On the Calabi-Yau equation in the Kodaira-Thurston manifold

We review some previous results about the Calabi-Yau equation on the Kodaira-Thurston manifold equipped with an invariant almost-Kaehler structure and assuming the volume form invariant by the action of a torus. In particular, we observe that under some restrictions the problem is reduced to a Monge-Amp\`ere equation by using the ansatz $\tilde \omega=\Omega-dJdu+da$, where $u$ is a $T^2$-invariant function and $a$ is a $1$-form depending on $u$. Furthermore, we extend our analysis to non-invariant almost-complex structures by considering some basic cases and we finally take into account a generalization to higher dimensions.


Introduction
The Calabi-Yau problem in 4-dimensional almost-Kähler manifolds is a PDEs system arising from the generalization of the classical Calabi-Yau theorem to the non-Kähler setting.
An analogue problem still makes sense in the almost-Kähler case, when J is merely an almost-complex structure and Ω is a J-compatible symplectic form. It turns out that in this more general context, the PDEs system arising from (2) is overdetermined for n ≥ 3, while it is elliptic in dimension 4 (see [3]). Consequently, the Calabi-Yau problem is mainly studied in 4-dimensional almost-Kähler manifolds (see [1,2,10,11,12,15] and the references therein). The study of the problem is strongly motivated by a project of Donaldson involving compact symplectic 4-manifolds (see [3]). The project is based on a conjecture stated in [3] and partially confirmed by Taubes in [13].
In [15] Weinkove attacked the problem by introducing a symplectic potential. Indeed, given two almost-Kähler forms Ω andω on a compact almost-complex manifold (X, J) (ω − Ω) ∧ω = −dJdu ∧ω . In terms of u one can always writeω where a is a 1-form which can be assumed co-closed with respect to the co-differential induced byω (in this way a is unique up addiction of harmonic 1-forms).
Weinkove proved that in order to show the solvability of the Calabi-Yau problem (2) it's enough to provide an a priori estimate on the C 0 -norm of the almost-Kähler potential (see theorem 1 in [15]); that can be always done if the L 1 -norm of the Nijenhuis tensor of J is small enough (see theorem 2 in [15]).
In [12] Tosatti and Weinkove studied the Calabi-Yau problem on the Kodaira-Thurston manifold (M, Ω 0 , J 0 ) showing that under the assumption on the initial datum F to be invariant by the action of a 2-dimensional torus the problem has a unique solution. The Kodaira-Thurston manifold M is a 4-dimensional 2-step nilmanifold carrying a natural almost-Kähler structure and it can be viewed as a torus bundle over a torus (more precisely M is an S 1 -bundle over a 3-dimensional torus).
In [4] it is observed that if F is T 2 -invariant, then (2) on the Kodaira-Thrurston manifold M can be rewritten in terms of the Monge-Ampère equation xy = e F on the 2-dimensional torus T 2 xy and the Tosatti-Weinkove result in [12] can be alternatively obtained by applying a result of Y.Y. Li in [8]. A similar approach was then adopted in [1,4] in order to study the Calabi-Yau problem in every 4-dimensional torus bundle over a torus equipped with an invariant almost-Kähler structure. In this more general case the equation writes in terms of a "modified" Monge-Ampère equation which is still solvable. Furthermore, in [2] it is studied the equation on the Kodaira-Thurston manifold when F is S 1 -invariant (instead of T 2 -invariant as in the previous papers). It turns out that in this last case the Calabi-Yau problem writes as a PDE on the 3-dimensional torus T 3 xyt which is not of Monge-Ampère type anymore.
In this paper we review some results in [4] showing that when the projection is Lagrangian, the reduction of the Calabi-Yau problem on the Kodaira-Thurston manifold to a scalar PDE can be obtained by setting where γ 1 and γ 2 are suitable invariant forms depending on (Ω, J), u is in the same space of F and y is a coordinate on the base.
In section 3 we study the Calabi-Yau equation on (M, Ω 0 ) for S 1 -invariant almost complex structures J compatible to Ω 0 . Under some strong restrictions on J, the equation can be still reduced to a PDE in a single unknown function. In section 4 we prove the solvability of the arising equations in some special cases leaving the more general cases for an eventually future work.
In the last section we consider a generalization of the previous sections to 2-step nilmanifold in higher dimensions. A remark on the notation. If P is an m-torus bundle over an n-torus, we denote by T n the base of P and by T m the principal fiber, in order to distinguish the base and the fibers. during "The 4 th workshop on complex Geometry and Lie groups" hold in Nara from the 22nd to the 26th of March 2016. The author thanks Anna Fino, Ryushi Goto and Keizo Hasegawa for the kind invitation. Moreover, the author is very grateful to Ernesto Buzano, Giulio Ciraolo, Valentino Tosatti and Michela Zedda for useful conversations.

Calabi-Yau equations on the Kodaira-Thurston manifold
In this section we review some results in [1,2,4] about the Calabi-Yau equation on the Kodaira-Thurston manifold. The Kodaira-Thurston manifold is a compact 2-step nilmanifold M defined as the quotient M = Γ\G, where G is the Lie group given by R 4 in the variables (x 1 , x 2 , y 1 , y 2 ) with the multiplication and Γ is the co-compact lattice given by Z 4 with the induced multiplication. Alternatively M can be defined as the product M = Γ 0 \Nil 3 × S 1 , where Nil 3 is the 3-dimensional real Heisenberg group and Γ 0 is the lattice in Nil 3 of matrices having integers entries. M has a natural structure of principal S 1 -bundle over a 3-dimensional torus T 3 induced by the map [x 1 , x 2 , y 1 , y 2 ] → [x 1 , x 2 , y 1 ] and it is parallelizable. A global co-frame on M is for instance given by For such co-frame we have and its dual basis is given by {∂ x 1 , ∂ x 2 + x 1 ∂ y 2 , ∂ y 1 , −∂ y 2 }. Furthermore, M has the "natural"almost-Kähler structure (Ω 0 , J 0 ) given by the symplectic form and the Riemannian metric The following proposition is proved in [2] Proposition 2.1. Let u : M → R be an S 1 -invariant function and Then dα is of type (1, 1) and where I is the identity 2 × 2 matrix and Proof. Let u : M → R be an S 1 -invariant function. Then which is a form of type (1, 1) with respect to J 0 . Formula (5) follows from a straightforward computation. where u is an unknown S 1 -invariant map. In this way the Calabi-Yau problem reduces to the single equation (7) det(I + A(u)) − u 2 is given by (6). The main result in [2] is the following Special cases of equation (7) occur when we see M as a 2-torus bundle over a 2dimensional torus and we assume F depending only on the coordinates of the base. Those cases correspond to assume F depending either on (x 1 , x 2 ) or on (x 2 , y 1 ) (the case F = F (x 1 , y 1 ) is equivalent to F = F (x 2 , y 1 )).
If F = F (x 1 , x 2 ), we can assume u depending only on (x 1 , x 2 ) and (7) reduces to the' Monge-Ampère type equation This equation has a solution in view of a theorem of Y.Y. Li (see [8]). Note that in this case the solution u to (8) is an almost-Kähler potential ofω = Ω 0 + dα with respect to Ω 0 . Indeed, If F = F (x 2 , y 1 ), we assume u depending only on (x 2 , y 1 ) and (7) reduces to the "modified" Monge-Ampère equation The existence of a solution to this last equation was proved in [4]. Note that in this casẽ and u is not an almost-Kähler potential.
Next, we take into account the Calabi-Yau problem on M viewed as a 2-torus bundle over a 2-torus equipped with an invariant Lagrangian almost-Kähler structure (Ω, J) and we assume F defined on the base. Here by Lagrangian we mean that the fibers of the fibration are Lagrangian submanifolds.
where l 1 and l 2 are positive real numbers.

The equation for non-invariant almost-complex structures
As pointed out in [12] it is interesting to extend the results described in the previous section to torus-invariant almost complex structures on the Kodaira-Thurston manifold M which are compatible to "natural" symplectic form Ω 0 defined in (3). In this section we consider some basic cases. Let h = h(x 1 , y 1 ) be a function in C ∞ (T 2 x 1 y 1 ) and consider the family of Ω 0 -compatible almost-complex structures J h induced by the relations The following result is a generalization of proposition 2.1 to the family J h . Then dα is of type (1, 1) and (13) where I is the identity 2 × 2 matrix and Proof. Let u be an S 1 -invariant function. Then i.e., which is of type (1, 1) and In view of proposition 3.1, the Calabi-Yau equation on (M, Ω 0 , J h ), for an S 1 -invariant function F : M → R can be reduced to (15) det(I + A h (u)) − e −h u 2 x 2 y 1 = e F where A h is given by (14) and u : M → R is an unknown S 1 -invariant function. Note that for h = 0, equation (15) reduces to equation (7) studied in [2]. We consider the following special cases: If h = h(x 1 ) and F = F (x 1 , x 2 ) we may assume u depending only on (x 1 , x 2 ) and (15) reduces in the variables x = x 1 , y = x 2 to det 1 + e h u xx + e h h ′ u x u xy e h u xy 1 + u yy = e F on the 2-dimensional torus T 2 xy . Such an equation can be rewritten as If h = h(y 1 ) and F = F (x 2 , y 1 ), then we assume u depending only on (x 2 , y 1 ) and (15) reduces in the variables x = y 1 , y = x 2 to on T 2 xy . Such an equation can be rewritten as Both cases fit in the following class of equations on T 2 where h = h(x) is a smooth 1-periodic functions on R and c ∈ R. We will show the solvability of the last class of equations in the next section.

Solvability of the special cases
The aim of this section is to prove the following result Then equation has a solution u ∈ C ∞ (T 2 ).
Before proving theorem 4.1 we consider the following preliminary lemma which is a slight generalization of lemma 6.3 in [4].
where C is a constant depending only on c and h.
Since v(s 0 ) = 0, we have Proof of Theorem 4.1. Fix 0 < α < 1 and let C 2,α 0 (T 2 ) be the space of C 2,α -functions u on T 2 satisfying Then we consider the operator T : in order that u ∈ C 2,α 0 (T 2 ) solves (16) if and only if T (u, 1) = 0. Then we define the set S := {t ∈ [0, 1] : there exists u ∈ C 2,α 0 (T 2 ) such that T (u, t) = 0} . Note that S is not empty since u ≡ 0 satisfies T (u, 0) = 0. We will show that 1 ∈ S by proving that S is open and closed in [0,1]. In this way we get that (16) has a solution u in C 2,α (T 2 ) and theorem 3 in [9] implies that u is in fact C ∞ . Note that if (u, t) ∈ C 2 0 (T 2 ) × [0, 1] is such that T (u, t) = 0, then the matrix and by using (19) we get Hence u satisfies a C 1 a priori bound. Furthermore, if t ∈ [0, 1] is fixed, equation belongs to the class of equations studied in [7] and theorem 2 in [7] implies that if u ∈ C 2,α 0 (T 2 ) solves T (u, t) = 0 for some t and satisfies a priori C 1 bound, then it also satisfies a C 2,α bound. This implies that S is closed in [0, 1]. Indeed, let t n be a sequence in S converging tot in [0, 1]. To each t n corresponds a function u n ∈ C 2,α 0 (T 2 ) such that T (u n , t n ) = 0. The C 2,α a priori bound on solutions to T (u, t) = 0 implies that the sequence u n is bounded in C 2,α 0 (T 2 ) and so it admits a subsequence, which we still denote by u n , which converges in C 2 0 (T 2 ) to a functionū ∈ C 2 0 (T 2 ). Since T is continuos, T (ū,t) = 0 and so, in view of [7],ū in C 2,α (T 2 ). Hencet ∈ S and S is closed.

A direct computation yields that
and so L is uniformly elliptic. L is injective by maximum principle and it is surjective in view of elliptic theory (see e.g. [5]). Therefore the implicit function theorem implies that t has a open neighborhood contained in S, and so S is open, as required.
We then consider on M n the symplectic form Ω n = n k=1 α k ∧ β k and the Ω n -compatible almost-complex structure J n induced by Ω n and the natural metric In terms of the basis B = {e 1 , . . . , e n , f 1 , . . . , f n }, J n is defined by Let u be a T n+1 -invariant function on M n ; then where A(u) = (A ij ) is the n × n matrix A 11 = u x 1 x 1 + u y 1 y 1 − u y 1 , A ij = u x i x j , if (i, j) = (1, 1) .
In analogy to the case n = 2, we can obtain special cases by regarding M n as a principal T n -bundle over a T n and assuming F to be T n -invariant. It is not restrictive considering only the following two cases: F = F (x 1 , . . . , x n ) , or F = F (x 2 , . . . , x n , y 1 ).
• In the first case F = F (x 1 , . . . , x n ), equation (21) reduces to the Monge-Ampère equation det(I + H(u)) = e F on the n-dimensional torus T n = R n /Z n , where H(u) is the Hessian metric of u. In this case the equation has a solution in view of [8].