobi fields case. By the similar argument as above, a pseudo-Jacobi field P decomposes into P = f + g, where f is an un-mixed pure polynomial in zi , z i , and g is a function on Xˆ (1) such that

E ′ (f ) = −E ′ (g ) = (c1 z4 + c2 z 4 + c3 ) + h3 h¯ 3 (c1′ z4 + c2′ z 4 + c3′ ) ∈ Q1 (4) ⊕ Q1 (4), for constants c1 , c2 , c3 , c1′ , c2′ , c3′ . Since the pseudo-Jacobi operator E ′ preserves the spectral weight and z4 , z 4 are pseudo-Jacobi fields, this implies that (up to adding constants to f , g),

E ′ (f ) = E ′ (g ) = 0. 

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6.3. Classification We give a proof of Theorem 6.1 by combining the results of Sections 6.1 and 6.2 with the classification of ‘‘Jacobi fields’’13 in [15,16]. 6.3.1. No even order Jacobi fields Lemma 6.14. There does not exist an even order weighted homogeneous polynomial Jacobi field in R. Proof. Let F = constant

3 x ). 2 0

F =

3 2

3 x z 2 0 2k

+ · · · ∈ R be an even order weighted homogeneous polynomial Jacobi field of weight 2k − 3 (for a

Consider the expansion,

x0 z2k

+z2k−1 (

x1 z4

)

+z2k−2 ( +z2k−3 ( +z2k−4 ( ... +z2k−(i−1) ( +z2k−i ( +z2k−(i+1) ( ... +zk+3 ( +zk+2 (

x2 z5 x3 z6 x4 z7

+y2 z42 ) +y3 z5 z4 + . . . ) +y4 z6 z4 + . . . )

xi−1 zi+2 xi zi+3 xi+1 zi+4

+yi−1 zi+1 z4 + . . . ) +yi zi+2 z4 + . . . ) +yi+1 zi+3 z4 + . . . )

xk−3 zk xk−2 zk+1

+yk−3 zk−1 z4 + . . . ) +yk−2 zk z4 + . . . ) +yk−1 zk2+1 z4 + . . . .

(6.15)

Here {xi , yi } are constant coefficients. The monomials {z2k , z2k−1 z4 , z2k−2 z5 , . . . zk+2 zk+1 } (with xi -coefficients) will be called the principal terms. Step 0. Consider the highest order z2k -term in E (F ). It implies

 x1 + Here aj,s =

a2k,0 + (j+2s+3) 2(j−1)



3 2

−

j −1 s+2



2k 3

 a3,0

x0 = 0.

(6.16)

.

In order to extract the compatibility equations imposed on the xi -coefficients only, we compute E (F ) modulo the curvature R = γ 2 − 2h3 h¯ 3 from now on, i.e., h3 h¯ 3 ≡

γ2 2

∈ R ⊕ (h3 h¯ 3 )R.

Step 1. It is easily checked by a direct computation that, under the Jacobi operator E , those terms not appearing in the above expansion do not have any contribution to the principal terms. Step 2. Since ∂ω z4 ≡ 0 mod R, the contributions from the remaining terms with yi -coefficients are eliminated by computing mod R. It follows that one may evaluate E ({principal terms}) mod R, and check only the principal terms in the image. This would yield a set of linear equations on the xi -coefficients. Step 3. Recall the following formulas. Tj+1 =

j −3 

aj,s hj−s ∂ξs R,

for j ≥ 3,

s=0

∂ξs R = δ0s γ 2 − 2h3+s h¯ 3 ,   j 2 dzj ≡ zj+1 − z4 zj ω + Tˆj r− 3 ω mod ˆI(∞) , 3

j ≥ 4.

13 The ‘‘Jacobi fields’’ in [15,16] correspond to the higher-order pseudo-Jacobi fields of the minimal Lagrangian system.

J.S. Wang / Journal of Geometry and Physics 114 (2017) 291–328

325

They imply, j

∂ξ ∂ξ (zj ) ≡ Tˆj+1 − (z4 Tˆj ) mod R, 3

Tˆj+1 ≡ −γ 2 aj,j−3 zj + (quadratic terms in zi ’s)

mod R.

Since the principal terms except z2k are quadratic in the balanced coordinates, the term −γ 2 aj,j−3 zj is the only form of contribution to the principal terms from ∂ξ zj+1 for j + 1 < 2k. Step 4. With this preparation, a direct computation yields the following formulas for the principal terms. Here we record only the relevant terms (for simplicity, we set γ 2 = 1 temporarily).

    2k 3 z2k + a2k,2k−4 + a2k,1 − a2k−1,2k−4 z4 z2k−1 −E (z2k ) ≡ a2k,2k−3 − 2

3

k−2  + (a2k,2k−j−3 + a2k,j ) zj+3 z2k−j , j =2

−E (z4 z2k−1 ) ≡

 ·   

a4,1 + a2k−1,2k−4 −

  

3 2

 z4 z2k−1

(a2k−2,2k−5 ) z5 z2k−2 ,

...  (aj+2,j−1 ) zj+2 z2k−(j−1)     3 −E (zj+3 z2k−j ) ≡ zj+3 z2k−j aj+3,j + a2k−j,2k−(j+3) −  2   (a2k−(j+1),2k−(j+4) ) zj+4 z2k−(j+1) , ...  (ak,k−3 ) zk zk+3     3 zk+1 zk+2 −E (zk+1 zk+2 ) ≡ 2ak+1,k−2 + ak+2,k−1 −  2   · 3 2

Note aj+3,j =

mod R.

for all j ≥ 0. Hence, except for the terms from E (z2k ) and the last term in E (zk+1 zk+2 ), all the terms have the

equal coefficient 32 . Step 5. Combining the results from Step

0 and Step 4, the equation

E (F ) ≡ 0 mod R implies the following system of three term equations on the xi -coefficients.

· ·

·

x1

x1

x1 +x 2

+x2 +x3

xj−1

+x j

+x j + 1

xk−4 xk−3

+xk−3 +xk−2

+xk−2 +xk−2

+(a2k,2k−3 + a2k,0 − k)x0 = 0, +(a2k,2k−4 + a2k,1 − k)x0 = 0, +(a2k,2k−5 + a2k,2 )x0 = 0, ... +(a2k,2k−j−3 + a2k,j )x0 = 0, ... +(a2k,k + a2k,k−3 )x0 = 0, +(a2k,k−1 + a2k,k−2 )x0 = 0.

(6.17)

Step 6. It is left to show that this system of k − 1 linear equations on the set of k − 1 coefficients {x0 , x1 , . . . xk−2 } has full rank. Set t0 = a2k,2k−3 + a2k,0 − k, t1 = a2k,2k−4 + a2k,1 − k, tj = a2k,2k−j−3 + a2k,j ,

for j ≥ 2.

A direct computation shows that the determinant χk of the (k − 1)-by-(k − 1) matrix for the set of linear equations is given by

±χk =

k−2  j =0

ϵkj tj ,

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J.S. Wang / Journal of Geometry and Physics 114 (2017) 291–328

where

ϵkj =



−2 when j ≡ k mod 3 +1 otherwise.

A computation mod 3 shows that

± χk =

 1 

when k ≡ 0 mod 3 2 1 otherwise. 

(6.18)

6.3.2. Proof of Theorem 6.1 (a) It is left to show that there exist no higher-order Jacobi fields of degree 0 mod 6. This follows from an induction using the recursion operators P , N in [16]. (b) As mentioned earlier, by Proposition 6.12 and the analysis in Section 4.4.4, the higher-order pseudo-Jacobi fields correspond to the ‘‘Jacobi fields’’ of the elliptic Tzitzeica equation in [15,16]. It follows from [16, Theorem 8.1].  7. Conservation laws Recall from Section 3.3 that the classical conservation laws are defined as the elements of the 1st characteristic cohomology of the original minimal Lagrangian system (X , I). Generalizing this, the conservation laws for the minimal Lagrangian system are defined as the elements of the 1-st characteristic cohomology of the infinitely prolonged differential (∞) system (Xˆ ∗∗ , ˆI(∞) ). In this section, we show that each of the canonical formal Killing fields X(p4 ), X(a5 ) generates an infinite sequence of conservation laws, Theorem 7.6. 7.1. Definition (∞)

(∞)

Let (Ω ∗ (Xˆ ∗∗ ), d) be the de-Rham complex of C-valued differential forms on Xˆ ∗∗ . Let (∞) ˆ(∞) (Ω ∗ = Ω ∗ (Xˆ ∗∗ )/I , d)

be the quotient space equipped with the induced differential d = d mod ˆI(∞) . By construction, ˆI(∞) is formally Frobenius, and

(∞) ˆ(∞) , I ) is by definition the cohomology (Ω ∗ , d) is a complex. The characteristic cohomology of the differential system (Xˆ ∗∗ ∗ of the quotient complex (Ω , d). Let

H q (Ω ∗ , d) denote the cohomology at Ω q , q = 0, 1, 2. (∞)

Definition 7.1. Let (Xˆ ∗∗ , ˆI(∞) ) be the triple cover of the infinite prolongation of the differential system for minimal (∞) Lagrangian surfaces. A conservation law is an element of the 1-st characteristic cohomology H 1 (Ω ∗ , d) of (Xˆ ∗∗ , ˆI(∞) ). The C-vector space of conservation laws is denoted by

C (∞) := H 1 (Ω ∗ , d). Note by definition that the classical conservation laws C (0) ⊂ C (∞) . 7.2. Spectral sequence (∞)

Consider the filtration of Ω ∗ (Xˆ ∗∗ ) by the subspaces (∞) (∞) ˆI(∞) . . . ∧ : Ω ∗ (Xˆ ∗∗ F p Ω q = Image{ˆI(∞) ∧  ) → Ω q (Xˆ ∗∗ )}.  p

From the associated graded, a standard construction yields the spectral sequence

(Erp,q , dr ),

dr has bidegree (r , 1 − r ),

r ≥ 0.

Consider the sub-complex 0 ,1

0 → E1

→ E11,1 → E12,1 ,

which is exact at

0 ,1 E1 ,

[26, p 562, Theorem 2 and Eq. (4)].

(7.2)

J.S. Wang / Journal of Geometry and Physics 114 (2017) 291–328

327

0,1

7.2.1. E1 By definition, the first piece is given by 0,1

E1

(∞) (∞) = {ϕ ∈ Ω 1 (Xˆ ∗∗ )|dϕ ≡ 0 mod ˆI(∞) }/{dΩ 0 (Xˆ ∗∗ ) + Ω 1 (ˆI(∞) )} (∞) ˆ(∞) = H 1 (Ω ∗ (Xˆ ∗∗ )/I , d)

= C (∞) . 1,1

7.2.2. E1

1,1

The second piece E1 fields, 1,1

E1

is called the space of cosymmetries, [24]. We claim that this is isomorphic to the space of Jacobi

≃ J(∞) . 0,1

1,1

The differential d1 : E1 ↩→ E1 can be considered as a symbol map. 1,1 We give a proof of this claim. From the structure equations (4.14), (4.16), a short analysis shows that a class in E1 can be represented by a 2-form Φ such that

Φ ≡ AΨ − θ0 ∧ σ

mod F 2 Ω 2

for a scalar coefficient A and a 1-form σ , where i Ψ = Im(θ1 ∧ ξ ) = − (θ1 ∧ ξ − θ 1 ∧ ξ ). 2 Note dΨ = 3iγ 2 θ0 ∧ (ξ ∧ ξ + θ1 ∧ θ 1 )

(7.3)

≡ 0 mod θ0 . Differentiating Φ , one gets 0 ≡ dA ∧ Ψ − dθ0 ∧ σ

mod θ0 , F 2 Ω 3 .

Since dθ0 = − (θ1 ∧ ξ + θ 1 ∧ ξ ), this implies 1 2

  σ ≡ −i (∂ξ A)ξ − (∂ξ A)ξ

mod ˆI(∞) .

With the given σ , the coefficient of θ0 ∧ ξ ∧ ξ -term in dΦ shows that

E (A) = 0, and A is a Jacobi field. 7.3. Conservation laws from canonical formal Killing fields From the analysis above, the question arises as to whether the symbol map 0 ,1

d1 : C (∞) ≃ E1

↩→ E11,1 ≃ J(∞)

is also surjective and hence an isomorphism. This would imply an infinitely prolonged version of Noether’s theorem for the minimal Lagrangian system. To this end, we note that there exists an infinite sequence of conservation laws represented by the 1-forms which are assembled from the coefficients of the canonical formal Killing fields. These 1-forms have the property that they are weighted homogeneous under the spectral weight, in such a way that matches the spectral weights of the higher-order Jacobi fields (6.2), (6.3). It is likely that the symbol map d1 is an isomorphism. [Formal Killing field X(p4 )] Recall the structure equation (5.8). Set

ϕn := b6n+5 ξ + s6n+3 ξ .

(7.4)

The structure equations show that dϕn ≡ 0

mod ˆI(∞) ,

and ϕn represents a conservation law. [Formal Killing field X(a5 )] Recall the structure equation (5.11). Set

ϕn′ := b6n+3 ξ + s6n+1 ξ . The structure equations show that dϕn′ ≡ 0

mod ˆI(∞) ,

and ϕn′ represents a conservation law.

(7.5)

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Theorem 7.6. Let X(p4 ), X(a5 ) be the canonical formal Killing fields, Section 5. The associated sequence of 1-forms

ϕn , ϕn′ ,

n = 0, 1, 2, . . . ,

represent conservation laws. It remains to verify that these conservation laws are nontrivial. Remark 7.7. Suppose the conservation laws [ϕn ], [ϕn′ ] are nontrivial. By an analysis of the differential d1 as in Section 7.2.2, the spectral weight count shows that d1 ([ϕn ]) = a6n+7 ,

d1 ([ϕn′ ]) = a6n+5 ,

up to constant scale. References [1] Robert L. Bryant, Minimal Lagrangian submanifolds of Kähler-Einstein manifolds, in: Differential Geometry and Differential Equations (SHanghai, 1985), in: Lecture Notes in Math., vol. 1255, 1987, pp. 1–12. [2] Robert C. McLean, Deformations of calibrated submanifolds, Comm. Anal. Geom. 6 (4) (1998) 705–747. [3] Marianty Ionel, Second order families of special Lagrangian submanifolds in C4 , J. Differential Geom. 65 (2) (2003) 211–272. [4] Dominic Joyce, Special Lagrangian submanifolds with isolated conical singularities. V. Survey and applications, J. Differential Geom. 63 (2) (2003) 279–347. [5] Adrian Butscher, Regularizing a singular special Lagrangian variety, Comm. Anal. Geom. 12 (4) (2004) 733–791. [6] Robert L. Bryant, Second order families of special Lagrangian 3-folds, Perspectives in Riemannian geometry, in: CRM Proc. Lecture Notes, vol. 40, Amer. Math. Soc., Providence, RI, 2006, pp. 63–98. [7] Dominic Joyce, Riemannian Holonomy Groups and Calibrated Geometry, Oxford University Press, Oxford, 2007. [8] Mark Haskins, Nikolaos Kapouleas, Special Lagrangian cones with higher genus links, Invent. Math. 167 (2) (2007) 223–294. [9] Yng-Ing Lee, The metric properties of Lagrangians, in: Surveys in Geometric Analysis and relativity. Dedicated to Richard Schoen in Honor of His 60th Birthday, Higher Education Press, Beijing, 2011, pp. 327–341. [10] Mark Haskins, Nikolaos Kapouleas, Closed twisted products and SO(p) × SO(q)-invariant special Lagrangian cones, Comm. Anal. Geom. 20 (1) (2012) 95–162. [11] Tommaso Pacini, Special Lagrangian conifolds. II: Gluing constructions in Cm , Proc. Lond. Math. Soc. (3) 107 (2) (2013) 225–266. [12] R. Schoen, J. Wolfson, Minimizing area among Lagrangian surfaces: the mapping problem, J. Differential Geom. 58 (1) (2001) 1–86. [13] Emma Carberry, Ian McIntosh, Minimal Lagrangian 2-tori in CP2 come in real families of every dimension, J. Lond. Math. Soc. (2) 69 (2) (2004) 531–544. [14] John Loftin, Ian McIntosh, Minimal Lagrangian surfaces in CH2 and representations of surface groups into SU (2, 1), Geom. Dedicata 162 (2013) 67–93. [15] Daniel Fox, Oliver Goertsches, Higher-order conservation laws for the nonlinear Poisson equation via characteristic cohomology, Selecta Math. (N.S.) 17 (2011) 795–831. [16] Daniel Fox, Killing fields and conservation laws for rank-1 Toda field equations, 2012, arXiv:1208.2634v1. [17] Daniel Fox, Joe S. Wang, Conservation laws for surfaces of constant mean curvature in 3-dimensional space forms, 2013, arXiv:1309.6606. [18] Ulrich Pinkall, Ivan Sterling, On the classification of constant mean curvature tori, Ann. of Math. (2) 130 (2) (1989) 407–451. [19] Robert L. Bryant, Bochner-Kähler metrics, J. Amer. Math. Soc. 14 (2001) 623–715. [20] Robert L. Bryant, Shiing Shen Chern, Robert B. Gardner, Hubert L. Goldschmidt, Phillip A. Griffiths, Exterior DIfferential SYstems, in: Mathematical Sciences Research Institute Publications, vol. 18, Springer-Verlag, New York, 1991. [21] Lei Fu, On the boundaries of special Lagrangian submanifolds, Duke Math. J. 79 (2) (1995) 405–422. [22] Ian McIntosh, Special Lagrangian cones in C3 and primitive harmonic maps, J. Lond. Math. Soc. (2) 67 (3) (2003) 769–789. [23] Dominic Joyce, Special Lagrangian 3-folds and integrable systems, Surveys on geometry and integrable systems, Adv. Stud. Pure Math. 51 (2008) 189–233. [24] I.S. Krasil’shchik, A.M. Vinogradov, Nonlocal trends in the geometry of differential equations: symmetries, conservation laws, and Bäcklund transformations, Acta Appl. Math. 15 (1–2) (1989) 161–209. Symmetries of partial differential equations, Part I. [25] Robert L. Bryant, Phillip A. Griffiths, Lucas Hsu, Hyperbolic exterior differential systems and their conservation laws. I, Selecta Math. (N.S.) 1 (1) (1995) 21–112. [26] Robert L. Bryant, Phillip A. Griffiths, Characteristic cohomology of differential systems. I. General theory, J. Amer. Math. Soc. 8 (3) (1995) 507–596.