On the Calabi-Yau equation in the Kodaira-Thurston manifold
aa r X i v : . [ m a t h . DG ] S e p ON THE CALABI-YAU EQUATION IN THE KODAIRA-THURSTONMANIFOLD
LUIGI VEZZONI
Abstract.
We review some previous results about the Calabi-Yau equation on theKodaira-Thurston manifold equipped with an invariant almost-K¨ahler structure andassuming the volume form T -invariant. In particular, we observe that under somerestrictions the problem is reduced to a Monge-Amp`ere equation by using the ansatz˜ ω = Ω − dJdu + da , where u is a T -invariant function and a is a 1-form depending on u . Furthermore, we extend our analysis to non-invariant almost-complex structures byconsidering some basic cases and we finally take into account a generalization to higherdimensions. Introduction
The
Calabi-Yau problem in 4-dimensional almost-K¨ahler manifolds is a PDEs systemarising from the generalization of the classical Calabi-Yau theorem to the non-K¨ahlersetting.The Calabi-Yau theorem [14] states that on a compact K¨ahler manifold (
X, J,
Ω) forevery smooth function F : X → R such that(1) Z X e F Ω n = Z X Ω n there always exists a unique K¨ahler form ˜ ω on ( X, J ) satisfying(2) [˜ ω ] = [Ω] , ˜ ω n = e F Ω n . An analogue problem still makes sense in the almost-K¨ahler case, when J is merely analmost-complex structure and Ω is a J -compatible symplectic form. It turns out thatin 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 problemis mainly studied in 4-dimensional almost-K¨ahler manifolds (see [1, 2, 10, 11, 12, 15] andthe references therein).The study of the problem is strongly motivated by a project of Donaldson involvingcompact 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-K¨ahler forms Ω and ˜ ω on a compact almost-complex manifold ( X, J ) Date : July 29, 2018.2010
Mathematics Subject Classification. satisfying [Ω] = [˜ ω ] there always exists a function u , called the symplectic potential , suchthat (˜ ω − Ω) ∧ ˜ ω = − dJ du ∧ ˜ ω . In terms of u one can always write˜ ω = Ω − dJ du + da , where a is a 1-form which can be assumed co-closed with respect to the co-differentialinduced by ˜ ω (in this way a is unique up addiction of harmonic C -norm of the almost-K¨ahler potential(see theorem 1 in [15]); that can be always done if the L -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-Thurstonmanifold ( M, Ω , J ) showing that under the assumption on the initial datum F to beinvariant by the action of a 2-dimensional torus the problem has a unique solution. TheKodaira-Thurston manifold M is a 4-dimensional 2-step nilmanifold carrying a naturalalmost-K¨ahler structure and it can be viewed as a torus bundle over a torus (more precisely M is an S -bundle over a 3-dimensional torus).In [4] it is observed that if F is T -invariant, then (2) on the Kodaira-Thrurston manifold M can be rewritten in terms of the Monge-Amp`ere equation(1 + u xx )(1 + u yy ) − u xy = e F on the 2-dimensional torus T xy and the Tosatti-Weinkove result in [12] can be alternativelyobtained 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 overa torus equipped with an invariant almost-K¨ahler structure. In this more general case theequation writes in terms of a “modified” Monge-Amp`ere equation which is still solvable.Furthermore, in [2] it is studied the equation on the Kodaira-Thurston manifold when F is S -invariant (instead of T -invariant as in the previous papers). It turns out that inthis last case the Calabi-Yau problem writes as a PDE on the 3-dimensional torus T xyt which is not of Monge-Amp`ere type anymore.In this paper we review some results in [4] showing that when the projection is La-grangian, the reduction of the Calabi-Yau problem on the Kodaira-Thurston manifold toa scalar PDE can be obtained by setting˜ ω = Ω + d ( − J u + uγ + u y γ )where γ and γ are suitable invariant forms depending on (Ω , J ), u is in the same spaceof F and y is a coordinate on the base.In section 3 we study the Calabi-Yau equation on ( M, Ω ) for S -invariant almostcomplex structures J compatible to Ω . Under some strong restrictions on J , the equationcan be still reduced to a PDE in a single unknown function. In section 4 we prove thesolvability of the arising equations in some special cases leaving the more general casesfor an eventually future work.In the last section we consider a generalization of the previous sections to 2-step nil-manifold in higher dimensions. ALABI-YAU EQUATION ON THE KODAIRA-THURSTON MANIFOLD 3
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 thefibers. Acknowledgments.
The research of the present paper was originated by some discussionsduring “
The th workshop on complex Geometry and Lie groups ” hold in Nara from the22nd to the 26th of March 2016. The author thanks Anna Fino, Ryushi Goto and KeizoHasegawa for the kind invitation. Moreover, the author is very grateful to Ernesto Buzano,Giulio Ciraolo, Valentino Tosatti and Michela Zedda for useful conversations.2. Calabi-Yau equations on the Kodaira-Thurston manifold
In this section we review some results in [1, 2, 4] about the Calabi-Yau equation onthe Kodaira-Thurston manifold. The
Kodaira-Thurston manifold is a compact 2-stepnilmanifold M defined as the quotient M = Γ \ G , where G is the Lie group given by R in the variables ( x , x , y , y ) with the multiplication( x , x , y , y ) · ( x ′ , x ′ , y ′ , y ′ ) = ( x + x ′ , x + x ′ , y + y ′ , y + y ′ + x x ′ )and Γ is the co-compact lattice given by Z with the induced multiplication. Alternatively M can be defined as the product M = Γ \ Nil × S , where Nil is the 3-dimensional realHeisenberg group Nil = nh x z y i | x, y, z ∈ R o and Γ is the lattice in Nil of matrices having integers entries. M has a natural structureof principal S -bundle over a 3-dimensional torus T induced by the map [ x , x , y , y ] [ x , x , y ] and it is parallelizable. A global co-frame on M is for instance given by e = dx , e = dx , f = dy , f = dy − x dx . For such co-frame we have de = de = df = 0 , df = − e ∧ e and its dual basis is given by { ∂ x , ∂ x + x ∂ y , ∂ y , − ∂ y } . Furthermore, M has the“natural”almost-K¨ahler structure (Ω , J ) given by the symplectic form(3) Ω = e ∧ f + e ∧ f and the Riemannian metric(4) g = e ⊗ e + f ⊗ f + e ⊗ e + f ⊗ f . The following proposition is proved in [2]
Proposition 2.1.
Let u : M → R be an S -invariant function and α := − J du − ue . Then dα is of type (1 , and (5) (Ω + dα ) = (cid:0) det( I + A ( u )) − u x y (cid:1) Ω , LUIGI VEZZONI where I is the identity × matrix and (6) A ( u ) = (cid:18) u x x + u y y + u y u x x u x x u x x (cid:19) . Proof.
Let u : M → R be an S -invariant function. Then du = u x e + u x e + u y f and − J du = u x f + u x f − u y e and − dJ du = X i,j =1 u x i x j e i ∧ f j + u x y e ∧ e + u x y f ∧ f + u y y e ∧ f − u x e ∧ e . Therefore, if α = − J du − ue , we have dα = − dJ du − du ∧ e = X i,j =1 u x i x j e i ∧ f j + u x y e ∧ e + u x y f ∧ f + u y y e ∧ f + u y e ∧ f which is a form of type (1 ,
1) with respect to J . Formula (5) follows from a straightforwardcomputation. (cid:3) Proposition 2.1 is useful in the study of the Calabi-Yau problem on ( M, Ω , J ). Indeed,let F : M → R be an S -invariant function satisfying R M e F Ω = 1 and consider theCalabi-Yau equation (Ω + dα ) = e F Ω on ( M, Ω , J ). In view of proposition 2.1, wecan study the Calabi-Yau problem by introducing the ansatz α = − J du − ue where u is an unknown S -invariant map. In this way the Calabi-Yau problem reduces tothe single equation(7) det( I + A ( u )) − u x y = e F , on the 3-dimensional torus T x x y , where A ( u ) is given by (6). The main result in [2] isthe following Theorem 2.2.
Equation (7) has a solution for every S -invariant initial datum F : M → R satisfying R M e F Ω = 1 . Consequently the Calabi-Yau problem (Ω + dα ) = e F Ω hasa unique solution for every S -invariant function F : M → R . Special cases of equation (7) occur when we see M as a 2-torus bundle over a 2-dimensional torus and we assume F depending only on the coordinates of the base. Thosecases correspond to assume F depending either on ( x , x ) or on ( x , y ) (the case F = F ( x , y ) is equivalent to F = F ( x , y )).If F = F ( x , x ), we can assume u depending only on ( x , x ) and (7) reduces to the’Monge-Amp`ere type equation(8) (1 + u x x )(1 + u x x ) − u x x = e F ALABI-YAU EQUATION ON THE KODAIRA-THURSTON MANIFOLD 5 on the 2-dimensional torus T x x . This equation has a solution in view of a theorem ofY.Y. Li (see [8]). Note that in this case the solution u to (8) is an almost-K¨ahler potentialof ˜ ω = Ω + dα with respect to Ω . Indeed,˜ ω = (1 + u x x ) e ∧ f + (1 + u x x ) e ∧ f + u x x e ∧ f + u x x f ∧ e and ˜ ω − Ω = − dJ du + da where a = − ue . Hence da = u x e ∧ e and ˜ ω ∧ da = 0which implies (˜ ω − Ω ) ∧ ˜ ω = − dJ du ∧ ˜ ω . If F = F ( x , y ), we assume u depending only on ( x , y ) and (7) reduces to the“modified” Monge-Amp`ere equation(9) (1 + u y y + u y )(1 + u x x ) − u x y = e F on the 2-dimensional torus T x y . The existence of a solution to this last equation wasproved in [4]. Note that in this case˜ ω = (1 + u y y + u y ) e ∧ f + (1 + u x x ) e ∧ f + u x y e ∧ e + u x y f ∧ f and if u solves (9), then dα = − dJ du + da , where da = − u x e ∧ e − u y e ∧ f . Therefore da ∧ ˜ ω = 0and u is not an almost-K¨ahler potential.Next, we take into account the Calabi-Yau problem on M viewed as a 2-torus bundleover a 2-torus equipped with an invariant Lagrangian almost-K¨ahler structure (Ω , J ) andwe assume F defined on the base. Here by Lagrangian we mean that the fibers of thefibration are Lagrangian submanifolds.
Proposition 2.3.
Let (Ω , J ) be an invariant almost-K¨ahler structure on M . Then thereexist real numbers µ and µ and an invariant -form β such that if u = u ( x , x ) is asmooth function on M , then α = − J du + µ u e − µ u e − u y β is such that dα is of type (1 , . Moreover (10) (Ω + dα ) = 1 l l (cid:0) ( l + u x x )( l + u x x ) − ( u x x ) (cid:1) Ω . where l and l are positive real numbers. LUIGI VEZZONI
Proof.
We set x = x and x = y in order to simplify the notation. We can find aninvariant Hermitian coframe { α , α , β , β } on M such thatΩ = α ∧ β + α ∧ β and dx = Aα , dy = Bα + Cα . Note that dx ∧ dy = ACα ∧ α and we can write dβ = λ dx ∧ dy , dβ = λ dx ∧ dy for some λ , λ in R . Now du = u x dx + u y dy = ( Au x + Bu y ) α + Cu y α and − J du = ( Au x + Bu y ) β + Cu y β So − dJ du = Au xx dx ∧ β + Au xy dy ∧ β + Cu xy dx ∧ β + Cu yy dy ∧ β + d ( γ + Bu y β )where γ = λ Au dy − λ Cu dx .
Hence − dJ du = A u xx α ∧ β + ABu xy α ∧ β + ACu xy α ∧ β + ACu xy α ∧ β + BCu yy α ∧ β + C u yy α ∧ β + d ( Bu y α + γ ) . which implies that α = − J du − Bu y α − γ is such that dα is of type (1 , dα ) = (cid:0) (1 + A u xx )(1 + C u yy ) − ( ACu xy ) (cid:1) Ω = 1 l l (cid:0) ( l + u xx )( l + u yy ) − ( u xy ) (cid:1) Ω where l = 1 /A and l = 1 /C and the claim follows. (cid:3) Proposition 2.4.
Let (Ω , J ) be an invariant almost-K¨ahler structure on M which is Lagrangian with respect to the fibration [ x , x , y , y ] [ x , y ] . There exist invariant -forms γ , γ such that if u = u ( x , y ) is a smooth function on M , then α = − J du + uγ + u y γ is such that dα is of type (1 , . Moreover (11) (Ω + dα ) = 1 l l (cid:0) ( l + u x x )( l + u y y + m u x + m u y ) − ( u x y ) (cid:1) Ω where l , l , m , m ∈ R and l , l < . ALABI-YAU EQUATION ON THE KODAIRA-THURSTON MANIFOLD 7
Proof.
First of all we use that (Ω , J ) is an invariant almost-K¨ahler structure on M whichis Lagrangian with respect to [ x , x , y , y ] [ x , y ], then there exists an invariantHermitian co-frame { α , α , β , β } on M such thatΩ = α ∧ β + α ∧ β and α ∈ h e i , β ∈ h e , f i , α ∈ h e , e , f i (see lemma 5.1 in [4]). In this way dx = Aα , dy = Bα + Cβ , dβ = λ α ∧ β + µα ∧ β for some A, B, C, λ, µ ∈ R . In order to semplify the notation we set x = x and y = y .Then du = u x dx + u y dy = Au x α + u y ( Bα + Cβ ) = ( Au x + Bu y ) α + Cu y β and − J du = − ( Au x + Bu y ) β + Cu y α . So − dJ du = − Au xx dx ∧ β − Au xy dy ∧ β + Cu xy dx ∧ α + Cu yy dy ∧ α − ( Au x + Bu y ) (cid:0) λ α ∧ β + µα ∧ β (cid:1) − d ( Bu y β )i.e., − dJ du = − A u xx α ∧ β − BAu xy α ∧ β − ACu xy β ∧ β + ACu xy α ∧ α + CBu yy α ∧ α + C u yy β ∧ α − ( Au x + Bu y ) (cid:0) λ α ∧ β + µα ∧ β (cid:1) − d ( Bu y β ) . Now, ( Au x + Bu y ) (cid:0) λ α ∧ β + µα ∧ β (cid:1) = λ ( Au x + Bu y ) α ∧ β + d ( µu β )and we can write − dJ du = ( − C u yy − λAu x − λBu y ) α ∧ β + ( − A u xx − BAu xy ) α ∧ β − ACu xy β ∧ β − ( ACu xy + B u yy ) α ∧ α − d ( µuβ + Bu y β )which implies the first part of the statement.Moreover,(Ω + dα ) = (cid:0) (1 − A u xx )(1 − C u yy − λAu x − λBu y ) − ( ACu xy ) (cid:1) Ω = 1 l l (cid:0) ( l + u xx )( l + u yy + m u x + m u y ) − ( u xy ) (cid:1) Ω where l = − A , l = − C , m = − λ AC , m = − λ BC and the claim follows. (cid:3) LUIGI VEZZONI
From propositions 2.3 and 2.4 it follows that if we see M as 2-torus over a 2-torusand we fix an invariant Lagrangian almost-K¨ahler structure (Ω , J ) on M ; then for everygiven F defined on the base of M and satisfying R M e F Ω = R M Ω the correspondingCalabi-Yau equation can be written in terms of an unknown function u on the base T xy of M as 1 l l (cid:0) ( l + u xx )( l + u yy + m u x + m u y ) − ( u xy ) (cid:1) = e F where l , l , m , m ∈ R and l and l are both positive or negative. This kind of equationsare solvable in view of theorem 6.2 in [4].3. The equation for non-invariant almost-complex structures
As pointed out in [12] it is interesting to extend the results described in the previoussection to torus-invariant almost complex structures on the Kodaira-Thurston manifold M which are compatible to “natural” symplectic form Ω defined in (3). In this sectionwe consider some basic cases. Let h = h ( x , y ) be a function in C ∞ ( T x y ) and considerthe family of Ω -compatible almost-complex structures J h induced by the relations(12) J h ( e ) = − e h f J h ( e ) = − f . The following result is a generalization of proposition 2.1 to the family J h . Proposition 3.1.
Let u : M → R be an S -invariant function and α := − J h du − ue . Then dα is of type (1 , and (13) (Ω + dα ) = (cid:0) det( I + A h ( u )) − e − h u x y (cid:1) Ω where I is the identity × matrix and (14) A h ( u ) = (cid:18) e h u x x + e − h u y y + u y + e h h x u x − e − h h y u y u x x e h u x x u x x (cid:19) Proof.
Let u be an S -invariant function. Then − J h du = e h u x f + u x f − e − h u y e and − dJ h du = (e h u x ) x e ∧ f + e h u x x e ∧ f + u x x e ∧ f + u x x e ∧ f + u x y f ∧ f + e − h u x y e ∧ e + (e − h u y ) y e ∧ f − u x e ∧ e , i.e., − dJ h du = (cid:0) e h u x x + e − h u y y + e h h x u x − e − h h y u y + u y (cid:1) e ∧ f + u x x e ∧ f + e h u x x e ∧ f + u x x e ∧ f + u x y f ∧ f + (cid:0) e − h u x y − u x (cid:1) e ∧ e . ALABI-YAU EQUATION ON THE KODAIRA-THURSTON MANIFOLD 9
Therefore if α = − J h du − ue , then dα = (cid:0) e h u x x + e − h u y y + e h h x u x − e − h h y u y + u y (cid:1) e ∧ f + u x x e ∧ f + e h u x x e ∧ f + u x x e ∧ f + u x y f ∧ f + e − h u x y e ∧ e . which is of type (1 ,
1) and(Ω + dα ) = det( I + A h ( u )) − e − h u x y , as required. (cid:3) In view of proposition 3.1, the Calabi-Yau equation on ( M, Ω , J h ), for an S -invariantfunction F : M → R can be reduced to(15) det( I + A h ( u )) − e − h u x y = e F where A h is given by (14) and u : M → R is an unknown S -invariant function. Note thatfor h = 0, equation (15) reduces to equation (7) studied in [2]. We consider the followingspecial cases:If h = h ( x ) and F = F ( x , x ) we may assume u depending only on ( x , x ) and (15)reduces in the variables x = x , y = x todet (cid:18) h u xx + e h h ′ u x u xy e h u xy u yy (cid:19) = e F on the 2-dimensional torus T xy . Such an equation can be rewritten asdet (cid:18) e − h + u xx + h ′ u x u xy u xy u yy (cid:19) = e F − h . If h = h ( y ) and F = F ( x , y ), then we assume u depending only on ( x , y ) and (15)reduces in the variables x = y , y = x todet (cid:18) − h u xx + (1 − e − h h ′ ) u x u xy e − h u xy u yy (cid:19) = e F on T xy . Such an equation can be rewritten asdet (cid:18) e h + u xx + (e h − h ′ ) u x u xy u xy u yy (cid:19) = e F + h . Both cases fit in the following class of equations on T xy det (cid:18) e − h + u xx + ( c e − h + h ′ ) u x u xy u xy u yy (cid:19) = e F − h where h = h ( x ) is a smooth 1-periodic functions on R and c ∈ R . We will show thesolvability 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
Theorem 4.1.
Let h = h ( x ) be a smooth -periodic functions on R , c ∈ R and let F = F ( x, y ) ∈ C ∞ ( T ) be such that Z T e F dx ∧ dy = 1 . Then equation (16) det (cid:18) e − h + u xx + ( c e − h + h ′ ) u x u xy u xy u yy (cid:19) = e F − h has a solution u ∈ C ∞ ( T ) . Before proving theorem 4.1 we consider the following preliminary lemma which is aslight generalization of lemma 6.3 in [4].
Lemma 4.2.
Let h, v ∈ C ( R ) be -periodic functions satisfying e h v ′ + ( c + e h h ′ ) v > − . Assume there exists s ∈ [0 , such that v ( s ) = 0 ; then k v k C ≤ C , where C is a constant depending only on c and h .Proof. Let G be a primitive of c e − h + h ′ in R . Since v ′ + ( c e − h + h ′ ) v > − e − h , in terms of G we have e G ( v ′ + G ′ v ) > − e G − h , i.e. dds (e G v ) > − e G − h . Since v ( s ) = 0, we have Z ss dds (e G v ) ds > − Z ss e G − h dτ , for every s ≥ , which implies v ( s ) > − e − G ( s ) Z ss e G − h dτ , for every s ∈ [1 , . On the other hand Z s s dds (e G v ) ds > − Z s s e G − h dτ , for every s ≤ , which implies v ( s ) < e − G ( s ) Z s s e G − h dτ , for every s ∈ [ − , . The claim follows since v is 1-periodic. (cid:3) ALABI-YAU EQUATION ON THE KODAIRA-THURSTON MANIFOLD 11
Now we can prove theorem 4 . Proof of Theorem . . Fix 0 < α < C ,α ( T ) be the space of C ,α -functions u on T satisfying Z T u dx ∧ dy = 0 . Then we consider the operator T : C ,α ( T ) × [0 , → C ,α ( T ) defined by T ( u, t ) = det (cid:18) e − h + u xx + ( c e − h + h ′ ) u x u xy u xy u yy (cid:19) − e − h (cid:0) t e F + 1 − t (cid:1) in order that u ∈ C ,α ( T ) solves (16) if and only if T ( u,
1) = 0. Then we define the set S := { t ∈ [0 ,
1] : there exists u ∈ C ,α ( T ) such that T ( u, t ) = 0 } . Note that S is not empty since u ≡ T ( u,
0) = 0. We will show that 1 ∈ S byproving that S is open and closed in [0 , u in C ,α ( T ) and theorem 3 in [9] implies that u is in fact C ∞ . Note that if ( u, t ) ∈ C ( T ) × [0 ,
1] is such that T ( u, t ) = 0, then the matrix A h := det (cid:18) e − h + u xx + ( c e − h + h ′ ) u x u xy u xy u yy (cid:19) is positive-defined. Indeed, since R T e F dx ∧ dy = 0, then A h ( u ) is non-singular and at aminimum point of u all the eigenvalues of A h are positive.Now we prove that S is closed. First of all we observe that if u ∈ C ( T ) satisfies T ( u, t ) = 0 for some t ∈ [0 , h u xx + ( c + e h h ′ ) u x > − , (17) 1 + u yy > − . (18)Indeed, since (1 + e h u xx + ( c + e h h ′ ) u x )(1 + u yy ) > x , y ) where u reaches its minimum value. Lemma 4.2 then implies(19) k u x k C ≤ C and k u y k C ≤ C where C is a constant depending on c, h and k . Now we focus on the C estimate on u .Let ( x , y ) be a point in [0 , × [0 ,
1] where u vanishes, then u ( x, y ) = ( x − x ) Z u x ((1 − t ) x + tx , (1 − t ) y + ty ) dt +( y − y ) Z u y ((1 − t ) x + tx , (1 − t ) y + ty )) dt, and by using (19) we get | u ( x, y ) | ≤ C ( x − x ) + C ( y − y )which readily implies k u k C ≤ C .
Hence u satisfies a C a priori bound. Furthermore, if t ∈ [0 ,
1] is fixed, equation T ( u, t ) = 0belongs to the class of equations studied in [7] and theorem 2 in [7] implies that if u ∈ C ,α ( T ) solves T ( u, t ) = 0 for some t and satisfies a priori C bound, then it also satisfiesa C ,α bound. This implies that S is closed in [0 , t n be a sequence in S converging to ¯ t in [0 , t n corresponds a function u n ∈ C ,α ( T ) such that T ( u n , t n ) = 0. The C ,α a priori bound on solutions to T ( u, t ) = 0 implies that thesequence u n is bounded in C ,α ( T ) and so it admits a subsequence, which we still denoteby u n , which converges in C ( T ) to a function ¯ u ∈ C ( T ). Since T is continuos, T (¯ u, ¯ t ) =0 and so, in view of [7], ¯ u in C ,α ( T ). Hence ¯ t ∈ S and S is closed.Next we show that S is open . Let t ∈ S . Then there exists u ∈ C ,α ( T ) such that T ( u, t ) = 0. Let L : C ,α ( T ) → C ,α ( T ) be defined as L ( w ) := T ∗| ( u,t ) ( w, . A direct computation yields that(20) L ( w ) = ( w xx + ( c e − h + h ′ ) w x )(1 + u yy )+ (e − h + u xx + ( c e − h + h ′ ) u x )( w yy ) − u xy w xy and so L is uniformly elliptic. L is injective by maximum principle and it is surjective inview 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. (cid:3) A generalization to higher dimensions
In this section we consider a generalization of the Kodaira-Thurston manifold in di-mension greater than 4. Assume n ≥
3. Let G n be the Lie group ( R n , ∗ n ), where( x , . . . , x n , y , . . . , y n ) ∗ n ( x ′ , . . . , x ′ n , y ′ , . . . , y ′ n ) =( x + x ′ , . . . , x n + x ′ n , y + y ′ , y + y ′ − x x ′ , . . . , y n − + y ′ n − − x n x ′ )and let M n = Γ n \ G n , where Γ n is Z n with the multiplication induced by ∗ n . Then M n isa 2-step nilmanifold and the projection π : R n → R n +1 onto the first ( n + 1)-coordinatesinduces to M n a structure of principal ( n − n + 1)–torus T n +1 . M n is parallelizable and e i = dx i , i = 1 . . . , n, f j = dy j − x dx j , j = 1 . . . , n defines a global coframe which satisfies de k = 0 , k = 1 , . . . , n , df = 0 , df k = e k ∧ e , k = 2 , . . . , n . We then consider on M n the symplectic formΩ n = n X k =1 α k ∧ β k ALABI-YAU EQUATION ON THE KODAIRA-THURSTON MANIFOLD 13 and the Ω n -compatible almost-complex structure J n induced by Ω n and the natural metric g n = n X k =1 α k ⊗ α k + β k ⊗ β k . In terms of the basis B = { e , . . . , e n , f , . . . , f n } , J n is defined by J n e k = − f k , J n f k = e k . Let u be a T n +1 -invariant function on M n ; then du = n X s =1 u x s e s + u y f , − J n du = n X s =1 u x s f s − u y e and so − dJ n du = n X r,s =1 u x r x s e r ∧ f s − n X k =1 u x k y e k ∧ e + u x k y f ∧ f k + u y y e ∧ f + n X k =2 u x r e r ∧ e = n X r,s =1 u x r x s e r ∧ f s − n X k =1 u x k y e k ∧ e + u x k y f ∧ f k + ( u y y − u y ) e ∧ f + d ( ue ) , and so if α = − J n du − ue , then dα is of type (1 ,
1) with respect to J n . Furthermore,(Ω n + dα ) n = n X r,s =1 ( δ rs + u x r x s ) e r ∧ f s + (1 + u y y − u y ) e ∧ f ! n − n ! n X k,m =2 n Y r,s =2 , ( r,s ) =( k,m ) u x r x s u x k y u x m y e ∧ f ∧ · · · ∧ e n ∧ f n and if F is a given T n +1 -invariant map, the Calabi-Yau equation (Ω + dα ) n = e F Ω n readsin terms of u as(21) det( I + A ( u )) − n X k,m =2 n Y r,s =2 , ( r,s ) =( k,m ) u x r x s u x k y u x m y = e F , where A ( u ) = ( A ij ) is the n × n matrix A = u x x + u y y − u y , A ij = u x i x j , if ( i, j ) = (1 , . Example 5.1.
For n = 3, equation (21) reads asdet( I + A ( u )) − u x x u x y − u x x u x y − u x x u x y u x y = e F , this kind of equations has been considered in [6]. 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 consideringonly the following two cases: F = F ( x , . . . , x n ) , or F = F ( x , . . . , x n , y ) . • In the first case F = F ( x , . . . , x n ), equation (21) reduces to the Monge-Amp`ereequation 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]. • In the second case, F = F ( x , . . . , x n , y ), in the variables z = y , z = x , . . . , z n = x n equation (21) take the following expressiondet( I + B ( u )) = e F where B ( u ) = ( B ij ) is given by B = u z z + u z , B ij = u z i z j , if i, j = 1 . References [1] E. Buzano, A. Fino, L. Vezzoni, The Calabi-Yau equation for T -bundles over the non-Lagrangiancase. Rend. Semin. Mat. Univ. Politec. Torino (2011), no. 3, 281–298.[2] E. Buzano, A. Fino, L. Vezzoni, The Calabi-Yau equation on the Kodaira-Thurston manifold, viewedas an S -bundle over a 3-torus. J. Differential Geom. (2015), no. 2, 175–195.[3] S. K. Donaldson, Two-forms on four-manifolds and elliptic equations.
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