Spacelike submanifolds in semi-Riemannian product spaces: an approach via maximum principles, parabolicity and conditions of volume growth

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Programa Associado de P´ os-Gradua¸ c˜ ao em Matem´ atica Doutorado em Matem´ atica

Spacelike submanifolds in semi-Riemannian product spaces: an approach via maximum

principles, parabolicity and conditions of volume growth

por

Danilo Ferreira da Silva

Jo˜ao Pessoa-PB 2022

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Programa Associado de P´ os-Gradua¸ c˜ ao em Matem´ atica Doutorado em Matem´ atica

Spacelike submanifolds in semi-Riemannian product spaces: an approach via maximum

principles, parabolicity and conditions of volume growth

por

Danilo Ferreira da Silva

Tese apresentada ao Corpo Docente do Pro- grama Associado de Pós-Gradua¸cão em Matemática (PAPGM) da UFPB/UFCG como requisito parcial para a obten¸cão do t´ıtulo de Doutor em Matemática.

Este exemplar corresponde a vers˜ao final da tese defendida pelo aluno Danilo Ferreira da Silva e aprovada pela comiss˜ao julgadora.

Eraldo Almeida Lima J´unior (Orientador)

(Coorientador)

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S586s Silva, Danilo Ferreira da.

Spacelike submanifolds in semi-Riemannian product spaces : an approach via maximum principles,

parabolicity and conditions of volume growth / Danilo Ferreira da Silva. - João Pessoa, 2022.

73 f. : il.

Orientação: Eraldo Almeida Lima Júnior.

Tese (Doutorado) - UFPB/CCEN.

1. Matemática. 2. Geometria Riemanniana. 3.

Subvariedade tipo-espaço. 4. Princípio do máximo. 5.

Espaço Gaussiano. 6. Espaço pseudo-Gaussiano. I. Lima Júnior, Eraldo Almeida. II. Título.

UFPB/BC CDU 51(043)

Elaborado por RUSTON SAMMEVILLE ALEXANDRE MARQUES DA SILVA - CRB-15/0386

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Programa Associado de P´ os-Gradua¸c˜ ao em Matem´ atica Doutorado em Matem´ atica

Tese apresentada ao Corpo Docente do Programa Associado de Pós-Gradua¸cão em Matemática (PAPGM) da UFPB/UFCG como requisito parcial para a obten¸cão do t´ıtulo de Doutor em Matemática.

Area de Concentra¸c˜´ ao: Geometria/Topologia Aprovada em: 29 de Junho de 2022

Prof. Dr. Eraldo Almeida Lima J´unior Orientador

Universidade Federal da Para´ıba

Prof. Dr. M´arcio Silva Santos Universidade Federal da Para´ıba

Prof. Dr. Carlos Augusto Romero Filho Universidade Federal da Para´ıba

Prof. Dr. Allan George de Carvalho Freitas Universidade Federal da Para´ıba

Prof. Dr. F´abio Reis dos Santos Universidade Federal do Pernambuco

Prof. Dr. Alfonso Romero Sarabia Universidade de Granada

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Primeiramente agrade¸co a Deus pela sa´ude, disposi¸c˜ao e perseveran¸ca para realizar esse trabalho.

Agrade¸co a minha fam´ılia, minha mãe Jucileide, meus irmãos Diego, Douglas, Genilson e minha namorada Cristina por acreditarem em mim e sempre me apoiarem na busca desse sonho de ser doutor em matemática.

Agrade¸co aos amigos que fiz nessa jornada de estudo que tornaram os dias mais leves e compensador, quero citar aqui: Keyson, Franciery, Gisane, Ozana, Esaú, Isabella, durante a gradua¸cão cuja amizade se estende até hoje. Aos amigos do mestrado, Jhonatan, Vin´ıcius, Cindel, Thiago Silveira, Fernando, Alan Kardec, Alan Freitas, JR, Juan, Clebes, Airton, Cristiano, João, Leonardo, Alex, Milena, Alice, Renata, Guilherme, Téo, Wington e Lucas Queiroz. Aos amigos do doutorado, Adrian, Júlio, Andreia, João Felipe, Abraão, Ginaldo, André, Janiely, Lindinês, Maria do Desterro, Anselmo, Elizabeth, Joyce, Nadiel, Ramon, Francisco Calvi, Pádua, Renato, Rafael, Thiago, Ivan e Tony.

Agrade¸co profundamente aos professores responsáveis por minha forma¸cão para chegar nesse objetivo de defender uma tese em matemática, cito aqui os professores da gradua¸cão na Universidade Regional do Cariri (URCA): José Tiago, Zelálber, Luciana, Bárbara, Flávio Fran¸ca, Ricardo Carvalho, Thiago Alencar, Leidmar, Valéria, Jocel, Alex, Paulo César, Ednardo, Mario de Assis, Daniela, em especial aos professores José Tiago, Zelálber e Flávio Fran¸ca pelo incentivo em particular de prosseguir os estudos numa pós-gradua¸cão;

aos professores do Mestrado na Universidade Federal do Amazonas(UFAM): Inês Padilha, Juliana, Rosilene, Thiago Rodrigo, Stefan, Nazareno, Marcos Aurélio, Gérman, Dragomir, Alfredo (in memorian), Júlio, agrade¸co em particular a professora Inês Padilha por ter sido minha orientadora na produ¸cão da minha disserta¸cão; aos professores do doutorado na Universidade Federal da Para´ıba (UFPB): Eraldo, Henrique Fernandes, Alfonso Romero, Francisco Palomo, Mirian, Ricardo Burity, Alexandre Bustamantes, Manassés, Damião Júnio.

Agrade¸co em especial aos professor Eraldo por me orientar no doutorado por toda paciência, incentivo e apoio em cada momento dessa jornada. Agrade¸co por fazer a ponte com os professores Henrique Fernandes, Alfonso Romero e Francisco Palomo cujas contribui¸cões possibilitaram trabalhos que deram origem a essa tese. Agrade¸co a cada um, o conhecimento e experiência adquiridos durante o per´ıodo de pesquisa.

Agrade¸co imensamente aos professores e pesquisadores: Márcio Santos, Carlos Romero, Allan George, Fábio Reis e Alfonso Romero por aceitarem compor a banca de minha defesa de tese e por contribuir com suas valiosas corre¸cões e sugestões.

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Fágner Araruna por toda dedica¸cão ao programa de pós-gradua¸cão, especialmente na busca de bolsa de estudo que pra mim possibilitou a continuidade no doutorado.

Por fim, mas não menos importante a CAPES - Coordena¸cão de Aperfei¸coamento de Pessoal de N´ıvel superior e a FAPEAM - Funda¸cão de Amparo à Pesquisa do Estado do Amazonas, pelo apoio financeiro.

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nas coisas feitas pelo cora¸c˜ao?

E quem irá dizer que não existe razão?”

Renato Russo (1960-1996) Na can¸c˜aoEduardo e Mˆonica, de sua autoria.

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O objetivo principal dessa tese é o estudo de subvariedades imersas em certos espa¸cos produto semi-Riemanniano. Para isso, aplicando um princ´ıpio do máximo de Omori- Yau mais geral devido a Chen e Qiu e resultados devido a Al´ıas, Caminha e do Nascimento, obtemos novos princ´ıpios do máximo para o drift Laplaciano em variedades Riemannianas com tensor de Bakry-Émery-Ricci limitado inferiormente por uma fun¸cão cont´ınua ou com condi¸cão de crescimento de volume polinomial. Aplicamos esses novos princ´ıpios do máximo para obter diversos resultados de unicidade de hipersuperf´ıcie tipo- espa¸co em espa¸cos produto Lorentziano ponderado da forma −R × M_fⁿ e resultados análogos no espa¸co produto ponderado da forma R×M_fⁿ. Em ambos os casos, obtemos também resultados tipo Calabi-Bernstein para gráficos inteiro de fun¸cões definida na base RiemannianMⁿ. Determinamos resultados de rigidez e unicidade de subvariedade imersas com vetor curvatura média Gaussiano paralelo nos clássicos espa¸co Gaussiano e pseudo- Gaussiano. Por fim, usando parabolicidade, determinamos diversas condi¸cões suficientes de rigidez sobre superf´ıcies estacionária tipo-espa¸co imersa no espa¸co-tempo de Roberston- Walker generalizado e apresentamos alguns exemplos justificando a necessidade dessas condi¸cões.

Palavras-chave: Espa¸cos produto semi-Riemanniano, Subvariedade tipo-espa¸co, Cresci- mento de Volume, Princ´ıpio do m´aximo, Superf´ıcie estacion´aria, Parabolicidade, Espa¸co Gaussiano, Espa¸co pseudo-Gaussiano.

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The main objective of this thesis is the study of submanifolds immersed in certain semi- Riemannian products. For this, applying a more general Omori-Yau maximum principle due to Chen and Qiu and results due to Alias, Caminha and Nascimento, we obtain new principles of the maximum for the Laplacian drift in Riemannian manifolds with Bakry- Emery-Ricci tensor bounded from below by a continuous function or with polynomial´ volume growth condition. We apply these new maximal principles to obtain various uniqueness results of hypersurface in weighted Lorentzian product spaces of type−R×M_fⁿ and analogous results in weighted product space of the form R×M_fⁿ. In both cases, we also obtain Calabi-Bernstein type results for the entire graph of functions defined in the Riemannian basisMⁿ. We determined uniqueness and rigidity results to submanifold immersed with parallel Gaussian mean curvature vector in the classical Gaussian and pseudo-Gaussian spaces. Finally, using for parabolicity, we determine various rigidity conditions onto stationary spacelike surface into generalized Roberston-Walker spacetime and we present examples justifying the need for these conditions.

Key-words: Semi-Riemannian product space, Spacelike submanifold, Volume growth, Maximum principles, Stationary surface, Parabolicity, Gaussian space, Pseudo-Gaussian space.

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Introduction 1

1 The Geometric Setting 6

1.1 New Maximum principles for the drift Laplacian . . . 8

2 Uniqueness of spacelike hypersurfaces in weighted Lorentzian product spaces 13 2.1 Preliminaries . . . 13

2.2 Uniqueness of spacelike hypersurfaces in weighted Lorentzian product spaces 15 2.3 Calabi-Bernstein type results . . . 21

3 Uniqueness of hypersurfaces in weighted product spaces via maximum principles for the drift Laplacian 24 3.1 Preliminaries . . . 24

3.2 Main Results . . . 26

3.3 Entire graphs in a weighted product space . . . 31

4 Submanifolds with parallel weighted mean curvature vector in the Gaussian and pseudo-Gaussian space 33 4.1 Preliminaries . . . 33

4.2 Rigidity results in Gaussian space . . . 34

4.3 Further rigidity results . . . 37

4.4 Rigidity results in pseudo-Gaussian space . . . 38

5 Complete stationary spacelike surfaces in ann-dimensional Generalized Robertson-Walker spacetime 43 5.1 Preliminaries . . . 43

5.2 Curvature and Parabolicity . . . 47

5.3 Main results . . . 47

5.4 Examples . . . 53

A Some auxiliary technical results 55

Bibliography 60

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In recent decades, the study of submanifolds immersed in semi-Riemannian product spaces, has been of great interest to research in math and other areas of science. Much is due to the success of Einstein’s theory of General Relativity in explaining the structure and various phenomena of the universe. Certain semi-Riemannian product spaces, provides spacetime models for the universe. This is the case, for Einstein’s static universe, modeled by Minkowski space, the generalized Roberston-Walker spacetime, modeled by a generalized Lorentzian warped product, non-static models for universe (see, [17], [6]). In a Riemannian setting, as calledweighted manifolds or manifold with density, appear as a non-trivial generalization of the Riemannian manifolds, with several lines of research in increasing development. It is worth mentioning the study of self-shrinkers, gradient Ricci solitons, harmonic heat flow. For a throughout discussion about weighted manifolds, we refer the works [39, 60, 70].

The study of submanifolds in a Lorentzian environment has been motivated for example, by the Penrose singularity theorems [44, Theorem 1, Cap 8], where the existence of Cauchy’s surface or Trapped surface, ensures the formation of singularity in the spacetime. In the weighted Riemannian scenario, the Ricci solitons gradient has its mathematical relevance verified in the solution of the Poincar´e conjecture given by Perelman, since the Ricci gradient solitons appear as self-similar solutions of Hamilton’s Ricci flow (see, [63]).

In [42, Section 9.4.E], Gromov introduced the notion off-mean curvatureH_f, which is defined in (1.11). This concept has motivated several works where the authors investigate the geometry of hypersurfaces immersed with constant f-mean curvature in a weighted Riemannian manifold and, in particular, in a gradient Ricci soliton. See, for instance, [14, 23, 26, 27, 28, 29, 30, 31, 37, 59, 66, 69] and references therein. In this topic, two basic problems are uniqueness and rigidity results of submanifolds in certain spacetime.

See for instance, [2, 4, 6, 48, 8, 9, 49, 52, 32, 37, 43, 67].

In this work we obtain rigidity and uniqueness results for spacelike submanifolds in certain semi-Riemannian product spaces relevant, to know, the generalized Robertson- Walker spacetime, the called GRW spaces-time, weighted Lorentzian ( or Riemannian) product spaces of type −R×M_fⁿ (or R×M_fⁿ), (where, motivated by [25, Theorem 1:2], it is assumed that the weight function f does not depend on the parameter t ∈ R) and in the classicals case particular, Gaussian space G^n+p and pseudo-Gaussian space G^n+pp , as we will see in a more precise definition, the Euclidean space Rⁿ endowed with the Gaussian weight function f(x) = |x|²/4 with Riemannian metric and semi-Riemannian

1

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of index p, respectively. The approach is made through the application of new maximum principles for drift Laplacian, obtained through a more general Omori-Yau maximum principle due to Chen and Qiu [32] and results of Al´ıas, Caminha and do Nascimento [12, 15]. Furthermore, parabolicity and conditions of volume growth in submanifolds.

More specifically, in this thesis we obtain rigidity and uniqueness results in the following setting: In Chapter 2 for complete spacelike hypersurfaces immersed in a weighted Lorentzian space product of type−R×M_fⁿ. In Chapter 3, for complete two-sided hypersurfaces immersed in a weighted Riemannian space product of the formR×M_fⁿ. In Chapter 4, for a complete submanifold immersed with parallel Gaussian mean curvature vector ξ in the (n+p)−dimensional Gaussian space G^n+p and in Chapter 5 for complete (noncompact) stationary spacelike surface in GRW spacetime.

The study of the uniqueness of complete hypersurfaces immersed in a weighted manifold through the application of maximum principles for the drift Laplacian ∆_f, which is defined in (1.8), constitutes an interesting thematic in the scope of Differential Geometry. In this setting, H.F. de Lima jointly with Oliveira and Santos [49] used a weak version of the Omori-Yau’s generalized maximum principle to obtain uniqueness results related to complete spacelike hypersurfaces immersed in a weighted Lorentzian product space of the type −R×M_fⁿ, whose Bakry-´Emery-Ricci tensor of the Riemannian fiber Mⁿis nonnegative and the Hessian of f is bounded from below. Supposing the constance of thef-mean curvature and considering suitable constraints on the norm of the gradient of the height function, they proved that such a spacelike hypersurface must be a slice of the ambient space.

In [10], An et al. showed that the only complete f-maximal spacelike hypersurfaces immersed in −R × Gⁿ such that the hyperbolic angle function is bounded, are the spacelike hyperplanes {t} ×Gⁿ. More recently, the H.F de Lima jointly with Aquino and Baltazar [11] obtained other uniqueness results concerning complete spacelike hypersurfaces with constant f-mean curvature in −R×M_fⁿ, and applied them to get Calabi-Bernstein type results for spacelike entire graphs in such an ambient space.

Proceeding with this picture, in Chapter 1.1 we use recent results due to Chen and Qiu [32] and Al´ıas, Caminha and do Nascimento [15] to establish new maximum principles related to the drift Laplacian in weighted Riemannian manifolds. Afterwards, in Chapter 2, we apply these maximum principles to study the uniqueness of complete spacelike hypersurfaces immersed with constantf-mean curvature in a weighted Lorentzian product space −R×M_fⁿ, under suitable constraints on the Bakry-´Emery-Ricci tensor and on the norm of the gradient of the height function. Finally, in Section 2.3 we present new Calabi-Bernstein type results concerning spacelike entire graph function of the defined on the Riemannian base Mⁿ.

When the ambient space is a weighted Riemannian product space of the formR×M_fⁿ, whose potential functionf does not depend on the parameter t∈R, H.F de Lima jointly with Oliveira and Santos [52] studied the rigidity of entire graphs defined over the baseMⁿ having nonnegative Bakry-´Emery-Ricci tensor. Supposing the constancy of the f-mean curvature and assuming appropriate controls on the norm of the gradient of the smooth function uwhich determines such a graph Σ(u), they proved that u must be constant on Mⁿ. We point out that the assumption that the potential functionf does not depend on the parameter t ∈R is motivated by a splitting theorem due to Fang, Li and Zhang [41, Theorem 1.1], which guarantees that if a weighted product space (R×Mⁿ)_f endowed with

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a bounded potential function f is such that the corresponding Bakry-´Emery-Ricci tensor is nonnegative, then f is actually constant along R. In this direction, M. P. Cavalcante, H.F. de Lima and M.S. Santos in [22] using the Omori-Yau generalized maximum principle or extension of a result due to Yau that guarantees the vanishing of the Laplacian drift obtains some rigidity results in weighted warped products I ×ρM_fⁿ for complete two- sided hypersurfaces contained in a slab of I ×_ρM_fⁿ with an angle function or f-average curvature that does not change sign. Under certain restrictions of the weigarten operator and the gradient norm of the height function (which depends on the warping function) the hypersurface is a slice. Other rigidity results are also obtained for complete two-sided hypersurfaces with non-negative or bounded from below Bakry-´Emery-Ricci tensor.

Furthermore, similar H.F de Lima, A.M. Oliveira and M.A.L. Vel´asquez got in [50]

several rigidity results in some class of weighted warped product for complete two-sided hypersurfaces.

Using the new maximum principles established in Chapter 1, in Chapter 3, we extend the study initiated in [52] obtaining several new uniqueness results related to complete two-sided hypersurfaces immersed with constant f-mean curvature Hf in a weighted product space R×M_fⁿ, under suitable constraints on H_f and on the Bakry-´Emery-Ricci tensor of the hypersurface (see Section 3.2). In particular, we prove that the slices are the only complete two-sided f-minimal (that is, with H_f identically zero) hypersurfaces lying in a half-space ofR×M_fⁿ and such that the Bakry-´Emery-Ricci tensor is bounded from below (see Theorem 3.3). Furthermore, in Section 3.3 we present new Bernstein type results concerning to entire graphs defined on the base Mⁿ.

An important example of weighted Riemannian manifold is the so-called Gaussian space G^n+p, which corresponds to the Euclidean space R^n+p endowed with the Gaussian probability measure

dµ=e⁻^|x|

2

4 dx². (1)

In this context, the Gaussian mean curvature vector ξ of an immersed n-dimensional submanifold X :Mⁿ↬G^n+p is defined by

ξ =H⃗ + 1

2X^⊥. (2)

Here, H⃗ stands for the standard (nonnormalized) mean curvature vector of the immersion X : Mⁿ ↬ R^n+p and ( )^⊥ denotes the normal part of a vector field on R^n+p. When ξ vanishes identically,Mⁿis called aself-shrinkerof the mean curvature flow, which plays an important role in the study of the mean curvature flow because they describe all possible blow up at a given singularity of such a flow and, as it was pointed out by Colding and Minicozzi in [34], self-shrinkers are critical submanifolds for the entropy functional.

There exist in the literature many characterization and rigidity results of self-shrinkers under appropriate hypothesis. For instance, Ecker and Huisken [38] proved that if a self- shrinker is an entire graph with polynomial volume growth, then it is a hyperplane. Later on, the condition of polynomial volume growth was removed by Wang [69]. In [56], Le and Sesum showed that any smooth self-shrinker with polynomial volume growth and satisfying |A|² < ¹₂ is a hyperplane, where A denotes the second fundamental form of an immersion. Afterwards, Cao and Li [24] generalized this result to arbitrary codimension

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proving that any smooth complete self-shrinker with polynomial volume growth and

|A|² ≤ ¹₂ is either a round sphere, a circular cylinder or a hyperplane.

In [33], Cheng and Peng gave estimates on supremum and infimum of the squared norm of the second fundamental form of self-shrinkers without assumption on polynomial volume growth which, in particular, enabled them to obtain the rigidity theorem of [24]

without assumption on polynomial volume growth (see also [35, Theorem 1.1] concerning the 2-dimensional case). More recently, Wang, Xu and Zhao [67] proved a rigidity theorem for complete n-dimensional submanifolds with parallel Gaussian mean curvature vector in the Gaussian spaceG^n+p, under an integral curvature pinching condition, generalizing a previous rigidity result for self-shrinkers due to Ding and Xin [37].

Proceeding with this picture, in Chapter 4 we obtain several rigidity results for X :Mⁿ→G^n+p complete submanifold immersed with parallel Gaussian mean curvature vectorξ in the Gaussian space G^n+p, with norm of second fundamental form bounded by a number (γ1 or γ2, defined in (4.26) and (4.13), respectively), initially in the Theorem 4.1 with codimension p ≥ 2, and in the Theorem 4.2 the case p = 1. Our approach is based on a Nishikawa type maximum principle for the drift Laplacian, which is proved in Chapter 1. Furthermore, in Section 4.3 we use a maximum principle at infinity due to Al´ıas, Caminha and Nascimento [12] to get additional rigidity results, as well as a nonexistence result related to nonminimal submanifolds immersed with parallel weighted mean curvature vector in G^n+p (see Theorems 4.9, 4.4 and 4.5).

In Chapter 5, agreeing with [5], spacelike surfaces with zero mean curvature in ambient spacetimes of dimension greater than three are called stationary. Stationary spacelike surfaces in 4-dimensional spacetimes are a relevant role in mathematical Relativity.

Indeed, a stationary spacelike surface may be seem as a limit of trapped surfaces which are inside the horizon of events around a singularity. Trapped surfaces are usually considered to be compact. We are interested here in parabolic stationary surfaces, i.e., stationary spacelike surfaces Σ² such that the only nonnegative superharmonic functions on Σ² are the constants. Obviously, this property is satisfied in the compact case. The study of stationary spacelike surfaces has always been subject for many researchers in the past decades. For instance, in [2] Al´ıas and Albujer proved that in a Lorentzian product

−R×M² ≡ (R×M²,−dt²+g_M) with Gauss curvature of M² satisfying K_M ≥ 0, any complete maximal surface is totally geodesic, moreover ifM² is not flat then these surfaces are just the slices{t₀} ×M². Further in [3] they developed their results for surfaces with non-empty boundary. We also observe that in [1] Albujer constructed maximal surfaces in −R×H², where H² is the hyperbolic plane of constant Gauss curvature −1, in order to justify these curvature restriction on the Riemannian surface M². Moreover, new uniqueness properties of complete maximal surfaces in Lorentzian product spacetimes

−I×M², where KM ≥ −κ for some constant κ > 0 were obtained by the E.A de Lima and Romero in [48], by means of an extension of a well-known result by Nishikawa in [61].

In [65] the Romero and Rubio proved several uniqueness results for complete maximal surfaces in−R×fR² ≡(R×R²,−dt²+f(t)²g0) which is a 3-dimensional Robertson-Walker spacetime with fiber the Euclidean plane (R², g₀) and warping function f : R → R⁺ extending the uniqueness results of Latorre and the third author in [46].

It was proved by Al´ıas, Estudillo and Romero in [4] that the only compact stationary spacelike surfaces Σ² in a 3-dimensional GRW spacetime −I×_f M with Gauss curvature

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satisfying

K_Σ≥ f^′(τ)²+K_M

f(τ)² , (3)

where τ is the restriction of the projection π_I to Σ² and K_M stands for the sectional curvature of M restricted to Σ², are the totally geodesic ones. This result was obtained using a universal integral inequality whose equality characterizes the totally case, [4, Theorem 9]. As the Example 5.11 below shows, the assumption (3) does not imply the same conclusion if the hypothesis compact is changed to complete, in general. Thus, the following question arises in a natural way,

Under what assumptions a (non-compact) complete stationary spacelike surface in a GRW spacetime with Gauss curvature satisfying (3) must be totally geodesic?

Along Chapter 5, we will give several answers to this question, focusing our attention in natural assumptions on the Gauss curvature, that lead to the parabollicity of the spacelike surface, and a remarkable smooth function naturally defined on the surface, that satisfies a certain partial differential equation from the fact that the mean curvature vector field vanishes everywhere.

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Chapter 1 The Geometric Setting

In order to fix notation, in this chapter we present the general geometric scenario of this thesis and some fundamental equations used throughout the text.

Recall thatm-dimensional Riemannian is a pair (M^m, g) composed by am-dimensional smooth manifold M^m and a Riemannian metric g on M^m, that is, for every p ∈ M^m, g_p :T_pM^m×T_pM^m →Ris a symmetric nondegenerate bilinear form and positive definite i.e.,g_p is bilinear form in the space tangentT_pM^mand satisfies respectively the properties:

a) g_p(u, v) = g_p(v, u) for all u, v ∈T_pM^m; b) Ifg_p(u, v) = 0 for all v ∈T_pM^m, then u= 0;

c) Provided u̸= 0, gp(u, u)>0.

Weakening the definition of Riemannian metric to metric tensor g being a symmetric nondegenerate bilinear form g_p in each space tangent T_pM^m, allows the trichotomy g_p(u, u) <0, g_p(u, u) = 0, g_p(u, u)>0 for u̸= 0 in T_pM^m, such vector u is classified as timelike,null or lightlike and spacelike, respectively.

If g_p(u, u)<0 for all u∈T_pM^m\{0},g_p is said negative definite on T_pM^m. The index of metric tensor g is the largest integer that is dimension of a subspace W ⊂ T_pM^m, such that gp is negative definite in W.

Therefore, a m-dimensional semi-Riemannian manifold (or semi-Riemannian manifold) (M^m, g) of index q is a m-dimensional smooth manifold M^m furnished with metric tensor g of constant index q. Notice the peudo-Riemannian manifolds are a generalization of the Riemannian manifolds. A semi-Riemannian manifold of index q = 1, is called Lorentzian manifold. (see, [62, 17]).

A spacelike submanifold Σⁿ in a semi-Riemannian manifold (M^n+p, g) is a smooth immersion ψ : Σⁿ → M^n+p and the induced metric ψ^∗g on Σⁿ is a Riemannian metric.

The number p ≥ 1 is called the codimension of submanifold. When p = 1 a spacelike submanifold Σⁿ onMⁿ⁺¹ is calledspacelike hypersurface.

From now on, we will denote the metric tensor g by⟨,⟩, as well as the induced metric ψ^∗⟨,⟩ on Σⁿ.

Denoting byX(M) the set of the tangent vector fields toM^n+p, along ofψ(Σⁿ)⊂M^n+p each X∈X(M) admits decomposition

X =X^⊤+X^⊥, 6

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whereX^⊤andX^⊥ are respectively, the tangent component and normal component to Σⁿ. Analogously, X(Σ) and X(Σ)^⊥ denote respectively, the set of the tangent and normal vector fields to Σⁿ.

Denoting by∇ and∇ the Levi-Civita connection ofM^n+p and induced connection on Σⁿ, the Gauss and Weingarten formulas are given by

∇_XY =∇_XY +II(X, Y) (1.1)

where II(X, Y) = ∇_XY^⊥

is the second fundamental form of Σⁿ. A_ηX=− ∇_Xη^⊤

(1.2) for all tangent vector fieldX, Y ∈X(Σ) andη ∈X(Σ)^⊥, where the operator A_η onX(Σ) is shape operator defined by

⟨II(X, Y), η⟩=⟨A_ηX, Y⟩.

for all X, Y ∈X(Σ) and every η ∈X(Σ)^⊥. When the second fundamental form II of the a submanifold Σⁿ, is identically null, we say that Σⁿ is totally geodesic.

Denoting by R and R the curvature tensor of Σⁿ and M^n+p, the Gauss of equation is given by

⟨R(X, Y)Z, W⟩=⟨R(X, Y)Z, W⟩+⟨II(X, Z), II(Y, W)⟩ − ⟨II(X, W), II(Y, Z)⟩, (1.3) for all X, Y, Z, W ∈X(Σ).

For every normal vector field η on Σⁿ, the Codazzi equation is given by

⟨R(X, Y)Z, η⟩= (∇YII)(X, Z, η)−(∇XII)(Y, Z, η), (1.4) where ∇_YII is the derivative covariant of second fundamental formII given by

(∇_YII)(X, Z, η) =Y⟨II(X, Z), η⟩ − ⟨II(∇_YX, Z), η⟩ − ⟨II(X,∇_YZ), η⟩

− ⟨II(X, Z),∇^⊥_Yη⟩.

with ∇^⊥_Yη= ∇_Yη^⊥

the normal connection.

A proof of the Gauss and Codazzi equations can be found in [62, Chapter 4].

Consider an (n+ 1)-dimensionalLorentzian product space Mⁿ⁺¹ of the form−R×Mⁿ, where (Mⁿ, g) is an n-dimensional connected oriented Riemannian manifold and Mⁿ⁺¹ is endowed with the standard Lorentzian product metric

⟨,⟩=−π^∗_R(dt²) +π_M^∗ (g), (1.5) where π_R and π_M denote the canonical projections from −R×Mⁿ onto each factor. For a fixed t0 ∈R, we say that M_tⁿ₀ ={t0} ×Mⁿ is a slice of Mⁿ⁺¹.

Given a smooth functionf onMⁿ⁺¹ =−R×Mⁿ, endowing it with the volume element dM_f :=e^−fdM, wheredM is the standard volume element of (Mⁿ⁺¹,⟨, ⟩), we obtain the weighted Lorentzian product space(Mⁿ⁺¹,⟨ , ⟩, dM_f), which will be denoted byMⁿ⁺¹_f . In this setting, f is called the weight function of Mⁿ⁺¹_f .

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TheBakry- ´Emery-Ricci tensorRic_f ofMⁿ⁺¹_f is defined as being the following extension of the standard Ricci tensor Ric:

Ric_f = Ric + Hessf. (1.6)

For a hypersurface Σⁿ inMⁿ⁺¹_f , the f-divergence operator on Σⁿ is defined by

divf(X) =e^fdiv(e^−fX), (1.7)

for all tangent vector fieldXon Σⁿ. For a smooth functionu: Σⁿ →R, itsdrift Laplacian (or f-Laplacian) is given by

∆_fu= div_f(∇u) = ∆u− ⟨∇u,∇f⟩. (1.8) According to Gromov [42], the f-mean curvature H_f of Σⁿ is defined by

nH_f =nH− ⟨∇f, N⟩, (1.9)

where H = −¹_ntr(A) corresponds to the standard mean curvature of Σⁿ with respect to its future-pointing Gauss map N.

We say that Σⁿ is f-maximalwhen H_f vanishes identically on Σⁿ.

The weighted product space Mⁿ⁺¹_f = (R×Mⁿ)_f is defined analougly, as well as f- divergence operator and drift-Laplacian on hypersurfaces Σⁿ. Nonetheless, of Bakry- Emeri-Ricci tensor on´ Mⁿ⁺¹_f is

Ric_f = Ric−Hessf, (1.10)

and thef-mean curvature Hf of Σⁿ is defined by

nH_f =nH+⟨∇f, N⟩, (1.11)

where H = _n¹tr(A) corresponds to the standard mean curvature of Σⁿ with respect to its Gauss map N.

1.1 New Maximum principles for the drift Laplacian

This section is devoted to establish some maximum principles for the drift Laplacian.

For this we will present some crucial results.

Let (M^n+p_q , g) be an semi-Riemannian manifold of indexq. According to [32], given a vector fieldV on M^n+p_q , we define the Bakry- ´Emery-Ricci tensor Ric_V of M^n+p_q , as being the following extension of the standard Ricci tensor Ric:

Ric_V = Ric− 1

2L_Vg. (1.12)

whereL_Vg denote the Lie derivative of metric tensorg. Furthermore, given a submanifold immersed Mⁿ on M^n+p_q , onto function u∈C²(M) consider the operator

∆_Vu= ∆u− ⟨∇u, V⟩. (1.13)

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where ∆u is the standard Laplacian relative to Mⁿ. Notice for gradient vector field, i.e., V =∇f, the operator ∆V coincide with the drift Laplacian ∆f defined in (1.8).

In [32, Theorem 1] Chen and Qiu proved the following version of the Omori-Yau maximum principle.

Theorem 1.1. Let (M^m, g) be a complete Riemannian manifold, V a C¹ vector field on M^m. If Ric_V ≥ −F(r)g, where r is the distance function on M^m from a fixed point x₀ ∈M^m, F :R→R is a positive continuous function satisfying

φ(t) :=

Z t ρ0+1

dr Rr

ρ0F(s)ds+ 1 −→+∞ (t→+∞) (1.14) for some positive constant ρ₀ ∈ R. Let f ∈ C²(M) with limx→∞ f(x)

φ(r(x)) = 0, then there exist points {x_j} ⊂M^m, such that

j→∞lim f(xj) = supf, lim

j→∞|∇f|(xj) = 0, lim

j→∞∆Vf(xj)≤0. (1.15) From the Theorem 1.1, we get a sort of extension of Nishikawa’s result in [61] (see also [7, Lemma 3], [51, Lemma 4.1] and [11, Proposition 1]).

Proposition 1.2. Let (Mⁿ,⟨,⟩) be an n-dimensional complete Riemannian manifold and let V be smooth function on Mⁿ such that Ric_V ≥ −G(r)⟨,⟩, where r is the distance function on Mⁿ from a fixed point of it, G : R → R is a positive continuous function satisfying

φ(t) :=

Z t ρ0+1

dr Rr

ρ0G(s)ds+ 1 −→+∞ (t→+∞)

for some positive constant ρ₀ ∈R. If u∈ C²(M) is a nonnegative function on Mⁿ, such that

∆_Vu≥βu^1+α, (1.16)

for some positive constants α, β ∈R, then u is identically zero on Mⁿ.

Proof. Let u ∈ C²(M) be a nonnegative function. We consider on Mⁿ the function F given by

F = 1

(1 +u)^λ, (1.17)

for some constant λ > 0 which will be chosen later. We have that 0 < F ≤ 1 and, in particular, infF ≥0.

Moreover, given any tangent vector field X onM, from (1.17) we obtain

⟨∇F, X⟩=X

1 (1 +u)^λ

=−λ(1 +u)^−λ−1X(u)

=− λ

(1 +u)^λ+1⟨∇u, X⟩=⟨−λF^λ+1^λ ∇u, X⟩, that is,

∇F =−λF^λ+1^λ ∇u. (1.18)

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Consequently, using (1.18) we get

∆F = div(−λF^λ+1^λ ∇u) =−λ∇u(F^λ+1^λ )−λF^λ+1^λ ∆u

=−(λ+ 1)F^λ¹∇u(F)−λF^λ+1^λ ∆u (1.19)

=λ(λ+ 1)F^2+λ^λ |∇u|²−λF^λ+1^λ ∆u.

Thus, from (1.8) and (1.19) we have

∆VF =λ(λ+ 1)F^2+λ^λ |∇u|²−λF^λ+1^λ ∆u+λF^λ+1^λ ⟨V,∇u⟩ (1.20)

=−λF^λ+1^λ (∆u− ⟨V,∇u⟩) +λ(λ+ 1)F^2+λ^λ |∇u|². Hence, from (1.18) and (1.20) we reach at the following relation

λF∆VF =−λ²F^2λ+1^λ ∆Vu+ (λ+ 1)|∇F|². (1.21) On the other hand, since F is bounded on Mⁿ, we have

x→∞lim

F(x) φ(r(x)) = 0,

since _φ(r(x))¹ −→0 when r(x)−→+∞. Thus, taking into account our constraint on Ric_V, we can apply Theorem 1.1 to guarantee the existence of a sequence {x_m}m∈N ⊂Mⁿ such

that 





0≤infMF ≤F(xm)<infMF + _m¹,

|∇F|(x_m)< _m¹,

∆_VF(x_m)>−_m¹.

(1.22) Combining (1.16), (1.21) and (1.22), we obtain

−1

mλF(x_m)< λF(x_m)∆_VF(x_m)

=−λ²F^2λ+1^λ (x_m)∆_Vu(x_m) + (λ+ 1)|∇F|²(x_m)

≤ −λ²βu^1+α(x_m)F^2λ+1^λ (x_m) + (λ+ 1)|∇F|²(x_m) (1.23)

≤(λ+ 1)|∇F|²(x_m).

Using (1.17) and making m−→+∞in (1.23), we get 0 = lim

m→+∞F^2λ+1^λ (x_m)u^1+α(x_m) = lim

m→+∞

u^1+α(x_m)

(1 +u(x_m))^2λ+1. (1.24) At this point, taking 2λ=α, from (1.24) we obtain

m→∞lim

u(x_m) 1 +u(x_m)

1+α

= 0.

But, from (1.17) we also have that

m→+∞lim F(x_m) = inf

M F if and only if lim

m→+∞u(x_m) = 1−(inf_MF)^α² (inf_MF)^α² . Consequently, we get ^1−(inf^M^F⁾

2 α

(infMF)^α² = 0 and, hence, inf_MF = 1. Therefore, since F ≤1, we conclude that u must be identically zero on Mⁿ.

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Considering a Riemannian manifold (Mⁿ, g), we denote byB(x, r)⊂Mⁿ the geodesic ball of center x∈Mⁿ and radius r >0. According to [15, Section 2], given a polynomial functionσ : (0,+∞)→(0,+∞), we say thatMⁿ has polynomial volume growth like σ(r) if there existsx∈Mⁿ such that

vol(B(x, r)) =O(σ(r)),

as r → +∞, where vol denotes the canonical Riemannian volume of (Mⁿ, g). As it was already observed in [15, Section 2], if x, y ∈Mⁿare at distanced from each other, we can verify that

vol(B(x, r))

σ(r) ≥ vol(B(y, r−d))

σ(r−d) .σ(r−d) σ(r) .

Consequently, the choice of x in the notion of volume growth is immaterial, and we will just say that Mⁿ has polynomial volume growth.

For the statement of the coming result, consider a vector fieldX onMⁿ, and its flows {ψ_t}. A subset Ω of Mⁿ is stable under the flow of X if ψ_t(Ω) ⊂Ω, for every t ≥0. In particular, this also holds for Ω = Mⁿ. In [15], Al´ıas, Caminha and do Nascimento got the following result.

Theorem 1.3. Let M be a connected, oriented, complete noncompact Riemannian manifold, let X ∈ X(M) be a bounded vector field on M, with |X| ≤ c < ∞ , and K be a (possibly empty) compact subset ofM such thatM\K is stable under the flow of X. Assume that f ∈ C^∞(M) is such that ⟨∇f, X⟩ ≥ 0 on M and divX ≥ af on M\K, for some a >0.

(a) If M has polynomial volume growth, then f ≤0 on M\K.

(b) If M has exponential volume growth, say like e^βt, then f ≤ cβ

a on M\K.

As application of this result, we obtain the following maximum principle for the drift Laplacian.

Lemma 1.4. Let (Mⁿ, g) be a complete noncompact Riemannian manifold. Let f, u ∈ C^∞(M) such that ∆_fu≥αu, for some positive constant α∈R. Suppose in addition that f and|∇u| are bounded onMⁿ. If Mⁿhas polynomial volume growth, then u≤0on Mⁿ. Proof. Considering the tangent vector field X=e^−f∇u onMⁿ, we have that

g(X,∇u) = e^−f|∇u|² ≥0 and, taking into account the boundedness of f and |∇u|,

|X|=e^−f|∇u|<+∞.

Moreover, using again the boundedness of f jointly with the hypothesis that ∆_fu ≥ αu, we also get that

divX =e^−f∆_fu≥αu,˜

for some positive constant ˜α ∈R. Hence, since we are assuming that Mⁿ has polynomial volume growth, we can apply the Theorem 1.3 to conclude that u≤0 on Mⁿ.

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We can reason as in the proof of Lemma 1.4 to get our next maximum principle.

Lemma 1.5. Let (Mⁿ, g) be a complete noncompact Riemannian manifold. Let f, u ∈ C^∞(M) such that ∆_fu≥ αe^fu and |∇u| ≤ βe^f, for some positive constants α, β ∈R. If Mⁿ has polynomial volume growth, then u≤0 on Mⁿ.

Proof. Considering once more the tangent vector field X = e^−f∇u on Mⁿ, since we are assuming that ∆_fu≥αe^fufor some positive constant α, from (1.8) we get

divX =e^−f∆_fu≥αu.

Moreover, we also have that

g(X,∇u) = e^−f|∇u|² ≥0

and, from the hypothesis |∇u| ≤βe^f for some positive constant β,

|X|=e^−f|∇u| ≤β <+∞.

Therefore, since we are also supposing that Mⁿ has polynomial volume growth, we can apply again the Theorem 1.3 to conclude thatu≤0 on Mⁿ.

According to the classical terminology in linear potential theory, a weighted manifold M_fⁿ endowed with a weight function f is said to be f-parabolic if there does not exist a nonconstant, nonnegative, f-superharmonic function defined on M_fⁿ. In this context, from [8, Corollary 2] we have the following f-parabolicity criterion.

Lemma 1.6. Let Mⁿ⁺¹_f = −R × M_fⁿ be a weighted Lorentzian product space whose Riemannian base Mⁿ is complete with f-parabolic universal Riemannian covering and let ψ : Σⁿ → Mⁿ⁺¹_f be a complete spacelike hypersurface with bounded hyperbolic angle function η. Then, Σⁿ is f-parabolic.

Let Mⁿ be a complete noncompact Riemannian manifold and let d(·, o) : Mⁿ → [0,+∞) denote the Riemannian distance of Mⁿ, measured from a fixed point o ∈ Mⁿ. We say that a smooth functionu∈C^∞(M)converges to zero at infinity, when it satisfies the following condition

d(x,o)→+∞lim u(x) = 0. (1.25)

Keeping in mind this previous concept, the following maximum principle at infinity corresponds to item (a) of [12, Theorem 2.2].

Lemma 1.7. LetMⁿ be a complete noncompact Riemannian manifold and letX ∈X(M) be a vector field onMⁿ. Assume that there exists a nonnegative, non-identically vanishing function u ∈ C^∞(M) which converges to zero at infinity and such that ⟨∇u, X⟩ ≥ 0. If divX ≥0 on Mⁿ, then ⟨∇u, X⟩ ≡0 on Mⁿ.

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Chapter 2 Uniqueness of spacelike hypersurfaces in weighted Lorentzian product spaces

2.1 Preliminaries

Consider an (n+1)-dimensional Lorentzian product spaceMⁿ⁺¹ of the form−R×Mⁿ, where (Mⁿ, g) is an n-dimensional connected oriented Riemannian manifold and Mⁿ⁺¹ is endowed with the standard Lorentzian product metric

⟨,⟩=−π_R^∗(dt²) +π_M^∗ n(g), (2.1) whereπ_R and πMⁿ denote the canonical projections from −R×Mⁿ onto each factor. For a fixed t₀ ∈R, we say that M_tⁿ

0 ={t₀} ×Mⁿ is a slice of Mⁿ⁺¹.

Given an n-dimensional connected manifold Σⁿ, a smooth immersion ψ : Σⁿ →Mⁿ⁺¹ is said to be aspacelike hypersurfaceif Σⁿ, furnished with the metric induced from ⟨,⟩via ψ, is a Riemannian manifold. If this is so, we will always assume that the metric on Σⁿ is the induced one, which will also be denoted by ⟨,⟩. In this setting, it follows from the connectedness of Σⁿ that one can uniquely choose a globally defined timelike unit vector field N ∈X(Σ)^⊥, having the same time-orientation of∂_t, that is, such that⟨N, ∂_t⟩ ≤ −1.

One then says thatN is the future-pointing Gauss map of Σⁿ.

We will denote by ∇ and ∇ the gradients with respect to the metrics of Mⁿ⁺¹ and Σⁿ, respectively. A simple computation shows that the gradient of π_R onMⁿ⁺¹ is given by

∇π_R=−⟨∇π_R, ∂_t⟩∂_t=−∂_t. (2.2) Hence, denoting by ( )^⊤ the tangential component of a vector field in X(Mⁿ⁺¹) along Σⁿ and considering η =⟨N, ∂_t⟩ the hyperbolic angle function of Σⁿ, from (2.2) we conclude that the gradient of the (vertical) height function h= (π_R)|_Σ of Σⁿ is given by

∇h= (∇π_R)^⊤ =−∂_t^⊤=−∂t−ηN. (2.3) Consequently, from (2.3) we get the following relation

|∇h|² =η²−1, (2.4)

13

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where| |denotes the norm of a vector field on Σⁿ. We note that the orthonormal projection N^∗ of N onto the Riemannian baseMⁿ is given byN^∗ =N+η∂t. Consequently, we have that

|N^∗|² =η² −1 = |∇h|². (2.5)

We will also consider the Weingarten endomorphismA:X(Σ)→X(Σ) with respect to the future-pointing Gauss map N. The mean curvature function of Σⁿ is defined as been H =−_n¹tr(A). Moreover, the Gauss and Weingarten formulas are given, respectively, by

∇_XY =∇_XY − ⟨AX, Y⟩N (2.6) and

AX =−∇_XN, (2.7)

for all tangent vector field X, Y ∈X(Σ). Moreover, we have that

∇η=A∇h. (2.8)

Indeed, given any X ∈X(Σ), we have

⟨∇η, X⟩=X(η) = ⟨∇_XN, ∂_t⟩+⟨N,∇_X∂_t⟩

=⟨AX,−∂_t^⊤⟩=⟨A∇h, X⟩.

because, ∇_X∂_t = 0 and by (2.3).

Considering the height function h and the angle function η of Σⁿ, from (2.3), (2.6) and (2.7) we get

∇_X∇h =∇_X∇h+⟨AX,∇h⟩N

=−X(η)N −η∇_XN − ⟨AX, ∂_t^⊤⟩N

=−X(η)N +ηAX+⟨∇XN, ∂t⟩N

=ηAX,

for all tangent vector fieldX on Σⁿ, where it was also used the parallelism of∂_tonX(M).

Thus, the Hessian of height function is equal to Hessh(X, Y) = ⟨AX, Y⟩η. Hence, its Laplacian is given by

∆h= tr(Hessh) =−nHη. (2.9)

Consequently, from (1.8), (1.9) and (2.9) we obtain

∆_fh=−nH_fη+⟨∇f, ∂_t⟩. (2.10) Now, assuming that f does not depend on the parameter t ∈ R, which means that

⟨∇f, ∂_t⟩= 0, from (2.10) we reach at the following suitable formula

∆fh=−nHfη. (2.11)

We recall that [25, Theorem 1.2] guarantees that, if a weighted Lorentzian product spaceMⁿ⁺¹_f = (−R×Mⁿ)_f is endowed with a bounded weight functionf and if its Bakry- Emery-Ricci tensor is such that Ric´ _f(V, V) ≥ 0, for all timelike vector field V ∈ X(M), then f does not depend on the parameter t ∈ R, showing that this condition is natural.

When the weight functionf of such a weighted Lorentzian product space does not depend on the parameter t∈R, we will just denote it by −R×M_fⁿ.

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2.2 Uniqueness of spacelike hypersurfaces in weighted Lorentzian product spaces

In what follows, we will use the maximum principles for the drift Laplacian established in Section 1.1 to study the uniqueness of complete spacelike hypersurfaces immersed in

−R×M_fⁿ. In [21], this the authors established several uniqueness results for hypersurface in static GRW, with restrictions on the weight function, B´akry-Emery-Ricci tensor, gradient of the height function and Weingarten operator. The following result generalize the [21, Theorem 3].

Theorem 2.1. Let Mⁿ⁺¹_f = −R×M_fⁿ be a weighted Lorentzian product space and let ψ : Σⁿ → Mⁿ⁺¹_f be a complete spacelike hypersurface whose Bakry- ´Emery-Ricci tensor satisfiesRicf ≥ −α⟨, ⟩, for some positive constantα. Suppose that thef-mean curvature H_f of Σⁿ is a nonpositive (nonnegative) constant and that its Weingarten operator A is positive (negative) semi-definite. If the height function h of Σⁿ satisfies the inequality

|∇h|² ≤ |A|²

α , (2.12)

then Σⁿ must be a slice of Mⁿ⁺¹_f .

Proof. In order to apply Lemma 1.2, we will prove that the function |∇h|² satisfies inequality (1.16). Indeed, from the Bochner-Lichnerowicz’s formula for the drift Laplacian (see, for instance, [70]) we have that

1

2∆_f|∇h|² =|Hessh|²+ Ric_f(∇h,∇h) +⟨∇h,∇(∆_fh)⟩. (2.13) Since H_f is constant, from (2.11) we get

⟨∇h,∇(∆fh)⟩=−nHf⟨∇h,∇η⟩=−nHf⟨∇h, A(∇h)⟩. (2.14) On the other hand, we can deduce from (2.9) that

|Hessh|² =|A|²(1 +|∇h|²). (2.15) Moreover, from our assumption on the Bakry-´Emery-Ricci tensor of Σⁿ, we have

Ric_f(∇h,∇h)≥ −α|∇h|². (2.16) Thus, inserting (2.14), (2.15) and (2.16) into (2.13) and taking into account hypothesis (2.12), we obtain

1

2∆f|∇h|² ≥ |A|²(1 +|∇h|²)−α|∇h|²−nHf⟨A(∇h),∇h⟩. (2.17) Hence, since our constraints on the signs ofH_f andAguarantee thatH_f⟨A(∇h),∇h⟩ ≤0, from (2.17) we reach at the following estimate

∆f|∇h|² ≥2α(|∇h|²)².

Therefore, we can apply Proposition 1.2 to conclude that |∇h| vanishes identically on Σⁿ, which means that h is constant on it and, hence, it must be a slice of Mⁿ⁺¹_f .

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When the spacelike hypersurface ψ : Σⁿ→Mⁿ⁺¹_f isf-maximal hypersurfaces, that is, itsf-mean curvature is identically zero, Theorem 2.1 reads as the following.

Corollary 2.2. Let Mⁿ⁺¹_f = −R×M_fⁿ be a weighted Lorentzian product space and let ψ : Σⁿ → Mⁿ⁺¹_f be a complete f-maximal spacelike hypersurface whose Bakry- ´Emery- Ricci tensor satisfies Ricf ≥ −α⟨ , ⟩,for some positive constant α. If the height function h of Σⁿ satisfies the inequality |∇h|² ≤ ^|A|_α², then Σⁿ must be a slice of Mⁿ⁺¹_f .

It is not difficult to see that we also obtain the following consequence of Theorem 2.1.

Corollary 2.3. Let Mⁿ⁺¹_f = −R×M_fⁿ be a weighted Lorentzian product space and let ψ : Σⁿ → Mⁿ⁺¹_f be a complete spacelike hypersurface with nonnegative Bakry- ´Emery- Ricci tensor. Suppose that the f-mean curvature H_f of Σⁿ is a nonpositive (nonnegative) constant and that its Weingarten operator A is positive (negative) semi-definite. If the height function h of Σⁿ satisfies the inequality |∇h|² ≤ β|A|² for some positive constant β ∈R, then Σⁿ must be a slice of Mⁿ⁺¹_f .

In our next uniqueness result, we obtain a sort of extension of [11, Theorem 1], in the sense that we will deal with the Bakry-´Emery-Ricci tensor instead of the standard Ricci tensor.

Theorem 2.4. Let Mⁿ⁺¹_f =−R×M_fⁿ be a weighted Lorentzian product space, such that the Bakry- ´Emery-Ricci tensor Ricf_f of the Riemannian base (Mⁿ, g) satisfies Ricf_f ≥ βg for some positive constant β ∈ R. Let ψ : Σⁿ → Mⁿ⁺¹_f be a complete spacelike hypersurface with constant f-mean curvature H_f and whose Bakry- ´Emery-Ricci tensor satisfies Ric_f ≥ −G(r)⟨ , ⟩, where r is the distance function on Σⁿ from a fixed point of it and G:R →R is a positive continuous function satisfying (1.2). Then, Σⁿ must be a slice of Mⁿ⁺¹_f .

Proof. From [9, Corollary 8.2] we have the following formula

∆η = (Ric(Nf ^∗, N^∗) +|A|²)η+n⟨∇H, ∂_t⟩, (2.18) where N^∗ = N +η∂_t stands for the orthonormal projection of N onto Mⁿ. Thus, since we are assuming thatH_f is constant, from (2.4), (1.11) and (2.18) we obtain

1

2∆_f|∇h|² =η∆η−η⟨∇f,∇η⟩

= (Ric(Nf ^∗, N^∗) +|A|²)η²+n⟨∇H, ∂_t⟩η−η⟨∇f,∇η⟩

= (Ric(Nf ^∗, N^∗) +|A|²)η²+∂_t^⊤(⟨∇f, N⟩)η−η⟨∇f,∇η⟩. (2.19) On the other hand, using ∂_t^⊤=∂_t+ηN, we get

∂_t^⊤(⟨∇f, N⟩) = ⟨∇_∂^⊤

t ∇f, N⟩+⟨∇f,∇_∂^⊤

t N⟩

=⟨∇_∂_t∇f, N⟩+η⟨∇_N∇f, N⟩+⟨∇f,∇η⟩.

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Moreover, since f does not depend of the parameter t ∈Rand ∇_N∂_t= 0, we have

⟨∇_∂_t∇f, N⟩=⟨∇_N∇f, ∂_t⟩=N⟨∇f, ∂_t⟩ − ⟨∇f,∇_N∂_t⟩= 0.

Consequently, we reach at the following

∂_t^⊤(⟨∇f, N⟩) =ηHessf(N, N) +⟨∇f,∇η⟩. (2.20) But, we also have that

Hessf(N, N) = Hessf(N^∗, N^∗)−2ηHessf(N^∗, ∂t) +η²Hessf(∂t, ∂t)

and, using once more that f does not depend on t ∈ R and ∇_X∂_t = 0 for any X ∈X(Mⁿ⁺¹),

Hessf(N^∗, ∂_t) = Hessf(∂_t, ∂_t) = 0.

Thus, we conclude that

Hessf(N, N) = Hessf(N^∗, N^∗) (2.21) Hence, since Ricf_f ≥ βg for some constant β > 0, considering (2.20) and (2.21) into (2.19) we obtain

1

2∆f|∇h|² = (Ric(Nf ^∗, N^∗) +|A|²)η² + Hessf(N, N)η²

= (Ricf_f(N^∗, N^∗) +|A|²)η²

≥β|N^∗|²η² =β|∇h|²(1 +|∇h|²)≥β|∇h|⁴.

Therefore, we can apply again Proposition 1.2 to infer that |∇h| is identically zero and, consequently, Σⁿ must be a slice of Mⁿ⁺¹_f .

From Theorem 2.4 we obtain the following consequence.

Corollary 2.5. Let Mⁿ⁺¹_f =−R×M_fⁿ be a weighted Lorentzian product space, such that the Bakry- ´Emery-Ricci tensor Ricf_f of the Riemannian base (Mⁿ, g) satisfies Ricf_f ≥ βg for some positive constantβ ∈R. Letψ : Σⁿ→Mⁿ⁺¹_f be a complete spacelike hypersurface with constant f-mean curvature and nonnegative Bakry- ´Emery-Ricci tensor. Then, Σⁿ must be a slice of Mⁿ⁺¹_f .

Next, we quote [49, Lemma 3], which gives sufficient conditions to ensure that the Bakry-´Emery-Ricci tensor of a spacelike hypersurface immersed in a weighted Lorentzian product space −R×M_fⁿ is bounded from below.

Lemma 2.6. Let Mⁿ⁺¹_f = −R×M_fⁿ be a weighted Lorentzian product space, such that the sectional curvature K_M of the Riemannian base Mⁿ and the Hessian of the weight function f are bounded from below. Let ψ : Σⁿ→ Mⁿ⁺¹_f be a spacelike hypersurface such that its f-mean curvature H_f and hyperbolic angle function η are bounded. Then, the Bakry- ´Emery-Ricci tensor Ric_f of Σⁿ is bounded from below.

Using Lemma 2.6 jointly with Theorem 2.4, we obtain.