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Complex analytic properties of minimal Lagrangian submanifolds

In this article we study complex properties of minimal Lagrangian submanifolds in Kaehler ambient spaces, and how they depend on the ambient curvature. In particular, we prove that, in the negative curvature case, minimal Lagrangians do not admit fillings by holomorphic discs. The proof relies on a mix of holomorphic curve techniques and on certain convexity results.


Introduction
An immersed submanifold ι : L n → M 2n in a Kähler manifold (M, J, g, ω) is said to be minimal if the mean curvature vector field H is zero; equivalently, if it is a critical point of the Riemannian volume. The submanifold is Lagrangian if the induced Kähler form on L is zero, i.e. ι * ω ≡ 0. From now on we will identify L with its image ι (L).
Thanks to the compatibility condition g (·, ·) = ω (·, J·), L is Lagrangian if and only if It follows that a Lagrangian submanifold has a special linear-algebraic property with respect to J: it is totally real, i.e. T p L ∩ J (T p L) = {0} .
We are interested in the geometric properties of compact submanifolds that are simultaneously minimal and Lagrangian. Notice that these conditions involve only the Riemannian and symplectic ambient structures, and indeed up to now minimal Lagrangian manifolds have been studied mainly from the Riemannian point of view, such as the second variation formula and stability under the mean curvature flow. However, the above compatibility condition suggests that these submanifolds should also have interesting complex analytic properties.
Specifically, we will investigate the existence of holomorphic discs with boundary on a fixed minimal Lagrangian, and the existence of fillings by holomorphic discs. Holomorphic discs and disc fillings are a classical problem, cf. [1], [5], [7], [6], [18], but the literature generally focuses on symplectic or complex assumptions on the submanifold, such as the existence of complex points or being contained in the boundary of a pseudo-convex domain. By substituting those assumptions with the Riemannian condition of being minimal we are taking a novel direction, cf. also [4], which we expect may have developments beyond those studied here. We remark that the minimal Lagrangian condition is typically over-constrained, cf [8], unless one restricts to Kähler-Einstein ambient spaces. Accordingly, this will be our main focus.
In order to put the problem into context, consider the situation in dimension 1. In this case M is a Riemann surface with constant curvature and the compact minimal Lagrangians are exactly the closed geodesics. We then observe that the topological and complex analytic properties of L depend on the sign of the curvature, as follows.
• If M has positive curvature, i.e. M is the sphere S 2 , geodesics are maximal radius circles. In this case they are homotopically trivial (see Figure 1), thus bound a holomorphic disc.
• If M has zero curvature, i.e. M is the torus T 2 , geodesics are not homologically trivial, so they do not bound discs.
• If M has negative curvature, there exist examples of homologically trivial geodesics. The closed geodesic on the genus 2 surface Σ 2 in Figure 1, for example, is the boundary of a handle N . The Gauss-Bonnet theorem shows however that such curves cannot bound a disc. To understand the appropriate analogue of these properties in higher dimensions, consider the Kähler-Einstein manifold P 1 (C) × P 1 (C), with positive scalar curvature. The product γ × γ of maximal circles is a minimal Lagrangian L. Each γ ×{ * } is the boundary of a holomorphic disc; the product is homologically trivial, and in fact L admits a filling by holomorphic discs.
On the other hand, in the Ricci-flat (more precisely, Calabi-Yau) case, the theory of calibrations shows that minimal Lagrangians minimize volume in their homology class. They are thus never homologically trivial, and in particular they cannot be filled by holomorphic discs.
We are thus led to trying to understand whether the negative curvature case has a distinctive behaviour in regards to these properties, as in dimension 1.
We start by examining the case of products. In this situation, in Section 2 we show that minimal Lagrangians do not admit even a single holomorphic disc, so they clearly do not admit holomorphic disc fillings even though, in contrast with the Ricci-flat case, they may be homologically trivial.
In Section 3 we use the theory of J-holomorphic curves to prove that any holomorphic disc with boundary on a minimal Lagrangian submanifold is rigid. This implies the following result, in marked contrast with the positive case.
It does not seem possible to use the same technique in higher dimensions. To generalize Theorem 1.1 we thus go back to dimension 1 to prove a new convexity result for the length functional, cf. Section 4. In order to illustrate its use, we give an alternative proof of the non existence of holomorphic discs bounded by a closed geodesic on a non positively curved Riemann surface. Our main interest in the convexity result however is that it admits a generalization to higher dimensions, explained in Section 5: this is based on the theory of the J-volume functional, introduced in [2] and further studied in [13] and in [14]. Thanks to those results, in Section 6 we prove our main result which roughly speaking states: Theorem 1.2. Minimal Lagrangians in non-positive Kähler-Einstein manifolds do not admit fillings by holomorphic discs in any dimension.
We refer to Theorem 6.3 for the precise statement, which requires a few preliminary definitions. More generally, the same is true for the class of totally real J-minimal submanifolds, which generalize minimal Lagrangians to ambient spaces which are not necessarily Kähler-Einstein.
In Section 7, following a technique used in [3], we exhibit two examples of minimal Lagrangian tori in a (non-product) 2-dimensional negative Kähler-Einstein manifold. discussions and suggestions. I would like to thank also Jonny Evans, Jason Lotay, Luciano Mari and Kai Zehmisch for useful discussions, as well as Robert Bryant and Claude LeBrun who indirectly helped me via conversations with Tommaso Pacini.

A special case: products
The simplest setting in which we can study minimal Lagrangian submanifolds is that of products. Let Σ 1 , . . . , Σ n be Riemann surfaces and let γ i : S 1 → Σ i be closed geodesics, with i = 1, . . . , n. Then the product manifold M = Σ 1 × · · · × Σ n inherits a Kähler structure with respect to which the submanifold L = γ 1 × · · · × γ n is minimal Lagrangian. The ambient structure is Kähler-Einstein if the constant curvature is the same, for all surfaces. For example, let Σ 2 be the genus 2 surface with its standard hyperbolic structure and let γ : S 1 → Σ 2 be the closed geodesic of Figure  1. Then the submanifold L := γ × γ : S 1 × S 1 → M := Σ 2 × Σ 2 is a homologically trivial minimal Lagrangian torus, since it is the boundary of the 3-manifold N × γ, where N is the handle bounded by γ.
We want to show that in the embedded case such minimal Lagrangians do not bound holomorphic discs. We first prove two preliminary results providing different generalizations of the standard (embedded) Gauss-Bonnet theorem.
From now on we will denote with ∆ the open unit disc. Proof. If such a map f : ∆ → Σ existed, we could consider its lift to the universal cover Σ obtaining the following commutative diagram: Then γ := f (∂∆) would be the support of a geodesic in Σ which lifts γ and is closed because it bounds the region f (∆). Let α be a parametrization of γ; fixing a point p := α(t 0 ), consider the map It is well defined because Σ is complete with nonpositive scalar curvature and it's a smooth convex function by Theorem 2.3 in [17]. Restricting to γ we have: where ∇ α α = 0 because α is a geodesic. We would then have a non negative convex function defined on S 1 , equal to 0 for t = t 0 , which is necessarily constantly equal to 0, giving a contradiction as γ is nonconstant. Proof. Suppose by contradiction that such a holomorphic map f exists.
We claim that there exists an interior point x ∈ ∆ whose image f (x) is in the topological boundary ∂ f ∆ . This is trivially true if f ∆ has no interior points. Otherwise there exists x 0 ∈ ∆ such that p := f (x 0 ) is an interior point of f ∆ . Thanks to the above Proposition f (∂∆) γ, so we can fix a point q ∈ γ that is not the image of any point in ∂∆. We can go from p to q with a smooth arc α : [0, 1] → Σ \ f (∂∆): this is possible because f (∂∆) is an arc strictly contained in the embedded γ, hence it does not disconnect Σ. Thus we can consider the set It is not empty because s = 0 ∈ I. Set s := sup I. Then α (s) is a boundary point of f ∆ , so there exists x ∈ ∆ such that α (s) = f (x). However by construction α (s) / ∈ f (∂∆), so x ∈ ∆. Thus the holomorphic map f maps an interior point of ∆ to a boundary point of its image, which contradicts the Open Mapping Theorem.
Proof. Let us denote with π j : M −→ Σ j the projection onto the j-th surface, for j = 1 . . . , n. If such a holomorphic map f : ∆ −→ M existed, each composition π j • f : ∆ −→ Σ j would still be holomorphic and such that π j • f (∂∆) ⊆ γ j . As f is nonconstant, there would exist j ∈ {1, . . . , n} such that the projection π j • f is nonconstant, but this would contradict Proposition 2.2.
Remark. The above results fail for piecewise smooth geodesics.
Using analogue techniques we can also prove that any harmonic map any harmonic disc with boundary on L is entirely contained in L.

Rigidity of holomorphic discs in dimension 2
We now want to show that, in complex dimension 2, minimal Lagrangians do not admit holomorphic fillings. In this dimension we can achieve this using standard results from the theory of J-holomorphic curves.
Given that minimal submanifolds are automatically smooth, based on the regularity theory of holomorphic curves from now on we will always assume that our holomorphic discs are smooth up to the boundary. Proof. Let us identify ∆ with its image u (∆) and consider the normal bundle V := u * (T M/T ∆) over ∆ and the totally real subbundle V 0 := u * (T L/T ∂∆) over ∂∆. Denote by W k,q V 0 (∆, V ) the closure of the space of vector fields denote the ball of radius ε centred at the zero section.
As in [11], there exists a differentiable map such that D −1 (0) consists of all holomorphic curves (∆, ∂∆) → (M, L) sufficently near u and whose linearization D 0 at 0 is exactly the canonical operator According to the axiomatic definition of the Maslov index in [16] and µ (T ∆, T ∂∆) = 2. Furthermore, [4] proves that each minimal Lagrangian submanifold L in a Kähler-Einstein ambient M is monotone; specifically, where λ is the Kähler-Einstein constant. Thus, in our case, Since the bundle V has complex rank 1 and µ (V, V 0 ) is negative, we can apply Theorem C.1.10, (iii) in [16], thus the operator D 0 is injective. Since we are in dimension 4 and ker D 0 = { infinitesimal transversal holomorphic deformations of u} , the holomorphic disc does not admit any transversal holomorphic deformation.
In particular, Theorem 1.1 follows. In arbitrary dimensions, the theory of J-holomorphic curves allows us to prove the same result only in the case of regular discs, i.e. discs for which D 0 is surjective. Thanks to the Riemann-Roch theorem, as stated in [16], Theorem C.1.10, the linearized operator D 0 is Fredholm, and Since the dimension of an analytic filling is n + 1 and the dimension of a disc is 2, it must be dim M ≥ n − 1, so which contradicts the fact that L is nonpositive monotone, as proved in [4].
However, in some cases one can prove that holomorphic discs are certainly not regular, because the index is negative. For example, if M is negative Kähler-Einstein and since it is even and negative.
Thus in dimension 3 and 4 we can not use the theory of J-holomorphic curves to prove general results regarding the non existence of holomorphic disc fillings. In order to generalize the above results new techniques are required.

Convexity of the length functional
In this Section we are going to prove a convexity result for the Riemannian volume functional in dimension 1, i.e. the length functional. It is an interesting result in view of its generalization in higher dimensions and it gives also an alternative proof of the non existence of holomorphic discs with geodesic boundary. We will need the following classical isothermal coordinates theorem which shows that a Riemannian structure on an oriented surface induces a complex structure, and that the metric is locally conformal to the Euclidean one. See [9] for the proof.
Theorem 4.1. Let (Σ, g) be an oriented Riemannian surface. Then Σ can be made into a Riemann surface, i.e. it admits a complex structure compatible with g. Local holomorphic coordinates are given by smoothly invertible solutions of the differential equation where µ is a complex valued function which depends on the metric g.
In such coordinates the metric has the form e σ dw ⊗ dw and the scalar curvature is K = − 1 e σ ∆σ, where ∆ is the Laplacian operator. Thanks to the complex structure J compatible with the metric g, it is possible to define the symplectic form ω (·, ·) = g (J·, ·) , and the hermitian metric h := g − iω.
From now on, J will denote this complex structure on (Σ, g). df | (r,·) (∂θ) g dθ is nondecreasing and convex with respect to the variable log r.
Proof. Denote with α r the circle re iθ ⊆ ∆ and consider where h is the hermitian metric induced by g.
We want to show that u : ∆ −→ R is subharmonic. Thanks to Theorem 4.1, for each point of Imm (f ) we can consider a holomorphic chart ϕ defined on a neighborhood of the point. Thus we have that Since K ≤ 0, we have that ∆σ ≥ 0. The function σ is subharmonic; since ϕ and f are holomorphic it follows that σ is locally subharmonic, so is locally subharmonic too. Covering Imm(f ) with local holomorphic charts, we have that u(z) is subharmonic. We claim that L is nondecreasing: choose 0 < r 1 < r 2 < 1 and let k(z) be the function harmonic in |z| < r 2 , continuous in |z| ≤ r 2 and equal to u(z) on α r 2 . Then u(z) ≤ k(z) in |z| ≤ r 2 , so L(r 1 ) ≤ Furthermore, the functional L is convex with respect to the variable log r: let k(z) be the harmonic function in the annulus A = {r 1 < |z| < r 2 }, continuous in r 1 ≤ |z| ≤ r 2 and equal to u(z) on the two boundary components of A. Then u(z) ≤ k(z) for r 1 ≤ |z| ≤ r 2 and d dr where dσ = rdθ is the element of arc length and ∂k ∂n denotes the normal derivative, i.e. < grad k, n >, where n is the normal vector.
As k is a harmonic map and thanks to the Divergence Theorem we have that:

The J-Volume functional
In this Section we introduce an alternative volume functional defined in [2] and we state its properties, proved by J. Lotay and T. Pacini in [13] and [14], which generalize the convexity of the length functional proved in in Section 4. where e 1 , . . . , e n is a positive orthonormal basis for the totally real n-plane π.
Fix a totally real submanifold ι : L → M . We can define the J-volume form as where vol g is the standard Riemannian volume on L.
Observe that ρ J (π) ≤ 1 and that the equality holds if and only if π is Lagrangian.
We can extend the map ρ J to the Grassmannian of n-planes, setting ρ J (π) = 0 and σ [π] = 0, when π contains a complex line.
We will say that a totally real submanifold is J-minimal if it is a critical point for the J-volume functional. See [2], [15] for examples of J-minimal submanifolds. The following results hold (cf. [13], [14]): Proof. The first part follows from the fact that ρ J ≤ 1 with equality if and only if L is Lagrangian.
To prove the second statement, let {L t } be a one-parameter family of totally real submanifolds such that L 0 is Lagrangian. Consider the real functions f (t) := Vol g (L t ) and g(t) := Vol J (L t ); we have that g − f ≥ 0. Since it is null for t = 0, this is minimum, hence it is a critical point, so f (0) = g (0) = 0. Interesting properties hold for the J-volume functional adding assumptions on the ambient manifold.
Definition 5.4. Let T be the space of totally real immersions of L into M which are homotopic, through totally real immersions, to the given ι, up to orientation preserving diffeomorphisms of L.
A one parameter family {ι t : L → M } t∈R in T is a geodesic in T if and only if there exists a fixed vector field X ∈ Λ 0 (T L) such that We say that a functional f : T −→ R is convex (respectively, strictly convex) if and only if it restricts to a convex function (respectively, strictly convex function) in one variable along any geodesic in T .
Such a family is called geodesic in T because there exists a connection on T with respect to which it is a geodesic; cf [13], Section 2.
In [13] the following result about the convexity of the J-volume functional is proved.
Observe that in dimension 1 any totally real curve is Lagrangian, so the J-volume functional, the Riemannian volume functional and the length functional coincide. In this case the statement above generalizes Theorem 4.2.

Non existence of a holomorphic discs filling
In Section 3 we proved the non existence of a filling by holomorphic discs for minimal Lagrangian tori, as stated in Theorem 1.1.
In this Section we will generalize that result to higher dimensions. Let L a compact, oriented n-dimensional manifold such that L admits a locally trivial S 1 -fibre bundle structure, i.e. there exists a S 1 -fibre bundle π : L → B. Observe that π : L → B is a restriction of a complex fiber line bundle (N, h) → B, the restrictions to the circles of radius r on each fiber are circle bundles themselves and that the restriction N → B to the unit discs is a disc bundle.
For example, the n−dimensional torus S 1 n , seen as the trivial S 1bundle over the (n − 1)-torus, is the restriction of the complex trivial line bundle C × S 1 n−1 → S 1 n−1 . Also the Hopf fibration is the restriction of the tautological bundle C n+1 → P n (C) to the sphere S 2n+1 .
In this setting, we want to generalize the concept of filling by holomorphic discs, used in Theorem 1.1. Definition 6.1. Using the notation above, let L → M be an immersion which extends to an immersion ι : N → M . If its restriction to each fiber ∆ p = π −1 (p) is holomorphic, we will say that it is a filling by holomorphic discs.
Definition 6.2. Let M be a 2n−dimensional Kähler manifold; an immersed submanifold S has complex rank 1 if, for each p ∈ S, the tangent space T p S contains exactly one complex line.
We can now prove our main result, which generalizes Theorem 1.1 to higher dimensions, since a solid torus in a 4-dimensional Kähler-Einstein manifold has complex rank 1. Theorem 6.3. Let M be a 2n-dimensional Kähler manifold with Ric ≤ 0 and let L be an oriented compact n-manifold which admits a locally trivial S 1 -fibre bundle structure. If ι : L → M is a J-minimal immersion, it does not admit any complex rank 1 filling by holomorphic discs.
Proof. The punctured disc ∆ \ {0} (with the standard complex structure as a subset of the complex plane C) is biholomorphic to the cylinder S 1 × R + , endowed with the complex structure given by where {∂θ, ∂t} is the standard oriented basis. The biholomorphism is given by For each t ∈ [0, +∞) we have that ι t : L → M is a totally real submanifold: if a tangent space T p L contained a complex line, it would contradict that ι : N → M has complex rank 1.
We claim that {ι t } is a geodesic in T (see Definition 5.4): considering the vector field X = ∂θ we have that dι t (X) = ∂ι t ∂θ = ∂ι ∂θ , and ∂ι t ∂t = ∂ι ∂t , so the equality (1) holds because it is equivalent to which is the Cauchy-Riemann equation for each holomorphic disc. Applying Theorem 5.5 we have that the J-volume functional restricted to the curve {ι t } ⊂ T is convex because Ric ≤ 0. Thanks to Theorem 5.2 we have that t = 0 is a critical point because ι 0 is J-minimal, and Vol J (ι 0 ) > 0. Furthermore it tends to 0 for t tending to +∞ because the Riemannian volume of ι t (L) tends to 0 and Vol J ≤ Vol (see Lemma 5.2), and it leads to a contradiction because the J-volume functional is convex.
Observe that, according to Theorem 5.3, the main theorem concerns exactly minimal Lagrangian submanifolds when the ambient space is negative Kähler-Einstein.
In Calabi-Yau manifolds the statement above is trivial for special Lagrangian and J-minimal totally real submanifolds since they are calibrated submanifolds, thus their homology classes are not null.

Examples
We want to look for nontrivial examples of minimal Lagrangians in a Kähler manifold (M, J, g, ω). Denoting with ρ the Ricci 2-form, the Lagrangian condition implies that d [ω (H, ·)] = ρ.
Thus a minimal Lagrangian submanifold L would have ρ| L = 0, and looking for examples of minimal Lagrangian submanifolds in Kähler ambients is an overconstrained problem. See [8] for details. This last condition is automatically verified on a Lagrangian submanifold L if M is a Kähler-Einstein manifold, i.e. if ρ (·, ·) = λω (·, ·), for some λ ∈ R.
Thus Kähler-Einstein manifolds are the suitable context in which look for minimal Lagrangian submanifolds. Very few examples are known; see [12] for results about existence of minimal Lagrangian tori in product spaces. We will exhibit a few more, following [3].
Let us consider a homogeneous polynomial p of degree d on C n+1 , with d > n + 1, and suppose that the zero locus is smooth. Denoting with J the induced complex structure, by the adjunction formula we have that its first Chern class is where ω is the pull back of the Fubini-Study metric on P n (C). Observe that c 1 (M, J) < 0.
A theorem of Yau guarantees that there is a unique Kähler-Einstein 2form ω on M in the class −c 1 (M, J). Let g denote the corresponding metric.
If p has real coefficients, M is invariant under complex conjugation on P n (C): its restriction to M gives a real structure c ∈ Diff (M ), i.e. an antiholomorphic involution such that c * J = −J.
The fixed point set L of c, called the real locus, if non empty, is a totally geodesic (thus minimal) Lagrangian submanifold of M , in fact ω p (v, w) = ω p (c(v), c(w)) = c * ω p (v, w) = −ω p (v, w) , for any p ∈ L and v, w ∈ T p L, thus ω| L ≡ 0.
The critical locus of q in R 3 is the circle C = (x, y, z) ∈ R 3 | x 2 + y 2 − 1 = 0, z = 0 plus the origin O = (0, 0, 0). Since q(C) = 0 and q(O) = 1, the regular values of q are all the real numbers except 0 and 1. Let ε be a real number in (0, 1) and consider the polynomial q ε := q − ε. Since 0 is not a critical value for q ε , the zero locus of q ε is smooth, and is a boundary component of the region q −1 [0, ε], which retracts onto C. Now let us consider the homogeneous polynomial Its singular points in P 3 (C) are the two nonreal points [±i : 1 : 0 : 0], hence M = {Q = 0} ⊂ P 3 (C) is a compact Kähler orbifold, so the Calabi conjecture holds; cf. [10]. Thus, away from the singular points, M admits a negative Kähler-Einstein structure.
Since Q(X, Y, Z, W ) = W = 0 only at the origin, the real locus is smooth and is given by It is a minimal Lagrangian torus as above.
It is possible to apply Proposition 3.1 to the two examples, in particular they do not admit any filling by holomorphic discs.