Almost Kahler metrics with non-positive scalar curvature which are Euclidean away from a compact set

ALMOST K ¨ AHLER METRICS WITH NON-POSITIVE SCALAR CURVATURE WHICH ARE

EUCLIDEAN AWAY FROM A COMPACT SET

Yutae Kang and Jongsu Kim

Abstract. OnR²ⁿ,n ≥ 2, with the standard symplectic structure we construct compatible almost K¨ahler metrics with negative scalar curvature on a polydisc which are Euclidean away from the polydisc.

1. Introduction

An almost K¨ ahler structure on a smooth manifold M is a triple (g, ω, J ) where ω is a symplectic form, J an almost complex structure, and g is an ω-compatible Riemannian metric, i.e. ω(x, y) = g(J x, y) for tangent vectors x, y to M . A manifold with an almost K¨ ahler structure is called an almost K¨ ahler manifold.

With Gromov’s theory of pseudoholomorphic curves [3] and Taubes’

recent works on Seiberg-Witten invariants [6], the study of almost K¨ ahler manifolds is becoming a more interesting subject than ever. However, it suffered for a long time from a lack of effective approaches with concrete interesting examples.

In Riemannian geometry one of the interesting problems associated to a curvature, for example the scalar, the Ricci or the sectional curvature, is whether there exists a metric on a Euclidean space R

which has the curvature negative on a ball and is Euclidean away from it. The nature of this question concerns the flexibility of the curvature under consideration. More interestingly it is empirically related to the harder problem of whether given a manifold of dimension ≥ 3 admits a metric with the curvature negative, as shown in Lohkamp’s recent works [4, 5].

Received April 10, 2003.

2000 Mathematics Subject Classification: 53C15.

Key words and phrases: almost K¨ahler metric, symplectic form, scalar curvature.

Supported by grant No. R14-2002-044-01000-0(2002) from the Korea Science and Engineering Foundation.

Motivated by this, in this paper we ask the above question in the almost K¨ ahler category as an approach to almost K¨ ahler metrics from a Riemannian viewpoint.

In the general Riemannian case it is not hard to construct a met- ric on R

which has the scalar curvature negative on a ball and is Euclidean outside the ball. Actually, in the afore-mentioned works Lohkamp proved a stronger result in Proposition 2.1 of [4] and the much harder case of Ricci curvature in Proposition 6.1 of [5]. In these works conformal deformation plays a technically essential role. But in a sharp contrast we cannot use conformal deformation in almost K¨ ahler cate- gory because a metric conformal to an almost K¨ ahler one is not almost K¨ ahler in general. Therefore to deal with even the scalar curvature one should find a way distinct from the general Riemannian case.

To construct the desired metrics on the even-dimensional Euclidean space R

, n ≥ 2, we considered almost K¨ahler metrics with n-dimen- sional torus group symmetry. We managed to choose metrics which are Euclidean outside a polydisc, i.e. the product of 2-dimensional discs, and have the scalar curvature non-positive inside the polydisc. But the scalar curvature is zero precisely along some thin subset of the polydisc.

Thus an elaborating argument was needed to perturb the metrics near this thin subset to get the scalar curvature negative everywhere inside the polydisc. In sum we proved;

Theorem 1. On R

, n ≥ 2, with the standard symplectic struc- ture there exist compatible almost K¨ ahler metrics which have the scalar curvature negative on a polydisc and are Euclidean outside of it. Fur- thermore these metrics can be chosen to be invariant under the n- dimensional torus group.

The paper is organized as follows. In section 2 we provide prelimi- naries for almost K¨ ahler metrics and scalar curvature. In section 3 we construct almost K¨ ahler metrics on R

which have non-positive scalar curvature and are Euclidean outside a polydisc. But these metrics have zero scalar curvature along a thin subset inside the polydisc. In section 4 we deform the metrics of section 3 near this thin subset to have the scalar curvature negative inside the polydisc. In section 5 we explain how to get similar metrics in higher dimensions.

2. Preliminaries

In this section we explain definitions and formulas which will be

needed in later sections.

A symmetric (2, 0)-tensor h (from now on we simply write (2, 0)- tensor as 2-tensor) on an almost K¨ ahler manifold can be decomposed as h = h

+ h

, where h

and h

are symmetric 2-tensors defined by h

(x, y) =

{h(x, y)+h(Jx, Jy)} and h

(x, y) =

{h(x, y)−h(Jx, Jy)}

for two vectors x and y tangent to M . A symmetric 2-tensor h is called J -invariant or J-anti-invariant if h = h

or h = h

respectively.

Given a symplectic form ω on a smooth manifold M , we denote by Ω

the set of all ω-compatible Riemannian metrics. It is well known that Ω

is naturally an infinite dimensional Fr´ echet manifold. According to Blair [2], for a smooth curve g

in Ω

with the corresponding curve J

of almost complex structures, h =

|

is J

-anti-invariant. Conversely, for g in Ω

with the corresponding J , any J -anti-invariant symmetric 2- tensor h is tangent to a smooth curve in Ω

. More precisely, ge

is such a smooth curve in Ω

, where ge

is defined by ge

(x, y) = g(x, e

y).

Here h and so e

is understood as a (1, 1)-tensor lifted with respect to g. So ge

(x, y) = g(x, y) + g(x,

k=1h^k k!

y).

We denote by ∇, R, r and s the Levi-Civita connection, the Rie- mannain curvature tensor, the Ricci tensor and the scalar curvature of a Riemannian manifold (M, g). For tangent vector fields X, Y, Z, W , the Riemannian curvature tensor R is defined by R(X, Y )Z = ∇

∇

Z −

∇

∇

Z − ∇

Z.

For a local tangent frame {e

}

of M , m = dimM , we shall adopt the usual notational convention: R

= g(R(e

, e

)e

, e

) and r

= r(e

, e

).

In a Riemannian manifold (M, g) the differential at g , in the direction of a symmetric 2-tensor h, of its scalar curvature s is given in [1], page 63, by

s

(h) = ∆

(tr

h) + δ

(δ

h) − g(r

, h),

where r

is the Ricci curvature tensor of g, ∆

is the Laplacian operator, tr

h is the trace of h with respect to g, δ

h is the divergence of h which can be written in local coordinates as (δh)

= −∇

h

and finally δ

( ·) on a 1-form is the formal adjoint of the (usual) differential on functions.

3. Almost-K¨ ahler metrics on R

with non-positive scalar curvature

In this section we shall find almost K¨ ahler metrics on R

which have

non-positive scalar curvature and are Euclidean outside a poydisc. But

these metrics have zero scalar curvature along a thin subset inside the polydisc.

We consider a metric on R

of the form (3.1) g

= f

dr

+ r

f

dθ

+ h

dρ

+ ρ

h

dσ

,

where (r, θ), (ρ, σ) are the polar coordinates for each summand of R

:=

R

× R

respectively and f , h are smooth positive functions on R

, which are functions of r and ρ only. Let e

=

, e

=

, e

=

, e

=

. A smooth almost complex structure J

on R

is defined by J

(e

) = e

, J

(e

) = −e

, J

(e

) = e

, J

(e

) = −e

. With the standard symplectic structure ω

on R

the triple (g

, ω

, J

) is an almost-K¨ ahler structure on R

. The Riemannian curvature components of g

of the form R

are computed as follows;

R

= − 3f

rf

+ 3f

f

− f

f

− f

f

h

, R

= f

f h

− f

h

f h

+ h

f

h − f

h

f

h , R

= − h

hf

+ h

f

h + 2f h

f

h

+ f

ρf h

− f

h

f h

, R

= − h

rf

h + h

f

f

h − f

ρf h

+ f

h

f h

, R

= − 3h

ρh

− h

h

+ 3h

h

− h

f

h

, R

= − f

f h

+ f

(2f

h + f h

) f

h

+ h

rf

h − f

h

f

h , where f

=

, f

=

, etc.. The scalar curvature is

s

= 2(R

+ R

+ R

+ R

+ R

+ R

)

= 2( f

f

+ 3f

rf

− 3f

f

− f

h

f

+ h

h

+ 3h

ρh

− 3h

h

− h

h

f

).

The crucial observation in order to show s

≤ 0 is that s

can be expressed as follows;

s

= −{(f

)

+ 3

r (f

)

+ (h

)

+ 3

ρ (h

)

} − 2f

h

f

− 2h

h

f

.

Setting F = f

and H = h

, we shall find F and H which satisfy

(3.2) F

+ 3

r F

+ H

+ 3

ρ H

= 0.

Set F

+

F

= α(r)β(ρ) where α, β are smooth functions on R which satisfy at least that

α(r) = 0 for r ≥ 1, β(ρ) = 0 for ρ ≥ 1.

These two functions α and β will be specified more below.

Since (r

F

)

= r

F

+ 3r

F

= r

α(r)β(ρ), we do integration with proper boundary conditions on F to get

F (r, ρ) = β(ρ)



( 1

y



0

x

α(x) dx) dy + 1.

We now specify the function α as follows; first consider a smooth function k(y) on R such that

(3.3)

⎧ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎪ ⎪

⎩

a) k(y) = 0 for y ≤ 0, y ≥ 1, b) |k

(y) |

≤ |y

|

,

c) 

0 k(y)

y³

dy = 0, d) 0 < 

0 k(y)

y³

dy < 1 for any r with 0 < r < 1 and then define α(y) =

(y)

y³

. Here we may choose one such function k so that the derivative α

of α is zero at exactly three points in the open interval (0, 1) as in Fig.1. Similarly we set

H(r, ρ) = −α(r)



0

( 1 y



0

x

β(x) dx) dy + 1,

where β(y) =

and the function ˜ k(y) is a smooth function on R satisfying (3.3). We may choose ˜ k similarly to k (or equally if one prefers) so that the derivative β

of β is zero at three points in (0, 1).

Hence the graph of β is similar to that of α.

Then the functions F (r, ρ) and H(r, ρ) satisfy the equation (3.2) and F, H ≡ 1 for r ≥ 1 or ρ ≥ 1,

F, H > 0 for r < 1 and ρ < 1.

For such functions F and H, we have s

= − H

2F

β

(ρ)

{



0

( 1 y



0

x

α(x) dx) dy }

− F

2H

α

(r)

{



0

( 1 y



0

x

β(x) dx) dy }

.

From the choices of α and β , the derivatives α

(r) and β

(ρ) are zero at r

= 0, r

, r

, r

and ρ

= 0, ρ

, ρ

, ρ

, where 0 ≤ r

, ρ

< 1, respectively.

It is now simple to check

s

(r, ρ) =

⎧ ⎪

⎨

⎪ ⎩

0 if r ≥ 1 or ρ ≥ 1,

0 at (r

, ρ

), i, j = 0, 1, 2, 3,

negative if r, ρ ∈ (0, 1) with r = r

or ρ = ρ

.

−0.2 0.2

r

r

r

1

0

Fig.1. The graph of α (The graph of β is similar).

Let B = {(r, θ, ρ, σ)|0 ≤ r, ρ < 1, 0 ≤ θ, σ < 2π} which is a polydisc and let D

= {(r

, θ, ρ

, σ) |0 ≤ θ, σ < 2π} for each i = 0, 1, 2, 3 and j = 0, 1, 2, 3. The metric g

is Euclidean on B

, the complement of B, and s

is negative on B \ ∪

D

but s

= 0 on ∪

D

. So to get a metric with the further property that s

< 0 on B, it is natural to deform the metric g

near ∪

D

. This will be done in section 4.

4. Almost-K¨ ahler metrics with negative scalar curvature on a polydisc, which is Euclidean outside the polydisc In this section we denote by (g

, ω

, J

) the almost-K¨ ahler structure constructed in section 3. Suppose that we have any symmetric 2-tensor η which satisfies the following;

1) η is J

-anti-invariant,

2) s

(η) = ∆

(tr

η) + δ

(δ

η) − g

(r

, η) < 0 on D := ∪

D

,

3) the support of η is contained in B.

Then the metric defined by g

= g

e

lies in Ω

by condition 1) as explained in section 2. Since s

is negative on B \ D and zero on D, by 2) and 3) we have s

< 0 on B for small t by the Taylor series expansion argument for its scalar curvature; s

= s

+ s

(η)t + s

(η)

+ · · · . Therefore by 3) we will have ω

-compatible almost-K¨ ahler metrics g

, which are Euclidean on B

with s

< 0 on B for small t. To find such an η, first we consider the expression of ∆(trη) + δ(δη) at a point p ∈ D via a local coordinate system. Let p be a point in a component D

of D, i, j = 1, 2, 3, and choose a local coordinate x = (x

, x

, x

, x

) near p such that x

= f (p)r, x

=

θ, x

= h(p)ρ, x

=

σ. Suppose that ˜ η is a J

-anti-invariant symmetric 2-tensor such that it is expressed in the coordinate x by ˜ η = ˜ η

dx

⊗ dx

where ˜ η

= ˜ η(

s

,

t

) are functions of x

and x

only. If we express ˜ η as a matrix ˜ η = (˜ η

) , then

˜

η satisfies the condition 1) if and only if it is of the following form ;

(4.1) η = ˜

⎛

⎜ ⎜

⎝

˜

η

η ˜

η ˜

η ˜

˜

η

−a

η ˜

a

b˜ η

−a

b

η ˜

˜

η

a

b˜ η

η ˜

η ˜

˜

η

−a

b

η ˜

η ˜

−b

η ˜

⎞

⎟ ⎟

⎠ ,

where a =

and b =

. As ˜ η is determined by six func- tions ˜ η

, ˜ η

, ˜ η

, ˜ η

, ˜ η

, ˜ η

, we shall only find these six ˜ η

’s. Note that at p, ˜ η

’s have the following relation:

˜

η

(p) = −˜η

(p), ˜ η

(p) = −˜η

(p), ˜ η

(p) = ˜ η

(p), ˜ η

(p) = −˜η

(p).

We can see that at p

(4.2) ∆(tr ˜ η) |

+ δ(δ ˜ η) |

= ˜ η

(p) + ˜ η

(p) + 2˜ η

(p) + L(p),

where ˜ η

=

k∂x_l

and L consists of lower order terms. If we choose

˜

η

(p) = −˜η

(p) to be a nonzero number c and all other ˜ η

(p) = 0 then

at p

g

(r

, ˜ η) = c(r

− r

)

= c(R

+ R

+ R

− R

− R

− R

)

= −2c f

f h

(f

= h

= 0 at p)

= c H F β

(ρ)



0

( 1 y



0

x

α(x) dx) dy

= 0 (Recall the graph of β).

We will take the value c to be 1 or −1 so that g

(r

, ˜ η) > 0. For these values ˜ η

(p) = −˜η

(p) = ±1, ˜η

(p) = ˜ η

(p) = ˜ η

(p) = ˜ η

(p) =

˜

η

(p) = 0 and any values

k

(p), ˜ η

(p) and ˜ η

(p), we choose the value ˜ η

(p) so that (4.2) becomes zero. Then we can choose smooth functions ˜ η

(x

, x

) on R

, extending these values at p, with support in a small neighborhood U ⊂ B of p. Note that we may and will choose U to be a neighborhood of D

because ˜ η

’s are functions of x

and x

only.

Since g

(r

, ˜ η) and the equation ∆(tr ˜ η) + δ(δ ˜ η) = 0 are S

× S

- invariant, these functions ˜ η

’s determine a J

-anti-invariant symmetric 2-tensor ˜ η with support contained in a neighborhood U ⊂ B of D

satisfying ∆(tr ˜ η) + δ(δ ˜ η) − g

(r

, ˜ η) < 0 along D

.

In the case that p ∈ D

where i = 0 or j = 0, ˜ η = cr ˜

dr

−

dθ

+ cρ

−

dσ

for some constant c satisfies ∆(tr ˜ η) + δ(δ ˜ ˜ η) ˜ → 4c as r → 0 or ρ → 0. Multiplying ˜˜η by a smooth cut off function with support in a small neighborhood of D

and choosing an appropriate c we obtain a J

-anti-invariant symmetric 2-tensor ˜ η with support contained in a neighborhood of D

satisfying s

(˜ η) < 0 along D

.

So far for each component D

, i, j = 0, 1, 2, 3 we have chosen a neighborhood U

⊂ B of D

and a J

-anti-invariant symmetric 2-tensor

˜

η

on R

such that

i) ˜ η

satisfies s

(˜ η

) < 0 along D

. ii) The support of ˜ η

is contained in U

. Moreover U

’s could be chosen small so that

iii)U

’s are disjoint each other.

Then η =

i,j=0

η ˜

is the desired J

-anti-invariant symmetric 2-tensor.

Remark. In the above we have chosen ˜ η

(p) = −˜η

(p) = ±1 and all other ˜ η

(p) = 0. But in fact, whatever values ˜ η

(p),

k

(p), ˜ η

(p)

and ˜ η

(p) we choose we can choose ˜ η

(p) so that ∆(tr ˜ η) |

+ δ(δ ˜ η) |

− g

(r

, ˜ η) |

to be negative. Extending these values at p, we can also obtain a J

-anti-invariant symmetric 2-tensor ˜ η we want.

Since f and h are functions of r and ρ only and the J

-anti-invariant symmetric 2-tensor η is S

× S

-invariant, the metric g

= g

e

is also S

× S

-invariant. Therefore we proved the following

Proposition 1. Let B

⊂ R

be an open ball of radius 1 and B = B

× B

⊂ R

. With the standard symplectic structure on R

, there exists an S

× S

-invariant compatible almost-K¨ ahler metric g on R

which has the scalar curvature negative on B and are Euclidean on B

, the complement of B.

Remark. It is not hard to show that the S

× S

-invariant metrics on R

of Proposition 1 can also be chosen to have the form of (3.1). So it is interesting to find such a form of metrics instead of going through the perturbation of the section 4.

5. Generalization to R

, n ≥ 3

We generalize Proposition 1 to higher dimensions. The method is similar to 4-dimensional case.

Let g

be a metric on R

, n ≥ 3 , defined by g

=



(f

dr

+ r

f

dθ

),

where (r

, θ

), i = 1, . . . , n, are the polar coordinates on R

and f

’s are smooth positive functions on R

depending only on the variables r

, . . . , r

. Let e

=

i

∂

∂r_i

, e

=

i

∂

∂θ_i

and J

be the almost complex structure defined by J

(e

) = e

, J

(e

) = −e

. Then with the standard symplectic form ω

on R

the triple (g

, ω

, J

) is an almost- K¨ ahler structure on R

. The scalar curvature of g

is

s

2 =



( f

f

+ 3 f

r

f

− 3 f

f

) − 

i<j

f

+ f

f

f

= − 1 2



{(f

)

+ 3

r

(f

)

} − 

i<j

f

+ f

f

f

,

where f

=

j

, f

=

k∂r_j

. Set F

= f

, i = 1, . . . , n. We shall find the functions F

so that they satisfy

(5.1)



(F

+ 3

r

F

) = 0.

Set F

+

i

F

= α

(r

) · · · α

(r

), i = 1, . . . , n −1 and F

+

n

F

=

−

i=1

α

(r

) · · · α

(r

), where α

(r

), i = 1, . . . , n −1, j = 1, . . . , n are smooth functions on R which satisfy at least

α

(r

) = 0 for r

≥ 1.

The functions α

’s need to be specified more. Let k

(t) be smooth func- tions on R satisfying the properties a), c), d) of (3.3) and |

(t) |

≤

n−1√1

n−1

|t

|

. Set α

(t) =

(t). Moreover we choose such functions k

’s so that each derivative

of α

is zero at exactly three points in the open interval (0, 1) and that the sets A

= {t ∈ (0, 1)|α

(t) = 0 } and B

= {t ∈ (0, 1)|

(t) = 0 } are disjoint each other, i.e. A

∩ A

= φ , B

∩ B

= φ for any pairs (i, j) = (k, l) and A

∩ B

= φ for any pairs (i, j), (k, l).

Define the functions F

, i = 1, . . . , n − 1 , and F

by F

(r

, . . . , r

) = α

(r

) · · · α

(r

)

α

(r

)



0

( 1 y



0

x

α

(x) dx) dy + 1, F

(r

, . . . , r

) = −

n−1



i=1

α

(r

) · · · α

(r

)



0

( 1 y



0

x

α

(x) dx) dy + 1.

Then the functions F

, i = 1, . . . , n − 1, and F

satisfies the equation (5.1) and

F

, F

≡ 1 if r

≥ 1 for some k, F

, H

> 0 if r

< 1 for all k.

The scalar curvature becomes s

(r

, . . . , r

) = − 1

2



i<j

( F

F

F

+ F

F

F

),

which is zero if r

≥ 1 for some k and non-positive if r

∈ (0, 1) for all k.

In order s

to be zero at (r

, . . . , r

) with all r

∈ (0, 1), F

= 0 and F

= 0 for all i < j. It is not difficult to check that this is impossible because we have chosen α

’s so that the sets A

’s and B

’s are disjoint each other. Therefore s

becomes zero only if one of r

’s vanishes. At these points, by a similar argument in Section 4, we can perturb the metric g

so that its scalar curvature becomes negative.

Let B = {(r

, θ

, . . . , r

, θ

) |0 ≤ r

< 1, 0 ≤ θ

< 2π, k = 1, . . . , n } be a polydisc and let T

=

n−times

  

S

× · · · × S

⊂ R

be the n-torus group.

Our metric constructed above is T

-invariant, Euclidean on B

and its scalar curvature is negative on B.

In summary, we proved

Proposition 2. Let B

⊂ R

be an open ball of radius 1 and B ⊂ R

, n ≥ 3, be the n-product of B

. With the standard symplectic structure on R

there exists an T

-invariant compatible almost-K¨ ahler metric g on R

which has the scalar curvature negative on B and are Euclidean on B

.

Proof of Theorem 1. In Proposition 1 and 2 we proved the statement of Theorem 1 when the polydisc is standard, i.e. the product of the 2-dimensional discs of radius 1. But our construction of metrics just works for general polydiscs which are the products of discs of variable radii. This finishes the proof.

In this article we have studied the scalar curvatures of almost K¨ ahler metrics only on the Euclidean space with the standard symplectic struc- ture. One might try to generalize this to an arbitrary almost K¨ ahler manifold. An interesting question along this line is whether every closed symplectic manifold admits an almost K¨ ahler metrics with negative scalar curvature.

Acknowledgement. The second-named author is very grateful to Claude LeBrun for discussions and comments on this work. He would like to thank the hospitality of the Department of Mathematics, SUNY at Stony Brook during the preparation of this work.

References

[1] A. L. Besse, Einstein manifolds, Ergeb. Math. Grenzgeb. (3) Folge, Band 10, Springer-Verlag, 1987.

[2] D. E. Blair, On the set of metrics associated to a symplectic or contact form, Inst.

Math. Acad. Sinica11 (1983), no. 3, 297–308.

[3] M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math.82 (1985), no. 2, 307–347.

[4] J. Lohkamp, Scalar curvature and hammocks, Math. Ann.313 (1999), no. 3, 385–

407.

[5] , Metrics of negative Ricci curvature, Ann. of Math.140 (1994), no. 3, 655–683.

[6] C. Taubes, The Seiberg-Witten invariants and symplectic forms, Math. Res. Lett.

1 (1994), 809–822.

Department of Mathematics Sogang University

Seoul 121-742, Korea

E-mail: lubo@math.sogang.ac.kr

jskim@ccs.sogang.ac.kr