Complete spacelike hypersurfaces in a Robertson-Walker spacetime

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Complete spacelike hypersurfaces in a Robertson-Walker spacetime

Alma L. Albujer

(joint work with Fernanda E. C. Camargo and Henrique F. de Lima)

University of C´ordoba

GELOSP’ 2013 S˜ao Paulo, July 22∼26, 2013

Partially supported by MICINN/FEDER project MTM2009-10418 and Fundaci´on S´eneca project 04540/GERM/06, Spain

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Calabi-Bernstein theorem (1970)

Non-parametric version

The only entire maximal graphs inL³are the spacelike planes.

Equivalently, the only entire solutions to the maximal surface equation

Div Du

p1− |Du|²

!

= 0, |Du|²<1

inR²are the affine functions.

Parametric version

The only complete maximal surfaces inL³ are the spacelike planes.

? Cheng and Yau (1976) extended this theorem to general dimension.

? Since then several authors have approached this result from different points of view: McNertey (1980), Kobayashi (1983), Estudillo and Romero (1991, ’92 y ’94), Romero (1996), Al´ıas and Palmer (2001), Albujer and Al´ıas (2009), Romero and Rubio (2010), Caballero, Romero and Rubio (2010).

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Calabi-Bernstein theorem (1970)

Div Du

p1− |Du|²

!

= 0, |Du|²<1

Parametric version

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Calabi-Bernstein theorem (1970)

Div Du

p1− |Du|²

!

= 0, |Du|²<1

Parametric version

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Calabi-Bernstein theorem (1970)

Div Du

p1− |Du|²

!

= 0, |Du|²<1

Parametric version

? Since then several authors have approached this result from different points of view: McNertey (1980), Kobayashi (1983), Estudillo and Romero (1991, ’92 y ’94), Romero (1996), Al´ıas and Palmer (2001), Albujer and Al´ıas (2009), Romero and Rubio (2010), Caballero,

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Two generalizations of the Calabi-Bernstein theorem

Aiyama (1992), Xin (1991)

Let Σⁿ be a complete spacelike hypersurface with constant mean curvature immersed intoLⁿ⁺¹. Suppose thatthe hyperbolic image of Σⁿ is contained in a geodesic ball ofHⁿ. Then Σⁿ is a spacelike hyperplane.

? Hⁿ is a spacelike hypersurface inLⁿ⁺¹with constant mean curvature which is not a spacelike hyperplane.

Albujer and Al´ıas (2009)

LetM²be a (necessarily complete) Riemannian surface with

non-negative Gaussian curvature,KM≥0. Then, any complete maximal surface Σ²in−R×M² is totally geodesic. Moreover, ifKM>0 at some point onM, then Σ is a slice{t0} ×M, {t0} ∈R.

? This result is no longer true in−R×H².

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Two generalizations of the Calabi-Bernstein theorem

Aiyama (1992), Xin (1991)

? Hⁿis a spacelike hypersurface inLⁿ⁺¹with constant mean curvature which is not a spacelike hyperplane.

non-negative Gaussian curvature,KM≥0. Then, any complete maximal surface Σ²in−R×M² is totally geodesic. Moreover, ifKM>0 at some point onM, then Σ is a slice{t0} ×M, {t0} ∈R.

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Two generalizations of the Calabi-Bernstein theorem

Aiyama (1992), Xin (1991)

non-negative Gaussian curvature,KM≥0. Then, any complete maximal surface Σ²in−R×M² is totally geodesic. Moreover, ifKM >0 at some point onM, then Σ is a slice{t0} ×M, {t0} ∈R.

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Two generalizations of the Calabi-Bernstein theorem

Aiyama (1992), Xin (1991)

non-negative Gaussian curvature,KM≥0. Then, any complete maximal surface Σ²in−R×M² is totally geodesic. Moreover, ifKM >0 at some point onM, then Σ is a slice{t0} ×M, {t0} ∈R.

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In this talk...

... we study constant mean curvature spacelike

hypersurfaces in a Robertson-Walker spacetime under certain energy conditions.

... Our main tool is the Omori-Yau maximum principle.

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In this talk...

... we study constant mean curvature spacelike

hypersurfaces in a Robertson-Walker spacetime under certain energy conditions.

... Our main tool is the Omori-Yau maximum principle.

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The Omori-Yau maximum principle

Let f : [a,b]→Rbe a continuous function. Thenf attains its maximum at some pointp0∈[a,b].

? Ifp₀∈(a,b) and f has continuous second derivative in a neighbourhood ofp₀, then

f⁰(p₀) = 0 and f⁰⁰(p₀)≤0.

Consider now a compact Riemannian manifoldM (without boundary) and consider any smooth function f ∈ C²(M). Thenf attains its maximum at some point p0∈M and

|∇f(p0)|= 0 and ∆f(p0)≤0. When M is not compact, a given functionf ∈ C²(M) with sup_Mf <+∞does not necessarily attains its supremum. Omori-Yau maximum principle (Omori, 1967 and Yau, 1975)

LetMⁿ be ann-dimensional complete Riemannian manifold whose Ricci curvature is bounded from below and considerf :Mⁿ→Ra smooth function which is bounded from above onMⁿ. Then there is a sequence of points{p_k}_k_∈_N⊂Mⁿ such that

k→∞lim f(pk) = sup

M

f , lim

k→∞|∇f(pk)|= 0 and lim

k→∞∆f(pk)≤0.

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The Omori-Yau maximum principle

f⁰(p₀) = 0 and f⁰⁰(p₀)≤0.

|∇f(p0)|= 0 and ∆f(p0)≤0. When M is not compact, a given functionf ∈ C²(M) with sup_Mf <+∞does not necessarily attains its supremum. Omori-Yau maximum principle (Omori, 1967 and Yau, 1975)

M

f , lim

k→∞∆f(pk)≤0.

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The Omori-Yau maximum principle

f⁰(p₀) = 0 and f⁰⁰(p₀)≤0.

|∇f(p0)|= 0 and ∆f(p0)≤0.

When M is not compact, a given functionf ∈ C²(M) with sup_Mf <+∞does not necessarily attains its supremum. Omori-Yau maximum principle (Omori, 1967 and Yau, 1975)

M

f , lim

k→∞∆f(pk)≤0.

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The Omori-Yau maximum principle

f⁰(p₀) = 0 and f⁰⁰(p₀)≤0.

|∇f(p0)|= 0 and ∆f(p0)≤0.

When M is not compact, a given functionf ∈ C²(M) with sup_Mf <+∞does not necessarily attains its supremum.

Omori-Yau maximum principle (Omori, 1967 and Yau, 1975)

M

f , lim

k→∞∆f(pk)≤0.

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The Omori-Yau maximum principle

f⁰(p₀) = 0 and f⁰⁰(p₀)≤0.

|∇f(p0)|= 0 and ∆f(p0)≤0. When M is not compact, a given functionf ∈ C²(M) with sup_Mf <+∞does not necessarily attains its supremum.

Omori-Yau maximum principle (Omori, 1967 and Yau, 1975)

k→∞lim f(p_k) = sup

M

f , lim

k→∞|∇f(p_k)|= 0 and lim

k→∞∆f(p_k)≤0.

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Robertson-Walker spacetimes

Let (Mⁿ,h,iM) be a Riemannian manifold of constant sectional curvature κ, I ⊆Ra real interval and f >0 a positive smooth function onI:

ARobertson-Walker spacetimeis the product manifold I ×Mⁿendowed with the Lorentzian metric

h,i=−dt²+f²h,i_M.

? We denote it by−I ×_f Mⁿ.

−I ×_f Mⁿ has constant sectional curvatureκif and only ifMⁿ has constant sectional curvatureκand

f⁰⁰

f =κ=(f⁰)²+κ f² .

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Robertson-Walker spacetimes

h,i=−dt²+f²h,iM.

?We denote it by −I ×_f Mⁿ.

f⁰⁰

f =κ=(f⁰)²+κ f² .

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Robertson-Walker spacetimes

h,i=−dt²+f²h,iM.

?We denote it by −I ×_f Mⁿ.

f⁰⁰

f =κ=(f⁰)²+κ f² .

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Robertson-Walker spacetimes

Some energy conditions

A spacetime Mⁿ⁺¹ satisfies thetimelike convergence condition (TCC)ifRic(Z,Z)≥0, for all timelike vector Z.

A spacetime Mⁿ⁺¹ satisfies thenull convergence condition (NCC)ifRic(Z,Z)≥0, for all lightlike vectorZ.

? For the case of a Robertson-Walker spacetime:

TCC ⇔

f⁰⁰≤0 κ≥sup

I

(ff⁰⁰−f⁰²)

NCC ⇔ κ≥sup

I

(ff⁰⁰−f⁰²)

If−I ×f Mⁿ has constant sectional curvature it satisfiesNCC.

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Robertson-Walker spacetimes

TCC ⇔

I

(ff⁰⁰−f⁰²)

NCC ⇔ κ≥sup

I

(ff⁰⁰−f⁰²)

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Robertson-Walker spacetimes

TCC ⇔

I

(ff⁰⁰−f⁰²)

NCC ⇔ κ≥sup

I

(ff⁰⁰−f⁰²)

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Robertson-Walker spacetimes

TCC ⇔

I

(ff⁰⁰−f⁰²)

NCC ⇔ κ≥sup

I

(ff⁰⁰−f⁰²)

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Robertson-Walker spacetimes

TCC ⇔

I

(ff⁰⁰−f⁰²)

NCC ⇔ κ≥sup

I

(ff⁰⁰−f⁰²)

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Spacelike hypersurfaces in −I ×

_f

M

ⁿ

Let Σⁿ be aspacelike hypersurfaceimmersed into−I ×f Mⁿ.

? Since∂t = (∂/∂t)(t,x) is a unitary timelike vector field globally defined on−I×f Mⁿ, then there exists a unique timelike unitary normal vector field Nglobally defined on Σⁿ such that

hN, ∂_ti ≤ −1<0 on Σⁿ. N is called the future-pointing Gauss map ofΣⁿand

coshθ=−hN, ∂timeasures thenormal hyperbolic angleof Σⁿ. Let hdenote theheight functionof Σⁿ,h= (πI)^>. It is not difficult to see that

∇h=−∂_t^>=−∂t+ coshθN and k∇hk²= cosh²θ−1. Givent0∈R, the spacelike hypersurface{t0} ×Hⁿ is called aslice. Slices are characterized byθ≡0. Or equivalently,h=c∈R.

? Slices are totally umbilical spacelike hypersurfaces withH= ^f_f⁰_(t^(t⁰⁾

0).

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Spacelike hypersurfaces in −I ×

_f

M

ⁿ

? Since∂t = (∂/∂t)_(t,x) is a unitary timelike vector field globally defined on−I×f Mⁿ, then there exists a unique timelike unitary normal vector field Nglobally defined on Σⁿ such that

hN, ∂_ti ≤ −1<0 on Σⁿ.

N is called the future-pointing Gauss map ofΣⁿand

coshθ=−hN, ∂timeasures thenormal hyperbolic angleof Σⁿ. Let hdenote theheight functionof Σⁿ,h= (πI)^>. It is not difficult to see that

0).

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Spacelike hypersurfaces in −I ×

_f

M

ⁿ

coshθ=−hN, ∂timeasures thenormal hyperbolic angleof Σⁿ.

Let hdenote theheight functionof Σⁿ,h= (πI)^>. It is not difficult to see that

0).

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Spacelike hypersurfaces in −I ×

_f

M

ⁿ

coshθ=−hN, ∂timeasures thenormal hyperbolic angleof Σⁿ. Let hdenote the height functionof Σⁿ,h= (πI)^>. It is not difficult to see that

∇h=−∂_t^>=−∂t+ coshθN and k∇hk²= cosh²θ−1.

Givent0∈R, the spacelike hypersurface{t0} ×Hⁿ is called aslice. Slices are characterized byθ≡0. Or equivalently,h=c∈R.

0).

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Spacelike hypersurfaces in −I ×

_f

M

ⁿ

Givent0∈R, the spacelike hypersurface{t0} ×Hⁿ is called aslice.

Slices are characterized byθ≡0. Or equivalently, h=c ∈R.

0).

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Spacelike hypersurfaces in −I ×

_f

M

ⁿ

Givent0∈R, the spacelike hypersurface{t0} ×Hⁿ is called aslice.

Slices are characterized byθ≡0. Or equivalently, h=c ∈R.

0).

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Spacelike hypersurfaces in −I ×

_f

M

ⁿ

Let A:X(Σ)→X(Σ) be theshape operatorof Σⁿ with respect to N and letκ₁, ..., κ_nbe the principal curvatures of Σⁿ.

Ther-mean curvature functionH_r of the spacelike hypersurface Σⁿ is defined by

n r

Hr(p) = (−1)^r X

i₁<...<i_r

κi₁(p)· · ·κi_r(p), 1≤r ≤n

? In the particular case when r = 1,H1=H=−¹_ntr(A) is the mean curvature of Σⁿ.

? In the caser = 2,H₂defines a geometric quantity related to the scalar curvature of the hypersurface.

Σⁿ is said to be contained in atimelike bounded region if it is contained in a slab

Σ⊂[t1,t2]×Mⁿ={(t,x)∈ −I×f Mⁿ:t1≤t≤t2}.

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Spacelike hypersurfaces in −I ×

_f

M

ⁿ

n r

Hr(p) = (−1)^r X

i₁<...<i_r

κi₁(p)· · ·κi_r(p), 1≤r ≤n

Σ⊂[t1,t2]×Mⁿ={(t,x)∈ −I×f Mⁿ:t1≤t≤t2}.

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Spacelike hypersurfaces in −I ×

_f

M

ⁿ

n r

Hr(p) = (−1)^r X

i₁<...<i_r

κi₁(p)· · ·κi_r(p), 1≤r ≤n

Σ⊂[t1,t2]×Mⁿ={(t,x)∈ −I×f Mⁿ:t1≤t≤t2}.

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Spacelike hypersurfaces in −I ×

_f

M

ⁿ

n r

Hr(p) = (−1)^r X

i₁<...<i_r

κi₁(p)· · ·κi_r(p), 1≤r ≤n

Σ⊂[t1,t2]×Mⁿ={(t,x)∈ −I×f Mⁿ:t1≤t≤t2}.

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Calabi-Bernstein type results in −I ×

_f

M

ⁿ

Theorem (Parametric version)

Let−I ×f Mⁿ be a Robertson-Walker spacetime whose Riemannian fiber Mⁿhas constant sectional curvature κand such that it obeysNCC. Let Σⁿ be a complete spacelike hypersurface of−I ×f Mⁿ with constant mean curvatureH6= 0 andcontained in a timelike bounded region. If

0≤Hsup

Σ

f⁰ f ◦h

≤H²

and

k∇hk²≤α

H−sup

Σ

f⁰ f ◦h

β

for some positive constantsαandβ, then Σⁿ is a slice.

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Sketch of the proof:

• MAIN IDEA: To apply the O-Y maximum principle to f(h) coshθ.

? f(h) coshθ is bounded from abovesince Σⁿis contained in a timelike bounded region andk∇hk²≤α

H−sup_Σ

f⁰ f ◦h

β

.

? Let {E1, ...En}be a local orthonormal frame of X(Σ). Then, by the Gauss equation, for any X ∈X(Σ) it holds that

Ric(X,X) =X

i

hR(X,Ei)X,Ei)i+

AX +nH 2 X

2

−n²H² 4 kXk² Ric(X,X) is bounded⇔P

ih,R(X,Ei)X,Eiiis bounded.

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H−sup_Σ

f⁰ f ◦h

β

.

Ric(X,X) =X

i

hR(X,Ei)X,Ei)i+

AX +nH 2 X

2

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H−sup_Σ

f⁰ f ◦h

β

.

Ric(X,X) =X

i

hR(X,Ei)X,Ei)i+

AX +nH 2 X

2

−n²H² 4 kXk²

Ric(X,X) is bounded⇔P

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H−sup_Σ

f⁰ f ◦h

β

.

Ric(X,X) =X

i

hR(X,Ei)X,Ei)i+

AX +nH 2 X

2

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We can compute X

i

hR(X,Ei)X,Eii= κ

f²(h)+f⁰²(h) f²(h)

(n−1)kXk²

+ κ

f²(h)−(lnf)⁰⁰(h)

(n−2)hX,∇hi² +

κ

f²(h)−(lnf)⁰⁰(h)

kXk²k∇hk²

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We can compute X

i

hR(X,Ei)X,Eii= κ

f²(h)+f⁰²(h) f²(h)

(n−1)kXk²

+

κ

f²(h)−(lnf)⁰⁰(h)

(n−2)hX,∇hi² +

κ

f²(h)−(lnf)⁰⁰(h)

kXk²k∇hk²

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We can compute X

i

hR(X,Ei)X,Eii= κ

f²(h)+f⁰²(h) f²(h)

(n−1)kXk²

+ κ

f²(h)−(lnf)⁰⁰(h)

(n−2)hX,∇hi² +

κ

f²(h)−(lnf)⁰⁰(h)

kXk²k∇hk²

≥ κ

f²(h)+f⁰²(h) f²(h)

(n−1)kXk²

? Sinceκis constant and Σⁿ is contained in a timelike bounded region P

ihR(X,Ei)X,Eiiis bounded from below.

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We can compute X

i

hR(X,Ei)X,Eii= κ

f²(h)+f⁰²(h) f²(h)

(n−1)kXk²

+ κ

f²(h)−(lnf)⁰⁰(h)

(n−2)hX,∇hi² +

κ

f²(h)−(lnf)⁰⁰(h)

kXk²k∇hk²

≥ κ

f²(h)+f⁰²(h) f²(h)

(n−1)kXk²

? Sinceκis constant and Σⁿ is contained in a timelike bounded region P

ihR(X,Ei)X,Eiiis bounded from below.

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There exists a sequence{p_k}_k∈_N on Σⁿsuch that

k→∞lim (f(h(pk)) coshθ(pk)) = sup

p∈Σⁿ

(f(h(p)) coshθ(p))

and

k→∞lim ∆ (f(h(pk)) coshθ(pk))≤0.

On the other hand,

∆(f(h) coshθ)=−nHf⁰(h)

+f(h) coshθ n²H²−n(n−1)H2

+ (n−1)f(h) coshθ κ

f²(h)−(lnf)⁰⁰(h)

k∇hk²

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p∈Σⁿ

(f(h(p)) coshθ(p))

and

On the other hand,

f²(h)−(lnf)⁰⁰(h)

k∇hk²

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p∈Σⁿ

(f(h(p)) coshθ(p))

and

On the other hand,

f²(h)−(lnf)⁰⁰(h)

k∇hk²

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There exists a sequence{pk}_k∈N on Σⁿsuch that

p∈Σⁿ

(f(h(p)) coshθ(p))

and

On the other hand,

∆(f(h) coshθ)≥ −ncoshθf(h)Hsup

Σ

f⁰ f ◦h

+f(h) coshθ n²H²−n(n−1)H2 + (n−1)f(h) coshθ

κ

f²(h)−(lnf)⁰⁰(h)

k∇hk²

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p∈Σⁿ

(f(h(p)) coshθ(p))

and

On the other hand,

∆(f(h) coshθ)≥−ncoshθf(h)Hsup

Σ

f⁰ f ◦h

+f(h) coshθ n²H²−n(n−1)H2 + (n−1)f(h) coshθ

κ

f²(h)−(lnf)⁰⁰(h)

k∇hk²

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p∈Σⁿ

(f(h(p)) coshθ(p))

and

On the other hand,

∆(f(h) coshθ)≥f(h) coshθnH

H−sup

Σ

f⁰ f ◦h

+n(n−1)f(h) coshθ H²−H2 + (n−1)f(h) coshθ

κ

f²(h)−(lnf)⁰⁰(h)

k∇hk²

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p∈Σⁿ

(f(h(p)) coshθ(p))

and

On the other hand,

H−sup

Σ

f⁰ f ◦h

κ

f²(h)−(lnf)⁰⁰(h)

k∇hk²

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p∈Σⁿ

(f(h(p)) coshθ(p))

and

k→∞lim ∆ (f(h(pk)) coshθ(pk))≤0

On the other hand,

H−sup

Σ

f⁰ f ◦h

κ

f²(h)−(lnf)⁰⁰(h)

k∇hk²

≥0.

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Therefore,

0≥ lim

k→∞(∆ (f(h(pk)) coshθ(pk)))≥0.

H

H−sup_Σ

f⁰

f ◦h

= 0.

? SinceH6= 0, H−sup_Σ

f⁰ f ◦h

= 0 k∇hk²= 0 Σⁿ is a slice.

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Therefore,

0≥ lim

H

H−sup_Σ

f⁰

f ◦h

= 0.

f⁰ f ◦h

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Therefore,

0≥ lim

H

H−sup_Σ

f⁰

f ◦h

= 0.

f⁰ f ◦h

= 0

k∇hk²= 0 Σⁿ is a slice.

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Therefore,

0≥ lim

H

H−sup_Σ

f⁰

f ◦h

= 0.

f⁰ f ◦h

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Calabi-Bernstein type results in −I ×

_f

M

ⁿ

Remarks:

We do not need to ask Σⁿ to be contained in a timelike bounded region. We just need thatf,f⁰ andf⁰⁰ are bounded on Σⁿand inf_Σf(h(p))>0.

In order to conclude thatH= sup_Σ

f⁰ f ◦h

we only need to ask the normal hyperbolic angle to be bounded.

Therefore, for the case of a static Robertson Walker spacetime: Corollary

Let−I ×Mⁿ be a static Robertson Walker spacetime whose Riemannian fiberMⁿhas non-negative constant sectional curvature. Let Σⁿ be a complete spacelike hypersurface in−I×Mⁿwith constant mean curvature and bounded normal hyperbolic angle. Then, Σⁿ is maximal.

? In this case

∆ coshθ≥ nH²+n(n−1)(H²−H2) + (n−1)κk∇hk²

coshθ≥0.

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Calabi-Bernstein type results in −I ×

_f

M

ⁿ

Remarks:

f⁰ f ◦h

Therefore, for the case of a static Robertson Walker spacetime: Corollary

? In this case

coshθ≥0.

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Calabi-Bernstein type results in −I ×

_f

M

ⁿ

Remarks:

f⁰ f ◦h

Therefore, for the case of a static Robertson Walker spacetime:

Corollary

? In this case

coshθ≥0.

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Calabi-Bernstein type results in −I ×

_f

M

ⁿ

Remarks:

f⁰ f ◦h

Therefore, for the case of a static Robertson Walker spacetime:

Corollary

? In this case

coshθ≥0.

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Calabi-Bernstein type results in −I ×

_f

M

ⁿ

In the particular case when Σⁿ is an immersed spacelike hypersurface in Lⁿ⁺¹, we haveN: Σⁿ→Hⁿ where

Hⁿ={x∈Lⁿ⁺¹:hx,xi=−1,x1≥1}.

? N(Σ) is called thehyperbolic image ofΣⁿ.

A geodesic ball B(a, %) inHⁿ of radius% >0 centered ata∈Hⁿ is B(a, %) ={p∈Hⁿ:−cosh%≤ hp,ai ≤ −1}.

The normal hyperbolic angle of Σⁿ is bounded if and only if N(Σ)⊆B(a, %) for certaina∈Hⁿ and% >0.

Corollary (Aiyama (1992), Xin (1991))

Let Σⁿ be a complete spacelike hypersurface with constant mean curvature immersed intoLⁿ⁺¹. Suppose that the hyperbolic image of Σⁿ is contained in a geodesic ball ofHⁿ. Then Σⁿ is a hyperplane.

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Calabi-Bernstein type results in −I ×

_f

M

ⁿ

Hⁿ={x∈Lⁿ⁺¹:hx,xi=−1,x1≥1}.

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Calabi-Bernstein type results in −I ×

_f

M

ⁿ

Hⁿ={x∈Lⁿ⁺¹:hx,xi=−1,x1≥1}.

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Calabi-Bernstein type results in −I ×

_f

M

ⁿ

Hⁿ={x∈Lⁿ⁺¹:hx,xi=−1,x1≥1}.

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Calabi-Bernstein type results in −I ×

_f

M

ⁿ

Hⁿ={x∈Lⁿ⁺¹:hx,xi=−1,x1≥1}.

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Entire spacelike vertical graphs in −I ×

_f

M

ⁿ

Anentire vertical graphin−I×f Mⁿ is determined by a smooth functionu∈ C^∞(M) and it is given by

Σⁿ(u) ={(u(x),x) :x∈Mⁿ} ⊂ −I ×_f Mⁿ.

? The metric induced onMⁿ from the Lorentzian metric of−I×f Mⁿ via the graph is

h,i=−du²+f²(u)h,iMⁿ.

? The graph Σⁿ(u) is a spacelike hypersurface iff|Du|²<f²(u). For any entire spacelike graph we have the following relation:

k∇hk²= |Du|² f²(u)− |Du|².

It is well known that an entire spacelike graph is not necessarily complete.

? In fact, there exist examples of entire non-complete graphs in

−R×Hⁿwhich are maximal (Albujer, 2008 and Albujer, Al´ıas, 2009) or spacelike with constant mean curvature (Alarc´on, Souam).

(66)

Entire spacelike vertical graphs in −I ×

_f

M

ⁿ

Σⁿ(u) ={(u(x),x) :x∈Mⁿ} ⊂ −I ×_f Mⁿ.

? The graph Σⁿ(u) is a spacelike hypersurface iff|Du|²<f²(u). For any entire spacelike graph we have the following relation:

k∇hk²= |Du|² f²(u)− |Du|².

(67)

Entire spacelike vertical graphs in −I ×

_f

M

ⁿ

Σⁿ(u) ={(u(x),x) :x∈Mⁿ} ⊂ −I ×_f Mⁿ.

? The graph Σⁿ(u) is a spacelike hypersurface iff|Du|²<f²(u).

For any entire spacelike graph we have the following relation: k∇hk²= |Du|²

f²(u)− |Du|².