In a similar vein, in this paper, we also study the existence and the completeness of some metric with prescribed scalar curvature functions on some Lorentzian warped product manifolds

https://doi.org/10.5831/HMJ.2017.39.2.187

THE EXISTENCE AND THE COMPLETENESS OF SOME METRICS ON LORENTZIAN WARPED PRODUCT MANIFOLDS WITH FIBER MANIFOLD OF

CLASS (B)

Yoon-Tae Jung, A-Ryong Kim, and Soo-Young Lee^∗

Abstract. In this paper, we prove the existence of warping func- tions on Lorentzian warped product manifolds and the completeness of the resulting metrics with some prescribed scalar curvatures.

1. Introduction

In [16, 17], Leung have studied the problem of scalar curvature func- tions on Riemannian warped product manifolds and obtained partial re- sults about the existence and nonexistence of Riemannian warped metric with some prescribed scalar curvature function([11]).

In a similar vein, in this paper, we also study the existence and the completeness of some metric with prescribed scalar curvature functions on some Lorentzian warped product manifolds.

Let (N, g) be a compact Riemannian manifold of dimension n without boundary and q = _n−2²ⁿ . Put µ = inf_φ[⁴⁽ⁿ⁻¹⁾_n−2 R

N|∇φ|²dN +R

NR(x)φ²dN ]·

||φ||⁻²_q on the set { φ ∈ H₁²(N ) | φ ≥ 0, φ 6≡ 0 }, where R(x) is the scalar curvature of the metric g on N and H₁²(N ) is a Hilbert space([1]).

The results of Kazdan and Warner([13, 14, 15]) imply that N belongs to one of the following three categories([1, p.137]):

Received November 23, 2016. Accepted May 16, 2017.

2010 Mathematics Subject Classification. 53C21, 53C50, 58C35, 58J05

Key words and phrases. warping function, Lorentzian warped product manifold, scalar curvature.

The first author was supported by Chosun University Research Fund 2016.

*Corresponding author

(A) N does not admit any metrics with µ = 0 or µ > 0. Then µ < 0 for every metric so the scalar curvature functions are precisely those which are negative somewhere.

(B) N does not admit any metrics with µ > 0, but does admit metrics with µ = 0 and µ < 0. This is identical with (A) case except that the zero function is also a scalar curvature.

(C) N has some metric g with µ > 0.

This completely answers the question of which smooth functions are scalar curvatures of Riemannian metrics on a compact manifold N ([8, 9, 10, 12]).

In [13, 14, 15], Kazdan and Warner also showed that there exists some obstruction of a Riemannian metric with positive scalar curvature (or zero scalar curvature) on a compact manifold([9]).

In [16, 17], the author considered the scalar curvature of some Rie- mannian warped product and its conformal deformation of warped prod- uct metric. And also in [5], authors considered the existence of a non- constant warping function on a Lorentzian warped product manifold such that the resulting warped product metric produces the constant scalar curvature when the fiber manifold has the constant scalar curva- ture. Ironically, even though there exists some obstruction of positive or zero scalar curvature on a Riemannian manifold, results of [5], say, Theorem 3.1, Theorem 3.5, and Theorem 3.7 of [5] show that there ex- ists no obstruction of positive scalar curvature on a Lorentzian warped product manifold. On the other hand, there may exist some obstruction of negative or zero scalar curvature([8, 10]).

In this paper, when N is a compact Riemannian manifold of class (B), we discuss the method of using warped products to construct timelike or null future complete Lorentzian metrics on M = [a, ∞)×_fN with specific scalar curvatures, where a is a positive constant. And we prove the existence of warping functions on Lorentzian warped product manifolds and the completeness of the resulting metrics with some prescribed scalar curvatures. These results of this paper are extensions of the results in [12].

2. Main Results

Let (N, g) be a Riemannian manifold of dimension n and let f : [a, ∞) → R⁺ be a smooth function, where a is a positive constant. A Riemannian warped product of N and [a, ∞) with warping function f is defined to be the product manifold ([a, ∞) ×fN, g⁰) with

(2.1) g⁰ = −dt²+ f²(t)g.

Let R(g) be the scalar curvature of (N, g). Then the scalar curvature R(t, x) of g⁰ is given by the equation

(2.2) R(t, x) = 1

f²(t){R(g)(x) + 2nf (t)f⁰⁰(t) + n(n − 1)|f⁰(t)|²} for t ∈ [a, ∞) and x ∈ N (For details, cf. [4] or [6]).

If we denote

u(t) = fⁿ⁺¹² (t), {t ∈ R | t > a}, then equation (2.2) can be rewritten as

(2.3) 4n

n + 1u⁰⁰(t) − R(t, x)u(t) + R(g)(x)u(t)¹⁻ⁿ⁺¹⁴ = 0.

In this paper, we assume that the fiber manifold N is a non-empty, connected, and compact Riemannian n−manifold without boundary.

Then, by Theorem 3.1, Theorem 3.5, and Theorem 3.7 in [5], we have the following proposition.

Proposition 2.1. If the scalar curvature of the fiber manifold N is an arbitrary constant, then there exists a nonconstant warping function f (t) on [a, ∞) such that the resulting Lorentzian warped product metric on [a, ∞) ×_fN produces positive constant scalar curvature.

Proposition 2.1 implies that in case of a Lorentzian warped product there is no obstruction of the existence of metric with positive scalar curvature. However, the results of [7] show that there may exist some obstruction about the Lorentzian warped product metric with negative or zero scalar curvature even when the fiber manifold has constant scalar curvature.

Remark 2.2. Theorem 5.5 in [18] implies that all timelike geodesics are future (resp. past) complete on (−∞, +∞) ×_v N if and only if R+∞

t0

 v 1+v

¹₂

dt = +∞ (resp. Rt0

−∞

 v 1+v

¹₂

dt = +∞) for some t0. On the other hand, Remark 2.58 in [2] implies that all null geodesics are future (resp. past) complete if and only if R+∞

t0 v¹²dt = +∞ (resp.

Rt0

−∞v¹²dt = +∞) for some t0 (cf. Theorem 4.1 and Remark 4.2 in [3]. In this reference, the warped product metric is g⁰= −dt²+ v(t)g).

If N is in class (B). Then we assume that N admits a Riemannian metric of zero scalar curvature. In this case, equation (2.3) is changed into

(2.4) 4n

n + 1u⁰⁰(t) − R(t, x)u(t) = 0.

If N admits a Riemannian metric of zero scalar curvature, then we let u(t) = t^α in equation(2.4), where α ∈ (0, 1) is a constant, and we have

R(t, x) = − 4n

n + 1α(1 − α)1

t² < 0, {t ∈ R | t > a}.

There, from the above fact, Remark 2.2 implies the following:

Theorem 2.3. For n ≥ 3, let M = [a, ∞) ×_f N be the Lorentzian warped product (n + 1)-manifold with N compact n-manifold. If N is in class (B), then on M there is a future geodesically complete Lorentzian metric of negative scalar curvature outside a compact set.

We note that the term α(1 − α) achieves its maximum when α = ¹₂. And when u = t¹² and N admits a Riemannian metric of zero scalar curvature, we have

R = − 4n n + 1

1 4

1

t², {t ∈ R | t > a}.

If R(t, x) is the function of only t−variable, then we have the following proposition.

Proposition 2.4. If R(g) = 0 and R(t, x) is a function of only t−variable, then there is no positive solution to equation (2.4) with

R(t) ≤ − 4n n + 1

c 4

1

t² for {t ∈ R | t ≥ t0}, where c > 1 and t₀ > a are constants.

Proof. See Proposition 2.4 in [7]. 

In particular, if R(g) = 0, then using Lorentzian warped product it is impossible to obtain a Lorentzian metric of uniformly negative scalar curvature outside a compact subset. The best we can do is when u(t) = t¹², or f (t) = tⁿ⁺¹¹ , where the scalar curvature is negative but goes to zero at infinity.

Proposition 2.5. Suppose that R(g) = 0 and R(t, x) = R(t) ∈ C^∞([a, ∞)). Assume that for t > t₀, there exist an (weak) upper so- lution u+(t) and a (weak) lower solution u−(t) such that 0 < u−(t) ≤ u₊(t). Then there exists a solution u(t) of equation (2.4) such that for t > t0, 0 < u−(t) ≤ u(t) ≤ u+(t).

Proof. See Theorem 2.5 in [7]. 

Theorem 2.6. Suppose that R(g) = 0. Assume that R(t, x) = R(t) ∈ C^∞([a, ∞)) is a function such that

− 4n n + 1

c 4

1

t² < R(t) ≤ 4n n + 1b1

t² for {t ∈ R | t ≥ t0},

where t₀ > a, c < 1, and 0 < b < ^(n+1)(n+3)₄ are constants. Then equation (2.4) has a positive solution on [a, ∞) and on M the resulting Lorentzian warped product metric is a future geodesically complete one.

Proof. Since R(g) = 0, put u₊(t) = t^α. Then u⁰⁰₊(t) = α(α − 1)t^α−2. Hence, by choosing 0 < α < 1 so that 4α(α − 1) + c < 0, we have

4n

n + 1u⁰⁰₊(t) − R(t)u₊(t) ≤ 4n

n + 1u⁰⁰₊(t) + 4n n + 1

c 4

1 t²u₊(t)

= 4n

n + 1α(α − 1)t^α−2+ 4n n + 1

c 4

1 t²t^α

= 4n

n + 1 1

4t^α−2[4α(α − 1) + c]

< 0.

Therefore u₊(t) is our (weak) upper solution.

And put u−(t) = t^−β, where β is a positive constant. Then u⁰⁰₋(t) = β(β + 1)t^−β−2. Hence

4n

n + 1u⁰⁰₋(t) − R(t)u−(t) ≥ 4n

n + 1u⁰⁰₋(t) − 4n n + 1b1

t²u−(t)

≥ 4n

n + 1β(β + 1)t^−β−2− 4n n + 1b1

t²t^−β

= 4n

n + 1t^−β−2[β(β + 1) − b]

> 0

if β is sufficiently close to ⁿ⁺¹₂ with 0 < β < ⁿ⁺¹₂ .

Thus u−(t) is a (weak) lower solution and 0 < u−(t) < u(t) < u₊(t) for t > t1 for some large t1.

And since β is sufficiently close to ⁿ⁺¹₂ , −_n+1^2β + 1 > 0. Therefore

Z +∞

t1

 f (t)² 1 + f (t)²

¹₂ dt =

Z +∞

t1

u(t)ⁿ⁺¹² q

1 + u(t)ⁿ⁺¹⁴ dt

≥

Z +∞

t1

u−(t)ⁿ⁺¹² q

1 + u−(t)ⁿ⁺¹⁴ dt =

Z +∞

t1

t^−n+12β q

1 + t^−n+14β dt

≥ 1

√ 2

Z +∞

t1

t⁻

2β

n+1dt = +∞

and

Z +∞

t1

f (t)dt = Z +∞

t1

u(t)ⁿ⁺¹² dt

≥

Z +∞

t1

u−(t)ⁿ⁺¹² dt = Z +∞

t1

t^−n+12β dt = +∞,

which, by Remark 2.2, implies that the resulting warped product metric

is a future geodesically complete one. 

The following example shows that the solution of equation (2.4) is not unique and that there maybe exists a solution of equation (2.4) which is not between the lower solution and the upper solution.

Example 2.7. If R(g) = 0 and R(t) = _n+1^{4n n(n+2)}_{4 t}¹2, then there is a positive solution to equation (2.4)

(2.5) t²u⁰⁰(t) −n(n + 2)

4 u(t) = 0.

By Euler-Cauchy equation method, we put u(t) = t^m then m(m − 1)t^m−2t²−n(n + 2)

4 t^m= 0, (m²− m −n(n + 2)

4 )t^m= 0 and

(m −n + 2

2 )(m + n 2) = 0

so m = ⁿ⁺²₂ , −ⁿ₂. Thus u(t) = c1tⁿ⁺²² + c2t⁻ⁿ² is solution of equation (2.5), where c₁ and c₂ are constants.

Therefore u(t) = c₂t⁻ⁿ² is our (weak) solution in the sense of Theorem 2.5 such that 0 < u−(t) ≤ u(t) ≤ u+(t). And Remark 2.2 implies that the resulting Lorentzian warped product metric is a future geodesically complete one. Similarly, in case of u(t) = c1tⁿ⁺²² , it is trivial that the resulting Lorentzian warped product is a future geodesically complete

one. 

Theorem 2.8. Suppose that R(g) = 0. Assume that R(t, x) = R(t) ∈ C^∞([a, ∞)) is a function such that

4n

n + 1bt⁻² < R(t) < 4n n + 1dt^s, where b and d are positive constants.

If b > ^(n+1)(n+3)₄ , then equation (2.4) has a positive solution on [a, ∞) and on M the resulting Lorentzian warped product metric is not a future geodesically complete metric.

Proof. See Theorem 2.7 in [12]. 

Lemma 2.9. If α is a positive constant, then Z ∞

t0

e^−α

√

tdt < ∞, where t₀ is a positive constant.

Proof. Since R∞ t0 e^−α

√

tdt =R∞ t0

−2α√ t

−2α√ te^−α

√

tdt, by using the integra- tion by part we have

[−2√ t α e^−α

√t]^∞_t0 + Z ∞

t0

1 α

√1 te^−α

√tdt

= 2√ t₀ α e^−α

√t0 + [−2 α²e^−α

√t]^∞_t0

= 2√ t₀ α e^−α

√t0 + 2 α²e^−α

√t0 < ∞.

ThereforeR∞ t0 e^−α

√tdt < ∞, where t₀ and α are positive constants.

 Theorem 2.10. Suppose that R(g) = 0. Assume that R(t, x) = R(t) ∈ C^∞([a, ∞)) is a function such that

4n

n + 1dt^s≤ R(t) < 4n n + 1be^ct,

where b, c and d are positive constants. Then equation (2.4) has a posi- tive solution on [a, ∞) and on M the resulting Lorentzian warped prod- uct metric is not a future geodesically complete metric.

Proof. Since R(g) = 0, put u−(t) = e^−eβt, where β is a positive large constant. Then u⁰⁰₋(t) = −β²e^βte^−eβt+ β²e^2βte^−eβt.

Hence 4n

n + 1u⁰⁰₋(t) − R(t)u−(t) = 4n

n + 1[−β²e^βte^−eβt+ β²e^2βte^−eβt] − R(t)e^−eβt

≥ 4n

n + 1e^−eβt[−β²e^βt+ β²e^2βt− be^ct]

> 0

for large β. Thus, for large β, u−(t) is a (weak) lower solution.

Since R(g) = 0 and R(t) ≥ _n+1⁴ⁿ dt^s, we take the upper solution u₊(t) = e⁻

√

t and u⁰⁰₊(t) = ¹₄ ¹

t√ te⁻

√

t+_4t¹e⁻

√ t. Hence

4n

n + 1u⁰⁰₊(t) − R(t)u+(t) ≤ 4n

n + 1u⁰⁰₊(t) − 4n

n + 1dt^su+(t)

= 4n

n + 1[1 4

1 t√

te⁻

√ t+ 1

4te⁻

√

t− dt^se⁻

√ t]

= 4n

n + 1 1 4e⁻

√t[ 1 t√

t + 1

4t− 4dt^s]

≤ 0

for large t, where s > 0 is a constant. Thus u+(t) is a (weak) upper solution and 0 < u−(t) < u₊(t) for large t.

Hence Proposition 2.5 implies that equation (2.4) has a (weak) posi- tive solution u(t) such that 0 < u−(t) < u(t) < u+(t) for large t.

Therefore, by Lemma 2.9, Z +∞

t0

 f (t)² 1 + f (t)²

¹₂ dt =

Z +∞

t0

u(t)ⁿ⁺¹² q

1 + u(t)ⁿ⁺¹⁴ dt

≤

Z +∞

t0

u+(t)ⁿ⁺¹² q

1 + u₊(t)ⁿ⁺¹⁴ dt =

Z +∞

t0

e⁻

√t_n+1²

q 1 + e⁻

√t_n+1⁴

dt

≤

Z +∞

t0

e⁻

√t_n+1²

dt < +∞

and

Z +∞

t0

f (t)dt = Z +∞

t0

u(t)ⁿ⁺¹² dt

≤

Z +∞

t0

u₊(t)ⁿ⁺¹² dt = Z +∞

t0

e⁻

√

t_n+1² dt < +∞,

which by Remark 2.2 implies that the resulting warped product metric

is not a future geodesically complete one. 

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Yoon-Tae Jung

Department of Mathematics, Chosun University, Kwangju 61452, Korea.

E-mail: ytajung@chosun.ac.kr

A-Ryong Kim

Department of Mathematics,

Graduate School of Education, Chosun University, Kwangju 61452, Korea.

E-mial: kal27@nate.com Soo-Young Lee

Department of Mathematics, Chosun University, Kwangju 61452, Korea.

E-mai: skdmlskan@hanmail.net