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Tese_Márcio _Silva.pdf
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UNIVERSIDADE FEDERAL DE ALAGOAS
INSTITUTO DE MATEMÁTICA
PROGRAMA DE PÓS-GRADUAÇÃO EM MATEMÁTICA UFBA-UFAL
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 Pós-graduação em Matemática UFBAUFAL do Instituto de Matemática da Universidade Federal de Alagoas como requisito parcial 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
Pós-graduação em Matemática. Maceió, 2014.
Bibliografia: f. 60-65.
1. Variedades ponderadas. 2. Tensor Bakry-Émery Ricci. 3. Curvatura ƒ-média.
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 durante 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 Pós-graduação 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, aproximemo-nos 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 Galloway for the Bakry-Émery 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
height estimates for hypersurfaces immersed in a semi-Riemannian manifold of type εI ×ρ P f .
Keywords: Weighted manifolds, Bakry-Émery Ricci tensor, f -mean curvature, Bernstein type
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 Bakry-Émery Ricci. Como aplicação, nós obtemos
teoremas do fechamento para o espaço-tempo 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 semiRiemanniana ponderada do tipo εI ×ρ P f .
Palavras-chave: Variedades ponderadas, tensor Bakry-Émery Ricci, curvatura f -média, teoremas tipo Bernstein, estimativas de altura.
Contents
1
Preliminaries
1.1 Curvature and Maximum principle
1.2 Weighted mean curvature
1.3 Hypersurfaces in weighted semi-Riemannian manifolds
11
11
13
14
2
Compactness of weighted manifolds
2.1 Introduction
2.2 Weighted Ambrose’s Theorem
2.3 Weighted Galloway’s Theorems
2.4 Weighted Riemannian Yun-Sprouse’s Theorem
2.5 Weighted Lorentzian Yun-Sprouse’s Theorem
2.6 Closure Theorem via Galloway’s Theorem
2.7 Closure Theorem via Ambrose’s Theorem
15
15
16
18
21
23
25
30
3
Bernstein type properties of complete hypersurfaces in weighted semi-Riemannian
manifolds
32
3.1 Introduction
32
3.2 Uniqueness results in weighted warped products
32
3.3 Rigidity results in weighted product spaces
37
3.4 Uniqueness results in weighted GRW spacetimes
45
3.5 Weighted static GRW spacetimes
48
4
Height Estimate for weighted semi-Riemannian manifolds
4.1 Introduction
4.2 The Riemannian setting
4.3 The Lorentzian setting
55
55
55
57
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 [22], [66], [23], [24] and [25].
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 2, based in the papers [22], in collaboration with M. Cavalcante and J. Oliveira,
and [66], we prove six generalizations of Myers compactness theorem under different conditions on its generalized Ricci curvature tensor. As applications of the our weighted compactness 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 3, based in the papers [24] and [25], in collaboration with M. Cavalcante and H.
de Lima, our aim is to investigate Bernstein properties of hypersurfaces immersed in a weighted
semi-Riemannian manifold of the type εI ×ρ P, where ε = ±1. In this context, an important
point is to explore the Bakry-Émery-Ricci tensor of such a hypersurface in order to ensure the
existence of Omori-Yau type sequences. The existence of these sequences will constitute an
important tool in the study of the uniqueness of the hypersurfaces.
In Chapter 4, based in the paper [23], 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 Bakry-Émery Ricci curvature (cf. Theorems 4.2.1 and 4.3.1). As applications of
our estimates, we obtain nonexistence results related to complete noncompact hypersurfaces
properly immersed in these weighted product spaces (cf. Theorems 4.2.2 and 4.3.2).
10
C HAPTER 1
Preliminaries
Here we establish the notations, some basic definitions and results, concerning weighted manifolds, that will be used along this thesis.
1.1
Curvature and Maximum principle
Given a complete n-dimensional Riemannian manifold (M n , g) and a smooth function f : M n →
R, the weighted manifold M nf associated to M n and f is the triple (M n , g, dµ = e− f dM), where
dM denotes the standard volume element of M n . Appearing naturally in the study of selfshrinkers, 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 [70] of Wey and Willie. We point out that a theory of Ricci curvature for
weighted manifolds goes back to Lichnerowicz [48, 49] and it was later developed by Bakry
and Émery in the work [11].
In this thesis we use three notions of Ricci curvature for weighted manifolds. Firstly, given
a constant k > 0, the k-Bakry-Émery-Ricci tensor is given by
1
Rickf = Ric + Hess f − d f ⊗ d f ,
k
where Ric stands for the usual Ricci tensor of (M, g) (see [11]).
The second one is the ∞-Bakry-Émery-Ricci tensor or just Bakry-Émery-Ricci tensor. It is
given by
Ric f := Ric∞f = Ric + Hess 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 metric-dual vector field and let denote by LV g the Lie derivative. Given a constant k > 0, the modified Ricci tensor with respect to V and k (see [50]) is
defined as
1
RicVk = Ric + LV g − V ∗ ⊗V ∗ .
k
k
Of course, in the above notations, Ric4k
f = Ric ∇ f .
2
In order to exemplify these notions of Ricci curvature we recall that a Ricci soliton is just a
11
1.1 CURVATURE AND MAXIMUM PRINCIPLE
12
weighted manifold satisfying the equation bellow
Ric f = λ g,
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 selfsimilar 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 quasi-Einstein just by replacing
the Bakry-Émery Ricci tensor by Rickf , see [60] and [56]. In the next chapter we deal with a
bigger class of manifolds, because we use inequalities instead equalities.
Now, we define the f -divergence of a vector field X ∈ X(M n ) as
div f (X) = e f div(e− f X),
where div is the usual divergence on M. From this, the drift Laplacian it is defined by
∆ f u = div 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
∆ f |∇u|2 = |Hessu|2 + h∇u, ∇(∆ f u)i + Ric f (∇u, ∇u),
2
(1.1)
where u is a smooth function on M. See [70] for more details.
An important tool along this work is the Omori-Yau maximum principle for the f -Laplacian.
According to [61], we have the following definition
Definition 1.1.1. Let M f be a weighted manifold. We say that the full Omori-Yau maximum
principle for ∆ f holds if for any C2 function u : M −→ R satisfying supM u = u∗ < ∞ there
exists a sequence {xn } ⊂ M along which
(i) lim u(xn ) = sup u, (ii) lim |∇u(xn )| = 0 and (iii) lim sup ∆ f u(xn ) ≤ 0.
n
M
n
n
If the condition (ii) don’t hold, then we say just that the weak Omori-Yau maximum principle
for ∆ f holds on M.
From the classical Omori-Yau maximum principle (see [57] and [71]) we conclude that if
1.2 WEIGHTED MEAN CURVATURE
13
|∇ f | is bounded and Ric is bounded from below, then the full Omori-Yau maximum principle
for ∆ f holds on M.
On the other hand, using volume growth conditions and stochastic process on a weighted
manifold, Rimoldi [61], showed that the validity of the weak Omory-Yau maximum principle
is guaranteed by a lower bound on the Bakry-Émery Ricci tensor. Namely:
Theorem 1.1.1. Let M f be a weighted manifold. If Ric f ≥ λ for some λ ∈ R, then the weak
Omori-Yau maximum principle holds on M.
1.2
Weighted mean curvature
Here, we define a notion of mean curvature in the weighted context. Indeed, let Σn be a hypersurface immersed in a weighted Riemannian manifold M n+1
and denote by ∇ the gradient
f
n+1
with respect to the metric of M . According to Gromov [41], the weighted mean curvature,
or simply f -mean curvature, H f of Σn is given by
nH f = nH + hN, ∇ f i,
(1.2)
where H denotes the standard mean curvature of Σn with respect to its orientation N. In this
context, it is natural to consider the first variation for the weighted volume
Z
vol f (Ω) =
e− f dΩ,
Ω
where Ω is a bounded domain in Σ. According to Bayle [12] the first variation formula is given
by
Z
d
vol f (Ωt ) = H f hN,V ie− f dΩ,
dt t=0
Ω
where V is the variational vector field.
2
In the Euclidean space Rn+1 , taking f = |x|2 , the hypersurfaces with f -mean curvature
H f = 0 are the well known self-shrinkers, that is, a hypersurface immersed in Rn+1 satisfying
H = hx, Ni,
where x is the position vector in Rn+1 , see for instance [29].
1.3 HYPERSURFACES IN WEIGHTED SEMI-RIEMANNIAN MANIFOLDS
1.3
14
Hypersurfaces in weighted semi-Riemannian manifolds
In what follows, let us consider an (n + 1)-dimensional product space M̄ n+1 of the form I × Pn ,
where I ⊂ R is an open interval, Pn is an n-dimensional connected Riemannian manifold and
M̄ n+1 is endowed with the standard product metric
h, i = επI∗ (dt 2 ) + πP∗ (h, iP ),
where ε = ±1, πI and πP denote the canonical projections from I × Pn onto each factor, and
h, iP is the Riemannian metric on Pn . For simplicity, we will just write M̄ n+1 = εI × Pn and
h, i = εdt 2 + h, iP . In this setting, for a fixed t0 ∈ I, we say that Ptn0 = {t0 } × Pn is a slice of
M̄ n+1 .
In Chapter 3 and 4, we will consider a connected hypersurface Σn immersed into M̄ n+1 .
In the case where M̄ n+1 is Lorentzian (that is, when ε = −1) we will assume that Σn is a
spacelike hypersurface, that is, the metric induced on Σn via the immersion is a Riemannian
metric. Since ∂t is a globally defined timelike vector field on −I ×Pn , it follows that there exists
an unique unitary timelike normal field N globally defined on Σn and, therefore, the function
angle Θ = hN, ∂t i satisfies |Θ| ≥ 1. On the other hand, when M̄ n+1 is Riemannian (that is, when
ε = 1), Σn is assumed to be a two-sided hypersurface in M̄ n+1 . This condition means that there
is a globally defined unit normal vector field N.
e the gradients with respect to the metrics of εI ×ρ Pn , Σn and P,
Denoting by ∇, ∇ and ∇
respectively, a simple computation shows that the gradient of πI on M̄ n+1 is given by
∇πI = εh∇πI , ∂t i∂t = ε∂t .
(1.3)
So, from (1.3) we conclude that the gradient of the (vertical) height function h = (πI )|Σ of Σn
is given by
∇h = (∇πI )> = ε∂t> = ε∂t − ΘN,
(1.4)
where ( )> denotes the tangential component of a vector field in X(M̄ n+1 ) along Σn . Thus, we
get the following relation
|∇h|2 = ε(1 − Θ2 ).
(1.5)
In this setting, the weighted mean curvature H f of Σn is given by
nH f = nH + εh∇ f , Ni.
(1.6)
C HAPTER 2
Compactness of weighted manifolds
The results in this chapter are part of the works [22] and [66].
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 [8],
the condition on the lower bound for the Ricci tensor is replaced by a condition on its integral
along geodesics. Namely:
Theorem A (Ambrose). Suppose there exists a point p in a complete Riemannian manifold M
for which every geodesic γ(t) emanating from p satisfies
Z ∞
Ric(γ 0 (s), γ 0 (s))ds = ∞.
0
Then M is compact.
The second important generalization we want to mention here is due to Galloway [38]
where a perturbed version of Myers theorem were considered.
Theorem B (Galloway). Let M n be a complete Riemannian manifold, and γ a geodesic joining
two points of M. Assume that
Ric(γ 0 (s), γ 0 (s)) ≥ a +
dφ
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
diam(M) ≤
π
c+
a
q
c2 + a(n − 1) .
Finally, we recall an interesting result due to C. Sprouse (see [68]). Namely:
Theorem C (Sprouse). Let (M, g) be a complete Riemannian manifold of dimension n satisfying Ric(v, v) ≥ −a(n − 1) for all unit vectors v and some a > 0. Then for any R, δ > 0 there
15
2.2 WEIGHTED AMBROSE’S THEOREM
16
exists ε = ε(n, a, R, δ ) such that if
1
sup
x Vol(B(x, R))
Z
B(x,R)
max {(n − 1) − Ric− (x), 0} dvol < ε(n, k, R, δ ),
then M is compact with diam(M) ≤ π + δ .
These theorems above have applications in Relativity Cosmology (see [38], [37] and [73])
and in the theory of Ricci Solitons (see [35] and [58]).
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 [73]) 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 (1.1) it is easy to see that m f satisfies the following Riccati inequality (see Appendix A of [70]):
Rickf (∂ r, ∂ r) ≤ −m0f −
1
m2f ,
k+n−1
(2.1)
where k ∈ (0, ∞) and m0f stands for the derivative of m f with respect to r.
Now we are in position to state and prove our first result.
Theorem 2.2.1 (Theorem 2.1 [22]). Let M f be a complete weighted manifold. Suppose there
exists a point p ∈ M f such that every geodesic γ(t) emanating from p satisfies
Z ∞
0
Rickf (γ 0 (s), γ 0 (s)) ds = ∞,
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 f (t) is smooth for all t > 0 along γ(t).
Integrating the inequality (2.1) we obtain
Z t
1
Rickf (γ 0 (s), γ 0 (s))ds ≤
Z t
1
(−m0f (s) −
1
m2f (s))ds.
k+n−1
17
2.2 WEIGHTED AMBROSE’S THEOREM
So we conclude that
1
lim (−m f (t) −
t→∞
k+n−1
Z t
1
m2f (s)ds) = ∞.
(2.2)
In particular,
lim −m f (t) = ∞.
t→∞
In the following we show that there exists a finite number T > 0 such that lim −m f (t) = ∞
t→T −
which contradicts the smoothness of m f (t).
From (2.2), given c > k + n − 1 there exists t1 > 1 such that
1
− m f (t) −
k+n−1
Z t
m2f (s)ds ≥
1
c
> 1,
k+n−1
(2.3)
for all t ≥ t1 , where n = dim(M).
Let denote by α = (k + n − 1) and let us consider {t` } the sequence defined inductively by
t`+1 = t` + α
α `−1
c
,
for ` ≥ 1.
c/α
Notice that {t` } is an increasing sequence converging to T = t1 + α (c/α)−1
.
`
Given ` ∈ N we claim that −m f (t) ≥ αc for all t ≥ t` . In fact, from inequality (2.3) we
have that it is true for ` = 1. Now, assume that claim holds for all t ≥ t` and fix t ≥ t`+1 . Then
using inequality (2.3) again:
t`
t`+1
1
1
c
+
m2f (s)ds +
m2f (s)ds
−m f (t) ≥
k+n−1 k+n−1 1
k + n − 1 t`
Z t`+1
1
≥
m2f (s)ds
k + n − 1 t`
c `+1
α`
1
c2`
≥
=
.
(k + n − 1) (k + n − 1)2` c`−1
α
Z
Z
In particular, lim −m f (t) = ∞ which is the desired contraction.
t→T −
Using the same techniques we are able to prove Ambrose’s theorem for the ∞-Bakry-Émery
Ricci tensor. In this case a condition on f is required. See also [74]. Namely:
2.3 WEIGHTED GALLOWAY’S THEOREMS
18
Theorem 2.2.2 (Theorem 2.2 [22]). Let M f be a complete weighted manifold, where ddtf ≤ 0
along γ. Suppose there exists a point p ∈ M f such that every geodesic γ(t) emanating from p
satisfies
Z ∞
Ric f (γ 0 (s), γ 0 (s)) ds = ∞.
0
Then M is compact.
Applying this theorem to the universal cover of M we obtain the finiteness of the fundamental group.
Corollary 2.2.1. Let (M, g) be a Riemannian manifold in the conditions of either Theorem
2.2.1 or Theorem 2.2.2. 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.
Theorem 2.3.1 (Theorem 3.1 [22]). Let M n be a complete Riemannian manifold and let V be
a smooth vector field on M. Suppose that for every pair of points in M n and any normalized
minimizing geodesic γ joining these points the modified Ricci tensor satisfies
RicVk (γ 0 , γ 0 ) ≥ (n − 1)c +
dφ
,
dt
(2.4)
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
"
π
b
p
diam(M) ≤ p
+
(n − 1)c
(n − 1)c
s
#
b2
+ n − 1 + 4k .
(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 `. Consider a parallel orthonormal
frame {E1 = γ 0 , E2 , ...., En } along γ. Let h ∈ C∞ ([0, l]) such that h(0) = h(`) = 0 and set Vi (t) =
h(t)Ei (t) along γ. Firstly, the index formula implies
Z `
I(Vi ,Vi ) =
0
((h0 )2 − h2 R(γ 0 , Ei )γ 0 , Ei )dt,
i = 2, . . . n.
19
2.3 WEIGHTED GALLOWAY’S THEOREMS
So
n
S := ∑ I(Vi ,Vi ) =
i=2
Z `
(n − 1)(h0 )2 − h2 Ric(γ 0 , γ 0 ) dt.
(2.5)
0
From the condition (2.4) we have
Z `
Z `
0 2
d 0
dφ
1 0
2
2
2
S ≤ (n − 1) (h ) − h c dt + h − + 2
γ ,V − γ ,V
dt.
dt
dt
k
0
0
On the other hand, integrating by parts we obtain
Z `
2
h2
0
d 0
γ ,V dt ≤
dt
Z `
1
2
4k(h0 )2 + h2 γ 0 ,V dt.
k
0
So,
S ≤ (n − 1 + 4k)
Z `
0 2
(h ) dt − (n − 1)c
0
Z `
2
h dt −
0
Z `
h2
0
dφ
dt.
dt
(2.6)
Now we choose h(t) = sin πt` .
Then from (2.6), we have
`
`
π2
dφ
S ≤ (n − 1 + 4k) − (n − 1)c −
h2 dt.
2`
2
dt
0
Z
Integrating by parts once more we get
Z `
2 dφ
π
dt = −
h
dt
`
0
Z `
2πt
φ sin
`
0
dt ≥ −bπ.
Thus we have
1
−(n − 1)c`2 + 2bπ` + π 2 (n − 1 + 4k) .
2`
Finally, because γ is minimizing we have S ≥ 0 and therefore
S≤
s
"
#
π
b
b2
p
`≤ p
+
+ n − 1 + 4k .
(n − 1)c
(n − 1)c
(n − 1)c
It finishes the proof.
Following some ideas of [51] we also obtain an extension of Galloway’s Theorem for
weighted manifolds using the ∞-Bakry-Émery-Ricci tensor.
Theorem 2.3.2 (Theorem 3.2 [22]). Let M f be a weighted complete Riemannian manifold with
| 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 f and normalized minimizing geodesic γ joining these points we
have
dφ
Ric f (γ 0 , γ 0 ) ≥ (n − 1)c +
,
(2.7)
dt
where φ is a smooth function satisfying |φ | ≤ b. Then M is compact and
q
π
diam(M) ≤
b + b2 + c(n − 1)λ ,
c(n − 1)
(2.8)
√
where λ = 2 2a + (n − 1).
Proof. From (2.7) and (2.5) we have
Z `
Z `
Z `
0 2
dφ
2
2
0 0
S ≤ (n − 1) (h ) − h c dt+ h Hess f (γ , γ )dt− h2 dt.
dt
0
0
0
(2.9)
We note that
h2 Hess f (γ 0 , γ 0 ) = h2
d
∇ f , γ 0 dt
dt
d 2
(h ∇ f , γ 0 ) − 2hh0 ∇ f , γ 0
dt
d
d
d 2
0
0
0
(h ∇ f , γ ) − 2
( f hh ) − f (hh ) .
=
dt
dt
dt
=
Integrating the last identity above we have
Z `
0
2
0
0
h Hess f (γ , γ )dt = 2
Z `
f
0
d
(hh0 )dt,
dt
since h(0) = h(`) = 0.
Thus
2 ! 12
Z `
√
d
h2 Hess f (γ 0 , γ 0 )dt ≤ 2a `
(hh0 ) dt
.
0
0 dt
Z `
Choosing h(t) = sin( πt` ) we have
√ 2
2π
a
h2 Hess f (γ 0 , γ 0 )dt ≤
.
`
0
On the other hand a direct computation yelds
Z `
(2.10)
2.4 WEIGHTED RIEMANNIAN YUN-SPROUSE’S THEOREM
π 2 c`
−
.
(n − 1)(h − ch )dt = (n − 1)
2` 2
0
Z `
02
2
21
(2.11)
Therefore, from (2.9), (2.10) and (2.11) we have
S≤−
i
√
1 h
−2aπ 2 2 + c(n − 1)`2 − π 2 (n − 1) + πb
2`
(2.12)
Since S ≥ 0 we get
q
π
2
`≤
b + b + c(n − 1)λ ,
c(n − 1)
√
where λ = 2 2a + (n − 1) and so M is compact and satisfies (2.8).
2.4
Weighted Riemannian Yun-Sprouse’s Theorem
In this section, following the ideas of [73], we provide a weighted version of Theorem C as
follows.
Theorem 2.4.1 (Theorem 1.1 [66]). Let M nf be a weighted complete Riemannian manifold.
Then for any δ > 0, and a > 0, there exists an ε = ε(n, a, δ ) satisfying the following:
If there is a point p such that along each geodesic γ emanating from p, the Rickf curvature
satisfies
Z ∞
n
o
k 0 0
max (n − 1)a − Ric f (γ , γ ), 0 dt < ε(n, a, δ )
(2.13)
0
π
+δ.
then M is compact with diam(M) ≤ q (n−1)a
n+k−1
Proof. For any small positive ε < a2 to be determined later, consider the following sets
n
√ o
E1 = t ∈ [0, ∞); Rickf (γ 0 (t), γ 0 (t)) ≥ (n − 1)(a − ε)
and
n
√ o
k 0
0
E2 = t ∈ [0, ∞); Ric f (γ (t), γ (t)) < (n − 1)(a − ε) .
From the inequality (2.1) we have on E1
m0f
n+k−1
√ ≤ −1.
mf
ε)
2
( n+k−1 ) + (n−1)(a−
n+k−1
On the other hand, on E2 we have
(2.14)
2.4 WEIGHTED RIEMANNIAN YUN-SPROUSE’S THEOREM
22
√
m0f
a − ε − Rickf (γ 0 , γ 0 )/(n − 1)
n+k−1
√ ≤
√
.
mf
ε)
a− ε
( n+k−1
)2 + (n−1)(a−
n+k−1
(2.15)
Now, using the assumption (2.13) on the Bakry-Émery Ricci tensor we get
Z ∞
o
n
0
k 0
ε >
max (n − 1)a − Ric f (γ (t), γ (t)), 0
Z0 n
o
k 0
0
>
(n − 1)a − Ric f (γ (t), γ (t)) dt
E
Z 2
√
(n − 1)a − (n − 1)(a − ε) dt
E2
√
= µ(E2 )(n − 1) ε.
>
That is,
√
ε
µ(E2 ) <
,
n−1
where µ is the Lebesgue measure on R.
(2.16)
Using inequalities (2.14)−(2.16) we obtain
m0f
n+k−1
√ dt
0 ( m f )2 + (n−1)(a− ε)
n+k−1
n+k−1
Z r
≤
m0f
n+k−1
√ dt
[0,r]∩E1 ( m f )2 + (n−1)(a− ε)
n+k−1
n+k−1
0
m
Z
f
n+k−1
√ dt
+
[0,r]∩E2 ( m f )2 + (n−1)(a− ε)
n+k−1
n+k−1
Z
ε
√
(n − 1)(a − ε)
ε
√
≤ −r + µ {[0, r] ∩ E2 } +
(n − 1)(a − ε)
√
ε
ε
√ .
≤ −r +
+
n − 1 (n − 1)(a − ε)
≤ −µ {[0, r] ∩ E1 } +
√
ε
ε √
Define τ(ε) = n−1
+ (n−1)(a−
. The integral of the left hand side can be computed exε)
plicitly and therefore we get
m f (r)
arctan
(n + k − 1)a(ε)
π
≤ a(ε)(−r + τ(ε)) + ,
2
2.5 WEIGHTED LORENTZIAN YUN-SPROUSE’S THEOREM
where a(ε) =
q
23
√
(n−1)(a− ε)
.
n+k−1
So
m f (r) ≤ −(n + k − 1)a(ε) cot(a(ε)(−r + τ(ε))),
π
for any r such that τ(ε) < r < a(ε)
+ τ(ε).
π
In particular, m f (γ(r)) goes to −∞ as r → ( a(ε)
+ τ(ε))+ . It implies that γ can not be
π
π
minimal beyond a(ε)
+ τ(ε). Otherwise m f would be a smooth function at r = a(ε)
+ τ(ε).
π
π
Taking ε explicitly so that a(ε) + τ(ε) = q (n−1)a + δ and using the completeness of M we have
n+k−1
the desired result.
2.5
Weighted Lorentzian Yun-Sprouse’s Theorem
Now let us discuss the Lorentzian version of Theorem 2.4.1. Let M be a time-oriented 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 + (p) ∩ J − (q) is compact for all p and q joined by a
causal curve (see [13]). Mathematically, global hyperbolicity often plays a role analogous to
geodesic completeness in Riemannian geometry.
Let γ : [a, b] −→ M be a future-directed timelike unit-speed geodesic. Given {E1 , E2 , . . . , En }
an orthonormal frame field along γ, and for each i ∈ {1, . . . , n} we let Ji be the unique Jacobi
field along c such that Ji (a) = 0 and Ji0 (0) = Ei . Denote by A the matrix A = [J1 J2 . . . Jn ], where
each column is just the vector for Ji in the basis defined by {Ei } . In this situation, we have that
A(t) is invertible if and only if γ(t) is not conjugate to γ(a).
1
Now we define B f = A0 A−1 − n−1
( f ◦ γ)0 E wherever A is invertible, where E(t) is the identity map on (γ 0 (t))⊥ . The f -expansion function is a smooth function defined by θ f = trB f (see
[21, Definition 2.6]) Note that, if |θ f | → ∞ as t → t0 , where t0 ∈ [a, b], then γ(t0 ) is conjugate
to γ(a).
Recently, Case in [21] obtained the following relation between the Bakry-Emery Ricci Tensor and the f -expansion function θ f :
24
2.5 WEIGHTED LORENTZIAN YUN-SPROUSE’S THEOREM
Lemma 2.5.1. Under the above notations,
θ f0 ≤ −Rickf (γ 0 , γ 0 ) −
θ f2
k+n−1
.
(2.17)
The inequality (2.17) is called (k, f )-Raychaudhuri inequality. This inequality is a generalization of the well known Raychaudhuri inequality (see for instance [36]).
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.
Theorem 2.5.1 (Theorem [66]). Let M nf be a weighted globally hyperbolic spacetime. Then
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
p, with l(γ) = sup {t ≥ 0, d(p, γ(t)) = t} the Rickf curvature satisfies
Z l(γ)
0
n
o
k 0 0
max (n − 1)a − Ric f (γ , γ ), 0 dt < ε(n, a, δ ),
π
+δ.
then the timelike diameter satisfies diam(M) ≤ q (n−1)a
n+k−1
Proof. Let ε(n, a, δ ) be the explicit constant in the previous theorem. Assume by contradiction
π
that there are two points p and q with d(p, q) > q (n−1)a
+ δ . On the other hand, since M
n+k−1
is globally hyperbolic, there exists a maximal timelike geodesic γ joining p and q such that
`(γ) = d(p, q). Following the steps of the proof of Theorem 2.4.1 using the (k, f )-Raychaudhuri
inequality (2.17) we get
lim θ f (t) = −∞,
t→t0+
π
π
+ τ(ε). So, we conclude that γ cannot be maximal beyond q (n−1)a
+ δ which
where t0 = a(ε)
n+k−1
is a contradiction.
An immediate consequence of Theorem 2.5.1 is the Lorentzian version of the original
weighted Myers Theorem obtained by Qian in [59]. Namely:
2.6 CLOSURE THEOREM VIA GALLOWAY’S THEOREM
25
Corollary 2.5.1. Let M nf be a weighted globally hyperbolic spacetime. Let a be a positive
constant and assume that Rickf (v, v) ≥ (n − 1)a, for all unit timelike vector field v ∈ T M. Then,
π
the timelike diameter satisfies diam(M) ≤ q (n−1)a
.
n+k−1
2.6
Closure Theorem via Galloway’s Theorem
Let M̄ n+1 be an (n + 1)-dimensional spacetime manifold, that is, a smooth manifold endowed
with a pseudo-Riemannian metric ds2 of signature one. Assume that M̄ n+1 is complete and
time orientable.
An interesting problem in mathematical relativity is to determine whether space-like 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 M̄.
Let us denote by u a unit time-like vector field on M̄. 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 M̄ there exists a complete connected n-dimensional hypersurface M orthogonal to u. Physically, we may interpret the manifold M as the spatial universe
at some moment in time. Moreover, in each point of M n there exist a local coordinate neighborhood U n 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 Ut = {t} ×U is orthogonal to u.
In W , the spacetime metric is given by
n
2
2
2
ds = −ϕ dt +
∑ gαβ (x,t)dxα dxβ ,
α,β =1
∂
.
where xα are local coordinates introduced in U n and u = ϕ1 ∂t
Let X̄ be a vector field tangent to M̄. X̄ is called invariant under the flow if
[X̄, u] = ∇u X̄ − ∇X̄ u = 0.
Denote X̄ T by the projection of the vector field X̄ on M. So,
X̄ T = X + hX̄, 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
v(X) = ∇u X
T
and
T
a(X) = ∇u ∇u X .
The shape operator of M as a hypersurface of M̄ is defined by b(X) = −∇X u. Let denote
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 [62] §3.3.1 or [36], page 161.).
The following lemma is an application of Gauss equation for space-like submanifolds.
Lemma 2.6.1. Let γ(s) be a geodesic in a weighted manifold M f with unit tangent X. Then
Ric f (X, X) = Ric f (X, X) − ha(X), Xi + hv(X), XiH f + hv(X), Xi2 + h∇u u, Xi2
n
d
1 ∂ϕ 2
2
+
hX, ∇u ui + ∑ hv(X), e j i +
,
ds
ϕ ∂s
j=2
where {e1 = X, e2 , . . . , en } is an orthonormal basis of Tp M and H f = H + hu, ∇ f i denotes the
f -mean curvature of M f .
Proof. Let {ē1 (t) = X̄(t), ē2 (t), . . . , ēn (t)} be a set invariant under the flow generated by u,
where α = 1, . . . , n. Since [ēα , u] = 0 we get that
∇eTα u = ∇u eTα − h∇u u, eTα iu.
Therefore, a straightforward calculation shows that
hv(eα ), eα i = −B(eα , eα )
and
e eα ) = hv(X), eα i2 .
B(X,
From the above identities and the Gauss equation we get
n
Ric(X, X) =
n
o
e
K(X,
e
)−B(X,
X)B(e
,
e
)+
B(X,
e
)
j
j j
j
∑
j=2
n
n
e e j)
= ∑ K(X, e j )−B(X, X)(H − B(X, X))+ ∑ B(X,
j=2
j=2
n
e e j)
= Ric(X, X)−K(X, u)−B(X, X)(H − B(X, X))+ ∑ B(X,
j=2
n
= Ric(X, X)−K(X, u)+hv(X), XiH + hv(X), Xi2+ ∑ hv(X), e j i2 .
j=2
27
2.6 CLOSURE THEOREM VIA GALLOWAY’S THEOREM
On the other hand, note that
T
T
K(X, u) = −hR(X , u)u, X i
T
T
= h∇u ∇X T u, Xi − h∇X T ∇u u, X i + h∇[X T ,u] u, X i
d
1 ∂ϕ 2
2
= ha(X), Xi − h∇u u, Xi − hX, ∇u ui −
.
ds
ϕ ∂s
Note that we may write
D
E
∇ f = ∇ f + ∇ f , u u,
and, therefore, we have
Hess f (X, X) = Hess 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 [38] we can find a diameter estimate M, under some
restrictions.
Theorem 2.6.1 (Theorem 4.5 [22]). Let M̄ n+1
be a weighted space time with | f | ≤ a, where a
f
is a positive constant. Let u be an irrotational unit time-like vector field on M̄ n+1
and let M n
f
be a complete spatial hypersurface orthogonal to u. Suppose that the following conditions hold
on M n .
(i) At each point p ∈ M n , the flow generated by u is expanding in all directions, i.e,
hv(X), Xi ≥ 0,
where X is a vector field tangent to M.
(ii) At each point p ∈ M n , the rate of expansion is decreasing in all directions, i.e,
ha(X), Xi ≤ 0,
where X is a vector field tangent to M.
(iii) Ric f of M̄ n+1 satisfies
inf(Ric f (ζ , ζ ) −
n−1 2
H ) = c > 0,
2n
28
2.6 CLOSURE THEOREM VIA GALLOWAY’S THEOREM
where ζ ∈ Tp M, |ζ | = 1, hζ , ui = 0 and H = −div u.
(iv) The flow lines are of bounded geodesic curvature on M n , i.e,
sup |∇u u| = µ < ∞.
(v) On M n , we have that
D
E
∇ f , u = u( f ) ≥ 0.
Then M n is compact and
π
diam(M ) ≤
c
n
µ+
q
µ 2 + cλ ,
(2.18)
√
where λ = 2( 2a + 1).
Proof. Let us assume that {X = e1 , e2 . . . , en } is an orthonormal basis of M. Consider the notation vα = hv(eα ), eα i. From item (i), vα ≥ 0. Note that H = ∑ni=1 vi . Then by Schwartz inequality
−(hv(X), Xi2 + hv(X), XiH) ≤
≤
=
1
2
n−1
2n
!2
n
∑ vi
i=1
n
n
− ∑ v2i
i=1
!2
∑ vi
i=1
n−1 2
H .
2n
(2.19)
n − 1 2 dφ
H +
,
2n
ds
(2.20)
From (2.19), Lemma 2.6.1 and item (ii) we get
Ric(X, X) ≥ Ric(X, X) −
D
E
where φ = X, ∇u u .
Then
n−1 2
dφ
H +
+ hB(X, X), ui u( f ).
Ric f (X, X) ≥ Ric f (X, X) −
2n
ds
(2.21)
2.6 CLOSURE THEOREM VIA GALLOWAY’S THEOREM
29
Item (i) gives hB(X, X), ui ≤ 0. From (2.21), (iii) and (v) we get
Ric f (ζ , ζ ) ≥ c +
dφ
.
dt
Therefore by Theorem 2.3.2 we have that M is compact and satisfies (2.18).
Analogously to the previous theorem, we give a closure theorem for the k-Bakry-ÉmeryRicci tensor as follows.
Theorem 2.6.2 (Theorem 4.6 [22]). Let M̄ n+1
be a weighted space time. Let u be an irrotational
f
unit time-like vector field on M̄ n+1
and M n be a complete space-like hypersurface orthogonal
f
to u. Suppose the following conditions hold on M n .
(i) At each point p ∈ M n , the flow generated by u is expanding in all directions, i.e,
hv(X), Xi ≥ 0,
where X is avector field tangent to M.
(ii) At each point p ∈ M n , the rate of expansion is decreasing in all directions, i.e,
ha(X), Xi ≤ 0,
where X is a vector field tangent to M.
k
(iii) Ric f of M̄ n+1 satisfies
k
inf(Ric f (ζ , ζ ) −
n−1 2
H ) = c > 0,
2n
where k ∈ (0, ∞), ζ ∈ TpV, |ζ | = 1, hζ , ui = 0 and h = −divu.
(iv) The flow lines are of bounded geodesic curvature on M n , i.e,
sup |∇u u| = µ < ∞.
(v) On M n , we have that
D
E
∇ f , u = u( f ) ≥ 0.
2.7 CLOSURE THEOREM VIA AMBROSE’S THEOREM
30
Then M n is compact and
π
diam(M ) ≤
c
n
µ+
q
µ 2 + cλ
,
√
where λ = 2( 2a + 1).
2.7
Closure Theorem via Ambrose’s Theorem
Let M n be a spatial hypersurface in a weighted spacetime M n+1
f . The unit tangent vectors to the
n
future directed geodesics orthogonal to M define a smooth unit timelike vector field u at least
in a neighborhood of M n
Let X be a vector tangent to M. Extended X along the normal geodesic by making it invariant under the flow generated by u. Following the ideas of Frankel and Galloway [37] we obtain
the following closure theorem.
Theorem 2.7.1 (Theorem 4.2 [22]). Let M n be a spatial hypersurface in a weighted spacetime
(M̄ n+1 , f ). Let u be a unit normal vector field time-like to M. If there is a point q ∈ M n such
that along each geodesic γ of M n emanating from q, we have ddtf ≤ 0 and
Z ∞
0
Ric f (X, X) + H f hv(X), Xi − ha(X), Xi dt = ∞,
(2.22)
where t is arc length along γ and X is the unit tangent to γ, then M n is compact.
Proof. From Lemma (2.6.1) and identity (2.22) we get
Z ∞
0
Ric f (X, X)dt = ∞.
From Theorem 2.2.2 we conclude that M n is compact.
Analogously, we get the following closure theorem.
Theorem 2.7.2 (Theorem 4.3 [22]). Let M n be a spatial hypersurface in a weighted space time
M̄ nf . Let u be a unit normal vector field time-like to M. If there is a point q ∈ M n such that along
each geodesic γ of M n emanating from q, we have
Z ∞n
0
o
k
Ric f (X, X) + H f hv(X), Xi − ha(X), Xi dt = ∞,
where t is arc length along γ and X is the unit tangent to γ, then M n is compact.
(2.23)
2.7 CLOSURE THEOREM VIA AMBROSE’S THEOREM
31
Remark 2.7.1. Conditions (2.22) and (2.23) may be interpreted as the mass-energy density on
M n (see [37]). So, roughly speaking Theorems 2.7.1 and 2.7.2 say that spatial hypersurfaces
with mass-energy density sufficiently large are compact.
Now, using the same steps of the above theorem and Theorem 2.4.1 we get the following
closure theorem.
Theorem 2.7.3 (Theorem 1.3 [66]). Let M n be a space-like hypersurface in M̄ n+1
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 ∞
0
o
n
k
max (n − 1)a − Ric f (X, X)−hv(X), XiH f +ha(X), Xi, 0 dt < ε(n, a, δ ),
π
is satisfied, where X = γ 0 , then M is compact and diam(M) ≤ q (n−1)a
+δ.
n+k−1
C HAPTER 3
Bernstein type properties of complete
hypersurfaces in weighted semi-Riemannian
manifolds
The results of this chapter are part of [24] and [25].
3.1
Introduction
The last years have seen a steadily growing interest in the study of the Bernstein-type properties concerning complete hypersurfaces immersed in a warped product of the type I ×ρ Pn ,
where Pn is a connected n-dimensional oriented Riemannian manifold, I ⊂ R is an open interval and ρ is a positive smooth function defined on I. In this context, an important question to
pay attention is the uniqueness of complete hypersurfaces in I ×ρ Pn , under reasonable restrictions on their mean curvatures. Along this branch, we may cite, for instance, the papers [5]
to [10], [17], [20], [28] and [55]. 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 Robertson-Walker
(GRW) spacetimes, that is, Lorentzian warped products with 1-dimensional negative definite
base and Riemannian fiber of type −I ×ρ Pn . In this setting, we may cite, for instance, the
works [1, 4, 6, 7, 15, 16, 18, 32, 63, 64].
The goal of the Chapter 3 is to study the unicity of hypersurfaces Σ immersed in a semiRiemannian manifold of type εI ×ρ Pn .
3.2
Uniqueness results in weighted warped products
It follows from a splitting theorem due to Fang, Li and Zhang (see [34], Theorem 1.1) that if a
product manifold I × P with bounded weighted function f is such that Ric f ≥ 0, then f must
be constant along R. So, motivated by this result, along this section, we will consider weighted
products Pn × R whose weighted function f does not depend on the parameter t ∈ I, that is
h∇ f , ∂t i = 0 and, for sake of simplicity, we will denote them by Pnf × R.
In order to prove our Bernstein type theorems in weighted warped products of the type
32
3.2 UNIQUENESS RESULTS IN WEIGHTED WARPED PRODUCTS
33
I ×ρ P f , we will need some auxiliary lemmas. The first one is an extension of Proposition 4.1
of [6].
Lemma 3.2.1. Let Σn be a hypersurface immersed in a weighted warped product I×ρ Pnf , with
height function h. Then,
(i) ∆ f h = (log ρ)0 (h)(n − |∇h|2 ) + nΘH f ;
(ii) ∆ f σ (h) = n(ρ 0 (h) + ρ(h)ΘH f );
where σ (t) = tt0 ρ(s)dt.
R
Proof. Let us prove item (i). Taking into account that f is constant along R, from (1.3) we get
that
(3.1)
h∇ f , ∇hi = −Θh∇ f , Ni.
On the other hand, from Proposition 4.1 of [6] (see also Proposition 3.2 of [20]) we have
∆h = (log ρ 0 (h)(n − |∇h|2 ) + nHΘ.
(3.2)
Hence, from (1.6), (3.1) and (3.2) we obtain
∆ f h = (log ρ)0 (h)(n − |∇h|2 ) + nHΘ − h∇ f , ∇hi
= (log ρ)0 (h)(n + |∇h|2 ) + nHΘ + Θh∇ f , Ni
= (log ρ)0 (h)(n + |∇h|2 ) + nΘH f .
Moreover, since Proposition 4.1 of [6] also gives that
∆σ (h) = n(ρ 0 (h) + ρ(h)ΘH),
in a similar way we also prove item (ii).
Let us denote by L f1 (Σ) the space of the integrable functions on Σn , in relation to weighted
volume element dµ = e− f dΣ. Since div f (X) = e f div(e− f X), it is not difficult to see that from
Proposition 2.1 of [19] we get the following extension of a result due to Yau in [72].
Lemma 3.2.2. Let u be a smooth function on a complete weighted Riemannian manifold Σn
with weighted function f , such that ∆ f u does not change sign on Σn . If |∇u| ∈ L f1 (Σ), then
∆ f u vanishes identically on Σn .
34
3.2 UNIQUENESS RESULTS IN WEIGHTED WARPED PRODUCTS
We recall that a slab of a warped product I ×ρ Pn is a region of the type
[t1 ,t2 ] × Pn = {(t, q) ∈ I ×ρ Pn : t1 ≤ t ≤ t2 }.
Now, we are in position to state and prove our first result.
Theorem 3.2.1 (Theorem 1 [24]). Let Σn be a complete two-sided hypersurface which lies in
a slab of a weighted warped product I×ρ Pnf . Suppose that Θ ≤ 0 and that H f satisfies
0 < H f ≤ inf(log ρ)0 (h).
(3.3)
Σ
If |∇h| ∈ L f1 (Σ), then Σn is a slice {t} × P.
Proof. From Lemma 3.2.1, since we are assuming that −1 ≤ Θ ≤ 0, we get
∆ f σ (h) = nρ(h) (log ρ)0 (h) + ΘH f
≥ nρ(h) (log ρ)0 (h) − H f
0
≥ nρ(h) inf(log ρ) (h) − H f .
(3.4)
Σ
Thus, taking into account our hypothesis (3.3), from (3.4) we have that ∆ f σ (h) ≥ 0.
On the other hand, since Σn lies in a slab of I×ρ Pn , we have that exists a positive constant
C such that
|∇σ (h)| = ρ(h)|∇h| ≤ C|∇h|.
Consequently, our hypothesis |∇h| ∈ L f1 (Σ) implies that |∇σ (h)| ∈ L f1 (Σ).
So, we can apply Lemma 3.2.2 to assure that ∆ f σ (h) = 0 on Σn . Hence, returning to (3.4)
we get
(log ρ)0 (h) = −ΘH f .
(3.5)
Consequently, using once more hypothesis (3.3), from (3.5) we have that
H f ≤ inf(log ρ)0 (h) ≤ (log ρ)0 (h) = −ΘH f .
(3.6)
Σ
Therefore, from (3.6) we conclude that Θ = −1 and, hence, Σn must be a slice {t} × P.
Remark 3.2.1. Concerning Theorem 3.2.1, we note that if Σn is locally a graph over Pn , then
its angle function Θ is either Θ < 0 or Θ > 0 along Σ. Hence, the assumption that Θ does
not change sign is generally weaker than that of Σn being a local graph. Moreover, as it was
3.2 UNIQUENESS RESULTS IN WEIGHTED WARPED PRODUCTS
35
already observed by Espinar and Rosenberg [33] 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 [55] and relation (1.6), if we consider on the slice
{t} × P of I ×ρ P f the orientation given by N = −∂t , then its f -mean curvature is given by
H f (t) = H(t) = (log ρ)0 (t).
Consequently, the differential inequality (3.3) means that, at each point (t, x) of the hypersurface Σn , the weighted mean curvature of Σn can be any value less than or equal to the value
of the weighted mean curvature of the slice {t} × Pn , with respect to the orientation given by
−∂t . Hence, we only suppose here a natural comparison inequality between weighted mean
curvature quantities, without to require that the weighted mean curvature of Σn be constant. In
this sense, (3.3) 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 3.2.1 we obtain
Corollary 3.2.1. Let Σn be a complete two-sided hypersurface which lies in a slab of a weighted
warped product I×ρ Pnf . Suppose that Θ ≤ 0 and that H f satisfies
0 < H f ≤ inf(log ρ)0 (h).
Σ
If Σn is f -parabolic, then Σn is a slice {t} × P.
Remark 3.2.2. As it was observed by Impera and Rimoldi in Remark 3.8 of [46], the f parabolicity of Σn holds if it has finite f -volume. On the other hand, in the case that Σn is
complete noncompact with Ric f nonnegative, from Theorem 1.1 of [69] a sufficient condition
for Σn to have finite f -volume is that the space of L2f harmonic one-forms be nontrivial.
From the proof of Theorem 3.2.1 we also get the following
Corollary 3.2.2. Let Σn be a complete two-sided hypersurface which lies in a slab of a weighted
warped product I×ρ Pnf . Suppose that Θ ≤ 0 and that H f is constant and satisfies
0 ≤ H f ≤ inf(log ρ)0 (h).
Σ
3.2 UNIQUENESS RESULTS IN WEIGHTED WARPED PRODUCTS
36
If either |∇h| ∈ L f1 (Σ) or Σn is f -parabolic, then Σn is either a f -minimal hypersurface or a
slice {t} × P.
Extending ideas of [6], [9], [10] and [55], in our next result we will assume that the ambient
space is a weighted warped product I×ρ Pnf which obeys the following convergence condition
KP ≥ sup((ρ 0 )2 − ρρ 00 ),
(3.7)
I
where KP stands for the sectional curvature of the fiber Pn .
Theorem 3.2.2 (Theorem 2 [24]). Let I ×ρ Pnf be a weighted warped product which satisfies
the convergence condition (3.7). Let Σn be a complete two-sided hypersurface which lies in a
slab of I ×ρ Pnf . Suppose that ∇ f and the Weingarten operator A of Σn are bounded. If Θ does
not change sign on Σn and
|∇h| ≤ inf (log ρ)0 (h) − |H f | ,
(3.8)
Σ
then Σn is a slice {t} × P.
Proof. From Lemma 3.2.1 we get
∆ f h = (log ρ)0 (h)(n − |∇h|2 ) + nH f Θ
≥ (log ρ)0 (h)(n − |∇h|2 ) − n|H f |
(3.9)
≥ n((log ρ)0 (h) − |H f |) − (log ρ)0 (h)|∇h|2 .
On the other hand, since from the Cauchy-Schwarz inequality we get nH 2 ≤ |A|2 , we have
that H is also bounded. Thus, since we are also assuming that Σn lies in a slice of I ×ρ Pnf , we
can apply Proposition 3.1 of [9] to assure that the Ricci curvature of Σn is bounded from below.
Hence, since |∇ f | is bounded on Σn , from maximum principle of Omori-Yau, there exists a
sequence {pk } in Σn such that
lim h(pk ) = sup h, lim |∇h(pk )| = 0 and lim sup ∆ f h(pk ) ≤ 0.
k
Σ
k
k
Thus, from inequality (3.9) we get
0 ≥ lim sup ∆ f h(pk ) ≥ lim((log ρ)0 (h) − |H f |)(pk ) ≥ 0.
k
k
3.3 RIGIDITY RESULTS IN WEIGHTED PRODUCT SPACES
37
Therefore, we have that limk ((log ρ)0 (h) − |H f |)(pk ) = 0 and, taking into account our hypothesis (3.8), we conclude that Σn 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
I× Pnf . We start with the following uniqueness result.
Theorem 3.3.1 (Theorem 3 [24]). Let Σn be a complete two-sided hypersurface which lies in
a slab of the weighted product I × Pnf . Suppose that Θ and H f do not change sign on Σn . If
|∇h| ∈ L f1 (Σ), then Σn is a slice {t} × P.
Proof. From Lemma 3.2.1 we have that
∆ f h = nH f Θ.
(3.10)
Since we are supposing that Θ and H f do not change sign on Σn , we get that ∆ f h also does not
change sign on Σn . Thus, since |∇h| ∈ L f1 (Σ), we can apply Lemma 3.2.2 to get that ∆ f h = 0
on Σn .
Consequently, we obtain
∆ f h2 = 2h∆ f h + 2|∇h|2 = 2|∇h|2 ≥ 0.
(3.11)
But, since h is bounded and using once more that |∇h| ∈ L f1 (Σ), Lemma 3.2.2 guarantees
also that ∆ f h2 = 0 on Σn . Therefore, returning to (3.11) we conclude that Σn must be a slice
{t} × P.
From the proof of Theorem 3.3.1 we get
Corollary 3.3.1. Let Σn be a complete two-sided hypersurface which lies in a slab of the
weighted product I × Pnf . Suppose that Θ and H f do not change sign on Σn . If Σn is f -parabolic,
then Σn is a slice {t} × P.
Consider the weighted product space R × Gn . Recall that Gn corresponds to the Euclidean
|x|2
space Rn endowed with the Gaussian measure dµ = e− 4 dx2 . Hieu and Nam extended the
classical Bernstein’s theorem [14] showing that the only graphs with weighted mean curvature
identically zero given by a function u(x2 , . . . , xn+1 ) = x1 over Gn are the hyperplanes x1 =
3.3 RIGIDITY RESULTS IN WEIGHTED PRODUCT SPACES
38
constant (cf. [43], Theorem 4). In this setting, with a straightforward computation we can
verify that
1
N=p
(Du − ∂t )
(3.12)
1 + |Du|
gives an orientation on Σn (u) such that −1 ≤ Θ < 0, where Du stands for the gradient of the
function u on Gn . Thus, since |N ∗ | = |∇h|, from (3.12) we deduce that
|∇h|2 =
|Du|2
.
1 + |Du|2
(3.13)
So, taking into account relation (3.13), from the proof of Theorem 3.3.1 jointly with Theorem 4 of [43] we obtain
Corollary 3.3.2. Let Σn (u) be a complete graph of a function u(x2 , . . . , xn+1 ) = x1 over the
Gaussian space Gn . Suppose that the weighted mean curvature of Σn (u) does change sign. If
|Du| ∈ L 1 (G), then Σn (u) is a hyperplane x1 = constant.
To prove our next result, we will need of the following auxiliary lemma.
Lemma 3.3.1. Let Σn be a hypersurface with constant f -mean curvature in a weighted product
manifold Pnf × R. Then
f f (N ∗ , N ∗ ))Θ,
∆ f Θ = −(|A|2 + Ric
(3.14)
f f stands for the Bakry-Émery-Ricci tensor
where A denotes the Weingarten operator of Σ, Ric
of the fiber P and N ∗ = N − Θ∂t is the orthonormal projection of N onto P.
Proof. It is well known that
∇Θ = −A∂tT − (∇N ∂t )T
(3.15)
f ∗ , N ∗ ) + |A|2 ),
∆Θ = −n∂tT (H) − Θ(Ric(N
(3.16)
and
see for instance [20]. Taking into account that h∂t , ∇ f i = 0 we get that
n∂tT (H) = ∂tT (nH f − h∇ f , Ni)
= −∂tT h∇ f , Ni
= −hHess f (∂t ), Ni + ΘHess f (N, N) + hA∂tT , ∇ f i.
(3.17)
3.3 RIGIDITY RESULTS IN WEIGHTED PRODUCT SPACES
39
On the other hand, from (3.15) we get that
h∇Θ, ∇ f i = −hA∂tT + (∇N ∂t )T , ∇ f i
= −hA∂tT , ∇ f i − h∇N ∂t − h∇N ∂t , NiN, ∇ f i
= −hA∂tT , ∇ f i + h∂t , ∇N ∇ f i
(3.18)
= −hA∂tT , ∇ f i + h∂t , Hess f (N)i.
From (3.17) and (3.18) we have
n∂tT (H) = ΘHess f (N, N) − h∇Θ, ∇ f i.
(3.19)
Now, taking into account once more that h∂t , ∇ f i = 0, it is not difficult to verify that
g f (N ∗ , N ∗ ).
Hess f (N, N) = Hess
(3.20)
Putting (3.16) and (3.20) into (3.19) we have the desired result.
Theorem 3.3.2 (Theorem 4 [24]). Let Σn be a complete two-sided hypersurface immersed in
f f ≥ 0, A is bounded, Θ has strict sign H f does
a weighted product I × Pnf . Suppose that Ric
f f > 0, then
change sign on Σn . If |∇h| ∈ L f1 (Σ), then Σn is totally geodesic. Moreover, if Ric
Σn is a slice {t} × P.
Proof. Since we are supposing that Θ has strict sign, H f does not change sign on Σn and that
|∇h| ∈ L f1 (Σ), from and applying Lemma 3.2.2 we get that Σ is f -minimal, that is, H f = 0 on
Σn .
Note that
|∇Θ| ≤ |A||∇h| ∈ L f1 (Σ).
(3.21)
f f ≥ 0, from (3.14) and (3.21) we have that ∆ f Θ = 0 on
So, since we are also supposing that Ric
Σn .
Hence, returning to equation (3.14) we conclude that Σn is totally geodesic. Furthermore,
f f is strictly positive, then N ∗ = 0 on Σn and, therefore, Σn is a slice {t} × P.
if Ric
When the fiber of the weighted product space is compact, we have
Theorem 3.3.3 (Theorem 5 [24]). Let Σn be a complete two-sided hypersurface immersed in a
weighted product I × Pnf , whose fiber Pn is compact with positive sectional curvature, and such
that the weighted function f is convex. Suppose that A is bounded and H f is constant. If Θ is
π
3π
such that either 0 ≤ arccos Θ ≤ or
≤ arccos Θ ≤ π, then Σn is a slice {t} × Pn .
4
4
40
3.3 RIGIDITY RESULTS IN WEIGHTED PRODUCT SPACES
Proof. Firstly, we claim that the Bakry-Émery Ricci tensor of Σn , Ric f , is bounded from below.
Indeed, since we are assuming that Pn is compact with positive sectional curvature, from Gauss
equation it follows that
Ric(X, X) ≥ (n − 1)κ 1 − |∇h|2 |X|2 + nHhAX, Xi − hAX, AXi,
(3.22)
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
Hess f (X, X) = Hess f (X, X) + h∇ f , NihAX, Xi
(3.23)
g f (X ∗ , X ∗ ) + h∇ f , NihAX, Xi
= Hess
≥ h∇ f , NihAX, Xi,
for all X ∈ X(Σ).
Thus, from (3.22) and (3.23) we get
Ric f (X, X) ≥ (n − 1)κ 1 − |∇h|2 |X|2 + nH f hAX, Xi − hAX, AXi.
(3.24)
Consequently, from (3.24) we obtain
Ric f (X, X) ≥ ((n − 1)κ 1 − |∇h|2 − (n|H f ||A| + |A|2 ))|X|2 ,
(3.25)
for all X ∈ X(Σ).
1
We also note that our restriction on Θ amounts to |∇h|2 ≤ . Hence, since A is also bounded
2
and H f is constant, from (3.25) we conclude that Ric f is bounded from below.
On the other hand, using once more that H f is constant, from Lemma 3.2.1 we have
∇∆ f h = nH f ∇Θ.
(3.26)
∇Θ = −A(∇h).
(3.27)
Recall that
Consequently, from (3.26) and (3.27) we get
∇∆ f h = −nH f A(∇h).
(3.28)
3.3 RIGIDITY RESULTS IN WEIGHTED PRODUCT SPACES
41
From (1.4) we have that
∇X ∇h = ∇X (∂t> ) = AXΘ.
(3.29)
|Hess h|2 = |A|2 Θ2 .
(3.30)
Thus, from (3.29) we obtain
Consequently, from (1.5) and (3.30) we get
|Hess h|2 = |A|2 − |∇h|2 |A|2 .
(3.31)
Now, from Bochner’s formula (1.1) we also have that
1
∆ f |∇h|2 = |Hess h|2 + Ric f (∇h, ∇h) + h∇∆ f h, ∇hi.
2
(3.32)
But, from (3.24) it follows that
Ric f (∇h, ∇h) ≥ (n − 1)κ(1 − |∇h|2 )|∇h|2 +nH f hA∇h, ∇hi−hA∇h, A∇hi
≥ (n − 1)κ(1 − |∇h|2 )|∇h|2 + nH f hA∇h, ∇hi − |A|2 |∇h|2 .
(3.33)
Hence, considering (3.28)), (3.31) and (3.33) into (3.32) we get
1
∆ f |∇h|2 ≥ (n − 1)κ(1 − |∇h|2 )|∇h|2 + |A|2 (1 − 2|∇h|2 ).
2
(3.34)
Consequently, from (1.5) and (3.34) jointly with our hypothesis on Θ we obtain
1
∆ f |∇h|2 ≥ (n − 1)κ(1 − |∇h|2 )|∇h|2 .
2
(3.35)
Since Ric f is bounded from below on Σn , from Theorem 1.1.1 we have that there exists a
sequence of points (pk )k≥1 in Σn such that
lim |∇h|2 (pk ) = sup |∇h|2 and lim sup ∆ f |∇h|2 (pk ) ≤ 0.
k
Σ
k
Thus, from (3.35) we have
0 ≥ lim sup ∆ f |∇h|2 (pk ) ≥ (n − 1)κ(1 − sup |∇h|2 ) sup |∇h|2 ≥ 0.
k
Σ
(3.36)
Σ
Consequently, taking into account once more our hypothesis on Θ, from (3.36) we conclude
that supΣ |∇h|2 = 0. Therefore, h is constant on Σn and, hence, Σn is a slice {t} × Pn .
3.3 RIGIDITY RESULTS IN WEIGHTED PRODUCT SPACES
42
Corollary 3.3.3. Let Σn be a f -parabolic complete two-sided hypersurface immersed in a
weighted product I × Pnf , whose fiber Pn is complete with nonnegative sectional curvature,
and such that the weighted function f is convex. Suppose that A is bounded and H f is constant.
It holds the following:
(a) If either 0 ≤ arccos Θ <
π
3π
or
< arccos Θ ≤ π, then Σn is totally geodesic.
4
4
π
3π
(b) If KP is positive and either 0 ≤ arccos Θ ≤ or
≤ arccos Θ ≤ π, then Σn is a slice
4
4
{t} × Pn .
Proof. In a very similar way of that was made in order to prove inequality (3.34), taking a local
orthonormal frame {E1 , . . . , En } on Σ, we obtain that
1
∆ f |∇h|2 ≥ (n − 1) min KP ((∇h)∗ , Ei∗ )(1 − |∇h|2 )|∇h|2 + |A|2 (1 − 2|∇h|2 ).
i
2
(3.37)
If we assume the hypothesis of item (a), since we are also supposing that KP is nonnegative,
from (3.37) we get that ∆ f |∇h|2 ≥ 0. Hence, since we are also assuming that Σn is f -parabolic,
we have that |∇h| is constant on Σ. Therefore, returning to (3.37) we conclude that |A| = 0 on
Σ, that is, Σ is totally geodesic.
Now, we assume the hypothesis of item (b). Since κ := mini KP ((∇h)∗ , Ei∗ ) > 0, from (3.37)
we also get that
1
∆ f |∇h|2 ≥ (n − 1)κ(1 − |∇h|2 )|∇h|2 ≥ 0.
(3.38)
2
Hence, using once more that Σn is f -parabolic and noting that |∇h| < 1, from (3.38) we conclude that |∇h| = 0 on Σn , that is, Σn must be a slice {t} × Pn .
Proceeding, we obtain an extension of Theorem 3.1 of [31].
Theorem 3.3.4 (Theorem 6 [24]). Let Σn be a complete two-sided hypersurface immersed in
a weighted product I × Pnf , whose fiber Pn has sectional curvature KP satisfying KP ≥ −κ and
g f ≥ −γ, for some positive constants κ and γ. Suppose that A is bounded, Θ is
such that Hess
bounded away from zero and H f is constant. If the height function h of Σn satisfies
|∇h|2 ≤
α
|A|2 ,
(n − 1)κ + γ
(3.39)
for some constant 0 < α < 1, then Σn 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
43
Moreover, since we are also assuming that the sectional curvature KP of the base Pn is such
that KP ≥ −κ for some κ > 0, from Gauss equation and with a straightforward computation
we get
f ∗ , N ∗ ) ≥ −(n − 1)κ|N ∗ |2 = −(n − 1)κ|∇h|2 .
Ric(N
(3.40)
g f jointly with equation 1.5 give
On the other hand, our restriction on Hess
g f (N ∗ , N ∗ ) ≥ −γ|N ∗ |2 = −γ|∇h|2 .
Hess
(3.41)
Thus, from (3.40) and (3.41) we get
f f (N ∗ , N ∗ ) ≥ −((n − 1)κ + γ)|∇h|2 .
Ric
(3.42)
Hence, from (3.42) and (3.14) we obtain
∆ f Θ ≤ −(|A|2 − ((n − 1)κ + γ)|∇h|2 )Θ.
(3.43)
Consequently, from (3.39) and (3.43) we have
∆ f Θ ≤ −(1 − α)|A|2 Θ.
(3.44)
Now, we claim that Ric f is bounded from below on Σn . Indeed, following similar ideas of
that in the proof of Theorem 3.3.3 we get
Hess f (X, X) ≥ h∇ f , NihAX, Xi − γ|X ∗ |2
(3.45)
= h∇ f , NihAX, Xi − γ(|X|2 − hX, ∂t i2 )
≥ h∇ f , NihAX, Xi − γ|X|2 .
From (3.22) and (3.45) we obtain
Ric f (X, X) ≥ ((n − 1)κ 1 − |∇h|2 − γ)|X|2 + nH f hAX, Xi − hAX, AXi.
(3.46)
Thus, since A and |∇h| are bounded and H f is constant, from (3.46) we conclude that Ric f is
bounded from below.
Now, we are in position to apply Theorem 1.1.1 and guarantee the existence of a sequence
3.3 RIGIDITY RESULTS IN WEIGHTED PRODUCT SPACES
44
of points pk ∈ Σn satisfying
lim inf ∆ f Θ(pk ) ≥ 0 and lim Θ(pk ) = inf Θ(p).
k→∞
p∈Σ
k→∞
Consequently, since we are assuming that A is bounded on Σn , from (3.44), up to a subsequence,
we get
0 ≤ lim inf ∆ f Θ(pk ) ≤ −(1 − α) lim |A|2 (pk ) inf Θ(p) ≤ 0.
(3.47)
k→∞
k→∞
p∈Σ
Thus, from (3.47) we obtain that limk→∞ |A|(pk ) = 0 and, using (3.39), we get
lim |∇h|(pk ) = 0.
k→∞
(3.48)
Therefore, from (1.5) and (3.48) we conclude that inf p∈Σ Θ(p) = 1 and, hence, Θ ≡ 1, that is,
Σn is a slice {t} × P.
To close our section, we will apply a Liouville type result due to Huang et al. [45] in order
to prove the following:
Theorem 3.3.5 (Theorem 7 [24]). Let Σn be a complete two-sided hypersurface which lies in
a slab of a weighted product I × Pnf , whose fiber Pn has sectional curvature KP bounded from
below and such that ∇ f is bounded. Suppose that A and H f are bounded on Σn . If Θ is not
adhere to 1 or −1, then infΣ H f = 0. Moreover, if H f is constant and Ric f is nonnegative, then
Σn is a slice {t} × P.
Proof. Following similar steps of the proof of Theorem 3.3.3, our restriction on the sectional
curvature of the fiber Pn jointly with our hypothesis on A, H f and ∇ f guarantee that the Ricci
curvature of Σn is bounded from below.
Now, suppose for instance that H f ≥ 0 on Σn . Thus, since Σn lies between two slices of
R × Pn , from Lemma 3.2.1 and maximum principle of Omori-Yau, we obtain a sequence of
points pk ∈ Σn such that
0 ≥ lim sup ∆ f h(pk ) = n lim sup H f Θ (pk ).
k→∞
k→∞
Moreover, from equation (1.5) we also have that
0 = lim |∇h|(pk ) = 1 − lim Θ2 (pk ).
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(pk ) = 1.
k→∞
Consequently,
0 ≥ lim sup ∆ f h(pk ) = n lim sup H f (pk ) ≥ 0
k→∞
k→∞
and, hence, we conclude that
lim sup H f (pk ) = 0.
k→∞
If H f ≤ 0, from Lemma 3.2.1 and (1.5), we can apply once more Omori-Yau’s generalized
maximum principle in order to obtain a sequence qk ∈ Σn such that
0 ≤ lim inf ∆ f h(qk ) = n lim inf H f Θ (qk )
k→∞
k→∞
and, supposing once more that Θ is not adhere to −1, we get
0 ≤ lim inf ∆ f h(pk ) = n lim inf H f (pk ) ≤ 0.
k→∞
k→∞
Consequently, from the above inequality we have that lim infk→∞ H f (pk ) = 0. Hence, in this
case, we also conclude that infΣ H f = 0.
When H f is constant, we have that, in fact, H f vanishes identically on Σn . Thus, since Σn is
contained in a slab of R × Pnf , there exists a constant β such that h − β is a positive harmonic
function of the f -Laplacian on Σn .
On the other hand, since ∇ f = ∇ f − h∇ f , N ∗ iN and |N ∗ | = |∇h|, we obtain
|∇ f |2 ≤ |∇ f |2 (1 − |∇h|2 ).
Consequently, since we are assuming that ∇ f is bounded, we have that ∇ f is also bounded on
Σn . Hence, if Ric f is nonnegative, then we can apply Corollary 1.4 of [45] to conclude that h
is constant on Σn , that is, Σn is a slice {t} × Pn .
3.4
Uniqueness results in weighted GRW spacetimes
We observe that it follows from a splitting theorem due to Case (cf. [21], 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,
along this work we will consider weighted GRW spacetimes −I ×ρ P whose weight function
f does not depend on the parameter t ∈ I, that is h∇ f , ∂t i = 0. For simplicity, we will denote
them by −I ×ρ P f .
In order to prove our Calabi-Bernstein’s type results in weighted GRW spacetimes of the
type −I ×ρ P f , we will need some auxiliary lemmas. The first one is an extension of Lemma
4.1 of [4], and its demonstration follows the same steps of 3.2.1.
Lemma 3.4.1. Let Σn be a spacelike hypersurface immersed in a weighted GRW spacetime
−I×ρ P f , with height function h. Then,
(i) ∆ f h = −(log ρ)0 (h)(n + |∇h|2 ) − nH f Θ;
(ii) ∆ f σ (h) = −n(ρ 0 (h) + ρ(h)ΘH f ),
where σ (t) = tt0 ρ(s)ds.
R
In what follows, a slab [t1 ,t2 ] × M n = {(t, q) ∈ −I ×ρ Pn : t1 ≤ t ≤ t2 } is called a timelike
bounded region of the weighted GRW spacetime −I ×ρ Pnf . Now, we are in position to state
and prove our first result.
Theorem 3.4.1 (Theorem 1 [25]). Let Σn be a complete spacelike hypersurface which lies in
a timelike bounded region of a weighted GRW spacetime −I ×ρ Pnf . Suppose that the f -mean
curvature H f of Σn satisfies the following inequality
H f ≥ sup(log ρ)0 (h) > 0.
(3.49)
Σ
If |∇h| ∈ L f1 (Σ), then Σn is a slice {t} × P.
Proof. From Lemma 3.4.1, for σ (t) = tt0 ρ(s)ds, we get
R
∆ f σ (h) = −nρ(h) (log ρ)0 (h) + ΘH f
≥ nρ(h) H f − (log ρ)0 (h)
0
≥ nρ(h) H f − sup(log ρ) (h) .
(3.50)
Σ
Thus, taking into account our hypothesis (3.49), from (3.50) we have that ∆ f σ (h) ≥ 0 on Σn .
On the other hand, since Σn is contained in a slab of −I ×ρ Pnf , we have that exists a positive
constant C such that
|∇σ (h)| = ρ(h)|∇h| ≤ C|∇h|.
3.4 UNIQUENESS RESULTS IN WEIGHTED GRW SPACETIMES
47
Consequently, the hypothesis |∇h| ∈ L f1 (Σ) implies that |∇σ (h)| ∈ L f1 (Σ).
So, we can apply Lemma 3.2.2 to assure that ∆ f σ (h) = 0 on Σn . Thus, returning to (3.50)
we get
(log ρ)0 (h) = −ΘH f .
Consequently, we have that
H f = sup(log ρ)0 (h) ≥ (log ρ)0 (h) = −ΘH f .
Σ
Therefore, Θ = −1 on Σn and, hence, we conclude that Σn is a slice {t} × P.
From the proof of Theorem 3.4.1 we also get the following.
Corollary 3.4.1. Let Σn be a complete spacelike hypersurface immersed in a timelike bounded
region of a weighted GRW spacetime −I×ρ Pnf . Suppose that Σn has constant f -mean curvature
H f and that it holds the following inequality
H f ≥ sup(log ρ)0 (h) ≥ 0.
Σ
If |∇h| ∈ L f1 (Σ), then Σn is either a f -maximal hypersurface or a slice {t} × P.
According to the terminology established by Alías and Colares in [4], we say that a GRW
spacetime −I ×ρ Pn obeys the strong null convergence condition when the sectional curvature
KP of its Riemannian fiber P satisfies the following inequality
KP ≥ sup( f 2 (log f )00 ).
(3.51)
I
Theorem 3.4.2 (Theorem 2 [25]). Let −I ×ρ Pn be a GRW spacetime obeying (3.51). Let Σn
be a complete spacelike hypersurface which lies in a timelike bounded region of the weighted
GRW spacetime −I ×ρ Pnf . Suppose that |∇ f | is bounded on Σn and that the f -mean curvature
H f of Σn satisfies
(log ρ)0 (h) ≤ H f ≤ α,
(3.52)
for some constant α. If
|∇h| ≤ inf H f − (log ρ)0 (h) ,
Σ
then Σn is a slice {t} × P.
(3.53)
3.5 WEIGHTED STATIC GRW SPACETIMES
48
Proof. From Lemma 3.4.1 we have
∆ f h = −(log ρ)0 (h)(n + |∇h|2 ) − nH f Θ.
Since N is a future-directed timelike vector field, we get
∆ f h ≥ n(H f − (log ρ)0 (h)) − (log ρ)0 (h)|∇h|2 .
(3.54)
We claim that the mean curvature of Σn is bounded. Indeed, we have that
n|H| = n|H f | + |h∇ f , Ni|
= n|H f | + |h∇ f , N ∗ i|.
(3.55)
On the other hand, taking into account that N ∗ = N + Θ∂t , we easily verify that |N ∗ | = |∇h|.
Thus, from (3.55) we get
(3.56)
n|H| ≤ n|H f | + |∇ f ||∇h|.
Consequently, since H f , ∇ f and ∇h are supposed to be bounded, it follows from (3.56) that H
is also bounded on Σn .
Hence, since we are assuming that ∇ f is bounded then we can apply Proposition 3.1 of [18]
jointly with generalized maximum principle of Omori and Yau to guarantee that there exists a
sequence {pk } in Σn such that
lim h(pk ) = sup h, lim |∇h(pk )| = 0 and lim sup ∆ f h(pk ) ≤ 0.
k
Σ
k
k
Thus, from inequality (3.54) we get
0 ≥ lim sup ∆ f h(pk ) ≥ lim(H f − (log ρ)0 (h))(pk ) ≥ 0.
k
k
Therefore, we have that limk (H f − (log ρ)0 (h))(pk ) = 0 and, taking into account our hypothesis
(3.53), 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
Lemma 3.5.1. Let Σn be a spacelike hypersurface with constant f -mean curvature H f in a
weighted static GRW spacetime −I × P f . Then,
f f (N ∗ , N ∗ ))Θ,
∆ f Θ = (|A|2 + Ric
(3.57)
where A denotes the Weingarten operator of Σn with respect to the future-pointing Gauss map
f f stands for the Bakry-Émery Ricci tensor of the fiber P.
N of Σn and Ric
The proof of Lemma 3.5.1 follows the sames steps of Lemma 3.3.1 and, therefore, it will
be omitted. Now, we can return to our uniqueness results.
Theorem 3.5.1 (Theorem 3 [25]). Let Σn be a complete spacelike hypersurface in a weighted
static GRW spacetime −I × P f , such that its f -mean curvature H f does not change sign. If
|∇h| ∈ L f1 (Σ), then Σn is f -maximal. In addition, we also have the following:
f f ≥ 0, then Σn is totally geodesic. Moreover, if Ric
f f > 0, then Σn
(i) If A is bounded and Ric
is a slice {t} × P.
(ii) If Σn lies in a timelike bounded region of −I × P f , then Σn is a slice {t} × P.
Proof. From Lemma 3.4.1 we have
∆ f h = −nH f hN, ∂t i.
(3.58)
Since we are supposing that H f does not change sign on Σn , from equation (3.58) we get that
∆ f h also does not change sign on Σn . Thus, since |∇h| ∈ L f1 (Σ), we can apply Lemma 3.2.2
to conclude that ∆ f h vanishes identically on Σn . Hence, returning to equation (3.58) we obtain
that Σn is f -maximal.
Assuming the hypothesis of item (i), we have
|∇Θ| ≤ |A||∇h| ∈ L f1 (Σ).
f f ≥ 0, then from Lemmas 3.2.2 and equation (3.57) we get that ∆ f Θ = 0 on Σn .
So, if Ric
f f > 0, then (3.57)
Hence, from (3.57) we conclude that Σn is totally geodesic. Moreover, if Ric
also gives that N ∗ = 0 on Σn , that is, Σn is a slice {t} × P.
On the other hand, since Σn is f -maximal, we have
∆ f h2 = 2h∆ f h + 2|∇h|2 = 2|∇h|2 .
50
3.5 WEIGHTED STATIC GRW SPACETIMES
Now, since |∇h| ∈ L f1 (Σ) and assuming that Σn lies in a timelike bounded region of −I × P f ,
then from Lemma 3.2.2 we get that ∆ f h2 = 0 on Σn . Hence, we obtain that |∇h| = 0 on Σn ,
which proves item (ii).
When the fiber of the ambient spacetime is compact, we have
Theorem 3.5.2 (Theorem 4 [25]). Let −R × Pnf be a static weighted GRW spacetime, whose
fiber Pn is compact with positive sectional curvature and such that the weighted function f is
convex. Let Σn be a complete spacelike hypersurface with constant f -mean curvature H f in
−R × Pnf . If |∇h| is bounded, then Σn is a slice {t} × P.
Proof. Using the fact that Pn is compact with KP > 0, it follows from inequalities (3.3) and
(3.4) of [32] that there exists a positive constant κ such that
Ric(X, X) ≥ κ (n − 1)|X|2 + |∇h|2 |X|2 + (n − 2)hX, ∇hi2
+nHhAX, Xi + |AX|2 .
(3.59)
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
Hess f (X, X) ≥ −h∇ f , NihAX, Xi.
(3.60)
From (3.59) and (3.60) we get the following lower bound for Ric f
Ric f (X, X) ≥ κ (n − 1)|X|2 + |∇h|2 |X|2 + (n − 2)hX, ∇hi2
+nH f hAX, Xi + |AX|2 .
(3.61)
Inequality (3.61) provides us
Ric f (∇h, ∇h) ≥ (n − 1)κ|∇h|2 (1 + |∇h|2 )
+nH f hA(∇h), ∇hi + |A(∇h)|2 .
(3.62)
Since H f is constant, we have
∇∆ f h = −nH f A(∇h).
(3.63)
3.5 WEIGHTED STATIC GRW SPACETIMES
51
Moreover, from Bochner’s formula (1.1), we have again (3.32). Consequently, from (3.62),
(3.63) and (3.32) we get
1
∆ f |∇h|2 ≥ (n − 1)κ|∇h|2 (1 + |∇h|2 ).
2
(3.64)
Now, we observe that we can write
2
2
nH f 2 n H f
X −
|X|2 .
nH f hAX, Xi + |AX| = AX +
2
4
2
(3.65)
Thus, from (3.61) and (3.65) we obtain that
Ric f (X, X) ≥ −
n2 H 2f
4
|X|2 ,
(3.66)
for all X ∈ X(Σ).
Consequently, from equation (3.66) we conclude that the Bakry-Émery Ricci tensor of Σn
is bounded from below. Hence, from Theorem 1.1.1 we have that there exists a sequence of
points (pk )k≥1 in Σn such that
lim |∇h|2 (pk ) = sup |∇h|2 and lim sup ∆ f |∇h|2 (pk ) ≤ 0.
k
Σ
k
Now, returning to (3.64), we conclude that
0 ≥ lim sup ∆ f |∇h|2 (pk ) ≥ κ sup |∇h|2 ≥ 0.
k
Σ
Consequently, we obtain that supΣ |∇h|2 = 0 and, hence, h is constant on Σn . Therefore, Σn is a
slice {t} × Pn .
Proceeding, we obtain the following
Theorem 3.5.3 (Theorem 5 [25]). Let −R × Pnf be a weighted static GRW spacetime, whose
sectional curvature KP of its fiber Pn is such that KP ≥ −κ for some positive constant κ and
such that the weighted function f is convex. Let Σn be a complete spacelike hypersurface with
constant f -mean curvature H f in −R × Pnf and with bounded second fundamental form A. If
|∇h|2 ≤
α
|A|2 ,
κ(n − 1)
for some constant 0 < α < 1, then Σn is a slice {t} × Pn .
(3.67)
52
3.5 WEIGHTED STATIC GRW SPACETIMES
Proof. From equation (3.67) we get inf p∈Σ Θ(p) exists and it is negative.
On the other hand, since the weighted function f is convex, |N ∗ | = |∇h| and taking a local
orthonormal frame {E1 , . . . , En } on Pn , we have that
f f (N ∗ , N ∗ ) ≥ Ric(N
f ∗, N ∗)
Ric
=
∑hRP(N ∗, Ei)N ∗, EiiP
i
=
∑ KP(N ∗, Ei) hN ∗, N ∗iP − hN ∗, Eii2P
(3.68)
i
≥ −κ ∑ hN ∗ , N ∗ iP − hN ∗ , Ei i2P
i
= −κ(n − 1)|∇h|2 ,
where we also have used our restriction on the sectional curvature KP of Pn .
Thus, from equation (3.57) jointly with (3.67) and (3.68), we get
|A|2 − κ(n − 1)|∇h|2 Θ
∆f Θ ≤
≤ (1 − α)|A|2 Θ ≤ 0.
On the other hand, following the same ideas of Theorem 3.5.2, we can verify that the BakryÉmery Ricci tensor of Σn is bounded from below and, hence, from Theorem 1.1.1 there exists a
sequence of points pk ∈ Σn such that
lim inf ∆ f Θ(pk ) ≥ 0
k
and
lim Θ(pk ) = inf Θ.
k
p∈Σ
Consequently,
lim Θ2 (pk ) = sup Θ2 .
k
p∈Σ
Thus,
0 ≤ lim inf ∆ f Θ(pk ) ≤ (1 − α) lim inf |A|2 (pk ) inf Θ ≤ 0.
k
k
p∈Σ
Up to a subsequence, it follows that limk |A|2 (pk ) = 0. Now, by using hypothesis (3.67), we
obtain that limk |∇h|2 (pk ) = 0 and therefore sup p∈Σ Θ2 = limk Θ2 (pk ) = 1. But Θ2 ≥ 1, hence,
Θ2 = 1 on Σn and, therefore, Σn is a slice {t} × Pn .
From the proof of Theorem 3.5.3 it is not difficult to see that we also get
3.5 WEIGHTED STATIC GRW SPACETIMES
53
Corollary 3.5.1. Let −R × Pnf be a weighted static GRW spacetime, such that KP ≥ −κ and
Hess f ≥ −γ for some positive constants κ and γ. Let Σn be a complete spacelike hypersurface
with constant f -mean curvature H f in −R × Pnf and with bounded second fundamental form A.
If
α
|∇h|2 ≤
|A|2 ,
κ(n − 1) + γ
for some constant 0 < α < 1, then Σn is a slice {t} × Pn .
We close our chapter, with the following result
Theorem 3.5.4 (Theorem 6 [25]). Let −R × Pnf be a weighted static GRW spacetime, such
that KP ≥ 0 and |∇ f | is bounded. Let Σn be a complete spacelike hypersurface immersed in a
timelike bounded region of −R × Pnf . If |∇h| and H f are bounded and H f does not change sign
on Σn , then H f is not globally bounded away from zero. In particular, if f is convex and H f is
constant, then Σn is a slice {t} × Pn .
Proof. Taking into account our restriction on the sectional curvature of the fiber Pn jointly with
the hypothesis that |∇ f |, |∇h| and H f are bounded on Σn , as in the proof of Theorem 3.4.2 we
can apply Proposition 3.2 of [32] to guarantee that the Ricci curvature of Σn is bounded from
below.
Now, suppose for instance that H f ≥ 0 on Σn . Thus, since Σn lies between two slices
of −R × Pn , from Lemma 3.4.1 and the generalized maximum principle of Omori [57] and
Yau [71] we obtain a sequence of points pk ∈ Σn such that
0 ≤ lim inf ∆ f (−h)(pk ) = n lim inf H f Θ (pk ).
k
k
On the other hand, note that
0 = lim |∇h|(pk ) = lim Θ2 (pk ) − 1.
k
k
Thus, since Θ ≤ −1,
lim Θ(pk ) = −1.
k
Consequently,
0 ≤ lim inf ∆ f (−h)(pk ) = −n lim inf H f (pk ) ≤ 0
k
k
and, hence, we conclude that
lim inf H f (pk ) = 0.
k
3.5 WEIGHTED STATIC GRW SPACETIMES
54
If H f ≤ 0, with the aid of Lemma 3.4.1 and applying once more the generalized maximum
principle of Omori [57] and Yau [71], we get a sequence qk ∈ Σn such that
0 ≤ lim inf ∆ f h(qk ) = −n lim inf H f Θ (qk )
k
k
and
lim Θ(qk ) = −1.
k
Therefore, we conclude again that H f is not globally bounded away from zero.
When H f is constant, we have that, in fact, H f vanishes identically on Σn . Thus, since Σn is
contained into a timelike bounded region of −R × Pnf , there exists a constant β such that h − β
is a positive harmonic function of the drifting Laplacian on Σn . Moreover, since f is supposed
convex, from inequality (3.61) to get that Ric f ≥ 0.
On the other hand, since ∇ f = ∇ f + h∇ f , N ∗ iN and |N ∗ | = |∇h|, we obtain
|∇ f |2 ≤ |∇ f |2 (1 + |∇h|2 ).
Consequently, since we are assuming that |∇ f | and |∇h| are bounded, we have that ∇ f is also
bounded on Σn .
Therefore, we are in position to apply Corollary 1.4 of [45] and conclude that h is constant
on Σn , that is, Σn is a slice {t} × Pn .
C HAPTER 4
Height Estimate for weighted semi-Riemannian
manifolds
The results of this chapter are part of [23].
4.1
Introduction
In 1954 Heinz [42] proved that a compact graph of positive constant mean curvature H in the
(n + 1)-dimensional Euclidean space Rn+1 with boundary on a hyperplane can reach at most
height H1 from the hyperplane. A hemisphere in Rn+1 of radius H1 shows that this estimate
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, [2, 3, 26, 39, 44, 53]).
Concerning the Lorentzian setting, López [52] obtained a sharp estimate for the height of
compact constant mean curvature spacelike surfaces with boundary contained in a spacelike
plane of the 3-dimensional Lorentz-Minkowski space L3 . Later on, de Lima [30] 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)dimensional Lorentz-Minkowski space Ln+1 . As in [52], through the computation of the height
of the hyperbolic caps of Ln+1 , he showed that his estimate is sharp. Afterwards, he jointly
with Colares [27] generalized the results of [30] to the context of the Lorentzian product spaces
−R × Pn .
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.
Theorem 4.2.1 (Theorem 1 [23]). Let I × Pnf be a weighted Riemannian product space with
Ric f ≥ 0 and let Σn be a compact hypersurface with boundary contained into the slice {s} × Pn ,
for some s ∈ I, and whose angle function Θ does not change sign. If Σn has nonzero constant
55
4.2 THE RIEMANNIAN SETTING
56
f -mean curvature such that nH 2f ≤ |A|2 , where A denotes the Weingarten operator of Σn with
respect to its unit normal vector field N, then the height function h of Σn satisfies
1
.
|H f |
(4.1)
ϕ = H f h + Θ.
(4.2)
f f (N ∗ , N ∗ )).
∆ f ϕ = −Θ(|A|2 − nH 2f + Ric
(4.3)
|h − s| ≤
Proof. Define on Σn the function
From (3.14) and (3.10) we get that
f f (N ∗ , N ∗ ) = Ric f (N, N) ≥ 0, nH 2 ≤ |A|2 and choosing N such that
Consequently, since Ric
f
−1 ≤ Θ ≤ 0, from (4.3) we get that ∆ f ϕ ≥ 0. Thus, we conclude from the maximum principle
that ϕ ≤ ϕ|∂ Σ and, hence, from (4.2) we have that
H f h − 1 ≤ H f h + Θ ≤ H f s.
(4.4)
We then consider the two possible cases. In the case that H f > 0, from (3.10) we have
∆ f h ≤ 0 and, from the maximum principle, h ≥ s on Σn . Thus, from (4.4) we conclude that
h−s ≤
1
.
Hf
(4.5)
Finally, in the case that H f < 0, from (3.10) we have ∆ f h ≥ 0 and, again from the maximum
principle, h ≤ s on Σn . Thus, from (4.4) we must have
s−h ≤ −
1
.
Hf
(4.6)
Therefore, estimate (4.1) follows from (4.5) and (4.6).
Remark 4.2.1. We point out that the hypothesis nH 2f ≤ |A|2 is automatically satisfied in the case
that the weighted function f is constant. Furthermore, taking into account Heinz’s estimate [42]
already commented in the introduction, we see that our estimate (4.1) is optimal.
From Theorem 4.2.1 we obtain the following half-space type result
Theorem 4.2.2 (Theorem 2 [23]). Let R × Pnf be a weighted Riemannian product space with
Ric f ≥ 0 and Pn compact. Let Σn be a complete noncompact two-sided hypersurface properly
57
4.3 THE LORENTZIAN SETTING
immersed in R × Pnf , whose angle function Θ does not change sign. If Σn has nonzero constant
f -mean curvature such that nH 2f ≤ |A|2 , then Σn cannot lie in a half-space of R×P. In particular,
Σn must have at least one top and one bottom end.
Proof. Suppose by contradiction that, for instance, Σn ⊂ (−∞, τ] × P, for some τ ∈ R. Thus,
for each s < τ we define
n
Σ+
s = {(t, x) ∈ Σ : t ≥ s} .
Since P is compact and Σn is properly immersed in R × Pnf , we have that Σ+
s is a compact
hypersurface contained in a slab of width τ − s and with boundary in {s} × P. Thus, we can
1
apply Theorem 4.2.1 to get that Σ+
s is contained in a slab of width |H | , so that it must be
f
τ − s ≤ |H1 | . Consequently, choosing s sufficiently small we violate this estimate, reaching to a
f
contradiction.
Analogously, if we suppose that Σn ⊂ [τ, +∞) × P with τ ∈ R, for each s > τ we define Σ−
s
by
Σ−
s = {(t, x) ∈ Σ;t ≤ s} .
Hence, since Σ−
s is a compact hypersurface with boundary in {s} × P, we can reason as in the
previous case and obtain another contradiction.
4.3
The Lorentzian setting
We proceeding with our second height estimate.
Theorem 4.3.1 (Theorem 3 [23]). Let −I × Pnf be a weighted Lorentzian product space with
Ric f ≥ 0 and let Σn be a compact spacelike hypersurface with boundary contained into the slice
{s} × Pn , for some s ∈ I. If Σn has nonzero constant f -mean curvature such that nH 2f ≤ |A|2 ,
where A denotes the Weingarten operator of Σn with respect to its future-pointing unit normal
vector field N, then the height function h of Σn satisfies
|h − s| ≤
max∂ Σ |Θ| − 1
.
|H f |
(4.7)
Proof. Analogously, we define on Σn the function
ϕ = −H f h − Θ.
(4.8)
4.3 THE LORENTZIAN SETTING
58
From equation 3.57 and Lemma 3.4.1 we get that
f f (N ∗ , N ∗ )).
∆ f ϕ = −Θ(|A|2 − nH 2f + Ric
f f (N ∗ , N ∗ ) = Ric f (N, N) ≥ 0, nH 2 ≤ |A|2 and choosing N futureConsequently, since Ric
f
pointing (that is, Θ ≤ −1), we get that ∆ f ϕ ≥ 0. Thus, we conclude from the maximum
principle that ϕ ≤ ϕ|∂ Σ and, hence, from (4.8) we have that
− H f h + 1 ≤ −H f h − Θ ≤ −H f s + max |Θ|.
(4.9)
∂Σ
We then consider the two possible cases. In the case that H f > 0, note that ∆ f h ≥ 0 and,
from the maximum principle, h ≤ s on Σn . Thus, from (4.9) we conclude that
s−h ≤
max∂ Σ |Θ| − 1
.
Hf
(4.10)
Finally, in the case that H f < 0, we have ∆ f h ≤ 0 and, again from the maximum principle, h ≥ s
on Σn . Thus, from (4.9) we must have
h−s ≤
1 − max∂ Σ |Θ|
.
Hf
(4.11)
Therefore, estimate (4.7) follows from (4.10) and (4.11).
Remark 4.3.1. Taking into account the height estimate of [30] mentioned in the introduction,
we see that our estimate (4.7) is also sharp.
Finally, reasoning as in the proof of Theorem 4.2.2, from Theorem 4.3.1 we get the following
Theorem 4.3.2 (Theorem 4 [23]). Let −R × Pnf be a weighted Lorentzian product space with
Ric f ≥ 0 and Pn compact. Let Σn be a complete noncompact spacelike hypersurface properly
immersed in −R × Pnf , with bounded angle function Θ. If Σn has nonzero constant f -mean
curvature such that nH 2f ≤ |A|2 , then Σn cannot lie in a half-space of −R × P. In particular, Σn
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 ∂t is called a
comoving observer and, for a fixed point p ∈ Σn , ∂t (p) is called an instantaneous comoving
observer. In this setting, among the instantaneous observers at p, ∂t (p) and N(p) appear naturally. From the orthogonal decomposition N(p) = N ∗ (p) − Θ(p)∂t (p), we have that |Θ(p)|
4.3 THE LORENTZIAN SETTING
59
corresponds to the energy e(p) that ∂t (p) measures for the normal observer N(p). Furthermore, the speed |υ(p)| of the Newtonian velocity υ(p) := e−1 (p)N ∗ (p) that ∂t (p) measures
for N(p) satisfies the equation |υ(p)|2 = tanh(cosh−1 |Θ(p)|). Hence, the boundedness of the
angle function Θ of the spacelike hypersurface Σn means, physically, that the speed of the Newtonian velocity that the instantaneous comoving observer measures for the normal observer do
not approach the speed of light 1 on Σn (cf. [65], Sections 2.1 and 3.1). In this direction, as
it was already observed by Latorre and Romero [47], 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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