Let Rn+pp be an (n+p)-dimensional pseudo-Euclidean of index p. The indefinite flat metric on Rn+pp is defined by
ds2 =
n
X
i=1
(dxi)2−
n+p
X
α=n+1
(dxα)2.
The (n+p)-dimensional pseudo-Gaussian space, denoted byGn+pp , correspond to pseudo-Euclidean space Rn+pp endowed with the Gaussian probability measure
dµ=e−|x|
2
4 dx2. (4.20)
Let X : Mn ↬Gn+pp be an n-dimensional connected spacelike submanifold immersed in the (n+p)-dimensional pseudo-Gaussian space Gn+pp . The Gaussian mean curvature vector ξ is still defined by
ξ =H⃗ + 1
2X⊥. (4.21)
whereH⃗ stands for the standard (nonnormalized) mean curvature vector of the immersion X : Mn ↬ Gn+pp and ( )⊥ denotes the normal part of a vector field on Gn+p. When ξ vanishes identically, Mn is called spacelike self-shrinker of Gn+pp .
In this context, the Gauss equation is given by Rijkl =−X
α
(hαikhαjl−hαilhαjk) (4.22) and the Ricci equation is
Rαβkl=X
m
(hαmihβmj −hαmjhβmi) (4.23)
Hence, denoting by Hα the components of the mean curvature vector, that is, H⃗ =X
α
Hαeα =X
α
X
k
hαkk
! eα,
it is not difficult to verify from (4.22) that the components of the Ricci tensorRik satisfy Rik =−X
α
Hαhαik+X
α,j
hαijhαjk. (4.24)
Similar to Theorem 4.1, we obtain for codimension p ≥ 1 any, the following rigidity result in pseudo-Gaussian space .
Theorem 4.7. Let X : Mn ↬Gn+pp be a complete spacelike submanifold immersed with parallel Gaussian mean curvature vector ξ, with |ξ|2 ≥ 2/p in the (n+p)-dimensional pseudo-Gaussian space Gn+pp . If the second fundamental form A of Mn satisfies
sup
M
|A|< γ1, (4.25)
where
γ1 = p 2
|ξ|2− r
|ξ|2− 2 p
, (4.26)
then Mn is a hyperplane of Gn+pp .
Proof. From (4.1), considering V = 12X⊤ = (∇f)⊤, and since we are assuming that ξ is parallel in the normal bundle, from [68, Lemma 2.1] we have that
∆V|A|2 = 2|∇A|2+|A|2−2⟨ξ, Aik⟩⟨Ajk, Aij⟩+ 2|R⊥|2+ 2X
α,β
S2αβ,
where R⊥ denotes the curvature of the normal bundle, Sαβ = P
i,jhαijhβij and Aij =
∇eiej
⊥
=P
αhαijeα. Since
X
α,β
Sαβ2 ≥X
α
Sαα2 ≥ 1 p
X
α
Sαα
!2
= 1 p|A|4. and using the reverse Cauchy-Schwarz inequality
∆V|A|2 ≥ 2
p|A|4+|A|2−2|ξ||A|3
= 2
p|A|2−2|ξ||A|+ 1
|A|2. (4.27)
At this point, we note that the constant γ1 defined in (4.26) is the smallest positive root of the function g(t) = 2pt2 −2|ξ|t + 1. So, from hypothesis (4.25) we can take a positive constantγ such that supM|A|< γ < γ1 and, considering the behavior ofg(t) for 0≤t≤γ1, we get
2
p|A|2−2|ξ||A|+ 1 =g(|A|)≥g(γ)|A|
γ . (4.28)
Hence, from (4.27) and (4.28) we reach at the following estimate
∆V|A|2 ≥β |A|21,5
, (4.29)
where β = g(γ) γ =
2
pγ2−2|ξ|γ+ 1 γ >0.
Moreover, from (4.24) we get that the boundedness of |A| = q P
α,i,j(hαij)2 implies, in particular, that the Ricci tensor of Mn is bounded from below. But, from V = (∇f)⊤ = ∇f, and since 12LV⟨Y, Y⟩ = Hessf(Y, Y) for any Y vector field tangent to Mn. Considering {ξ1, ..., ξp}normal orthonormal frame to Mn, we obtain
Hessf(Y, Y) = Hessf(Y, Y) +X
α
⟨X⊥, ξα⟩⟨AξαY, Y⟩,
but ∇f = 12X⊤,hence Hessf(Y, Y) = 12|Y|2 and by (4.21),
−Hessf(Y, Y) = −1
2|Y|2+ 2⟨A(Y, Y), ⃗H−ξ⟩.
Using the Reverse Cauchy-Schwarz inequality, we obtain
−Hessf(Y, Y)≥ −1
2|Y|2−2(|ξ|+|H|)|A||Y⃗ |2,
Since |ξ|, |H|⃗ are bounded, because ξ is parallel and A is bounded, from (1.12) we have that the Bakry-´Emery-Ricci tensor of Mn is also bounded from below.
Therefore, we are in position to apply Proposition 1.2 to conclude that |A| vanishes identically on Mn, which means thatMn is an n-dimensional hyperplane of Gn+pp .
We obtain also apply the Proposition 1.2 the following.
Theorem 4.8. Let X : Mn ↬Gn+pp be a complete spacelike submanifold immersed with parallel Gaussian mean curvature vector ξ, such that |ξ|2 <2/pin the(n+p)-dimensional pseudo-Gaussian space Gn+pp . If the second fundamental form A of Mn is bounded then Mn is a hyperplane of Gn+pp .
Proof. Similarly, we obtain the estimative
∆V|A|2 ≥ 2
p|A|2−2|ξ||A|+ 1
|A|2.
From |ξ|2 < 2p, the function g(t) = 2pt2 −2|ξ|t+ 1, is non-negative and since g admit a minimum global, let’s say g∗ >0, by |A| ≤C, for some constantC, we obtain
∆V|A|2 ≥g(|A|)|A|2 ≥ g∗
CC|A|2 ≥ g∗
C|A|3. Thus
∆V|A|2 ≥β(|A|2)1,5 for β = gC∗ >0.
Therefore by Proposition 1.2, |A| vanishes identically, hence Mn is an n-dimensional hyperplane of Gn+pp .
Considering the converge to zero at infinity and using the Lemma 1.7 we obtain our next rigidity result.
Theorem 4.9. Let X : Mn ↬ Gn+pp be a complete noncompact submanifold immersed with parallel Gaussian mean curvature vector ξ, such that |ξ|2 ≥ 2/p in the (n + p)-dimensional pseudo-Gaussian space Gn+pp . If the second fundamental form A of Mn is such that |A| converges to zero at infinity and |A| ≤γ1, where γ1 is the positive constant defined in (4.26), then Mn is a hyperplane of Gn+pp .
Proof. Let us suppose by contradiction that Mn is not a hyperplane of Gn+pp or, equivalently, that |A| does not vanish identically on Mn. Taking the vector field X = e−f∇|A|2, since we are assuming that |A| ≤ γ1, from (1.8) and (4.27) we obtain that
divX =e−f∆V|A|2 ≥0. (4.30)
Moreover, choosing the smooth function u=|A|2 we also have that
⟨∇u, X⟩=e−f|∇|A|2|2 ≥0. (4.31) Consequently, since we are also supposing that |A| converges to zero at infinity, we can apply Lemma 1.7 to get that ⟨∇u, X⟩ ≡ 0 on Mn. So, returning to (4.18) we conclude that |A| must be constant on Mn. Therefore, from (1.25) we have that |A| is identically zero onMn and, hence, we reach at a contradiction.
Chapter 5
Complete stationary spacelike surfaces in an n-dimensional Generalized Robertson-Walker spacetime
5.1 Preliminaries
Let (Mn−1, g) be an (n−1)-dimensional Riemannian manifold and let I ⊂ R be an open interval of the real lineR. Ann-dimensional Generalized Robertson-Walker (GRW) spacetime Mn:=−I×f Mn−1is the warped product with base (I,−dt2), fiber (Mn−1, g) and warping function f :I →R+. Thus, it is a time orientable Lorentzian manifold with the metric
⟨ , ⟩=−πI∗dt2+f2(πI)π∗Mg, (5.1) where, as usual, πI, πM denote the corresponding projections onto I,Mn−1 respectively.
Thus,Mnbecomes a spacetime when it is endowed of the time orientation defined by the timelike vector field ∂/∂t [6]. In the case f = 1 the GRW spacetime is called static and it is denoted by −I×Mn−1, in other words, a static GRW spacetime is the Lorentzian product of a negative definite open interval and a Riemannian manifold.
Let x : Σ2 → Mn be a spacelike surface1, i.e, x is an immersion and the induced metric x∗⟨ , ⟩ on Σ2 is Riemannian. The timelike vector field T :=f(πI)∂/∂t∈ X(Mn), decomposes along x as follows
T =T⊤+TN (5.2)
where, at any point of Σ2,T⊤is the tangent component and TN is the normal component of T. The vector field T satisfies ∇XT = f′(πI)X, for all X ∈ X(Mn), where ∇ is the Levi-Civita connection of the Lorentzian metric (5.1).
Indeed, writting X =−x0∂t+Xv, the decomposition in components horizontal and
1Any spacelike surface will be assumed to be connected.
43
vertical, by [62, Proposition 7.35]
∇XT =−∇x0∂tf(πI)∂t+∇Xvf(πI)∂t
=−x0(∂t(f(πI))∂t+f(πI)∇∂t∂t) +f(πI) 1
f(πI)∂t(f(πI))Xv.
=f′(πI)(−x0∂t+Xv) =f′(πI)X.
Moreover, denoting by LT⟨ , ⟩ the Lie derivative of tensor metric ⟨ , ⟩ LT⟨X, Y⟩=T⟨X, Y⟩ − ⟨LTX, Y⟩ − ⟨X,LTY⟩
=⟨∇XT, Y⟩+⟨X,∇YT⟩
=f′(πI)⟨X, Y⟩+f′(πI)⟨X, Y⟩
= 2f′(πI)⟨X, Y⟩.
i.e, it is conformal withLT⟨, ⟩= 2f′(πI)⟨, ⟩and closed in the sense that the associated 1-formωT(X) =⟨T, X⟩is a 1-form closed, i.e dωT = 0. This property onMn is translated, via the Gauss and Weingarten formulas (see [62, Chapter 4] for instance), to Σ2 as it follows
∇XT⊤=ATNX+f′(τ)X, ∇⊥XTN =−σ(T⊤, X) (5.3) whereX ∈X(Σ),∇is the Levi-Civita connection of the induced metric, which is denoted by the same symbol as in (5.1), ATN is the Weingarten endomorphism corresponding to TN ∈X⊥(Σ), τ :=πI ◦x, ∇⊥ is the normal connection and σ is the second fundamental form. In fact,
∇XT =∇XT⊤+∇XTN
=∇XT⊤−ATN(X) + ∇XT⊤N
+∇⊥XTN. By ∇XT =f′(πI)X, we have
f′(πI)X =∇XT⊤−ATN(X) and ∇⊥XTN +σ(T⊤, X) = 0.
Therefore we obtain (5.3).
Let us consider the smooth function u:=−⟨TN, TN⟩=f(τ)2+|T⊤|2 ≥f(τ)2 >0 on Σ2.From the second equation in (5.3) we get the following expression for the gradient of u, ∇u= 2ATNT⊤. Therefore
|∇u|2 = 4⟨A2TNT⊤, T⊤⟩. (5.4) In the case the mean curvature vector field H⃗ of Σ2 vanishes identically, we call Σ2 a stationary spacelike surface. Since H⃗ = 12tr(σ), so H⃗ = 0 implies that the trace of ATN, tr(ATN) = 0. The Cayley-Hamilton Theorem’s forATN ensures
A2TN = tr(ATN)ATN −det(ATN)I,
where I is the identity transformation. Since the determinant of ATN satisfies det(ATN) = 1
2 tr(ATN)2−tr(A2TN) ,
By tr(ATN) = 0, we have
A2TN = 1
2tr(A2TN).
Hence, in stationary spacelike surface, the formula (5.4) is written as
|∇u|2 = 2tr(A2TN)|T⊤|2. (5.5) As shown in [4], from the Gauss and Codazzi equations, when H⃗ = 0, formulas (5.3) and the expression of the curvature tensor R ofMn in terms of the warping function and the curvature of M [62, Prop. 7.42], we get for the Laplacian of u
∆u= 2
KΣ− f′′(τ) f(τ)
|T⊤|2+ 2tr(A2TN). (5.6) A proof for the formula of the Laplacian of u is presented in the Appendix, in the Proposition A.1 and as consequence the equation (5.6) in Corollary A.2.
A direct computation from (5.5) and (5.6) gives
Lemma 5.1. Let x : Σ2 → Mn be a stationary spacelike surface. For the function u=−⟨TN, TN⟩>0 on Σ2 we have
∆ logu= 2
KΣ− f′′(τ) f(τ)
− 2f(τ)2 u2
KΣ− f′′(τ) f(τ)
u−tr(A2TN)
. (5.7) In the order to go further, around any p ∈ Σ2 consider a local orthonormal normal frame {ξ1, ..., ξn−2} where ξn−2 is, at any point, collinear to TN. Thus, we have
⟨ξi, ξj⟩=δij, 1≤i, j ≤n−3, ⟨ξn−2, ξn−2⟩=−1.
Consider {e1, e2} a local orthonormal frame tangent at p∈Σ2. By Gauss equation
⟨R(e1, e2)e1, e2⟩=⟨R(e1, e2)e1, e2⟩ − ⟨σ(e1, e2), σ(e1, e2)⟩+⟨σ(e1, e1), σ(e2, e2)⟩ (5.8) Writting
σ(e1, e2) =
n−3
X
i=1
⟨σ(e1, e2)ξi, ξi⟩ − ⟨σ(e1, e2), ξn−2⟩ξn−2
σ(e1, e1) =
n−3
X
j=1
⟨σ(e1, e1)ξj, ξj⟩ − ⟨σ(e1, e1), ξn−2⟩ξn−2
σ(e2, e2) =
n−3
X
k=1
⟨σ(e2, e2)ξk, ξk⟩ − ⟨σ(e2, e2), ξn−2⟩ξn−2
We have,
2KΣ = 2K−2
n−3
X
i=1
⟨Aξk(e1), e2⟩⟨Aξk(e1), e2⟩+ 2⟨Aξk(e1), e2⟩2
+ 2
n−3
X
i=1
⟨Aξk(e1), e1⟩⟨Aξk(e2), e2⟩ −2⟨Aξn−2(e1), e1⟩⟨Aξn−2(e2), e2⟩
= 2K−
n−3
X
i=1
tr(A2ξk) + tr(A2ξn−2) +
n−3
X
i=1
(trAξk)2−(trAξn−2)2
Under the assumption that Σ2 is stationary and since TN = √
uξn−2, that implies, tr(Aξ2
n−2) = 1utr(A2TN), the Gauss equation becomes 2KΣ= 2K−
n−3
X
i=1
tr(A2ξ
i) + 1
utr(A2TN) (5.9)
where K is, at any point q ∈ Σ2, the sectional curvature in Mn of the spacelike tangent plane dxq(TqΣ)⊂ Tx(q)Mn. Next, K may be expressed in terms of the warping function and the sectional curvature KM of the fiber as it follows [4, Lemma 2].
Lemma 5.2. Let Σ2 be a spacelike surface in Mn. Then the sectional curvature in Mn of the spacelike plane tangent to Σ2 is given by
K = f′′(τ)
f(τ) +f′(τ)2−f′′(τ)f(τ)
f(τ)4 u+ u
f(τ)4KM.
where KM stands for the sectional curvature in fibre Mn−1 of the tangent plane to Σ2 projected on the fibre.
The proof of the Lemma 5.2 is presented in the Appendix in Lemma A.3.
Now using the previous result, the Gauss equation (5.9) can be rewritten to get KΣ = f′′(τ)
f(τ) +f′(τ)2−f′′(τ)f(τ)
f(τ)4 u+ u
f(τ)4KM −1 2
n−3
X
i=1
tr(A2ξi) + 1
2utr(A2TN).
Taking into account that the gradient on Σ2 of τ satisfies ∇τ = −∂t⊤, we obtain
|∇τ|2 = (u−f(τ)2)/f(τ)2. Therefore, KΣ =f′(τ)2+KM
f(τ)2 +
KM
f(τ)2 −(logf)′′(τ)
|∇τ|2
+ 1
2utr(A2TN)− 1 2
n−3
X
i=1
tr(A2ξi). (5.10)
Now we can rewrite the conclusion of Lemma 5.1 with the help of the formula (5.10) as it follows.
Lemma 5.3. Let x : Σ2 → Mn be a stationary spacelike surface. For the function u=−⟨TN, TN⟩>0 on Σ2 we have
∆ logu= 2
KΣ− f′(τ)2+KM f(τ)2
+f(τ)2 u2
(
tr(A2TN) +u
n−3
X
i=1
tr(A2ξ
i) )
. (5.11) Note that in the case u is constant, formula (5.11) gives
KΣ−f′(τ)2+KM
f(τ)2 ≤0 (5.12)
everywhere on Σ2, and the equality holds in (5.12) if, and only if, ATN = Aξi = 0, i = 1,2, ..., n−3, i.e., if, and only if, Σ2 is totally geodesic in Mn.