Scalar Curvature Functions of Almost-Kähler Metrics
aa r X i v : . [ m a t h . DG ] S e p SCALAR CURVATURE FUNCTIONS OF ALMOST-K ¨AHLERMETRICS
JONGSU KIM AND CHANYOUNG SUNG
Abstract.
For a closed smooth manifold M admitting a symplecticstructure, we define a smooth topological invariant Z ( M ) using almost-K¨ahler metrics, i.e. Riemannian metrics compatible with symplecticstructures. We also introduce Z ( M, [[ ω ]]) depending on symplectic de-formation equivalence class [[ ω ]]. We first prove that there exists a 6-dimensional smooth manifold M with more than one deformation equiv-alence classes with different signs of Z ( M, [[ ω ]]). Using Z invariants, weset up a Kazdan-Warner type problem of classifying symplectic mani-folds into three categories.We finally prove that on every closed symplectic manifold ( M, ω ) ofdimension ≥
4, any smooth function which is somewhere negative andsomewhere zero can be the scalar curvature of an almost-K¨ahler metriccompatible with a symplectic form which is deformation equivalent to ω . Introduction
In Riemannian geometry, scalar curvature encodes certain information ofthe differential topology of a smooth closed manifold. There has been muchprogress on topological conditions for the existence of a metric of positivescalar curvature, and more generally which functions on a given manifoldcan be the scalar curvature of a Riemannian metric.The latter problem is known as the Kazdan-Warner problem. They proved[9] that the necessary and sufficient condition for a smooth function f ona closed manifold M of dimension ≥ • f is arbitrary, in case Y ( M ) > • f is identically zero or somewhere negative, in case Y ( M ) = 0 and M admits a scalar-flat metric, • f is negative somewhere, in the remaining case, Date : July 17, 2018.2010
Mathematics Subject Classification.
Key words and phrases. almost-K¨ahler, scalar curvature, symplectic manifold.The first author was supported by the National Research Foundation of Korea(NRF)grant funded by the Korea government(MOE) (No.NRF-2010-0011704). where Y ( M ) denotes the Yamabe invariant of M . For the Yamabe invariant,the readers are referred to [13].As an extension of the Kazdan-Warner problem, it is natural to pursue asimilar classification in some restricted class of Riemannian metrics.Let M be a smooth manifold with a symplectic form ω . An almost-complex structure J is called ω -compatible, if ω ( J · , J · ) = ω ( · , · ), and ω ( · , J · )is positive-definite. Thus a smooth ω -compatible J defines a smooth J -invariant Riemannian metric g ( · , · ) := ω ( · , J · ), which is called an ω -almost-K¨ahler metric. It is K¨ahler iff J is integrable.Due to the lack of a Yamabe-type theorem ([15]) which would producemetrics of constant scalar curvature, a Kazdan-Warner type result for sym-plectic manifolds is left open even without any conjectures. A general ex-istence result so far is in [10] stating that every symplectic manifold of di-mension ≥ ω and ω on M are called defor-mation equivalent , if there exists a diffeomorphism ψ of M such that ψ ∗ ω and ω can be joined by a smooth homotopy of sympelctic forms, [17]. Thereare a number of smooth manifolds which admit more than one deformationequivalence classes: see [22] or references therein. For a symplectic form ω ,its deformation equivalence class shall be denoted by [[ ω ]]. By abuse of nota-tion, we say that a metric g is in [[ ω ]] when g is compatible with a symplecticform ω in [[ ω ]]. Definition 1.
Let M be a smooth closed manifold of dimension n ≥ which admits a symplectic structure. Define Z ( M, [[ ω ]]) = sup g ∈ [[ ω ]] R M s g d vol g ( Vol g ) n − n , where s g is the scalar curvature of g , and define Z ( M ) = sup [[ ω ]] Z ( M, [[ ω ]]) . The denominator in R M s g d vol g (Vol g ) n − n was put for the invariance under a scalechange ω → c · ω with c >
0, and one can get the following inequality from
LMOST-K ¨AHLER SCALAR CURVATURE FUNCTIONS 3 the formulas (2.2) and (6.15) below;(1.1) Z ( M, [[ ω ]]) ≤ sup ω ∈ [[ ω ]] πc ( ω ) · [ ω ] n − ( n − ( [ ω ] n n ! ) n − n , where c ( ω ) is the first Chern class of ω .These Z numbers may take the value of ∞ and are different in nature fromthe Yamabe invariant which is bounded above in each dimension. Obviously Z ( M ) is a smooth topological invariant of M . Although the quantity inthe right hand side of (1.1) may serve usefully for many purposes, the Z value reflects almost-K¨ahler geometry better, so seems more relevant to ourpurpose. Of course, it would be very interesting to know if the equality in(1.1) always holds or not.To explain why possible scalar curvature functions may depend on [[ ω ]],we shall demonstrate a smooth manifold which admits two symplectic de-formation equivalence classes with distinct signs of Z ( M, [[ ω ]]). Theorem 1.1.
There exists a smooth closed 6-dimensional manifold withdistinct symplectic deformation equivalence classes [[ ω i ]] , i = 1 , such that Z ( M, [[ ω ]]) = ∞ and Z ( M, [[ ω ]]) < . Next, we use the method of [10, 11] to prove our main theorem;
Theorem 1.2.
Let ( M, [[ ω ]]) be a smooth closed manifold of dimension n ≥ with a deformation equivalence class of symplectic forms. Then any smoothfunction on M which is somewhere negative and somewhere zero is the scalarcurvature of some smooth almost-K¨ahler metric in [[ ω ]] . We speculate that any smooth somewhere-negative function on a closedmanifold M with [[ ω ]] might be the scalar curvature of some smooth almost-K¨ahler metric in [[ ω ]]. A key step to prove it should be to show that every( M, [[ ω ]]) has an almost-K¨ahler metric of negative constant scalar curvaturein [[ ω ]].With Z ready, one can see that Theorem 1.2 contributes to answering thefollowing question; Question 1.3.
Let M be a smooth closed manifold of dimension n ≥ admitting a symplectic structure.Is the (necessary and sufficient) condition for a smooth function f on M to be the scalar curvature of some smooth almost-K¨ahler metric as follows? (a) f is arbitrary, if < Z ( M ) ≤ ∞ , (b) f is identically zero or somewhere negative, if Z ( M ) = 0 and M admits a scalar-flat almost-K¨ahler metric, (c) f is negative somewhere, if otherwise. JONGSU KIM AND CHANYOUNG SUNG
Also, is the condition for a smooth function f on M to be the scalarcurvature of some smooth almost-K¨ahler metric in [[ ω ]] as follows? ( a ′ ) f is arbitrary, if < Z ( M, [[ ω ]]) ≤ ∞ , ( b ′ ) f is identically zero or somewhere negative, if Z ( M, [[ ω ]]) = 0 and M admits a scalar-flat almost-K¨ahler metric in [[ ω ]] , ( c ′ ) f is negative somewhere, if otherwise. This paper is organized as follows. In section 2, some computations of Z ( M ) are explained. Theorem 1.1 is proved in section 3. In section 4,Kazdan-Warner type argument in almost K¨ahler setting is explained. The-orem 1.2 is proved in section 5 and 6.2. Some computations of Z ( M )In this section we explain some basic properties and computations of Z invariant in relatively simple cases. Lemma 2.1.
Let M be a smooth closed manifold of dimension ≥ admittinga symplectic structure. If Y ( M ) ≤ , then Z ( M ) ≤ .Proof. Suppose not. Then there exists an almost K¨ahler metric g on M suchthat R M s g dvol g >
0. Then by the Yamabe problem [15], a conformal changeof g gives a metric of positive constant scalar curvature. This implies that Y ( M ) >
0, thereby yielding a contradiction. (cid:3)
Lemma 2.2.
Any compact minimal K¨ahler surface M of Kodaira dimension0 has Z ( M ) = 0 , and it is attained by a Ricci-flat K¨ahler metric.Proof. By the Kodaira-Enriques classification [3], such M is K3 or T ortheir finite quotients, and hence it admits a Ricci-flat K¨ahler metric. Thus Z ( M ) ≥
0. Since M cannot admit a metric of positive scalar curvature, (infact, Y ( M ) = 0), the above lemma forces Z ( M ) = 0. (cid:3) By using Lemma 2.1, one can show that if Y ( M ) = 0, and M admitsa “collapsing” sequence of almost-K¨aher metrics with bounded scalar cur-vature, then Z ( M ) = 0. For example, we consider the Kodaira-Thurstonmanifold M KT ; see Section 4. Lemma 2.3.
For the Kodaira-Thurston manifold M KT , Z ( M KT ) = 0 , andit is obtained as the limit by a collpasing sequence of almost-K¨ahler metricsof negative constant scalar curvature.Proof. M KT never admits a metric of positive scalar curvature; one can useSeiberg-Witten theory, see [18]. So, Y ( M KT ) ≤
0. Thus Z ( M KT ) ≤ LMOST-K ¨AHLER SCALAR CURVATURE FUNCTIONS 5
We let R = { ( x, y, z, t ) } endowed with the metric dx + dy + ( dz − xdy ) + dt . Then M KT with an almost-K¨ahler metric is obtained as thequotient of R by the group generated by isometric actions, γ ( x, y, z, t ) = ( x + 1 , y, y + z, t ) , γ ( x, y, z, t ) = ( x, y + 1 , z, t ) ,γ ( x, y, z, t ) = ( x, y, z + 1 , t ) , γ ( x, y, z, t ) = ( x, y, z, t + d ) , where d is any positive constant.For any d >
0, the scalar curvature is − . (See the curvature computa-tions in Lemma 6.2.) But by taking d > g on M KT such that R M KT s g d vol g / (Vol g ) is arbitrar-ily close to 0. Therefore Z ( M KT ) = 0. (cid:3) Now M KT , with Z ( M KT ) = 0, never admits a scalar-flat almost-K¨ahlermetric, because such a metric has to be a K¨ahler metric, which is not allowedon M KT with b ( M KT ) = 3. Therefore one can expect that M KT belongsto the category (c) in the classification of Question 1.3. Indeed this wasalready proved in [11]. Note that M KT is a nilmanifold. One may expect tofind more examples of symplectic solvmanifolds with vanishing Z value andprove them to be in the category (c).Now let us give an example of Z ( M ) > Lemma 2.4.
For the complex projective plane C P , Z ( C P ) = 12 √ π , andit is attained by a K¨ahler Einstein metric.Proof. First we claim that C P with the reversed orientation does not sup-port any almost complex structure. Suppose it does. Recall that any closedalmost complex 4-manifold satisfy c = 2 χ + 3 τ, where χ and τ respectively denote Euler characteristic and signature. Thenthis formula gives c = 3, which is impossible because of the fact that c ∈ H ( C P ; Z ). Thus c must be 9 so that c is 3[ H ] or − H ], where [ H ] isthe hyperplane class.For any symplectic form ω on C P , [ ω ] must be a nonzero multiple of [ H ].We apply the Blair formula (6.15) to get R C P s g dvol g (Vol g ) ≤ R C P ( s g + s ∗ g ) dvol g (Vol g ) = 4 πc ( ω ) · [ ω ]( [ ω ] · [ ω ]2 ) = ± √ π, JONGSU KIM AND CHANYOUNG SUNG for any ω -almost-K¨ahler metric g on C P . In the above inequality, we useda relation between s and the star-scalar curvature s ∗ [2];(2.2) s ∗ − s = 12 |∇ J | ≥ . Therefore Z ( C P , [[ ω ]]) ≤ √ π . In fact the Fubini-Study metric with ω F S saturates this upper bound, so we finally get Z ( C P ) = Z ( C P , [[ ω F S ]]) = 12 √ π. (cid:3) Remark 2.5.
In the above we only treated a few simple examples. How-ever, we expect that Z invariant is fairly computable under some symplecticsurgeries. For instance, one can see that Z ( ˆ T ) = 0, where ˆ T is the blow-up at one point of the K¨ahler 4-torus. In fact, one only needs to check thatLeBrun’s argument in the proof of theorem 3 of [14] still works in almostK¨ahler context. This gives Z ( ˆ T ) ≥
0. Together with Seiberg-Witten the-ory one gets Z ( ˆ T ) = 0. A similar argument, albeit in K¨ahler case, may befound in [23].3. Symplectic deformation classes on a manifold with distinctsigns of Z ( · , [[ ω ]])In this section we shall prove Theorem 1.1.We use one of the examples in [21]. Let W be a complex Barlow surface,which is a minimal complex surface of general type homeomorphic, but notdiffeomorphic, to R , the blown-up complex surface at 8 points in generalposition in the complex projective plane. By a small deformation of complexstructure we may assume that W has ample canonical line bundle [5]. Thenby Yau’s solution of Calabi conjecture, W and R admit a K¨ahler Einsteinmetric of negative (and positive, respectively) scalar curvature. Ruan showedthat for a compact Riemann surface Σ, R × Σ and W × Σ are diffeomorphicbut their natural symplectic structures are not deformation equivalent.We prove;
Proposition 3.1.
Let W be a Barlow surface with ample canonical linebundle and Σ be a Riemann surface of genus 2. Consider a K¨ahler Einsteinmetric of negative scalar curvature on W with K¨ahler form ω W on W anda K¨ahler form ω Σ on Σ with constant negative scalar curvature.Then Z ( W × Σ , [[ ω W + ω Σ ]]) = − π , and it is attained by a K¨ahlerEinstein metric.Proof. We recall a few results about W from [21, Section 4]; there is ahomeomorphism of W onto R which preserves the Chern class c and thereis a diffeomorphism of W × Σ onto R × Σ which preserves c . LMOST-K ¨AHLER SCALAR CURVATURE FUNCTIONS 7
Then, the first Chern class of W can be written as c ( W ) = 3 E − P i =1 E i ∈ H ( W, R ) ∼ = R , where E i , i = 0 , · · ·
8, is the Poincare dualof a homology class ˜ E i , i = 0 , · · · E i , i = 0 , · · ·
8, form a basis of H ( W, Z ) ∼ = Z and their intersections satisfy ˜ E i · ˜ E j = ǫ i δ ij , where ǫ = 1and ǫ i = − i ≥
1. So, in this basis the intersection form becomes I = · · − · · . . · · . . · ·
00 0 0 − . We have the orientation of W induced by the complex structure and thefundamental class [ W ] ∈ H ( W, Z ) ∼ = Z . As ω W is K¨ahler Einstein of neg-ative scalar curvature, we may get [ ω W ] = − E + P i =1 E i by scaling ifnecessary.Now a compact Riemann surface Σ of genus 2 has its fundamental class[Σ] ∈ H (Σ , Z ) ∼ = Z . Let c be the generator of H (Σ , Z ) ∼ = Z such that thepairing h c, [Σ] i = 1. Then c (Σ) = − c . We consider a K¨ahler form ω h withconstant negative scalar curvature such that [ ω h ] = c ∈ H (Σ , R ) ∼ = R .Set M = W × Σ. By K¨unneth theorem, H ( M, R ) ∼ = π ∗ H ( W ) ⊕ π ∗ H (Σ) ∼ = R ⊕ R , where π i are the projection of M onto the i-th fac-tor. Then, c ( M ) = π ∗ c ( W ) + π ∗ c (Σ) = π ∗ (3 E − X i =1 E i ) − π ∗ c ∈ H ( M, R ) . Consider any smooth path of symplectic forms ω t , 0 ≤ t ≤ δ , on M suchthat ω = ω W + ω h . We may write[ ω t ] = n ( t ) π ∗ E + X i =1 n i ( t ) π ∗ E i + l ( t ) π ∗ c ∈ H ( M, R )for some smooth functions n i ( t ) , l ( t ), i = 0 , · · · ,
8. As they are connected,their first Chern class c ( ω t ) = c ( M ). We have;[ ω t ] ([ W × Σ]) = [ n ( t ) π ∗ E + P i =1 n i ( t ) π ∗ E i + l ( t ) π ∗ c ] ([ W × Σ])(3.3) = 3 { n ( t ) − P i =1 n i ( t ) } l ( t ) > . As l (0) = 1 > l ( t ) >
0. So, n ( t ) > P i =1 n i ( t ). From above, we knowthat n (0) = − <
0. As n ( t ) >
0, we get n ( t ) < c · [ ω t ] ([ W × Σ]) = − { n ( t ) − X i =1 n i ( t ) } + 2 l ( t ) { n ( t ) + X i =1 n i ( t ) } . JONGSU KIM AND CHANYOUNG SUNG
Since n ( t ) > P i =1 n i ( t ) and | P i =1 n i ( t ) | ≤ √ qP i =1 n i , we get3 n ( t ) + X i =1 n i ( t ) ≤ n ( t ) + 2 √ qP i =1 n i ( t )(3.5) < n ( t ) + 2 √ p n ( t ) = (3 − √ n ( t ) < . So, c · [ ω t ] ([ W × Σ]) <
0. Putting A = n ( t ) − P i =1 n i ( t ) and B =3 n ( t ) + P i =1 n i ( t ), we have c [ ω t ] [ ω t ] / = 23 / { − A + l ( t ) BA / l ( t ) / } = 23 / { − A / l ( t ) / + l ( t ) / BA / } . For a, b < x >
0, set h ( x ) := / ( ax + xb ). Since h ′ ( x ) = / ( x − abx b ), h ( x ) has maximum when x = (2 ab ) / . So we get h ( x ) ≤ ( ab ) .With a = − A / , b = A / B and x = l ( t ) / , this gives c [ ω t ] [ ω t ] / ≤ − ( B A ) , and from (3.5) B A ≥ { n ( t ) + 2 √ qP i =1 n i ( t ) } n ( t ) − P i =1 n i ( t ) = (3 − √ √ y ) − y where y = P i =1 n i ( t ) n ( t ) . By calculus, (3 − √ √ y ) − y ≥ y ∈ [0 ,
1) withequality at y = .We have c [ ω t ] [ ω t ] / ≤ − ; the equality is achieved exactly when n ( t ) = − n i ( t ) = 1, i = 1 , · · · , l ( t ) = 2 modulo scaling, i.e. when [ ω t ] is apositive multiple of − c ( M ). The K¨ahler form of a product K¨ahler Einsteinmetric on M = W × Σ belongs to this class.As the expression πc ( ω ) · [ ω ] n − n − ( [ ω ] nn ! ) n − n is invariant under a change ω φ ∗ ( ω ) bya diffeomorphism φ , so from (1.1), Z ( M, [[ ω ]]) ≤ sup ω ∈ [[ ω ]] π · / c [ ω ] [ ω ] / ≤ − π. As the equality is attained by a K¨ahler Einstein metric, Z ( M, [[ ω ]]) = − π . (cid:3) The next corollary yields Theorem 1.1.
LMOST-K ¨AHLER SCALAR CURVATURE FUNCTIONS 9
Corollary 3.2.
There exists a smooth closed 6-d manifold with distinct sym-plectic deformation equivalence classes [[ ω ]] and [[ ω ]] such that Z ( M, [[ ω ]]) = ∞ which is obtained by a sequence of K¨ahler metrics of positive constantscalar curvature, and Z ( M, [[ ω ]]) < .Proof. Consider V × Σ and W × Σ where V = R , W and Σ are as in Propo-sition 3.1. They are diffeomorphic but their natural symplectic structuresare not deformation equivalent. Let ω be the K¨ahler form of a productK¨ahler Einstein metric on V × Σ. One can easily get Z ( M, [[ ω ]]) = ∞ byscaling on one factor of the product. Let ω be the ω W + ω Σ in Proposition3.1. (cid:3) Remark 3.3.
Theorem 1.1 and its proof hint that much more examplesmay be obtained in a similar way. Just for another instance, consideringthe example M = R × R of Catanese and LeBrun in [5], we could checkthat the smooth 8-dimensional manifold M admits distinct symplectic de-formation equivalence classes [[ ω i ]], i = 1 , Z ( M, [[ ω ]]) = ∞ and Z ( M, [[ ω ]]) < Kazdan-Warner method adapted to almost K¨ahler metrics
Our method to prove Theorem 1.2 is an adaptation of ordinary scalarcurvature theory to an almost-K¨ahler setting, and recall and modify thematerial explained in [12, 11].Let M denote the space of Riemannian metrics on a given smooth manifold M of real dimension 2 n , and we regard M as the completion of smoothmetrics with respect to L p -norm for p > n . Given a symplectic form ω on M , let Ω ω be the subspace of ω -almost-K¨ahler metrics on M . The space Ω ω is a smooth Banach manifold with the above norm, and its tangent space T g Ω ω at g := ω ( · , J · ) consists of symmetric (0 ,
2) tensors h which are J -anti-invariant, i.e. h + ( X, Y ) := 12 ( h ( X, Y ) + h ( J X, J Y )) = 0for all
X, Y ∈ T M .The space Ω ω admits a natural parametrization by the exponential map;for g ∈ Ω ω , define E g : T g Ω ω → Ω ω by E g ( h ) = g · e h with g · e h ( X, Y ) = g ( X, e ˆ h ( Y )) = g ( X, Y + ∞ X k =1 k ! ˆ h k Y ) , where X, Y ∈ T M , and ˆ h is the (1 , h with respectto g . Clearly we have d { g · e th } dt | t =0 = h. Given g ∈ Ω ω with corresponding J , any other metric ˜ g in Ω ω can be ex-pressed as(4.6) ˜ g = g · e h , where h is a J -anti-invariant symmetric (0 , S ω : Ω ω → L p ( M ), the derivative at g is given by D g S ω ( h ) = δ g δ g ( h ) − g ( r g , h )for h ∈ T g Ω ω , where r g is the Ricci curvature of g , and its formal adjoint isgiven by ( D g S ω ) ∗ ( ψ ) = ( ∇ dψ ) − − r − g ψ, where A − for a symmetric (0 ,
2) tensor A denotes the J -anti-invariant part ( A ( · , · ) − A ( J · , J · )).The followings are key facts for the Kazdan-Warner type problem in thealmost-K¨ahler setting. Lemma 4.1. [12] , [11, Lemma 1] If D g S ω is surjective for a smooth g ∈ Ω ω ,then S ω is locally surjective at g , i.e. there exists an ǫ > such that for any f ∈ L p ( M ) with || f − S ω ( g ) || < ǫ there is an L p almost-K¨ahler metric ˜ g ∈ Ω ω satisfying f = S ω (˜ g ) . Furthermore if f is C ∞ , so is ˜ g . Recall that a diffeomorphism φ is said to be isotopic to the identity mapif there is a homotopy of diffeomorphisms φ t , 0 ≤ t ≤
1, such that φ = id and φ = φ . The following lemma was proved in [8, Theorem 2.1] withoutthe isotopy clause. Here we add a few arguments in their argument to verifythe isotopy part. Lemma 4.2.
Suppose dim M ≥ and f ∈ C ( M ) . Then an L p function f on M belongs to the L p closure of { f ◦ φ | φ is a diffeomorphism of M, isotopic to the identity map } if and only if inf f ≤ f ≤ sup f almost everywhere.Proof. Let ε > C ( M ) is dense in L p ( M ), we may assume f ∈ C ( M ). We triangulate M into n-simplexes ∆ i so that M = ∪ ∆ i andthat max x,y ∈ ∆ i | f ( x ) − f ( y ) | < δ with 2 δ = ε (2vol M ) p . Choose b i in theinterior of ∆ i .There exist disjoint open balls V i ⊂ M such that | f ( y ) − f ( b i ) | < δ for y ∈ V i and for each i . One chooses a neighborhood Ω of the ( n − M ( n − of M , disjoint from the b i , so small that(max M | f | + max M | f | ) p Vol(Ω) < ε p . LMOST-K ¨AHLER SCALAR CURVATURE FUNCTIONS 11
For each i , let U i be a small ball neighborhood of b i , disjoint from Ω, andchoose open sets O and O , so that M − Ω ⊂ O ⊂ ¯ O ⊂ O ⊂ ¯ O ⊂ M − M ( n − . There is a homotopy of diffeomorphisms φ t , 0 ≤ t ≤
1, such that φ = id and φ ( U i ) ⊂ V i , for each i . And there is a homotopy of diffeomorphisms ψ t , 0 ≤ t ≤
1, with ψ = id such that ψ satisfies ψ | M − O = id and that ψ ( O ∩ ∆ i ) ⊂ U i , for each i . Let Φ t = φ t ◦ ψ t . Then, we get k f ◦ Φ − f k pp = ( Z Ω + Z M − Ω )( | f ◦ Φ − f | p dvol ) < ε p X i Z O ∩ ∆ i | f ◦ Φ ( y ) − f ( b i ) + f ( b i ) − f ( y ) | p dvol< ε p X i p δ p V ol (∆ i ) = ε p . This proves if part, and only if part should be clear. (cid:3)
Proposition 4.3. If D g S ω is surjective for a smooth g ∈ Ω ω , then anysmooth function f with inf f ≤ S ω ( g ) ≤ sup f is the scalar curvature of asmooth almost-K¨ahler metric compatible with φ ∗ ω for a diffeomorphism φ of M isotopic to the identity.Proof. By the above two lemmas, there exists a diffeomorphism ˜ φ isotopicto the identity such that f ◦ ˜ φ = S ω (˜ g ) for a smooth ω -almost-K¨ahler metric˜ g . Thus f = S ( ˜ φ − ) ∗ ω (( ˜ φ − ) ∗ ˜ g ). (cid:3) Lemma 4.4.
The principal part of the fourth-order linear partial differentialoperator ( D g S ω ) ◦ ( D g S ω ) ∗ : C ∞ ( M ) → C ∞ ( M ) is equal to that of ∆ g ◦ ∆ g ,where ∆ g is the g -Laplacian.Proof. Let e , J e , · · · , e n , J e n be a local orthonormal frame satisfying J e i − = e i for i = 1 , · · · , n . The fourth order differentiation only occurs at δ g δ g ( ∇ dψ ) − . A direct computation shows that δ g δ g (2 ∇ dψ ) − ( ψ ) = n X i =1 2 n X j =1 e i ( e j ( ∇ dψ ( e i , e j ) − ∇ dψ ( J ( e i ) , J ( e j )))) + l.o.t.= n X i =1 2 n X j =1 { e i ( e j ( e i ( e j ψ ))) − e i ( e j ( J e i ( J e j ψ ))) } + l.o.t.= n X i =1 2 n X j =1 { e i ( e i ( e j ( e j ψ ))) − e i ( J e i ( e j ( J e j ψ ))) } + l.o.t.= ∆ ψ − { n X i =1 ( e i − J e i − + e i J e i )( n X j =1 e j ( J e j ψ )) } + l.o.t.= ∆ g ψ − { n X i =1 ( e i − e i − e i e i − )( n X j =1 e j ( J e j ψ )) } + l.o.t.= ∆ g ψ + l.o.t. , where l.o.t denotes the terms of differentiations up to the 3rd order and ∆is the Euclidean Laplacian. (cid:3) Almost-K¨ahler Surgery and deformation
Here we consider a left-invariant almost-K¨ahler metric on the 4-dimensionalKodaira-Thurston nil-manifold [1]. The metric can be written on R = { ( x, y, z, t ) | x, y, z, t ∈ R } as g KT = dx + dy + ( dz − xdy ) + dt and the left-invariant symplectic form is ω KT = dt ∧ dx + dy ∧ dz . The almostcomplex structure J is then given by J ( e ) = e , J ( e ) = − e , J ( e ) = e , J ( e ) = − e , where e = ∂∂x , e = ∂∂y + x ∂∂z , e = ∂∂z , e = ∂∂t which form anorthonormal frame for the metric.Now we consider a metric g n on the product manifold R × R n − , n ≥ g n = g KT + g Euc , where g Euc is the Euclidean metric on R n − .This manifold has the symplectic form ˜ ω = ω KT + ω , where ω is thestandard symplectic structure on R n − and g n is ˜ ω -almost-K¨ahler .Given any symplectic manifold ( M, ω ) with an almost-K¨ahler metric g andthe corresponding almost complex structure J g , we pick a point p . Thereexists a Darboux coordinate neighborhood ( U, x i ) of p with x ( p ) = 0 sothat ω = P ni =1 dx i − ∧ dx i . Assume that U contains a ball B gδ ( p ) of g -radius δ with center at p . By considering the (coordinates-rearranging)local diffeomorphism φ ( x , · · · , x n ) = ( x , x , x , x , x · · · , x n ), φ : U → LMOST-K ¨AHLER SCALAR CURVATURE FUNCTIONS 13 R × R n − , i.e. identifying x = x, x = y, x = z, x = t · · · , we can getthe pulled-back metric of g n via φ , which we still denote by g n . Note that φ ∗ ˜ ω = ω . We may express g n = g · e h on U for a unique smooth symmetric J g -anti-invariant tensor h from (4.6), because g and g n are both ω -almost-k¨ahler. Let η ( r ) be a smooth cutoff function in C ∞ ( R > , [0 , η ≡ r < δ and η ≡ r ≥ δ .We define a new ω -almost-K¨ahler metric h on M by(5.7) h := g on M \ B g δ (p) ,g n · e η ( r g ) h on B g δ (p) \ B g δ (p) g n on B g δ (p)where r g is the distance function of g from p .In [10], using a Lohkamp type argument [16] adapted to symplectic mani-folds, the first author proved on any closed symplectic manifold of dimension ≥ Proposition 5.1. [10, Theorem 3]
Let S be a closed subset in a smoothclosed symplectic manifold ( M, ω ) of dimension n ≥ , and U ⊃ S be anopen neigborhood. Then for any smooth ω -almost-K¨ahler metric g on U with s ( g ) < , there exists a smooth ω -almost-K¨ahler metric g on M suchthat g = g on S and s ( g ) < on M . We now apply proposition 5.1 to the metric h of (5.7) with U = B g δ ( p )and S = B g δ ( p ). We get; Corollary 5.2.
On any smooth closed symplectic manifold ( M, ω ) of dimen-sion n ≥ , there exists a smooth ω -almost-K¨ahler metric g with negativescalar curvature such that g is isometric to the product metric of g KH andthe Euclidean metric on an open subset B ǫ of M . Proof of Theorem 1.2
Theorem 6.1.
Let ( M, ω ) be a smooth closed symplectic manifold of dimen-sion n ≥ . Then any smooth function on M which is somewhere negativeand somewhere zero is the scalar curvature of some smooth almost-K¨ahlermetric associated to cϕ ∗ ω , where c > is a constant and ϕ is a diffeomor-phism of M , isotopic to the identity.Proof. On (
M, ω ), let’s take the ω -almost-K¨ahler metric g constructed inCorollary 5.2. By Lemma 6.2 below, we have that D g S ω is surjective. If f ∈ C ∞ ( M ) is somewhere negative and somewhere zero, then for a sufficientlylarge constant c >
0, inf cf ≤ S ω ( g ) ≤ sup cf. Then by Proposition 4.3, cf is the scalar curvature of an almost-K¨ahlermetric G compatible with φ ∗ ω for some diffeomorphism φ of M isotopic tothe identity, and hence f is the scalar curvature of the almost-K¨ahler metric c · G compatible with cφ ∗ ω . (cid:3) Theorem 1.2 follows from the above theorem 6.1, since cφ ∗ ω is deformationequivalent to ω for any constant c > Lemma 6.2.
For the metric g constructed in Corollary 5.2, the kernel of ( D g S ω ) ∗ is { } .Proof. We look into the computations in [11, Section 4], where any global so-lution on Kodaira-Thurston compact manifold was shown to be zero, whereaswe shall now improve to show any local solution on it is zero.Let’s first compute the curvature of g on B ǫ . Since it is the product ofthe Kodaira-Thurston metric and the Euclidean metric, we will only listcomponents for i = 1 , · · · ,
4. (Recall the frame e , e , e , e in Section 5.)From the formula2 h∇ X Y, Z i = X h Y, Z i + Y h X, Z i− Z h X, Y i−h X, [ Y, Z ] i−h Y, [ X, Z ] i + h Z, [ X, Y ] i , one can compute ∇ e e = −∇ e e = 12 e , ∇ e e = ∇ e e = − e , ∇ e e = ∇ e e = 12 e , ∇ e i e i = ∇ e e i = ∇ e i e = 0 . Thus letting ∇ e i = X j ω ij e j , we have ω ij = 12 dz − xdy dy − dz + xdy − dx − dy dx , LMOST-K ¨AHLER SCALAR CURVATURE FUNCTIONS 15 and the Cartan structure equation givesΩ ij = dω ij + X k ω ik ∧ ω kj = 14 − dx ∧ dy dx ∧ ( dz − xdy ) 03 dx ∧ dy dy ∧ dz dz − xdy ) ∧ dx − dy ∧ dz . Denoting the sectional curvature of the plane spanned by e i and e j by K ij ,we have K = − , K = 14 , K = 14 , K i = 0for any i . Then the Ricci tensor ( r ij ) is given by r = − , r = − , r = 12 , r = 0 , r ij = 0 for i = j so that r − = − , r − = − , r − = 12 , r − = 14 , r − ij = 0 for i = j. Denoting ( ∇ dψ )( e i , e j ) = e i ( e j ψ ) − ( ∇ e i e j ) ψ for ψ ∈ C ∞ ( B ǫ ) by ∇ dψ ij ,one can easily get ∇ dψ = ψ xx , ∇ dψ = ψ yy + 2 xψ yz + x ψ zz , ∇ dψ = ψ zz , ∇ dψ = ψ tt , ∇ dψ = ψ xy + xψ yz + 12 ψ z , ∇ dψ = ψ xz + 12 ψ y + x ψ z , ∇ dψ = ψ xt , ∇ dψ = ψ yz + xψ zz − ψ x , ∇ dψ = ψ yt + xψ zt , ∇ dψ = ψ zt . We list only ∇ dψ ij for i, j = 1 , · · · ,
4, because that’s enough for our purpose.Also denoting ( ∇ dψ ) − ( e i , e j ) = ( ∇ dψ ( e i , e j ) − ∇ dψ ( J e i , J e j )) simply by ∇ − dψ ij , one can get2 ∇ − dψ = ψ xx − ψ tt , ∇ − dψ = ψ yy + 2 xψ yz + ( x − ψ zz , ∇ − dψ = ψ xy + xψ xz + 12 ψ z + ψ zt , ∇ − dψ = ψ xz + 12 ψ y + 12 xψ z − ψ yt − xψ zt , ∇ − dψ = 2( ψ yz + xψ zz − ψ x ) , ∇ − dψ = 2 ψ xt . Now suppose ψ ∈ ker( D g S ω ) ∗ , i.e.(6.8) ∇ − dψ − ψr − = 0 . Then from the above, we get the following 6 equations of ψ on B ǫ : ψ xx − ψ tt = − ψ, (6.9) ψ yy + 2 xψ yz + ( x − ψ zz = − ψ, (6.10) ψ xy + xψ xz + 12 ψ z + ψ zt = 0 , (6.11) ψ xz + 12 ψ y + 12 xψ z − ψ yt − xψ zt = 0 , (6.12) ψ yz + xψ zz − ψ x = 0 , (6.13) ψ xt = 0 . (6.14)In order to deduce ψ = 0 on B ǫ out of these 6 equations, let’s writethe local coordinate ( x, y, z, t, x , · · · , x n ) on B ǫ as ( x, y, z, t ) × w so that w = ( x , · · · , x n ). We will first show ψ t = 0 and then ψ x = 0, whichtogether imply ψ = 0 by (6.9).From (6.14), ψ ( x, y, z, t, w ) can be written as a ( x, y, z, w ) + b ( y, z, t, w ).Substituting it into (6.9) gives a xx − b tt = −
12 ( a + b ) . Then the LHS of a xx + a = b tt − b is a function of x, y, z, w , whereas itsRHS is a function of y, z, t, w . Thus both sides are functions of y,z, and wonly. Differentiating the RHS with respect to t gives b ttt − b t = 0 . Solving this ODE, we get b t = b ( y, z, w ) e t √ + b ( y, z, w ) e − t √ so that b = √ b ( y, z, w ) e t √ − √ b ( y, z, w ) e − t √ + b ( y, z, w ) . Now plugging a + b into (6.11), and picking up only t terms, we get12 ( √ ∂b ∂z e t √ − √ ∂b ∂z e − t √ ) + ∂b ∂z e t √ + ∂b ∂z e − t √ = 0so that ∂b ∂z = ∂b ∂z = 0, and hence we can conclude that b and b are functionsof y and w only.Plugging new a + b into (6.12), and picking up only t terms, we get12 ( √ ∂b ∂y e t √ − √ ∂b ∂y e − t √ ) − ( ∂b ∂y e t √ + ∂b ∂y e − t √ ) = 0 , which implies that ∂b ∂y = ∂b ∂y = 0, and hence b and b are functions of w only. LMOST-K ¨AHLER SCALAR CURVATURE FUNCTIONS 17
Again plugging this new a + b into (6.10), and picking up only t terms,we get 0 = − ( √ b e t √ − √ b e − t √ ) , which finally implies that b = b = 0 and hence ψ t = 0.Taking ∂∂x to (6.11) produces ψ xxy + xψ xxz + 32 ψ xz = 0 . Applying ψ xx = − ψ from (6.9), this becomes − ψ y − xψ z + 32 ψ xz = 0 . Comparing it with ψ xz + ψ y + xψ z = 0 from (6.12) gives ψ y + xψ z = 0so that ψ yz + xψ zz = 0 . Combing it with (6.13), we get desired ψ x = 0.Finally we have ψ = 0 on B ǫ . ψ is a solution of a linear elliptic equation( D g S ω ) ◦ ( D g S ω ) ∗ ψ = 0 whose principal part is bi-Laplacian from Lemma4.4. So we get ψ = 0 on M by the unique continuation principle for abi-Laplace type operator , finishing the proof. (cid:3) Remark 6.3.
In our argument, a particular metric g KT on Kodaira-Thurstonmanifold is used, as it guarantees the surjectivity of the derivative of scalarcurvature map at the constructed metric. One may guess, reasonably, thata generic almost K¨ahler metric satisfies this surjectivity, as in the Riemann-ian case [4, 4.37]. However, the equation (6.8) is not readily understood.For this matter, the article [6, Theorem 7.4] on local deformation of scalarcurvature is interesting; generic (local) surjectivity of scalar curvature wasshown by some method. In that argument a crucial part was the existenceof one real analytic metric without local KIDs , i.e. with (locally) surjectivederivative of scalar curvature functional.Here we did not try to prove such generic surjectivity. Rather we bypassedit; we found one almost K¨ahler metric without local KIDs, and then weimbedded it onto any symplectic manifold, obtaining a no-global-KID metric.We hope to address this generic surjectivity issue in near future. For example, one can apply Protter’s theorem [20] which states that if a real-valued function u defined in a domain D ⊂ R m containing 0 satisfies that | ∆ n u | ≤ f ( x, u, Du, · · · , D k u ) for Lipschitzian f and k ≤ [ n ], and e r − β u → r := p x + · · · + x m → β >
0, then u vanishes identically in D . Remark 6.4.
For an almost K¨ahler version of Kazdan-Warner theory, onemay try the Hermitian scalar curvature ( s + s ∗ ) [2] rather than the usualscalar curvature, where s ∗ is the star scalar curvature. However, although theHermitian scalar curvature is natural in almost K¨ahler geometry, Kazdan-Warner theory goes better with usual one. To see this, recall the Blair’sformula [7] for g ∈ Ω ω :(6.15) Z
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