UNIVERSIDADE FEDERAL DE ALAGOAS INSTITUTO DE MATEMÁTICA PROGRAMA DE PÓSGRADUAÇÃO EM MATEMÁTICA UFBAUFAL MÁRCIO SILVA SANTOS
Livre
UNIVERSIDADE FEDERAL DE ALAGOAS
INSTITUTO DE MATEMÁTICA
PROGRAMA DE PốSGRADUAđấO EM MATEMÁTICA UFBAUFAL
MÁRCIO SILVA SANTOS
On the Geometry of Weighted Manifolds
Tese de Doutorado
Maceió
2014 MÁRCIO SILVA SANTOS On the Geometry of Weighted Manifolds
Tese de Doutorado apresentada ao Programa de Pósgraduação em Matemática UFBA UFAL do Instituto de Matemática da Universi dade Federal de Alagoas como requisito par cial para obtenção do grau de Doutor em Matemática.
Orientador: Prof. Marcos Petrúcio de A. Cavalcante
Maceió 2014
Catalogação na fonte Universidade Federal de Alagoas Biblioteca Central Divisão de Tratamento Técnico Bibliotecário Responsável: Valter dos Santos Andrade S237o Santos, Márcio Silva.
On the geometry of weighted manifolds / Márcio Silva Santos. – Maceió, 2014.
65 f.Orientador: Marcos Petrúcio de Almeida Cavalcante.
Tese (Doutorado em Matemática) – Universidade Federal de Alagoas. Instituto
de Matemática. Doutorado Interinstitucional UFAL/UFBA. Programa de Pósgraduação em Matemática. Maceió, 2014.Bibliografia: f. 6065.
1. Variedades ponderadas. 2. Tensor BakryÉmery Ricci. 3. Curvatura ƒmédia.
4. Teoremas tipo Bernstein. 5. Estimativas de altura. I. Título.
CDU: 517
To my beloved wife Thatiane.
Acknowledgement
Ao meu Deus pela graça e misericórdia.Ao meu orientador, Professor Marcos Petrúcio de A. Cavalcante, que me deu toda a ajuda necessária ao longo desta jornada. Obrigado pelo incentivo e pelas oportunidades que me proporcionou.
Aos membros da banca de defesa de tese, pelas valiosas sugestões e discussões em prol da melhoria deste trabalho. Um agradecimento especial ao Professor Feliciano Vitório, meu orientador durante o primeiro ano do curso de doutorado. Obrigado pelo incentivo e confiança no meu trabalho. Aos Professores Henrique de Lima (UFCG) e Jorge de Lira (UFC), pela colaboração du rante a fase de pesquisa de tese. Agradeço por esta honra e pela oportunidade singular. À minha esposa Thatiane, por permancer ao meu lado nos momentos mais difíceis, desde o início da minha vida acadêmica. Obrigado pelo apoio incondicional e, acima de tudo, pelo amor que dedicas a mim.
Aos meus pais, Leônidas e Elenice, que são a fonte de minha maior motivação. Obrigado por tudo o que fizeram por mim. A todos os amigos dos departamentos de Matemática da UFC e UFCG, pela hospitalidade e pelo incentivo. A todos os amigos da Pósgraduação em Matemática da UFBA e UFAL, pela amizade e apoio ao longo destes anos. À Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), pelo suporte financeiro.
Assim, aproximemonos do trono da graça com toda a confiança, a fim de recebermos misericórdia e encontrarmos graça que nos ajude no momento da necessidade. —HEBREUS 4:16
Abstract
In this thesis we present contributions to the study of weighted manifolds in the intrinsic and extrinsic setting.
Firstly, we prove generalizations of Myers compactness theorem due to Ambrose and Gal loway for the BakryÉmery Ricci tensor. As application we obtain closure theorems for the weighted spacetime.
After that, using maximum principles for f Laplacian, we obtain results type Bernstein and P . height estimates for hypersurfaces immersed in a semiRiemannian manifold of type εI × ρ f
Weighted manifolds, BakryÉmery Ricci tensor, f mean curvature, Bernstein type Keywords: theorems, height estimates.
Resumo
Nesta tese, nós apresentamos contribuições para o estudo das variedades ponderadas no sentido intrínseco e extrínseco.
Primeiramente, nós provamos generalizações do teorema de compacidade de Myers, devido a Ambrose and Galloway, para o tensor BakryÉmery Ricci. Como aplicação, nós obtemos teoremas do fechamento para o espaçotempo ponderado.
Depois disso, usando pricípios do máximo para o f Laplaciano, nós obtemos resultados tipo Bernstein e estimativas de altura para hipersuperfícies imersas em uma variedade semi P . Riemanniana ponderada do tipo εI × ρ f
Variedades ponderadas, tensor BakryÉmery Ricci, curvatura f média, teo Palavraschave: remas tipo Bernstein, estimativas de altura.
Contents
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Introduction
Many problems lead us to consider Riemannian manifolds endowed with a measure that has a smooth positive density with respect to the Riemannian one. As consequence, the interest in the study of the weighted manifolds have been growing up.
In this thesis, we obtain intrinsic and extrinsic results for weighted manifolds, most of them generalizing well known results in the standard Riemannian and Lorentzian cases. Our results are part of the papers
In the first chapter, we fix some notations, give some basic definitions, and state results that we will use in the other chapters. In Chapter we prove six generalizations of Myers compactness theorem under different condi tions on its generalized Ricci curvature tensor. As applications of the our weighted compact ness theorems, we present new closures theorems, that is, theorems which has as conclusion the finiteness of the spatial part of a spacetime manifold.
In Chapter in collaboration with M. Cavalcante and H. de Lima, our aim is to investigate Bernstein properties of hypersurfaces immersed in a weighted
P
ρ
semiRiemannian manifold of the type εI × , where ε = ±1. In this context, an important point is to explore the BakryÉmeryRicci tensor of such a hypersurface in order to ensure the existence of OmoriYau type sequences. The existence of these sequences will constitute an important tool in the study of the uniqueness of the hypersurfaces.
In Chapter in collaboration with M. Cavalcante and H. de Lima, we prove height estimates concerning compact hypersurfaces with nonzero constant weighted mean curvature and whose boundary is contained into a slice of weighted product spaces of nonnegative BakryÉmery Ricci curvature (cf. Theorems . As applications of our estimates, we obtain nonexistence results related to complete noncompact hypersurfaces properly immersed in these weighted product spaces (cf. Theorems
HAPTER
C
1 Preliminaries
Here we establish the notations, some basic definitions and results, concerning weighted man ifolds, that will be used along this thesis.
1.1 Curvature and Maximum principle
n n
, g) and a smooth function f : M Given a complete ndimensional Riemannian manifold (M
→
n n n − f
R , the weighted manifold M associated to M , g, dµ = e dM), where and f is the triple (M
f n
dM denotes the standard volume element of M . Appearing naturally in the study of self shrinkers, Ricci solitons, harmonic heat flows and many others, weighted manifolds are proved to be important nontrivial generalizations of Riemannian manifolds and, nowadays, there are several geometric investigations concerning them. For a brief overview of results in this scope, we refer the article and it was later developed by Bakry and Émery in the work
In this thesis we use three notions of Ricci curvature for weighted manifolds. Firstly, given a constant k > 0, the kBakryÉmeryRicci tensor is given by
1
k
Ric = Ric + Hess
f f − d f ⊗ d f ,
k where Ric stands for the usual Ricci tensor of (M,g) (see The second one is the ∞BakryÉmeryRicci tensor or just BakryÉmeryRicci tensor. It is given by
∞
Ric := Ric = Ric + Hess .
f f f
We also consider a more general notion to the Ricci curvature as follows. Given a smooth
∗
vector field V on M let V be its metricdual vector field and let denote by L g the Lie deriva
V
tive. Given a constant k > 0, the modified Ricci tensor with respect to V and k (see is defined as
1
k ∗ ∗
Ric = Ric + L V .
V
V g − ⊗Vk
4k k
Of course, in the above notations, Ric = Ric ∇ .
f
f2 In order to exemplify these notions of Ricci curvature we recall that a Ricci soliton is just a
1.1 CURVATURE AND MAXIMUM PRINCIPLE
12 weighted manifold satisfying the equation bellow Ric = λ g,
f
where λ ∈ R. Moreover, a Ricci soliton is called shrinking, steady, or expanding when λ > 0, λ = 0 or λ < 0, respectively. Ricci solitons are a generalization of Einstein manifolds and its importance is due to Perelman’s solution of Poincaré conjecture. They correspond to self similar solutions to Hamilton’s Ricci flow and often arise as limits of dilations of singularities developed along the Ricci flow. Moreover one defines a metric quasiEinstein just by replacing
k
the BakryÉmery Ricci tensor by Ric , see In the next chapter we deal with a
f bigger class of manifolds, because we use inequalities instead equalities. n
) as Now, we define the f divergence of a vector field X ∈ X(M
f
− fdiv X), (X) = e div(e
f
where div is the usual divergence on M. From this, the drift Laplacian it is defined by ∆ u = div
f f
(∇u) = ∆u − h∇u,∇ f i, where u is a smooth function on M. We will also refer such operator as the f Laplacian of M. From the above definitions we have a Bochner type formula for weighted manifolds. Namely:
1
2
2
∆ (∇u, ∇u), (1.1)
f f f
∇u = Hessu + h∇u,∇(∆ u)i + Ric
2 where u is a smooth function on M. See for more details. An important tool along this work is the OmoriYau maximum principle for the f Laplacian. According to we have the following definition
Let M be a weighted manifold. We say that the full OmoriYau maximum Definition 1.1.1.
f
2 ∗
principle for ∆ holds if for any C u = u < ∞ there
f function u : M −→ R satisfying sup M
exists a sequence {x n } ⊂ M along which
∆ u(x u, (ii)lim u(x (i) lim n ) = sup n n
f ∇u(x ) = 0 and (iii)limsup ) ≤ 0. n n n
M
If the condition (ii) don’t hold, then we say just that the weak OmoriYau maximum principle for ∆ holds on M.
f
From the classical OmoriYau maximum principle (see we conclude that if
1.2 WEIGHTED MEAN CURVATURE
13 ∇ f  is bounded and Ric is bounded from below, then the full OmoriYau maximum principle for ∆ holds on M.
f
On the other hand, using volume growth conditions and stochastic process on a weighted manifold, Rimoldi showed that the validity of the weak OmoryYau maximum principle is guaranteed by a lower bound on the BakryÉmery Ricci tensor. Namely: Theorem 1.1.1. Let M be a weighted manifold. If Ric
f f
≥ λ for some λ ∈ R, then the weak OmoriYau maximum principle holds on M.
1.2 Weighted mean curvature
n
Here, we define a notion of mean curvature in the weighted context. Indeed, let Σ be a hy
n+1
persurface immersed in a weighted Riemannian manifold M and denote by ∇ the gradient
f n+1
with respect to the metric of M . According to Gromov the weighted mean curvature,
n
or simply f mean curvature, H of Σ is given by
f
nH (1.2)
f
= nH + hN,∇ f i,
n
where H denotes the standard mean curvature of Σ with respect to its orientation N. In this context, it is natural to consider the first variation for the weighted volume
Z
− fvol (Ω) = e dΩ,
f Ω
where Ω is a bounded domain in Σ. According to Bayle the first variation formula is given by
Z
d
− f
vol H dΩ, (Ω ) =
f t f
hN,V ie
t=0 Ω
dt where V is the variational vector field.
2 x n+1
In the Euclidean space R , the hypersurfaces with f mean curvature , taking f =
2 n+1
H satisfying
= 0 are the well known selfshrinkers, that is, a hypersurface immersed in R
f
H = hx,Ni,
n+1 where x is the position vector in R , see for instance
1.3 HYPERSURFACES IN WEIGHTED SEMIRIEMANNIAN MANIFOLDS
14
1.3 Hypersurfaces in weighted semiRiemannian manifolds
n+1 n
M , In what follows, let us consider an (n + 1)dimensional product space ¯ of the form I ×P
n
is an ndimensional connected Riemannian manifold and where I ⊂ R is an open interval, P
n+1
¯ M is endowed with the standard product metric
2
∗ ∗
_{P}I (h,i n
(dt ) + π P ), h,i = επ
and π P onto each factor, and where ε = ±1, π
I denote the canonical projections from I × P _{P is the Riemannian metric on P . For simplicity, we will just write ¯ M and} n n+1 n
h,i = εI × P
2 _{P . In this setting, for a fixed t is a slice of} n n
h,i = εdt + h,i ∈ I, we say that P t = {t } × P
n+1
¯ M .
n n+1 In Chapter we will consider a connected hypersurface Σ immersed into ¯ M . n+1 n
In the case where ¯ M is a is Lorentzian (that is, when ε = −1) we will assume that Σ
n
spacelike hypersurface , that is, the metric induced on Σ via the immersion is a Riemannian
n
metric. Since ∂ , it follows that there exists
t
is a globally defined timelike vector field on −I ×P
n
an unique unitary timelike normal field N globally defined on Σ and, therefore, the function
n+1
M is Riemannian (that is, when
t
angle Θ = hN,∂ i satisfies Θ ≥ 1. On the other hand, when ¯
n n+1
ε = 1), Σ is assumed to be a twosided hypersurface in ¯ M . This condition means that there is a globally defined unit normal vector field N.
n n
∇ P
Denoting by ∇, ∇ and e , Σ and P, the gradients with respect to the metrics of εI × ρ
n+1
respectively, a simple computation shows that the gradient of π on ¯ is given by M
I
∇ (1.3) π = ε , ∂ = ε∂ .
I I t t t
h∇π i∂
n
of Σ So, from we conclude that the gradient of the (vertical) height function h = (π
I Σ
) is given by
⊤ ⊤
∇ h = (∇π ) = ε∂ = ε∂ (1.4)
I t t − ΘN, n+1 n ⊤
M ) along Σ . Thus, we where ( ) denotes the tangential component of a vector field in X( ¯ get the following relation
2
2
= ε(1 ). (1.5) ∇h − Θ
n
In this setting, the weighted mean curvature H of Σ is given by
f
nH (1.6)
f = nH + εh∇ f ,Ni.
HAPTER
C
2 Compactness of weighted manifolds The results in this chapter are part of the works
2.1 Introduction The Myers Theorem has been generalized and its deepness is shown in its many applications.
In the following we recall some important generalizations. In the first one, due to Ambrose the condition on the lower bound for the Ricci tensor is replaced by a condition on its integral along geodesics. Namely:
(Ambrose). Suppose there exists a point p in a complete Riemannian manifold M Theorem A for which every geodesic γ(t) emanating from p satisfies
Z ∞ ′ ′
Ric(γ (s), γ (s))ds = ∞. Then M is compact.
The second important generalization we want to mention here is due to Galloway where a perturbed version of Myers theorem were considered.
n
Theorem B (Galloway). Let M be a complete Riemannian manifold, and γ a geodesic joining two points of M. Assume that dφ
′ ′
Ric(γ (s), γ (s)) ≥ a + dt holds along γ, where a is a positive constant and φ is any smooth function satisfying φ ≤ c.
Then M is compact and its diameter is bounded from above by q π
2
c + c . diam(M) ≤
 a(n − 1) a
Finally, we recall an interesting result due to C. Sprouse (see Namely: Theorem C
(Sprouse). Let (M,g) be a complete Riemannian manifold of dimension n satis fying Ric(v,v) ≥ −a(n − 1) for all unit vectors v and some a > 0. Then for any R,δ > 0 there
2.2 WEIGHTED AMBROSE’S THEOREM
16 exists ε = ε(n,a,R,δ ) such that if
Z
1 sup (x), 0 max {(n − 1) − Ric
− }dvol < ε(n,k,R,δ ),
Vol(B(x,R)) B(x,R)
x
then M is compact with diam(M) ≤ π + δ.
These theorems above have applications in Relativity Cosmology (see In this chapter we generalize Theorems A, B and C to the context of weighted manifolds, actually, we generalize an improved versions of Theorem C due to Yun (see for complete
Riemannian manifolds and for globally hyperbolic spacetimes. Our results have applications for closure theorems of spatial hypersurfaces in mathematical relativity.
2.2 Weighted Ambrose’s Theorem
We denote by m f the weighted Laplacian of the distance function from a fixed point p ∈ M. From the Bochner equality it is easy to see that m satisfies the following Riccati inequal
f
ity (see Appendix A of
1
k
2 ′
Ric (2.1)
(∂ r, ∂ r) m ,
f ≤ −m f − f
k + n − 1
′ stands for the derivative of m with respect to r. f
where k ∈ (0,∞) and m
f Now we are in position to state and prove our first result.
Theorem 2.2.1 (Theorem 2.1 . Let M be a complete weighted manifold. Suppose there
f
such that every geodesic γ(t) emanating from p satisfies exists a point p ∈ M f
Z ∞ k
′ ′
Ric (γ (s), γ (s)) ds = ∞,
f where k ∈ (0,∞). Then M is compact.
Proof.
Suppose by contradiction that M is not compact and let us assume that γ(t) is a unit speed ray issuing from p. Then, the function m (t) is smooth for all t > 0 along γ(t).
f
Integrating the inequality we obtain
Z Z t t
1
k 2 ′ ′ ′
Ric m (γ (s), γ (s))ds.
f (s))ds ≤ (−m f (s) − f
1
1
k + n − 1
2.2 WEIGHTED AMBROSE’S THEOREM
Z t
From given c > k + n − 1 there exists t
1
> 1 such that − m
f
(t) −
1 k + n − 1
1
(t) = ∞ which contradicts the smoothness of m
m
2 f (s)ds ≥
c k + n − 1 > 1,
(2.3) for all t ≥ t
1 , where n = dim(M).
Let denote by α = (k + n − 1) and let us consider {t ℓ } the sequence defined inductively by t
f (t).
f
= t
1
ℓ−1
17 So we conclude that lim
t→∞
(−m f (t) −
1 k + n − 1
Z t
m
−m
2 f
(s)ds) = ∞. (2.2) In particular, lim
t→∞
−m
f
(t) = ∞.In the following we show that there exists a finite number T > 0 such that lim
t→T −
α c
 α
ℓ
 α
2ℓ
Z t
ℓ+1 t
ℓ
m
2 f
(s)ds ≥
1 (k + n − 1) c
2ℓ
(k + n − 1)
ℓ
α
(s)ds ≥
c
ℓ−1
= c α
ℓ+1 .
In particular, lim
t→T −
−m
f (t) = ∞ which is the desired contraction.
Using the same techniques we are able to prove Ambrose’s theorem for the ∞BakryÉmery Ricci tensor. In this case a condition on f is required. See also Namely:
1 k + n − 1
m
2 f
f
, for ℓ ≥ 1. Notice that {t ℓ
} is an increasing sequence converging to T = t
1
c/α (c/α)−1
f
(t) ≥
c α
ℓ
for all t ≥ t ℓ . In fact, from inequality we have that it is true for ℓ = 1. Now, assume that claim holds for all t ≥ t ℓ and fix t ≥ t ℓ+1
. Then using inequality again: −m
(t) ≥ c k + n − 1
ℓ+1
Z
t
ℓ
1
m
2 f
(s)ds +
1 k + n − 1
Z t
ℓ+1 t
 1 k + n − 1
ℓ
. Given ℓ ∈ N we claim that −m
2.3 WEIGHTED GALLOWAY’S THEOREMS
18
d f
Theorem 2.2.2 (Theorem 2.2 . Let M be a complete weighted manifold, where
f
≤ 0
dt
such that every geodesic γ(t) emanating from p along γ. Suppose there exists a point p ∈ M f satisfies
Z ∞
′ ′
Ric (γ (s), γ (s)) ds = ∞.
f Then M is compact.
Applying this theorem to the universal cover of M we obtain the finiteness of the funda mental group. Corollary 2.2.1. Let (M,g) be a Riemannian manifold in the conditions of either Theorem Then the first fundamental group of M is finite.
2.3 Weighted Galloway’s Theorems Our first result in this section is Galloway’s Theorem for the modified Ricci tensor.
n
(Theorem 3.1 . Let M be a complete Riemannian manifold and let V be Theorem 2.3.1
n
a smooth vector field on M. Suppose that for every pair of points in M and any normalized minimizing geodesic γ joining these points the modified Ricci tensor satisfies dφ
k ′ ′
Ric (2.4)
(γ , γ ,
V ) ≥ (n − 1)c +
dt where k and c are positive constants and φ is a smooth function such that φ ≤ b for some b ≥ 0. So M is compact and s
" #
2
π b b . diam(M) ≤
 p p + n − 1 + 4k
(n − 1)c (n − 1)c (n − 1)c Proof.
Let p,q be distinct points in M and let γ be a normalized geodesic that minimizes distance between p and q. Assume that the length of γ is ℓ. Consider a parallel orthonormal
∞
′1 2 n i
} along γ. Let h ∈ C h(t)E (t) along γ. Firstly, the index formula implies
= γ , E , ...., E ([0, l]) such that h(0) = h(ℓ) = 0 and set V (t) = frame {E
i Z
ℓ
2
2 ′ ′ ′
I(V R(γ i = 2,...n.
,V ) = ((h ) , E )γ , E )dt,
i i i i
− h
2.3 WEIGHTED GALLOWAY’S THEOREMS
19 So
n Z ℓ
2
2 ′ ′ ′
S I(V dt. (2.5) := ,V ) = ) Ric(γ , γ )
∑ i i
(n − 1)(h − h
i=2
From the condition we have
Z Z ℓ ℓ
dφ d
1
2
2
2
2
′ ′ ′(h ) c dt + h + 2 γ ,V γ ,V dt. S ≤ (n − 1) − h − − dt dt k
On the other hand, integrating by parts we obtain
Z Z ℓ ℓ
d
1
2
2
2
2 ′ ′ ′
2 h h dt.
 dt k
γ ,V 4k(h ) γ ,V dt ≤
So,
Z Z Z ℓ ℓ ℓ
dφ
2
2
2 ′
(2.6) (h ) h h dt. S ≤ (n − 1 + 4k) dt − (n − 1)c dt − dt
πt Now we choose h(t) = sin .
ℓ
Then from we have
Z
2 ℓ
π ℓ dφ
2 h dt.
S ≤ (n − 1 + 4k) − (n − 1)c 2 − dt 2ℓ
Integrating by parts once more we get
Z Z ℓ ℓ
dφ 2πt π
2
h φ sin dt = − dt ≥ −bπ. dt ℓ ℓ
Thus we have
1
2
2 + 2bπℓ + π .
S ≤ −(n − 1)cℓ (n − 1 + 4k) 2ℓ
Finally, because γ is minimizing we have S ≥ 0 and therefore s " #
2
b b π . p p
ℓ ≤ + n − 1 + 4k
 (n − 1)c
(n − 1)c (n − 1)c It finishes the proof.
Following some ideas of we also obtain an extension of Galloway’s Theorem for weighted manifolds using the ∞BakryÉmeryRicci tensor. Theorem 2.3.2 (Theorem 3.2 . Let M be a weighted complete Riemannian manifold with
f
 f  ≤ a, where a is a positive constant. Suppose there exist constants c > 0 and b ≥ 0 such that
2.3 WEIGHTED GALLOWAY’S THEOREMS
20 for every pair of points in M and normalized minimizing geodesic γ joining these points we
f
have dφ
′ ′
Ric (2.7)
(γ , γ ,
f
) ≥ (n − 1)c + dt where φ is a smooth function satisfying φ ≤ b. Then M is compact and q
π
2
b + b (2.8) , diam(M) ≤
 c(n − 1)λ c(n − 1)
√ where λ = 2 2a + (n − 1).
Proof. From we have
Z Z Z ℓ ℓ ℓ
dφ
2
2
2
2
′ ′ ′
c dt+ h Hess h dt. (2.9) (h ) (γ , γ
S ≤ f (n − 1) − h )dt− dt
We note that d
2
2 ′ ′ ′
f
∇ h Hess (γ , γ ) = h f ,γ dt
dt d
2 ′ ′ ′
∇ ∇ f ,γ f ,γ = (h
) − 2hh dt d d d
2 ′ ′ ′
∇ = (h f ,γ ( f hh (hh ) .
) − 2 ) − f dt dt dt Integrating the last identity above we have
Z Z ℓ ℓ
d
2 ′ ′ ′
h Hess f (γ , γ )dt = 2 (hh )dt,
f
dt since h(0) = h(ℓ) = 0. Thus
1
!
2
ℓ ℓ 2 Z Z
√ d
2 ′ ′ ′
h Hess dt (γ , γ ℓ (hh ) .
f
)dt ≤ 2a dt
πt
) we have Choosing h(t) = sin(
ℓ
√
Z
2 ℓ
2π a
2
′ ′
h Hess (γ , γ . (2.10)
f
)dt ≤ ℓ
On the other hand a direct computation yelds
2.4 WEIGHTED RIEMANNIAN YUNSPROUSE’S THEOREM
21 Z
2 ℓ
π cℓ
2 ′2
(2.11) . (n − 1)(h − ch )dt = (n − 1)
2 2ℓ −
Therefore, from we have h i 1 √
2
2
2
(2.12)
−2aπ − π (n − 1) 2ℓ
 πb S ≤ − 2 + c(n − 1)ℓ
Since S ≥ 0 we get q π
2
b + b , ℓ ≤ + c(n − 1)λ c(n − 1)
√ where λ = 2 2a + (n − 1) and so M is compact and satisfies
2.4 Weighted Riemannian YunSprouse’s Theorem
In this section, following the ideas of we provide a weighted version of Theorem C as follows.
n Theorem 2.4.1 (Theorem 1.1 . Let M be a weighted complete Riemannian manifold. f
Then for any δ > 0, and a > 0, there exists an ε = ε(n,a,δ ) satisfying the following:
k
curvature If there is a point p such that along each geodesic γ emanating from p, the Ric
f
satisfies
Z ∞
n o
k
′ ′max dt < ε(n,a,δ ) (2.13) (γ , γ ), 0
(n − 1)a − Ric f
π q
 δ . then M is compact with diam(M) ≤
(n−1)a n+k−1
2 Proof. to be determined later, consider the following sets
For any small positive ε < a n o
√
k
′ ′
E = (γ (t), γ ε)
1 t ∈ [0,∞);Ric f (t)) ≥ (n − 1)(a −
and n o
√
k
′ ′
E = (γ (t), γ ε) .
2
t ∈ [0,∞);Ric (t)) < (n − 1)(a −
f
From the inequality we have on E
1 ′ m f n+k−1
(2.14)
√ ≤ −1. m ε) f (n−1)(a−
2
( + )
n+k−1 n+k−1
On the other hand, on E we have
2
2.4 WEIGHTED RIEMANNIAN YUNSPROUSE’S THEOREM
22
′ m
√
f k ′ ′
(γ , γ a − ε − Ric )/(n − 1)
f n+k−1 √ . (2.15)
≤ √
m f ε)
(n−1)(a−
2
 ( )
ε a −
n+k−1 n+k−1
Now, using the assumption on the BakryÉmery Ricci tensor we get
Z ∞
n o
k ′ ′
max ε > (γ (t), γ (t)), 0
(n − 1)a − Ric f
Z
n o
k ′ ′
> (γ (t), γ (t)) dt (n − 1)a − Ric f
E
2 Z
√ > ε) dt
(n − 1)a − (n − 1)(a −
E
2
√ = µ(E ε.
2
)(n − 1) That is,
√ ε
(2.16) µ(E ) < ,
2
n − 1 where µ is the Lebesgue measure on R. Using inequalities we obtain
′ ′ m m
f f
Z Z r n+k−1 n+k−1
√ √ dt
dt ≤
m m
f ε) f ε)
(n−1)(a− (n−1)(a−
2
2 [0,r]∩E
( + ) ( )
 1
n+k−1 n+k−1 n+k−1 n+k−1 ′ m f
Z n+k−1
dt
 √
m ε) f (n−1)(a−
2 [0,r]∩E
( )
 2
n+k−1 n+k−1
ε
1
≤ −µ {[0,r] ∩ E } + √ ε)
(n − 1)(a − ε
2 √
≤ −r + µ {[0,r] ∩ E } + ε)
(n − 1)(a − √
ε ε + . ≤ −r + √
ε) n − 1 (n − 1)(a −
√ ε ε √
Define τ(ε) = . The integral of the left hand side can be computed ex +
n−1 ε) (n−1)(a−
plicitly and therefore we get m (r) π
f
arctan ,
≤ a(ε)(−r + τ(ε)) +
2 (n + k − 1)a(ε)
2.5 WEIGHTED LORENTZIAN YUNSPROUSE’S THEOREM
23 q √
ε) (n−1)(a−
where a(ε) = .
n+k−1
So m
f
(r) ≤ −(n + k − 1)a(ε)cot(a(ε)(−r + τ(ε))),
π + τ(ε).
for any r such that τ(ε) < r <
a(ε)
 π
In particular, m (γ(r)) goes to + τ(ε)) . It implies that γ can not be
f
−∞ as r → (
a(ε) π π
minimal beyond + τ(ε). Otherwise m + τ(ε).
f would be a smooth function at r = a(ε) a(ε) π π q
Taking ε explicitly so that + τ(ε) = + δ and using the completeness of M we have
a(ε) (n−1)a n+k−1
the desired result.
2.5 Weighted Lorentzian YunSprouse’s Theorem
Now let us discuss the Lorentzian version of Theorem Let M be a timeoriented Lorentzian manifold. Given p ∈ M we set
 J (p) = {q ∈ M : there exist a future pointing causal curve from p to q},
−
called the causal future of p. The causal past J (p) is defined similarly. We say that M is
−
 globally hyperbolic if the set J(p,q) := J (q) is compact for all p and q joined by a
(p) ∩ J causal curve (see Mathematically, global hyperbolicity often plays a role analogous to geodesic completeness in Riemannian geometry.
, E , . . . , E Let γ : [a,b] −→ M be a futuredirected timelike unitspeed geodesic. Given {E
1 2 n
} be the unique Jacobi an orthonormal frame field along γ, and for each i ∈ {1,...,n} we let J i
′
field along c such that J J
i (a) = 0 and J (0) = E i . Denote by A the matrix A = [J
1 2 . . . J n ], where i
each column is just the vector for J
i i
in the basis defined by {E }. In this situation, we have that A(t) is invertible if and only if γ(t) is not conjugate to γ(a).
1 ′ −1 ′
Now we define B = A A E
f wherever A is invertible, where E(t) is the iden
− ( f ◦ γ)
n−1 ′ ⊥
f f
(t)) . The f expansion function is a smooth function defined by θ = trB (see tity map on (γ
, where t ) is conjugate
Definition 2.6]) Note that, if θ f  → ∞ as t → t ∈ [a,b], then γ(t to γ(a).
Recently, Case in obtained the following relation between the BakryEmery Ricci Ten sor and the f expansion function θ :
f
2.5 WEIGHTED LORENTZIAN YUNSPROUSE’S THEOREM
24 Lemma 2.5.1. Under the above notations,
2
θ
f k ′ ′ ′
(2.17) θ (γ , γ .
f ≤ −Ric f ) −
k + n − 1 The inequality
The distance between two timelike related points is the supremum of lengths of causal curves joining the points. It follows that the distance between any two timelike related points in a globally hyperbolic spacetime is the length of such a maximal timelike geodesic. The timelike diameter
, diam(M), of a Lorentzian manifold is defined to be the supremum of distances d(p,q) between points of M.
n
(Theorem . Let M be a weighted globally hyperbolic spacetime. Then Theorem 2.5.1
f
for any δ > 0, a > 0, there exists an ε = ε(n,a,δ ) satisfying the following: If there is a point p such that along each future directed timelike geodesic γ emanating from
k
curvature satisfies p, with l(γ) = sup{t ≥ 0,d(p,γ(t)) = t} the Ric
f Z l(γ)
n o
k
′ ′max (γ , γ ), 0 dt < ε(n,a,δ ),
(n − 1)a − Ric f
π q
 δ . then the timelike diameter satisfies diam(M) ≤
(n−1)a n+k−1
Proof.
Let ε(n,a,δ ) be the explicit constant in the previous theorem. Assume by contradiction
π q
that there are two points p and q with d(p,q) > + δ . On the other hand, since M
(n−1)a n+k−1
is globally hyperbolic, there exists a maximal timelike geodesic γ joining p and q such that ℓ(γ) = d(p, q). Following the steps of the proof of Theorem we get lim
θ
f
(t) = −∞,
 t→t π
π
where t
q
= + τ(ε). So, we conclude that γ cannot be maximal beyond + δ which
a(ε) (n−1)a n+k−1
is a contradiction.
An immediate consequence of Theorem Namely:
2.6 CLOSURE THEOREM VIA GALLOWAY’S THEOREM
25
n
Corollary 2.5.1. Let M be a weighted globally hyperbolic spacetime. Let a be a positive
f k
constant and assume that Ric
f (v, v) ≥ (n − 1)a, for all unit timelike vector field v ∈ T M. Then, π q .
the timelike diameter satisfies diam(M) ≤
(n−1)a
n+k−1
2.6 Closure Theorem via Galloway’s Theorem
n+1
Let ¯ M be an (n + 1)dimensional spacetime manifold, that is, a smooth manifold endowed
2 n+1
with a pseudoRiemannian metric ds of signature one. Assume that ¯ M is complete and time orientable.
An interesting problem in mathematical relativity is to determine whether spacelike slices of spacetime manifolds are compact. In this section we present some theorems in this direction, although our results are purely geometric. For the sake of simplicity we will omit the dimension super index and we always use a bar for geometric objects related to ¯ M .
Let us denote by u a unit timelike vector field on ¯ M . Let us assume that u is irrotational, that is the bracket of two vectors orthogonal to u is still orthogonal to u. In this case, from Frobenius theorem, at any point of ¯ M there exists a complete connected ndimensional hyper surface M orthogonal to u. Physically, we may interpret the manifold M as the spatial universe
n
at some moment in time. Moreover, in each point of M there exist a local coordinate neighbor
n
hood U and a local coordinate t with values in the interval (−ε,ε) such that W = (−ε,ε) ×U is an open neighborhood of a point in M and U =
t
{t} ×U is orthogonal to u.In W , the spacetime metric is given by
n
2
2
2 α β
ds dt g (x,t)dx dx , = −ϕ ∑
 αβ
α,β =1
1 ∂ α n
where x are local coordinates introduced in U and u = .
ϕ ∂ t
Let ¯ X be a vector field tangent to ¯ M. ¯ X is called invariant under the flow if ¯
X,u] = ∇ u [ ¯ u ¯ = 0.
X − ∇
X T
Denote ¯ X by the projection of the vector field ¯
X on M. So,
T
¯
X = X + h ¯X,uiu. Let X be a vector field tangent to M. Extend X along the flow by making it invariant under the flow generated by u. In this setting, we define the velocity and acceleration of X by the
2.6 CLOSURE THEOREM VIA GALLOWAY’S THEOREM
26 vector fields given respectively by
T T
u u u
∇ v(X) = ∇ X and a(X) = ∇ X .
The shape operator of M as a hypersurface of ¯ M u . Let denote
X
is defined by b(X) = −∇ by B the second fundamental form of b and by H its mean curvature function. We point out that H = −divu, that is, the averaged Hubble expansion parameter at points of M in relativistic cosmology (see page 161.).
The following lemma is an application of Gauss equation for spacelike submanifolds. Lemma 2.6.1.
Let γ(s) be a geodesic in a weighted manifold M f with unit tangent X. Then
2
2 Ric
(X, X) = Ric u
f f f u
(X, X) − ha(X),Xi + hv(X),XiH + hv(X),Xi + h∇ , Xi
2 n
d
1 ∂ ϕ
2 X,∇ u
 u ,
∑ j
i + hv(X),e i dsh ϕ ∂ s
j=2
and H = X, e , . . . , e M where {e
1 2 n p f
} is an orthonormal basis of T = H + hu,∇ f i denotes the f mean curvature of M .
f
Proof. (t) = ¯ X(t), ¯e (t), . . . , ¯e (t) Let { ¯e
1 2 n
} be a set invariant under the flow generated by u, where α = 1,...,n. Since [ ¯e , u] = 0 we get that
α T T
∇
T e
u = ∇ u u u , e
e α − h∇ α iu.
α
Therefore, a straightforward calculation shows that
2
and B(X,e e ), e , e ) .
α α α α α α
hv(e i = −B(e ) = hv(X),e i From the above identities and the Gauss equation we get
n
n o K(X,e B(X,e
Ric(X,X) = j j , e j )+ e j )
∑
)−B(X,X)B(e
j=2 n n
K(X,e B(X,e e =
)
∑ j ∑ j
)−B(X,X)(H − B(X,X))+
j=2 j=2 n
B(X,e e )
j
= Ric(X, X)−K(X,u)−B(X,X)(H − B(X,X))+ ∑
j=2 n
2
2
 j .
∑
= Ric(X, X)−K(X,u)+hv(X),XiH + hv(X),Xi hv(X),e i
j=2
2.6 CLOSURE THEOREM VIA GALLOWAY’S THEOREM
27 On the other hand, note that
T T
, u)u, X K(X,u) = −hR(X i
T T
∇ ∇
T T u , X T u , X u u
= h∇ u,Xi − h∇ i + h∇ i
X X [X ,u]
2
d
1 ∂ ϕ
2 X,∇ u u u u .
= ha(X),Xi − h∇ , Xi − i − dsh ϕ ∂ s Note that we may write
D E ∇ ∇ f = ∇ f + f ,u u , and, therefore, we have
Hess (X, X) = Hess
f f (X, X) − u( f )hv(X),Xi.
Thus, using the above equations we obtain the desired result. Now inspired by the ideas of Galloway we can find a diameter estimate M, under some restrictions.
n+1
(Theorem 4.5 . Let ¯ Theorem 2.6.1 M be a weighted space time with  f  ≤ a, where a
f n+1 n
is a positive constant. Let u be an irrotational unit timelike vector field on ¯ and let M M
f
be a complete spatial hypersurface orthogonal to u. Suppose that the following conditions hold
n
on M .
n
, the flow generated by u is expanding in all directions, i.e, (i) At each point p ∈ M hv(X),Xi ≥ 0, where X is a vector field tangent to M.
n
, the rate of expansion is decreasing in all directions, i.e, (ii) At each point p ∈ M ha(X),Xi ≤ 0, where X is a vector field tangent to M.
n+1
(iii) Ric of ¯ satisfies M
f
n − 1
2 H
inf(Ric ) = c > 0,
f
(ζ , ζ ) − 2n
2.6 CLOSURE THEOREM VIA GALLOWAY’S THEOREM
28 where ζ ∈ T p M, ζ = 1, hζ,ui = 0 and H = −div u.
n
(iv) The flow lines are of bounded geodesic curvature on M , i.e, u
u
sup ∇  = µ < ∞. n(v) On M , we have that D E
∇ f ,u = u( f ) ≥ 0.
n
Then M is compact and q π
n
2
(2.18) diam(M µ + µ + cλ , ) ≤ c
√ where λ = 2( 2a + 1). Proof.
, e . . . , e Let us assume that {X = e
1 2 n
} is an orthonormal basis of M. Consider the no
n
tation v v
), e . Then by Schwartz in
α α α α i
= hv(e i. From item (i), v ≥ 0. Note that H = ∑ i=1 equality
!
2 n n
1
2
2
v v
∑ i ∑
−(hv(X),Xi + hv(X),XiH) ≤ − i
2
i=1 i=1
!
2 n
n − 1 v
∑ i
≤ 2n
i=1
n − 1
2 H (2.19) = .
2n From and item (ii) we get dφ n − 1
2 H (2.20)
, Ric(X,X) ≥ Ric(X,X) −
 2n ds
D E X,∇ u where φ = u .
Then dφ n − 1
2 Ric Ric H (2.21) + + f f
(X, X) ≥ (X, X) − hB(X,X),uiu( f ).
2n ds
2.6 CLOSURE THEOREM VIA GALLOWAY’S THEOREM
29 Item (i) gives hB(X,X),ui ≤ 0. From (iii) and (v) we get dφ Ric .
f
(ζ , ζ ) ≥ c + dt Therefore by Theorem
Analogously to the previous theorem, we give a closure theorem for the kBakryÉmery Ricci tensor as follows.
n+1
Theorem 2.6.2 (Theorem 4.6 . Let ¯ M be a weighted space time. Let u be an irrotational
f n+1 n
unit timelike vector field on ¯ M and M be a complete spacelike hypersurface orthogonal
f n
to u. Suppose the following conditions hold on M .
n
, the flow generated by u is expanding in all directions, i.e, (i) At each point p ∈ M hv(X),Xi ≥ 0, where X is avector field tangent to M.
n
, the rate of expansion is decreasing in all directions, i.e, (ii) At each point p ∈ M ha(X),Xi ≤ 0, where X is a vector field tangent to M.
k n+1
(iii) Ric of ¯ M satisfies
f k
n − 1
2
inf(Ric H ) = c > 0,
f (ζ , ζ ) −
2n
p where k ∈ (0,∞), ζ ∈ T V, ζ = 1, hζ,ui = 0 and h = −divu. n
(iv) The flow lines are of bounded geodesic curvature on M , i.e,
u u
sup ∇  = µ < ∞.
n
(v) On M , we have that D E
∇ f ,u = u( f ) ≥ 0.
2.7 CLOSURE THEOREM VIA AMBROSE’S THEOREM
30
n
Then M is compact and q π
n
2
µ + µ + cλ , diam(M ) ≤ c
√ where λ = 2( 2a + 1).
2.7 Closure Theorem via Ambrose’s Theorem
n n+1
Let M be a spatial hypersurface in a weighted spacetime M . The unit tangent vectors to the
f n
future directed geodesics orthogonal to M define a smooth unit timelike vector field u at least
n
in a neighborhood of M Let X be a vector tangent to M. Extended X along the normal geodesic by making it invari ant under the flow generated by u. Following the ideas of Frankel and Galloway we obtain the following closure theorem.
n
Theorem 2.7.1 (Theorem 4.2 . Let M be a spatial hypersurface in a weighted spacetime
n+1 n
M such
( ¯ , f ). Let u be a unit normal vector field timelike to M. If there is a point q ∈ M
d f n
that along each geodesic γ of M emanating from q, we have ≤ 0 and
dt Z ∞
Ric dt = ∞, (2.22) (X, X) + H
f f
hv(X),Xi − ha(X),Xi
n where t is arc length along γ and X is the unit tangent to γ, then M is compact.
Proof. From Lemma we get
Z ∞
Ric (X, X)dt = ∞.
f n
From Theorem we conclude that M is compact.
Analogously, we get the following closure theorem.
n
Theorem 2.7.2 (Theorem 4.3 . Let M be a spatial hypersurface in a weighted space time
n n
¯ M such that along
f . Let u be a unit normal vector field timelike to M. If there is a point q ∈ M n
each geodesic γ of M emanating from q, we have
Z ∞
n o
k
Ric (X, X) + H dt = ∞, (2.23)
f
f hv(X),Xi − ha(X),Xi
n where t is arc length along γ and X is the unit tangent to γ, then M is compact.2.7 CLOSURE THEOREM VIA AMBROSE’S THEOREM
31 Remark 2.7.1. Conditions may be interpreted as the massenergy density on
n
M (see say that spatial hypersurfaces with massenergy density sufficiently large are compact.
Now, using the same steps of the above theorem and Theorem we get the following closure theorem.
n n+1
Theorem 2.7.3 (Theorem 1.3 . Let M be a spacelike hypersurface in ¯ M and assume
f
that M is complete in the induced metric. Then for any δ > 0, a > 0, there exists an ε = ε(n, a, δ ) satisfying the following:
If there is a point p ∈ M such that along each geodesic γ in M emanating from p, the condition
Z ∞
n o
k
max dt < ε(n,a,δ ),
f
(n − 1)a − Ric f (X, X)−hv(X),XiH +ha(X),Xi,0
π ′ q + δ .
is satisfied, where X = γ , then M is compact and diam(M) ≤
(n−1)a n+k−1 HAPTER
C
3 Bernstein type properties of complete
hypersurfaces in weighted semiRiemannian
manifolds
The results of this chapter are part of
3.1 Introduction
The last years have seen a steadily growing interest in the study of the Bernsteintype prop
n
P ,
ρ
erties concerning complete hypersurfaces immersed in a warped product of the type I ×
n
where P is a connected ndimensional oriented Riemannian manifold, I ⊂ R is an open inter val and ρ is a positive smooth function defined on I. In this context, an important question to
n
P , under reasonable restric pay attention is the uniqueness of complete hypersurfaces in I × ρ tions on their mean curvatures. Along this branch, we may cite, for instance, the papers A Lorentzian version of the problems discussed above, that was pursued by several authors more recently, was to treat the problem of uniqueness for complete constant mean curvature spacelike hypersurfaces of generalized RobertsonWalker (GRW) spacetimes, that is, Lorentzian warped products with 1dimensional negative definite
n
P . In this setting, we may cite, for instance, the base and Riemannian fiber of type −I × ρ works
The goal of the Chapter is to study the unicity of hypersurfaces Σ immersed in a semi
n
ρ
P .
Riemannian manifold of type εI ×
3.2 Uniqueness results in weighted warped products
It follows from a splitting theorem due to Fang, Li and Zhang (see Theorem 1.1) that if a
f
product manifold I × P with bounded weighted function f is such that Ric ≥ 0, then f must be constant along R. So, motivated by this result, along this section, we will consider weighted
n
products P × R whose weighted function f does not depend on the parameter t ∈ I, that is
n t
h∇ f ,∂ i = 0 and, for sake of simplicity, we will denote them by P × R.
f
In order to prove our Bernstein type theorems in weighted warped products of the type
3.2 UNIQUENESS RESULTS IN WEIGHTED WARPED PRODUCTS
33 P , we will need some auxiliary lemmas. The first one is an extension of Proposition 4.1
I × ρ f of
n n
P Lemma 3.2.1. Let Σ
, with be a hypersurface immersed in a weighted warped product I× ρ
f
height function h. Then,
2 ′
(i) ∆ h = (logρ) ) + nΘH ;
f f
(h)(n − ∇h
′
(ii) ∆ σ (h) = n(ρ (h) + ρ(h)ΘH );
f f R t where σ(t) = ρ(s)dt. t
Let us prove item (i). Taking into account that f is constant along R, from we get Proof. that
(3.1) h∇ f ,∇hi = −Θh∇ f ,Ni. On the other hand, from Proposition 4.1 of we have
2 ′
∆ h = (logρ (3.2) ) + nHΘ.
(h)(n − ∇h Hence, from we obtain
2 ′
∆ h = (logρ)
f
(h)(n − ∇h ) + nHΘ − h∇ f ,∇hi
2 ′
= (log ρ) (h)(n + ∇h ) + nHΘ + Θh∇ f ,Ni
2 ′
= (log ρ) ) + nΘH .
f
(h)(n + ∇h Moreover, since Proposition 4.1 of also gives that
′
∆ σ (h) = n(ρ (h) + ρ(h)ΘH), in a similar way we also prove item (ii).
1 n
Let us denote by L , in relation to weighted
(Σ) the space of the integrable functions on Σ
f f − f − f
. Since div volume element dµ = e dΣ (X) = e div(e X), it is not difficult to see that from
f Proposition 2.1 of n
Let u be a smooth function on a complete weighted Riemannian manifold Σ Lemma 3.2.2.
1 n
with weighted function f , such that ∆ u does not change sign on Σ (Σ), then
f . If ∇u ∈ L f n
∆ u vanishes identically on Σ .
f
3.2 UNIQUENESS RESULTS IN WEIGHTED WARPED PRODUCTS
34
n
P is a region of the type We recall that a slab of a warped product I × ρ
n n
P [t ,t : t
1 2 ρ
1
2 ] × P = {(t,q) ∈ I × ≤ t ≤ t }.
Now, we are in position to state and prove our first result.
n
(Theorem 1 . Let Σ be a complete twosided hypersurface which lies in Theorem 3.2.1
n
P satisfies a slab of a weighted warped product I× ρ . Suppose that Θ ≤ 0 and that H f
f ′
(3.3) 0 < H (log ρ) (h).
f
≤ inf
Σ
1 n(Σ), then Σ If ∇h ∈ L is a slice {t} × P.
f Proof.
From Lemma since we are assuming that −1 ≤ Θ ≤ 0, we get
′
∆ σ (h) = nρ(h) (log ρ) (h) + ΘH
f f ′
(3.4)
f
≥ nρ(h) (logρ) (h) − H
′ inf (log ρ) . f
≥ nρ(h) (h) − H
Σ
Thus, taking into account our hypothesis we have that ∆
f σ (h) ≥ 0. n n
P On the other hand, since Σ , we have that exists a positive constant
ρ
lies in a slab of I× C such that ∇σ(h) = ρ(h)∇h ≤ C∇h.
1
1 (Σ).
Consequently, our hypothesis ∇h ∈ L (Σ) implies that ∇σ(h) ∈ L
f f n
So, we can apply Lemma σ (h) = 0 on Σ
f
we get
′
(3.5) (log ρ) .
f
(h) = −ΘH Consequently, using once more hypothesis we have that
′ ′
H (3.6) (log ρ) .
f f
≤ inf (h) ≤ (logρ) (h) = −ΘH
Σ n
Therefore, from we conclude that Θ = −1 and, hence, Σ must be a slice {t}×P.
n n
Remark 3.2.1. Concerning Theorem we note that if Σ is locally a graph over P , then its angle function Θ is either Θ < 0 or Θ > 0 along Σ. Hence, the assumption that Θ does
n
not change sign is generally weaker than that of Σ being a local graph. Moreover, as it was
3.2 UNIQUENESS RESULTS IN WEIGHTED WARPED PRODUCTS
35 already observed by Espinar and Rosenberg when they made allusion to immersions into the Euclidean space, the condition that Θ does not change sign can also be regarded as the image of the Gauss map of the hypersurface lying in a closed hemisphere of the Euclidean sphere.
On the other hand, from Proposition 1 of , if we consider on the slice P
, then its f mean curvature is given by
ρ f t
{t} × P of I × the orientation given by N = −∂
′ H (t) = H(t) = (log ρ) (t). f
Consequently, the differential inequality means that, at each point (t,x) of the hypersur
n n
face Σ , the weighted mean curvature of Σ can be any value less than or equal to the value
n
, with respect to the orientation given by of the weighted mean curvature of the slice {t} × P . Hence, we only suppose here a natural comparison inequality between weighted mean
t
−∂
n
curvature quantities, without to require that the weighted mean curvature of Σ be constant. In this sense, is a mild hypothesis.
According to the classical terminology in linear potential theory, a weighted manifold Σ with weighted function f is said to be f parabolic if every bounded solution of ∆
f
u ≥ 0 must be identically constant. So, from Theorem we obtain
n
Let Σ be a complete twosided hypersurface which lies in a slab of a weighted Corollary 3.2.1.
n
P satisfies warped product I× ρ . Suppose that Θ ≤ 0 and that H f
f ′
0 < H (log ρ) (h).
f
≤ inf
Σ
n nIf Σ is f parabolic, then Σ is a slice {t} × P. Remark 3.2.2.
As it was observed by Impera and Rimoldi in Remark 3.8 of the f 
n n
parabolicity of Σ holds if it has finite f volume. On the other hand, in the case that Σ is complete noncompact with Ric nonnegative, from Theorem 1.1 of a sufficient condition
f n 2 for Σ to have finite f volume is that the space of L harmonic oneforms be nontrivial. f
From the proof of Theorem we also get the following
n
Corollary 3.2.2. Let Σ be a complete twosided hypersurface which lies in a slab of a weighted
n
P is constant and satisfies warped product I× ρ . Suppose that Θ ≤ 0 and that H f
f ′
(log ρ) (h). 0 ≤ H f
≤ inf
Σ
3.2 UNIQUENESS RESULTS IN WEIGHTED WARPED PRODUCTS
36
1 n n
is f parabolic, then Σ is either a f minimal hypersurface or a (Σ) or Σ
If either ∇h ∈ L
f slice {t} × P.
Extending ideas of in our next result we will assume that the ambient
n
ρ
space is a weighted warped product I×
P which obeys the following convergence condition
f
2 _{P} ′ ′′
K ((ρ ) ), (3.7) ≥ sup − ρρ
I n
where K P stands for the sectional curvature of the fiber P .
n
P Theorem 3.2.2 be a weighted warped product which satisfies
(Theorem 2 . Let I × ρ
f
nthe convergence condition Let Σ be a complete twosided hypersurface which lies in a
n n
P . Suppose that ∇ f and the Weingarten operator A of Σ are bounded. If Θ does slab of I × ρ
f n
not change sign on Σ and
′
(log ρ) , (3.8)
f
∇h ≤ inf (h) − H 
Σ n
then Σ is a slice {t} × P.
From Lemma we get Proof.
2 ′
∆ Θ h = (logρ) ) + nH
f f
(h)(n − ∇h
2 ′
(3.9)
f
≥ (logρ) (h)(n − ∇h ) − nH 
2 ′ ′
.
f
≥ n((logρ) (h) − H ) − (logρ) (h)∇h
2
2 On the other hand, since from the CauchySchwarz inequality we get nH , we have
≤ A
n n
ρ
lies in a slice of I ×
P that H is also bounded. Thus, since we are also assuming that Σ , we
f n
is bounded from below. can apply Proposition 3.1 of to assure that the Ricci curvature of Σ
n
, from maximum principle of OmoriYau, there exists a Hence, since ∇ f  is bounded on Σ
n
such that
k
sequence {p } in Σ ∆ lim h(p h, lim h(p
) = sup
k k f k
∇h(p ) = 0 and limsup ) ≤ 0. k k Σ kThus, from inequality we get
′
sup ∆ h(p ((log ρ) 0 ≥ lim f k f k
) ≥ lim (h) − H )(p ) ≥ 0.
k k
3.3 RIGIDITY RESULTS IN WEIGHTED PRODUCT SPACES
37
′
Therefore, we have that lim ((log ρ) ) = 0 and, taking into account our hypoth
k f k
(h) − H )(p
n
esis we conclude that Σ is a slice {t} × P.
3.3 Rigidity results in weighted product spaces
In this section, we will treat the special case when the ambient space is a weighted product
n . We start with the following uniqueness result.
I× P
f n
Theorem 3.3.1 (Theorem 3 . Let Σ be a complete twosided hypersurface which lies in
n n
. Suppose that Θ and H do not change sign on Σ . If
f
a slab of the weighted product I × P
f 1 n
(Σ), then Σ ∇h ∈ L is a slice {t} × P.
f
Proof. From Lemma we have that ∆ h = nH (3.10)
Θ.
f f
nSince we are supposing that Θ and H do not change sign on Σ , we get that ∆ h also does not
f f n
1
change sign on Σ h = 0
(Σ), we can apply Lemma to get that ∆ . Thus, since ∇h ∈ L
f f n on Σ .
Consequently, we obtain
2
2
2
∆ h = 2h∆ (3.11)
f f h + 2∇h = 2∇h ≥ 0.
1
(Σ), Lemma guarantees But, since h is bounded and using once more that ∇h ∈ L
f 2 n n
also that ∆ h . Therefore, returning to we conclude that Σ must be a slice = 0 on Σ
f {t} × P.
From the proof of Theorem we get
n
Corollary 3.3.1. Let Σ be a complete twosided hypersurface which lies in a slab of the
n n n
. Suppose that Θ and H do not change sign on Σ . If Σ is f parabolic, weighted product I ×P f
f n
then Σ is a slice {t} × P.
n n
. Recall that G corresponds to the Euclidean Consider the weighted product space R × G
x2 n
2 −
4
space R . Hieu and Nam extended the endowed with the Gaussian measure dµ = e dx classical Bernstein’s theorem showing that the only graphs with weighted mean curvature
n
over G are the hyperplanes x identically zero given by a function u(x , . . . , x ) = x =
2 n+1
1
1
3.3 RIGIDITY RESULTS IN WEIGHTED PRODUCT SPACES
38 constant (cf. Theorem 4). In this setting, with a straightforward computation we can verify that
1 N = ) (3.12)
t
p (Du − ∂ 1 + Du
n
gives an orientation on Σ (u) such that −1 ≤ Θ < 0, where Du stands for the gradient of the
n ∗
function u on G . Thus, since N
 = ∇h, from we deduce that
2
2 Du
(3.13) = . ∇h
2
1 + Du So, taking into account relation we obtain
n
Let Σ over the
Corollary 3.3.2. (u) be a complete graph of a function u(x , . . . , x ) = x
2 n+1
1 n n
Gaussian space G . Suppose that the weighted mean curvature of Σ (u) does change sign. If
1 n
(G), then Σ (u) is a hyperplane x = constant.
1
Du ∈ L To prove our next result, we will need of the following auxiliary lemma.
n
Lemma 3.3.1. Let Σ be a hypersurface with constant f mean curvature in a weighted product
n
manifold P × R. Then
f
2 ∗ ∗
∆
 f Ric (N , N ))Θ, (3.14)
f f
Θ = −(A Ric stands for the BakryÉmeryRicci tensor where A denotes the Weingarten operator of Σ, f f
∗
of the fiber P and N t is the orthonormal projection of N onto P.
= N − Θ∂ Proof. It is well known that
T T
(3.15) ∂ )
N t
∇Θ = −A∂ t − (∇ and
T
2 ∗ ∗
(3.16) Ric(N , N ),
∆Θ = −n∂ t (H) − Θ( f ) + A , ∇ f see for instance Taking into account that h∂ t i = 0 we get that
T T
n∂ (H) = ∂ (nH
f t t − h∇ f ,Ni) T
= −∂ t h∇ f ,Ni
T
(3.17) (∂ t
f f
= −hHess ), Ni + ΘHess (N, N) + hA∂ t , ∇ f i.3.3 RIGIDITY RESULTS IN WEIGHTED PRODUCT SPACES
39 On the other hand, from we get that
T T
 (∇ ∂ )
N t
h∇Θ,∇ f i = −hA∂ , ∇ f i
t T
, ∇ f ∂ ∂
N t N t
= −hA∂ t i − h∇ − h∇ , NiN,∇ f i
T
∇ , ∇ f , ∇ (3.18)
t N
= −hA∂ t i + h∂ f i
T
, ∇ f , Hess
t f = −hA∂ t i + h∂ (N)i.
From we have
T
n∂ (3.19)
(H) = ΘHess
f
t (N, N) − h∇Θ,∇ f i. t , ∇ fNow, taking into account once more that h∂ i = 0, it is not difficult to verify that
∗ ∗
Hess (N, N) = g Hess (N , N ). (3.20)
f f Putting we have the desired result. n
Theorem 3.3.2 (Theorem 4 . Let Σ be a complete twosided hypersurface immersed in
n
. Suppose that f Ric does a weighted product I × P f f
≥ 0, A is bounded, Θ has strict sign H
f n 1 n
change sign on Σ is totally geodesic. Moreover, if f Ric (Σ), then Σ > 0, then
. If ∇h ∈ L
f f n
Σ is a slice {t} × P.
n
Since we are supposing that Θ has strict sign, H does not change sign on Σ and that Proof.
f
1
(Σ), from and applying Lemma we get that Σ is f minimal, that is, H = 0 on
f
∇h ∈ L
f n
Σ .
Note that
1
(3.21) (Σ). ∇Θ ≤ A∇h ∈ L f
So, since we are also supposing that f Ric Θ = 0 on
f f
≥ 0, from we have that ∆
n
Σ .
n
Hence, returning to equation we conclude that Σ is totally geodesic. Furthermore,
n n ∗
if f Ric is strictly positive, then N and, therefore, Σ = 0 on Σ
f is a slice {t} × P.
When the fiber of the weighted product space is compact, we have
n
(Theorem 5 . Let Σ be a complete twosided hypersurface immersed in a Theorem 3.3.3
n n
, whose fiber P is compact with positive sectional curvature, and such weighted product I ×P
f
that the weighted function f is convex. Suppose that A is bounded and H is constant. If Θ is
f
3π π
n n
or . such that either 0 ≤ arccosΘ ≤ arccos Θ ≤ π, then Σ is a slice {t} × P 4 4 ≤
3.3 RIGIDITY RESULTS IN WEIGHTED PRODUCT SPACES
40
n Proof. Firstly, we claim that the BakryÉmery Ricci tensor of Σ , Ric , is bounded from below. f n
Indeed, since we are assuming that P is compact with positive sectional curvature, from Gauss equation it follows that
2
2
(3.22) Ric(X,X) ≥ (n − 1)κ 1 − ∇h X + nHhAX,Xi − hAX,AXi, for all X ∈ X(Σ) and some positive constant κ = κ(X).
On the other hand, taking into account that f is convex and constant along R, we have (3.23)
Hess f (X,X) = Hess f (X,X) + h∇ f ,NihAX,Xi
∗ ∗
g = Hess f (X , X
) + h∇ f ,NihAX,Xi ≥ h∇ f ,NihAX,Xi, for all X ∈ X(Σ).
Thus, from we get
2
2 Ric
(3.24)
 nH
f f (X, X) ≥ (n − 1)κ 1 − ∇h X hAX,Xi − hAX,AXi.
Consequently, from we obtain
2
2
2 Ric
(3.25) ,
f f
(X, X) ≥ ((n − 1)κ 1 − ∇h − (nH A + A ))X for all X ∈ X(Σ).
1
2
. Hence, since A is also bounded We also note that our restriction on Θ amounts to ∇h ≤
2 and H is constant, from we conclude that Ric is bounded from below.
f f
On the other hand, using once more that H is constant, from Lemma we have
f
∇∆ (3.26) h = nH ∇Θ.
f f
Recall that (3.27) ∇Θ = −A(∇h).
Consequently, from we get ∇∆
A(∇h). (3.28)
f h = −nH f
3.3 RIGIDITY RESULTS IN WEIGHTED PRODUCT SPACES
41 From we have that
⊤
∇ ∇ h = ∇ (3.29) (∂ ) = AXΘ.
X X t
Thus, from we obtain
2
2
2
Θ (3.30) .
Hessh = A Consequently, from we get
2
2
2
2
. (3.31) Hessh = A − ∇h A
Now, from Bochner’s formula we also have that
1
2
2
∆ (3.32)
 Ric
f f f ∇h = Hessh (∇h, ∇h) + h∇∆ h,∇hi.
2 But, from (3.24) it follows that
2
2 Ric
 nH
f f
(∇h, ∇h) ≥ (n − 1)κ(1 − ∇h )∇h hA∇h,∇hi−hA∇h,A∇hi
2
2
2
2
(3.33) + nH .
f
≥ (n − 1)κ(1 − ∇h )∇h hA∇h,∇hi − A ∇h Hence, considering we get
1
2
2
2
2
2
∆ (3.34) ).
f
∇h ≥ (n − 1)κ(1 − ∇h )∇h + A (1 − 2∇h
2 Consequently, from (1.5) and (3.34) jointly with our hypothesis on Θ we obtain
1
2
2
2
∆ (3.35) .
f
∇h ≥ (n − 1)κ(1 − ∇h )∇h
2
n
Since Ric is bounded from below on Σ , from Theorem we have that there exists a
f n
in Σ such that sequence of points (p )
k k≥1
2
2
2
lim and lim sup ∆ (p ) = sup (p
k f k ∇h ∇h ∇h ) ≤ 0. k k Σ
Thus, from we have
2
2
2
sup ∆ (3.36)
(p ) sup 0 ≥ lim f k ∇h ) ≥ (n − 1)κ(1 − sup ∇h ∇h ≥ 0.
k Σ Σ
Consequently, taking into account once more our hypothesis on Θ, from we conclude
2 n n n
that sup and, hence, Σ .
= 0. Therefore, h is constant on Σ
Σ
is a slice {t} × P ∇h
3.3 RIGIDITY RESULTS IN WEIGHTED PRODUCT SPACES
42
n
Corollary 3.3.3. Let Σ be a f parabolic complete twosided hypersurface immersed in a
n n
, whose fiber P is complete with nonnegative sectional curvature, weighted product I × P
f and such that the weighted function f is convex. Suppose that A is bounded and H is constant. f
It holds the following: π 3π
n or is totally geodesic.
(a) If either 0 ≤ arccosΘ < < arccos Θ ≤ π, then Σ
4
4 3π
π
n
(b) If K P or is a slice is positive and either 0 ≤ arccosΘ ≤ arccos Θ ≤ π, then Σ 4 4 ≤
n .
{t} × P Proof. In a very similar way of that was made in order to prove inequality taking a local
1 n
} on Σ, we obtain that
, . . . , E orthonormal frame {E
1
2
2
2
2
2 ∗ ∗
∆ _{P} (3.37) K ((∇h) , E ).
f
∇h ≥ (n − 1)min i )(1 − ∇h )∇h + A (1 − 2∇h
i
2 P
If we assume the hypothesis of item (a), since we are also supposing that K is nonnegative,
2 n
from we get that ∆ is f parabolic,
f
∇h ≥ 0. Hence, since we are also assuming that Σ we have that ∇h is constant on Σ. Therefore, returning to we conclude that A = 0 on Σ , that is, Σ is totally geodesic. _{P} ∗ ∗
K ((∇h) , E ) > 0, from Now, we assume the hypothesis of item (b). Since κ := min i
i
we also get that
1
2
2
2
∆ (3.38)
f ∇h ≥ (n − 1)κ(1 − ∇h )∇h ≥ 0.
2
n
Hence, using once more that Σ is f parabolic and noting that ∇h < 1, from we con
n n n , that is, Σ .
clude that ∇h = 0 on Σ must be a slice {t} × P Proceeding, we obtain an extension of Theorem 3.1 of
n
Theorem 3.3.4 (Theorem 6 . Let Σ be a complete twosided hypersurface immersed in
n n _{P P}
, whose fiber P has sectional curvature K satisfying K a weighted product I × P ≥ −κ and
f
such that g Hess f ≥ −γ, for some positive constants κ and γ. Suppose that A is bounded, Θ is
n
bounded away from zero and H is constant. If the height function h of Σ satisfies
f
α
2
2
(3.39) ,
∇h ≤ A (n − 1)κ + γ
n
for some constant 0 < α < 1, then Σ is a slice {t} × P. Proof. Since we are assuming that Θ is bounded away from zero, we can suppose that Θ > 0 and, consequently, inf Θ > 0.
3.3 RIGIDITY RESULTS IN WEIGHTED PRODUCT SPACES
is bounded from below on Σ
(3.45) = h∇ f ,NihAX,Xi − γ(X
2

∗
. Indeed, following similar ideas of that in the proof of Theorem we get Hess f (X,X) ≥ h∇ f ,NihAX,Xi − γX
n
f
− hX,∂
(3.44) Now, we claim that Ric
2 Θ.
Θ ≤ −(1 − α)A
f
∆
)Θ. (3.43) Consequently, from we have
2
2
t
2
2
is bounded from below. Now, we are in position to apply Theorem and guarantee the existence of a sequence
f
is constant, from we conclude that Ric
f
(3.46) Thus, since A and ∇h are bounded and H
f hAX,Xi − hAX,AXi.
− γ)X
i
2
(X, X) ≥ ((n − 1)κ 1 − ∇h
f
From we obtain Ric
2 .
) ≥ h∇ f ,NihAX,Xi − γX
2
− ((n − 1)κ + γ)∇h
Θ ≤ −(A
43 Moreover, since we are also assuming that the sectional curvature K ^{P} of the base P

∗
Hess f (N
(3.40) On the other hand, our restriction on g Hess f jointly with equation give g
2 .
= −(n − 1)κ∇h
2
∗
∗
) ≥ −(n − 1)κN
∗
, N
∗
Ric(N
is such that K ^{P} ≥ −κ for some κ > 0, from Gauss equation and with a straightforward computation we get f
n
, N
) ≥ −γN
f
∗
∆
(3.42) Hence, from we obtain
2 .
) ≥ −((n − 1)κ + γ)∇h
∗
, N
(N
∗
f
Ric f
. (3.41) Thus, from we get
2
= −γ∇h
2

 nH
3.3 RIGIDITY RESULTS IN WEIGHTED PRODUCT SPACES
44
n
of points p satisfying
k
∈ Σ ∆ lim inf Θ(p Θ(p ) = inf Θ(p).
f k ) ≥ 0 and lim k k→∞ k→∞ p∈Σ n
Consequently, since we are assuming that A is bounded on Σ , from up to a subsequence, we get
2
∆ Θ(p (p ) inf (3.47)
f 0 ≤ liminf k ) ≤ −(1 − α) lim A k Θ(p) ≤ 0. k→∞ k→∞ p∈Σ
Thus, from we get A(p k
k→∞
lim ) = 0. (3.48)
k
∇h(p
k→∞
Therefore, from we conclude that inf
p∈Σ Θ(p) = 1 and, hence, Θ ≡ 1, that is, n
Σ is a slice {t} × P.
To close our section, we will apply a Liouville type result due to Huang et al. in order to prove the following:
n
(Theorem 7 . Let Σ be a complete twosided hypersurface which lies in Theorem 3.3.5
n n
, whose fiber P has sectional curvature K P bounded from a slab of a weighted product I × P
f n
below and such that ∇ f is bounded. Suppose that A and H are bounded on Σ . If Θ is not
f
H is constant and Ric is nonnegative, then
Σ = 0. Moreover, if H
adhere to 1 or −1, then inf f f f
n
Σ is a slice {t} × P. Proof. Following similar steps of the proof of Theorem our restriction on the sectional
n
curvature of the fiber P jointly with our hypothesis on A, H and ∇ f guarantee that the Ricci
f n
curvature of Σ is bounded from below.
n n
Now, suppose for instance that H . Thus, since Σ lies between two slices of
f ≥ 0 on Σ n
R , from Lemma and maximum principle of OmoriYau, we obtain a sequence of
× P
n
points p such that
k ∈ Σ
∆ Θ h(p ) = n lim sup H (p ). 0 ≥ limsup f k f k
k→∞ k→∞
Moreover, from equation we also have that
2
k k
∇h(p ) = 1 − lim
Θ 0 = lim (p ).
k→∞ k→∞
3.4 UNIQUENESS RESULTS IN WEIGHTED GRW SPACETIMES
45 Thus, if we suppose, for instance, that Θ is not adhere to −1, we get lim Θ(p ) = 1.
k k→∞
Consequently, ∆ h(p ) = n lim sup H (p
f k f k
0 ≥ limsup ) ≥ 0
k→∞ k→∞
and, hence, we conclude that lim sup H (p ) = 0.
f k k→∞
If H
f ≤ 0, from Lemma we can apply once more OmoriYau’s generalized n
maximum principle in order to obtain a sequence q such that
k ∈ Σ
∆ Θ h(q ) = n lim inf H (q )
f k f k
0 ≤ liminf
k→∞ k→∞
and, supposing once more that Θ is not adhere to −1, we get ∆ h(p H
) = n lim inf (p 0 ≤ liminf f k f k ) ≤ 0.
k→∞ k→∞
Consequently, from the above inequality we have that lim inf H (p ) = 0. Hence, in this
f k k→∞
case, we also conclude that inf H Σ = 0.
f n n
When H is constant, we have that, in fact, H vanishes identically on Σ . Thus, since Σ is
f f n
contained in a slab of R × P , there exists a constant β such that h − β is a positive harmonic
f n
function of the f Laplacian on Σ .
∗ ∗
On the other hand, since ∇ f = ∇ f − h∇ f ,N iN and N  = ∇h, we obtain
2
2
2 ).
∇ f  ≤ ∇ f  (1 − ∇h Consequently, since we are assuming that ∇ f is bounded, we have that ∇ f is also bounded on
n
Σ . Hence, if Ric is nonnegative, then we can apply Corollary 1.4 of to conclude that h
f n n n
is constant on Σ , that is, Σ . is a slice {t} × P
3.4 Uniqueness results in weighted GRW spacetimes
We observe that it follows from a splitting theorem due to Case (cf. Theorem 1.2) that a weighted GRW spacetime whose weight function f is bounded and such that Ric
f
(V,V ) ≥ 0
3.4 UNIQUENESS RESULTS IN WEIGHTED GRW SPACETIMES
46 for all timelike vector field V , then f must be constant along R. So, motivated by this result, P whose weight function along this work we will consider weighted GRW spacetimes −I × ρ f
t
does not depend on the parameter t ∈ I, that is h∇ f ,∂ i = 0. For simplicity, we will denote P . them by −I × ρ f
In order to prove our CalabiBernstein’s type results in weighted GRW spacetimes of the P
, we will need some auxiliary lemmas. The first one is an extension of Lemma
ρ f
type −I × 4.1 of
n
Let Σ be a spacelike hypersurface immersed in a weighted GRW spacetime Lemma 3.4.1. P
, with height function h. Then,
ρ f
−I×
2 ′
Θ (i) ∆ ;
f f
h = −(logρ) (h)(n + ∇h ) − nH
′
(ii) ∆ (h) + ρ(h)ΘH ),
f f
σ (h) = −n(ρ
R t ρ(s)ds.
where σ(t) =
t n n
P : t
In what follows, a slab [t ,t
1 2 ρ
1
2
] × M = {(t,q) ∈ −I × ≤ t ≤ t } is called a timelike
n
P bounded region . Now, we are in position to state of the weighted GRW spacetime −I × ρ
f and prove our first result. n
Theorem 3.4.1 (Theorem 1 . Let Σ be a complete spacelike hypersurface which lies in
n
P . Suppose that the f mean
ρ
a timelike bounded region of a weighted GRW spacetime −I ×
f n
curvature H of Σ satisfies the following inequality
f ′
H (3.49) (log ρ) (h) > 0.
f
≥ sup
Σ 1 n
f R t
(Σ), then Σ If ∇h ∈ L is a slice {t} × P.
Proof. ρ(s)ds, we get From Lemma for σ(t) =
t ′
∆ (h) + ΘH (3.50)
f f
σ (h) = −nρ(h) (logρ)
′
(h)
f
≥ nρ(h) H − (logρ)
′
H (log ρ) (h) .
f
≥ nρ(h) − sup
Σ n
Thus, taking into account our hypothesis we have that ∆ .
f
σ (h) ≥ 0 on Σ
n n
P On the other hand, since Σ , we have that exists a positive
ρ
is contained in a slab of −I ×
f
constant C such that ∇σ(h) = ρ(h)∇h ≤ C∇h.
3.4 UNIQUENESS RESULTS IN WEIGHTED GRW SPACETIMES
47
1
1 (Σ).
Consequently, the hypothesis ∇h ∈ L
f (Σ) implies that ∇σ(h) ∈ L f n
So, we can apply Lemma σ (h) = 0 on Σ
f
we get
′ (log ρ) . f
(h) = −ΘH Consequently, we have that
′ ′ H = sup (log ρ) .
f f
(h) ≥ (logρ) (h) = −ΘH
Σ n n
and, hence, we conclude that Σ Therefore, Θ = −1 on Σ is a slice {t} × P. From the proof of Theorem we also get the following.
n
Corollary 3.4.1. Let Σ be a complete spacelike hypersurface immersed in a timelike bounded
n n
P . Suppose that Σ has constant f mean curvature
ρ
region of a weighted GRW spacetime −I×
f
H and that it holds the following inequality
f ′
H (log ρ)
f ≥ sup (h) ≥ 0.
Σ 1 n
f
(Σ), then Σ If ∇h ∈ L is either a f maximal hypersurface or a slice {t} × P.
According to the terminology established by Alías and Colares in we say that a GRW
n
P obeys the strong null convergence condition when the sectional curvature spacetime −I × ρ K P of its Riemannian fiber P satisfies the following inequality
2
_{P} ′′K (3.51) ( f (log f ) ). ≥ sup
I
n
nP Theorem 3.4.2 be a GRW spacetime obeying . Let Σ
ρ
(Theorem 2 . Let −I × be a complete spacelike hypersurface which lies in a timelike bounded region of the weighted
n n
P and that the f mean curvature GRW spacetime −I × ρ . Suppose that ∇ f  is bounded on Σ
f n
of Σ satisfies H
f ′
(3.52) (log ρ)
f
(h) ≤ H ≤ α, for some constant α. If
′
H (3.53)
(h) ,
f
∇h ≤ inf − (logρ)
Σ n
then Σ is a slice {t} × P.
3.5 WEIGHTED STATIC GRW SPACETIMES
48 Proof. From Lemma we have
2 ′
∆ Θ.
f h = −(logρ) (h)(n + ∇h ) − nH f
Since N is a futuredirected timelike vector field, we get
2 ′ ′
∆ (3.54) .
f f
h ≥ n(H − (logρ) (h)) − (logρ) (h)∇h
n
We claim that the mean curvature of Σ is bounded. Indeed, we have that nH = nH f  + h∇ f ,Ni
∗
(3.55)
f
= nH  + h∇ f ,N i. ∗ ∗On the other hand, taking into account that N = N + Θ∂
t , we easily verify that N  = ∇h.
Thus, from we get (3.56)
f
nH ≤ nH  + ∇ f ∇h.Consequently, since H , ∇ f and ∇h are supposed to be bounded, it follows from that H
f n
is also bounded on Σ .
Hence, since we are assuming that ∇ f is bounded then we can apply Proposition 3.1 of jointly with generalized maximum principle of Omori and Yau to guarantee that there exists a
n
such that sequence {p k } in Σ
) = sup
∆ lim h(p h, lim h(p
k k f k
∇h(p ) = 0 and limsup ) ≤ 0. k k Σ kThus, from inequality we get
′
sup ∆ h(p (H (h))(p
f k f k 0 ≥ lim ) ≥ lim − (logρ) ) ≥ 0. k k ′
Therefore, we have that lim (H (h))(p ) = 0 and, taking into account our hypothesis
k f k
−(logρ) we conclude the proof.
3.5 Weighted static GRW spacetimes
Along this section, we treat the case when the ambient space is a static GRW spacetime, that is, its warping function is constant, which, without loss of generality, can be supposed equal to
3.5 WEIGHTED STATIC GRW SPACETIMES
49
1. In this context, we will need of the following formula
n
Lemma 3.5.1. Let Σ be a spacelike hypersurface with constant f mean curvature H in a
f
. Then,
f
weighted static GRW spacetime −I × P
2 ∗ ∗
∆ Ric (3.57)
 f (N , N ))Θ,
f f
Θ = (A
n
where A denotes the Weingarten operator of Σ with respect to the futurepointing Gauss map
n N of Σ and f Ric stands for the BakryÉmery Ricci tensor of the fiber P. f
The proof of Lemma and, therefore, it will be omitted. Now, we can return to our uniqueness results.
n
Theorem 3.5.1 (Theorem 3 . Let Σ be a complete spacelike hypersurface in a weighted , such that its f mean curvature H does not change sign. If
f f
static GRW spacetime −I × P
1 n
is f maximal. In addition, we also have the following: (Σ), then Σ
∇h ∈ L
f n n
(i) If A is bounded and f Ric is totally geodesic. Moreover, if f Ric > 0, then Σ
f f
≥ 0, then Σ is a slice {t} × P.
n n
(ii) If Σ , then Σ lies in a timelike bounded region of −I × P f is a slice {t} × P.
Proof. From Lemma we have ∆
(3.58)
t
f h = −nH f hN,∂ i.
nSince we are supposing that H does not change sign on Σ , from equation we get that
f n
1
∆ h also does not change sign on Σ (Σ), we can apply Lemma
f
. Thus, since ∇h ∈ L
f
n
to conclude that ∆ vanishes identically on Σ . Hence, returning to equation we obtain h
f n
that Σ is f maximal.
Assuming the hypothesis of item (i), we have
1 (Σ).
∇Θ ≤ A∇h ∈ L f
n
So, if f Ric Θ = 0 on Σ .
f ≥ 0, then from Lemmas we get that ∆ f n
Hence, from
f n n ∗
also gives that N = 0 on Σ , that is, Σ is a slice {t} × P.
n
On the other hand, since Σ is f maximal, we have
2
2
2
∆ h = 2h∆ .
f f
h + 2∇h = 2∇h
3.5 WEIGHTED STATIC GRW SPACETIMES
50
1 n
, (Σ) and assuming that Σ
Now, since ∇h ∈ L lies in a timelike bounded region of −I × P f
f 2 n n
then from Lemma we get that ∆ h ,
= 0 on Σ
f . Hence, we obtain that ∇h = 0 on Σ which proves item (ii).
When the fiber of the ambient spacetime is compact, we have
n
Theorem 3.5.2 be a static weighted GRW spacetime, whose (Theorem 4 . Let −R × P
f
nfiber P is compact with positive sectional curvature and such that the weighted function f is
n
convex. Let Σ be a complete spacelike hypersurface with constant f mean curvature H in
f n n
. If ∇h is bounded, then Σ is a slice {t} × P. −R × P
f n
Using the fact that P is compact with K P Proof.
> 0, it follows from inequalities (3.3) and (3.4) of that there exists a positive constant κ such that
2
2
2
2 Ric(X,X) ≥ κ (n − 1)X
 ∇h X + (n − 2)hX,∇hi
2
(3.59) .
 nHhAX,Xi + AX Since we are supposing that the weighted function f is convex and taking into account that
Hess f (X,X) = Hess f (X,X) − h∇ f ,NihAX,Xi, we have (3.60) Hess f (X,X) ≥ −h∇ f ,NihAX,Xi. From we get the following lower bound for Ric
f
2
2
2
2 Ric f
(X, X) ≥ κ (n − 1)X + ∇h X + (n − 2)hX,∇hi
2
 nH . (3.61)
f hAX,Xi + AX
Inequality provides us
2
2 Ric
)
f
(∇h, ∇h) ≥ (n − 1)κ∇h (1 + ∇h
2
(3.62) +nH .
f
hA(∇h),∇hi + A(∇h) Since H is constant, we have
f
∇∆ A(∇h). (3.63)
f f
h = −nH
3.5 WEIGHTED STATIC GRW SPACETIMES
51 Moreover, from Bochner’s formula we get
1
2
2
2
∆ (3.64) ).
f
∇h ≥ (n − 1)κ∇h (1 + ∇h
2 Now, we observe that we can write
2
2 2 n H
nH
f f
2
2
nH X (3.65) = .
f
X hAX,Xi + AX − AX + 2 4 
Thus, from we obtain that
2
2
n H
f
2 Ric
(3.66) ,
f
(X, X) ≥ − X 4  for all X ∈ X(Σ).
n
Consequently, from equation we have that there exists a sequence of
n
in Σ such that points (p )
k k≥1
2
2
2
lim (p ) = sup and lim sup ∆ (p
f ∇h k ∇h ∇h k ) ≤ 0. k k Σ
Now, returning to we conclude that
2
2
sup ∆ (p 0 ≥ lim f k
∇h ) ≥ κ sup ∇h ≥ 0.
k Σ 2 n n
Consequently, we obtain that sup . Therefore, Σ is a
= 0 and, hence, h is constant on Σ
Σ
∇h
n .
slice {t} × P Proceeding, we obtain the following
n
Theorem 3.5.3 be a weighted static GRW spacetime, whose (Theorem 5 . Let −R × P
f
_{P P} nsectional curvature K of its fiber P is such that K ≥ −κ for some positive constant κ and
n
such that the weighted function f is convex. Let Σ be a complete spacelike hypersurface with
n
constant f mean curvature H and with bounded second fundamental form A. If
f
in −R × P
f
α
2
2
(3.67) ,
∇h ≤ A κ(n − 1)
n n .
for some constant 0 < α < 1, then Σ is a slice {t} × P
3.5 WEIGHTED STATIC GRW SPACETIMES
52 Proof. From equation we get inf Θ(p) exists and it is negative.
p∈Σ ∗
On the other hand, since the weighted function f is convex, N  = ∇h and taking a local
n
, we have that
1 , . . . , E n
orthonormal frame {E } on P
∗ ∗ ∗ ∗
f Ric (N , N , N )
f Ric(N
) ≥ f _{P P} ∗ ∗ = (N , E )N , E
i i ∑ hR i i
2 _{P P} ∗ ∗ ∗ ∗
K P (3.68) = (N , E , N , E
∑ i i
) hN i − hN i
i
2 ∗ ∗ ∗ _{P}
, N , E P
i
≥ −κ ∑ hN i − hN i
i
2
, = −κ(n − 1)∇h _{P} n where we also have used our restriction on the sectional curvature K of P .
Thus, from equation we get
2
2
∆ Θ
f
Θ ≤ A − κ(n − 1)∇h
2 ≤ (1 − α)A Θ ≤ 0.
On the other hand, following the same ideas of Theorem we can verify that the Bakry
n
Émery Ricci tensor of Σ is bounded from below and, hence, from Theorem there exists a
n
sequence of points p such that
k
∈ Σ ∆ lim inf Θ(p
f k
) ≥ 0
k
and lim Θ(p ) = inf Θ.
k k p∈Σ
Consequently,
2
2
Θ Θ lim (p ) = sup .
k k p∈Σ
Thus,
2
∆ Θ(p (p ) inf 0 ≤ liminf f k k
) ≤ (1 − α)liminf A Θ ≤ 0.
k k p∈Σ
2 Up to a subsequence, it follows that lim
(p ) = 0. Now, by using hypothesis we
k k
A
2
2
2
2
k k k k
Θ Θ obtain that lim (p ) = 0 and therefore sup = lim (p ) = 1. But Θ
∇h ≥ 1, hence,
p∈Σ 2 n n n
Θ and, therefore, Σ .
= 1 on Σ is a slice {t} × P From the proof of Theorem it is not difficult to see that we also get
3.5 WEIGHTED STATIC GRW SPACETIMES
53
n _{P}
Corollary 3.5.1. be a weighted static GRW spacetime, such that K Let −R × P
≥ −κ and
f n
be a complete spacelike hypersurface Hess f ≥ −γ for some positive constants κ and γ. Let Σ
n
with constant f mean curvature H and with bounded second fundamental form A. f in −R×Pf
If α
2
2
, ∇h ≤ A
κ(n − 1) + γ
n n .
for some constant 0 < α < 1, then Σ is a slice {t} × P We close our chapter, with the following result
n
Theorem 3.5.4 be a weighted static GRW spacetime, such (Theorem 6 . Let −R × P
f _{P} n
that K be a complete spacelike hypersurface immersed in a ≥ 0 and ∇ f  is bounded. Let Σ
n
are bounded and H does not change sign timelike bounded region of −R×P . If ∇h and H f f
f n
on Σ , then H is not globally bounded away from zero. In particular, if f is convex and H is
f f n n
constant, then Σ . is a slice {t} × P
n
Proof. Taking into account our restriction on the sectional curvature of the fiber P jointly with
n
are bounded on Σ , as in the proof of Theorem we the hypothesis that ∇ f , ∇h and H f
n
is bounded from can apply Proposition 3.2 of to guarantee that the Ricci curvature of Σ below.
n n
Now, suppose for instance that H . Thus, since Σ lies between two slices
f
≥ 0 on Σ
n
, from Lemma and of −R × P
n
Yau we obtain a sequence of points p such that
k ∈ Σ
∆ Θ ) = n lim inf H (p ).
f k f k
0 ≤ liminf (−h)(p
k k
On the other hand, note that
2
Θ 0 = lim ) = lim (p
k k ∇h(p ) − 1. k k
Thus, since Θ ≤ −1, lim Θ(p
k ) = −1. k
Consequently, ∆
H (p
f f
0 ≤ liminf (−h)(p k ) = −nliminf k ) ≤ 0
k k
and, hence, we conclude that lim inf H (p ) = 0.
f k k
3.5 WEIGHTED STATIC GRW SPACETIMES
54 If H
f
≤ 0, with the aid of Lemma and applying once more the generalized maximum
n
principle of Omori we get a sequence q such that
k
∈ Σ ∆ Θ h(q H (q )
f f
0 ≤ liminf k ) = −nliminf k
k k
and lim Θ(q
k
) = −1. k Therefore, we conclude again that H is not globally bounded away from zero. f n nWhen H is constant, we have that, in fact, H vanishes identically on Σ . Thus, since Σ is
f f n
contained into a timelike bounded region of −R × P , there exists a constant β such that h − β
f n
is a positive harmonic function of the drifting Laplacian on Σ . Moreover, since f is supposed convex, from inequality to get that Ric
f ≥ 0.
∗ ∗
On the other hand, since ∇ f = ∇ f + h∇ f ,N iN and N  = ∇h, we obtain
2
2
2 ).
∇ f  ≤ ∇ f  (1 + ∇h Consequently, since we are assuming that ∇ f  and ∇h are bounded, we have that ∇ f is also
n bounded on Σ .
Therefore, we are in position to apply Corollary 1.4 of and conclude that h is constant
n n n on Σ , that is, Σ .
is a slice {t} × P
HAPTER
C
4 Height Estimate for weighted semiRiemannian
manifolds
The results of this chapter are part of4.1 Introduction
In 1954 Heinz proved that a compact graph of positive constant mean curvature H in the
n+1
(n + 1)dimensional Euclidean space R with boundary on a hyperplane can reach at most
1 n+1
1
height from the hyperplane. A hemisphere in R of radius shows that this estimate
H H
is optimal. In particular, Heinz’s result motivated several authors to approach the problem of obtain a priori estimates for the height function of a compact hypersurface whose boundary is contained into a slice of a Riemannian product space (see, for instance,
Concerning the Lorentzian setting, López obtained a sharp estimate for the height of compact constant mean curvature spacelike surfaces with boundary contained in a spacelike
3
plane of the 3dimensional LorentzMinkowski space L . Later on, de Lima established a height estimate for compact spacelike hypersurfaces with some positive constant higher order mean curvature and whose boundary is contained in a spacelike hyperplane of the (n + 1)
n+1
dimensional LorentzMinkowski space L . As in through the computation of the height
n+1
of the hyperbolic caps of L , he showed that his estimate is sharp. Afterwards, he jointly with Colares to the context of the Lorentzian product spaces
n .
−R × P In this chapter, we prove height estimates concerning compact hypersurfaces with nonzero constant weighted mean curvature and whose boundary is contained into a slice of εI × P.
4.2 The Riemannian setting Now, we present our first height estimate.
n
be a weighted Riemannian product space with Theorem 4.2.1
(Theorem 1 . Let I × P
f n n
Ric ,
f be a compact hypersurface with boundary contained into the slice {s}×P
≥ 0 and let Σ
n
has nonzero constant for some s ∈ I, and whose angle function Θ does not change sign. If Σ
4.2 THE RIEMANNIAN SETTING
56
2
2 n
f mean curvature such that nH , where A denotes the Weingarten operator of Σ with
f ≤ A n
respect to its unit normal vector field N, then the height function h of Σ satisfies
1 (4.1) . h − s ≤
f
H 
n
Proof. Define on Σ the function (4.2) ϕ = H h + Θ.
f
From we get that
2
2 ∗ ∗
∆ Ric (4.3) + f (N , N )).
f f
ϕ = −Θ(A − nH f
2
2 ∗ ∗
Consequently, since f Ric and choosing N such that
(N , N ) = Ric
f f
(N, N) ≥ 0, nH f ≤ A
f
−1 ≤ Θ ≤ 0, from we have that that ϕ ≤ ϕ
∂ Σ
H s. (4.4)
f f f
h − 1 ≤ H h + Θ ≤ H We then consider the two possible cases. In the case that H
> 0, from we have
f n
∆ . Thus, from we conclude that
f h ≤ 0 and, from the maximum principle, h ≥ s on Σ
1 (4.5) . h − s ≤
H
f
Finally, in the case that H < 0, from we have ∆
f f h ≥ 0 and, again from the maximum n
. Thus, from we must have principle, h ≤ s on Σ
1 (4.6) . s − h ≤ −
H
f Therefore, estimate
2
2 We point out that the hypothesis nH is automatically satisfied in the case Remark 4.2.1.
≤ A
f
that the weighted function f is constant. Furthermore, taking into account Heinz’s estimate is optimal.
From Theorem we obtain the following halfspace type result
n
Theorem 4.2.2 be a weighted Riemannian product space with (Theorem 2 . Let R × P
f
n nRic compact. Let Σ be a complete noncompact twosided hypersurface properly
f
≥ 0 and P
4.3 THE LORENTZIAN SETTING
57
n n
, whose angle function Θ does not change sign. If Σ has nonzero constant immersed in R × P
f
2 2 n
f mean curvature such that nH , then Σ cannot lie in a halfspace of R×P. In particular,
f ≤ A n
Σ must have at least one top and one bottom end.
n
Proof. Suppose by contradiction that, for instance, Σ ⊂ (−∞,τ] × P, for some τ ∈ R. Thus, for each s < τ we define
n
 Σ
=
s {(t,x) ∈ Σ : t ≥ s}.
n n +
Since P is compact and Σ , we have that Σ is a compact is properly immersed in R × P s
f
hypersurface contained in a slab of width τ − s and with boundary in {s} × P. Thus, we can
1
 s H f 
apply Theorem to get that Σ is contained in a slab of width , so that it must be
1
. Consequently, choosing s sufficiently small we violate this estimate, reaching to a τ − s ≤
H f  contradiction. n
−
Analogously, if we suppose that Σ ⊂ [τ,+∞) × P with τ ∈ R, for each s > τ we define Σ s by
−
Σ =
s {(t,x) ∈ Σ;t ≤ s}.
−
Hence, since Σ is a compact hypersurface with boundary in {s} × P, we can reason as in the
s previous case and obtain another contradiction.
4.3 The Lorentzian setting We proceeding with our second height estimate.
n
Theorem 4.3.1 be a weighted Lorentzian product space with (Theorem 3 . Let −I × P
f
nRic be a compact spacelike hypersurface with boundary contained into the slice
f
≥ 0 and let Σ
n n
2
2
has nonzero constant f mean curvature such that nH , , for some s ∈ I. If Σ
{s} × P
f ≤ A
n
where A denotes the Weingarten operator of Σ with respect to its futurepointing unit normal
n
vector field N, then the height function h of Σ satisfies max
∂ Σ Θ − 1
(4.7) . h − s ≤
f
H 
n
Proof. Analogously, we define on Σ the function (4.8) f h − Θ. ϕ = −H
4.3 THE LORENTZIAN SETTING
58 From equation we get that
2
2 ∗ ∗
∆ + f Ric (N , N )).
f ϕ = −Θ(A − nH f f
2
2 ∗ ∗
Consequently, since f Ric (N , N ) = Ric and choosing N future
f f
(N, N) ≥ 0, nH ≤ A
f f
pointing (that is, Θ ≤ −1), we get that ∆ ϕ ≥ 0. Thus, we conclude from the maximum and, hence, from we have that principle that ϕ ≤ ϕ
∂ Σ
s + max (4.9)
f h + 1 ≤ −H f h − Θ ≤ −H f − H Θ.
∂ Σ
We then consider the two possible cases. In the case that H > 0, note that ∆
f f
h ≥ 0 and,
n
. Thus, from we conclude that from the maximum principle, h ≤ s on Σ max
∂ Σ Θ − 1
. (4.10) s − h ≤ H
f
Finally, in the case that H < 0, we have ∆
f f h ≤ 0 and, again from the maximum principle, h ≥ s n
on Σ . Thus, from we must have 1 − max ∂ Σ Θ . (4.11) h − s ≤
H
f Therefore, estimate
Taking into account the height estimate of mentioned in the introduction, Remark 4.3.1. we see that our estimate is also sharp.
Finally, reasoning as in the proof of Theorem we get the follow ing
n
be a weighted Lorentzian product space with Theorem 4.3.2
(Theorem 4 . Let −R × P
f
n nRic compact. Let Σ be a complete noncompact spacelike hypersurface properly
f
≥ 0 and P
n n
, with bounded angle function Θ. If Σ has nonzero constant f mean immersed in −R × P
f
2 2 n n
curvature such that nH , then Σ cannot lie in a halfspace of −R × P. In particular, Σ
f ≤ A must have at least one top and one bottom end.
Remark 4.3.2. We recall that an integral curve of the unit timelike vector field ∂ is called a
t n
, ∂ (p) is called an instantaneous comoving
t
comoving observer and, for a fixed point p ∈ Σ observer. In this setting, among the instantaneous observers at p, ∂ (p) and N(p) appear nat
t ∗
urally. From the orthogonal decomposition N(p) = N (p), we have that
t
(p) − Θ(p)∂ Θ(p)
4.3 THE LORENTZIAN SETTING
59 corresponds to the energy e(p) that ∂ (p) measures for the normal observer N(p). Further
t −1 ∗
(p)N (p) that ∂ (p) measures more, the speed υ(p) of the Newtonian velocity υ(p) := e t
2 −1
= tanh(cosh for N(p) satisfies the equation υ(p) Θ(p)). Hence, the boundedness of the
n
angle function Θ of the spacelike hypersurface Σ means, physically, that the speed of the New tonian velocity that the instantaneous comoving observer measures for the normal observer do
n
not approach the speed of light 1 on Σ (cf. the assumption of Θ be bounded on a complete spacelike hypersurface is a natural hypothesis to supply the noncompactness of it.
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