Research article Special Issues

Spacelike translating solitons of the mean curvature flow in Lorentzian product spaces with density

  • By applying suitable Liouville-type results, an appropriate parabolicity criterion, and a version of the Omori-Yau's maximum principle for the drift Laplacian, we infer the uniqueness and nonexistence of complete spacelike translating solitons of the mean curvature flow in a Lorentzian product space R1×Pnf endowed with a weight function f and whose Riemannian base Pn is supposed to be complete and with nonnegative Bakry-Émery-Ricci tensor. When the ambient space is either R1×Gn, where Gn stands for the so-called n-dimensional Gaussian space (which is the Euclidean space Rn endowed with the Gaussian probability measure) or R1×Hnf, where Hn denotes the standard n-dimensional hyperbolic space and f is the square of the distance function to a fixed point of Hn, we derive some interesting consequences of our uniqueness and nonexistence results. In particular, we obtain nonexistence results concerning entire spacelike translating graphs constructed over Pn.

    Citation: Márcio Batista, Giovanni Molica Bisci, Henrique de Lima. Spacelike translating solitons of the mean curvature flow in Lorentzian product spaces with density[J]. Mathematics in Engineering, 2023, 5(3): 1-18. doi: 10.3934/mine.2023054

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  • By applying suitable Liouville-type results, an appropriate parabolicity criterion, and a version of the Omori-Yau's maximum principle for the drift Laplacian, we infer the uniqueness and nonexistence of complete spacelike translating solitons of the mean curvature flow in a Lorentzian product space R1×Pnf endowed with a weight function f and whose Riemannian base Pn is supposed to be complete and with nonnegative Bakry-Émery-Ricci tensor. When the ambient space is either R1×Gn, where Gn stands for the so-called n-dimensional Gaussian space (which is the Euclidean space Rn endowed with the Gaussian probability measure) or R1×Hnf, where Hn denotes the standard n-dimensional hyperbolic space and f is the square of the distance function to a fixed point of Hn, we derive some interesting consequences of our uniqueness and nonexistence results. In particular, we obtain nonexistence results concerning entire spacelike translating graphs constructed over Pn.



    This paper is dedicated with great esteem and admiration to Rosario Mingione, on the occasion of his 50th birthday.

    Let Σn be an n-dimensional connected manifold and let ¯Mn+1 be a (n+1)-dimensional Lorentzian manifold. Furthermore, let ψ:Σn¯Mn+1 be a spacelike immersion, that is Σn, endowed by the metric induced by , via the map ψ, is a Riemannian manifold. In such a case the map ψ:Σn¯Mn+1 is said to be a spacelike hypersurface.

    The mean curvature flow Ψ:[0,T)×Σn¯Mn+1 of the spacelike immersion ψ:Σn¯Mn+1, satisfies Ψ(0,)=ψ() and the evolution equation

    Ψt=H,

    where H(t,) is the (non-normalized) mean curvature vector field of the spacelike hypersurface Σnt=Ψ(t,Σn) for every t[0,T). Roughly speaking, the family of hypersurfaces Σnt=Ψ(t,Σn) evolves by mean curvature flow Ψ if the velocity Ψt coincide with the mean curvature vector H at every point of [0,T)×Σn.

    Mean curvature flow in a Lorentzian manifold is an important thematic in the scope of Geometric Analysis and it has been extensively studied by several authors; see, among others, the papers [1,18,19,20,21,23,27] as well as [29,30,31,32,33,34]. This wide interest in the current literature is mainly due to the fact that spacelike translating solitons can be regarded as a natural way of foliating spacetimes by almost null-like hypersurfaces; for more details, see [19]. For the sake of clarity, we recall here that spacelike translating solitons are spacelike hypersurfaces ψ:Σn¯Mn+1 such that H=cV for some constant c, where V stands for a suitable timelike vector field globally defined on the (n+1)-dimensional Lorentzian manifold ¯Mn+1.

    Particular examples may give insight into the structure of certain spacetimes at null infinity and have possible applications in General Relativity; see [19] for comments and details.

    In the Riemannian setting, de Lira and Martín [17] have investigated solitons invariant with respect to the flow generated by a complete parallel vector field in a Riemannian manifold. A special case occurs when the ambient manifold is the Riemannian product space R×Pn and the complete parallel vector field is just the coordinate vector field t. In such a case, in analogy with the Euclidean framework, they preserve the term translating solitons. Moreover, when the metric of the base Pn is rotationally invariant and its sectional curvature is nonpositive, the authors characterize all the rotationally invariant translating solitons deducing several nonexistence results by using careful geometric analysis of these new families of barriers.

    On the other hand, it is well known that many problems lead us to consider Riemannian manifolds endowed with a measure that has a smooth positive density with respect to the Riemannian one. This turns out to be compatible with the metric structure of the manifold and the resulting spaces are the weighted manifolds, which are also called manifolds with density or smooth metric measure spaces in the literature. More precisely, given a complete n-dimensional Riemannian manifold (Pn,g) and a smooth function fC(Pn), the weighted manifold Pnf associated to Pn and f is the triple (Pn,g,dμ=efdP), where dP denotes the standard volume element of Pn.

    Appearing naturally in the study of self-shrinkers, Ricci solitons, harmonic heat flows and many others, weighted manifolds are proved to be important nontrivial generalizations of Riemannian manifolds and, nowadays, there are several geometric investigations concerning them. For a brief overview of results in this scope, we refer the articles of Morgan [37] and Wei-Wylie [42].

    We point out that a theory of Ricci curvature for weighted manifolds goes back to Lichnerowicz [35,36] and it was later developed by Bakry and Émery in the seminal work [10], where they introduced the Bakry-Émery-Ricci tensor Ricf of a weighted manifold Pnf as being the following extension of the standard Ricci tensor Ric of Pn:

    Ricf=Ric+Hessf.

    The Bakry-Émery-Ricci curvature tensor arises in scalar-tensor gravitation theories in the conformal gauge known as the Jordan frame; see [22,43] for more details. It is also worth mentioning that Case [13] has shown that a sign condition on timelike components of the Bakry-Émery-Ricci tensor, the so-called f-timelike convergence condition, will, in an analogous fashion to the Riemannian case, imply that singularity theorems and the timelike splitting theorem hold; see Remark 1.

    Motivated by this previous digression and adapting the concept of translating soliton established in [17] and [9], here we investigate the uniqueness and nonexistence of complete spacelike translating solitons of the mean curvature flow in a Lorentzian product space R1×Pnf endowed with a weight function f and whose Riemannian base Pn is supposed to be complete and with nonnegative Bakry-Émery-Ricci tensor. This is made through the applications of suitable Liouville-type results, an appropriate parabolicity criterion, and a version of the Omori-Yau's maximum principle for the drift Laplacian (see Theorems 1, 2, 3, 4 and 5). Applications to R1×Gn are also given, where Gn stands for the so-called n-dimensional Gaussian space (see Example 1 and Corollaries 1 and 4), as well as applications to R1×Hnf, where Hn denotes the standard n-dimensional hyperbolic space and f is the square of the distance function to a fixed point of Hn; see Examples 2 and 3, as well as Corollaries 2, 3 and 6. Furthermore, we also infer the nonexistence of entire spacelike translating graphs constructed over the Riemannian base Pn; see Theorems 6 and 7, as well as Corollaries 7, 8 and 9.

    Throughout this paper, let us consider an (n+1)-dimensional Lorentzian product space ¯Mn+1 of the form R1×Pn, where (Pn,,Pn) is an n-dimensional connected Riemannian manifold and ¯Mn+1 is endowed with the standard product metric

    ,=πR(dt2)+πPn(,Pn),

    where πR and πPn denote the canonical projections from R1×Pn onto each factor. For a fixed t0R, we say that Pnt0={t0}×Pn is a slice of ¯Mn+1.

    Given an n-dimensional connected manifold Σn, a smooth immersion ψ:Σn¯Mn+1 is said to be a spacelike hypersurface if Σn, furnished with the metric induced by , via ψ, is a Riemannian manifold. If this is so, we shall 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 NX(Σ), having the same time-orientation of t, i.e., such that N,t1. One then says that N is the future-pointing Gauss map of Σn.

    In this setting, we will consider its Weingarten operator A:X(Σ)X(Σ), which is given by A(X)=¯XN, where ¯ stands for the Levi-Civita connection of ¯Mn+1. So, the (non-normalized) future mean curvature function of Σn is defined as been H=tr(A). Moreover, we will also denote by ¯ and the gradients with respect to the metrics of ¯Mn+1 and Σn, respectively. A simple computation shows that the gradient of πR on ¯Mn+1 is given by

    ¯πR=¯πR,tt=t. (2.1)

    So, from (2.1) we conclude that the gradient of the (vertical) height function h=(πR)|Σ of Σn is given by

    h=(¯πR)=t=tΘN, (2.2)

    where () denotes the tangential component of a vector field in X(¯Mn+1) along Σn and Θ=N,t stands for the hyperbolic angle function of Σn. Thus, we get the following relation

    |h|2=Θ21, (2.3)

    where || denotes the norm of a vector field on Σn. Moreover, from (2.2) we deduce that the Hessian of h on Σ, Hessh:X(Σ)×X(Σ)C(Σ), is given by

    Hessh(X,Y)=Xt,Y=¯X(t+ΘN),Y=AX,YΘ. (2.4)

    Hence, from (2.4) we obtain that the Laplacian of h is

    Δh=HΘ. (2.5)

    Now, we consider a smooth function f defined on R1×Pn. The triple (R1×Pn,,,dμ=efdσ), where dσ denotes the canonical volume element associated to the metric ,, will be called a weighted Lorentzian product space and f its weight function. According to [10], the Bakry-Émery-Ricci tensor ¯Ricf of such a weighted manifold is defined as being the following extension of the standard Ricci tensor Ric

    ¯Ricf=¯Ric+¯Hessf, (2.6)

    where ¯Hess denotes the Hessian of a function defined on ¯M.

    For a spacelike hypersurface Σn immersed in a weighted Lorentzian product space, the f-divergence operator on Σn is defined by

    divf(X)=efdiv(efX), (2.7)

    where X is a tangent vector field on Σn and div stands for the standard divergence operator of Σn. From this, for all smooth function u:ΣnR, we define the drift Laplacian of u by

    Δfu=divf(u)=Δuu,f. (2.8)

    Following the ideas of Gromov [24,Section 9.4.E] and considering the future-pointing Gauss map N of Σn, its (non-normalized) future f-mean curvature Hf is defined by

    Hf=H¯f,N, (2.9)

    where H denotes the future mean curvature of Σn.

    Remark 1. As a consequence of a splitting theorem due to Case [13,Theorem 1.2], if a weighted Lorentzian product space R1×P is endowed with a bounded weight function f and if its Bakry-Émery-Ricci tensor is such that ¯Ricf(V,V)0, for all timelike vector field V on R1×P, then f must be constant along R. Motivated by this result, along this work we will always consider that the ambient space is a weighted Lorentzian product space R1×Pn whose weight function f does not depend on the parameter tR, that is, ¯f,t=0 and, for sake of simplicity, we will denote it by R1×Pnf. In this setting, we get from (2.9) that the slices Pnt0 are f-maximal, which means that Hf is identically zero. $

    Considering the observations done in Remark 1, hereafter we study hypersurfaces in manifolds of the kind R1×Pnf. In such setting and in a similar spirit of [17,Definition 2] or [9,Eq (2.6)], we say that a spacelike hypersurface ψ:Σn¯Mn+1 immersed in a Lorentzian product space ¯Mn+1=R1×Pn is a spacelike translating soliton of the mean curvature flow with respect to t and with soliton constant cR if its future mean curvature function satisfies

    H=cΘ. (3.1)

    So, we observe that the slices {t}×Pn are spacelike translating solitons of the mean curvature flow with respect to t and with soliton constant c=0.

    In order to establish our first result, we quote a Liouville-type result due to Pigola, Rigoli and Setti, which is a consequence of [40,Theorem 1.1]. For this, we will consider the following set

    Lpf(M):={u:MnR:M|u|p(x)ef(x)dM<+}.

    Before introduce some useful results, recall that a smooth function u is f-subharmonic if Δfu0 and u is semi-bounded whether u is bounded from above or from below.

    The first useful Liouville-type result reads as follows:

    Lemma 1. Let u be a nonnegative smooth f-subharmonic function on a complete Riemannian manifold Mn. If uLpf(M), for some p>1, then u is constant.

    The next lemma is a consequence of an extension of another Liouville-type result due to Yau in [45].

    Lemma 2. The only harmonic semi-bounded functions defined on an n-dimensional complete Riemannian manifold whose Ricci tensor is nonnegative are the constant ones.

    Before presenting our first uniqueness result concerning spacelike translating solitons, we recall that a spacelike hypersurface is said maximal when its mean curvature is identically zero.

    Theorem 1. Let R1×Pnf be a weighted Lorentzian product space such that its Riemannian base Pn is complete, with nonnegative Bakry-Émery-Ricci tensor. Let ψ:ΣnR1×Pnf be a complete spacelike translating soliton of the mean curvature flow with respect to t, having soliton constant c and constant future f-mean curvature. If HLqf(Σ), for some q>2, then Σn is maximal. Moreover, if in addition Pn has nonnegative sectional curvature and Σn lies in a vertical half-space of R1×Pnf, then Σn must be a slice Pnt0 for some t0R.

    Proof. Let ψ:ΣnR1×Pnf be such a spacelike translating soliton. If c=0 the first conlusion is immediate. So, assume that c0. From [5,Corollary 8.2] we have the following key formula

    ΔΘ=(~Ric(N,N)+|A|2)Θ+H,t. (3.2)

    where ~Ric is the standard Ricci tensor of Pn and N=N+Θt denotes the orthonormal projection of N onto Pn.

    Thus, since Hf is constant, from (2.9) and (3.2) we obtain

    12ΔΘ2=(~Ric(N,N)+|A|2)Θ2+H,tΘ+|Θ|2=(~Ric(N,N)+|A|2)Θ2+t(¯f,N)Θ+|Θ|2. (3.3)

    On the other hand, [39,Proposition 7.35] gives that

    ¯Xt=0, (3.4)

    for every tangent vector field X on the ambient space. Then, from (2.2) and (3.4) we have that

    X(Θ)=¯XN,t=A(h),X

    and, consequently,

    Θ=A(h). (3.5)

    Thus, using once more (3.4), from (2.2) and (3.5) we get that

    t(¯f,N)=¯t¯f,N+¯f,¯tN=¯Hessf(N,N)Θ+¯f,Θ. (3.6)

    Substituting (3.6) in (3.3), we obtain

    12ΔΘ2=(~Ric(N,N)+|A|2)Θ2+¯Hessf(N,N)Θ2+12¯f,(Θ2)+|Θ|2. (3.7)

    Consequently, since ¯Hessf(N,N)=~Hessf(N,N), where ~Hess stands for the Hessian computed in the metric of Pn, using (2.6) and (2.8) in (3.7) we reach at the following equation

    12ΔfΘ2=(~Ricf(N,N)+|A|2)Θ2+|Θ|2, (3.8)

    where ~Ricf stands for the Bakry-Émery-Ricci tensor of Pn.

    Hence, since we are assuming that the soliton constant c is nonzero, from (3.1) and (3.8) we get the following inequality

    12ΔfH2(~Ricf(N,N)+|A|2)H2. (3.9)

    Since we are also supposing that ~Ricf is nonnegative, from (3.9) we obtain that ΔfH20. So, since HLqf(Σ), for some q>2, we can apply Lemma 1 for p=q/2 to conclude that H is constant.

    Returning to (3.9), we also get that |A| vanishes identically on Σn, which means that Σn is totally geodesic. Consequently, H must be identically zero and thus Σn is maximal and the first conclusion follows.

    Moreover, considering XX(Σ) and α=infPnKPn0, where KPn stands for the sectional curvature of Pn, from the proof of [8,Lemma 3.1] we obtain the following suitable lower bound for the Ricci tensor of Σn

    Ric(X,X)(n1)α|X|2+α|h|2|X|2+(n2)αX,h2+|AX|2. (3.10)

    In particular, from (3.10) we get that the Ricci tensor of Σn is nonnegative.

    Therefore, since from (2.5) we have that h is an harmonic function on Σn, and if we also assume that Σn lies in vertical half-space of R1×Pnf (which means that h is semi-bounded), we can apply Lemma 2 conclude that h is constant, that is, Σn is a slice Pnt0 for some t0R.

    Since the drift Laplacian is an elliptic operator and taking into account inequality (3.9), we can apply the classical strong maximum principle due to Hopf [26] obtaining the following result; see also the classical book [41] due to Pucci and Serrin for related topics.

    Theorem 2. Let R1×Pnf be a weighted Lorentzian product space such that its Riemannian base Pn is complete, with nonnegative Bakry-Émery-Ricci tensor. Let ψ:ΣnR1×Pnf be a complete spacelike translating soliton of the mean curvature flow with respect to t, having soliton constant c and constant future f-mean curvature. If H2 attains its maximum on Σn, then Σn is maximal. Moreover, if in addition Pn has nonnegative sectional curvature and Σn lies in vertical half-space of R1×Pnf, then Σn must be a slice Pnt0 for some t0R.

    Example 1. An important example of weighted Riemannian manifold is the so-called Gaussian space Gn, which corresponds to the Euclidean space Rn endowed with the Gaussian probability measure

    efdx2=(2π)n2e|x|22dx2. (3.11)

    Concerned with the weighted Lorentzian product space R1×Gn, An et al extended the classical Bernstein's theorem [11] showing that the only entire f-maximal graphs Σn(u) of functions u(x2,,xn+1)=x1 defined over Gn, with supΣ(u)|Du|Rn<1 (where ||Rn is the standard norm of Rn), are the hyperplanes x1=constant; see [7,Theorem 4].

    Since the Bakry-Émery-Ricci tensor of Gn is positive, from Theorem 1 and Example 1 we obtain the following consequence.

    Corollary 1. The only complete spacelike translating solitons ψ:ΣnR1×Gn of the mean curvature flow with respect to t, having constant future f-mean curvature (where f is the Gaussian probability measure defined in (3.11)), lying in a vertical half-space of R1×Gn and such that either H2 attains its maximum on Σn or HLqf(Σ), for some q>2, are the slices {t}×Gn.

    Example 2. Let Hn={(x1,,xn)Rn:xn>0} be the n-dimensional hyperbolic space endowed with its standard complete metric

    ,Hn=1x2n(dx21++dx2n)

    and let f:HnR be the weight function given by

    f(x)=(d(x,x0))2, (3.12)

    where d(x,x0) denotes the distance in Hn between x and a fixed point x0Hn. According to [42,Example 7.2], we have that the Bakry-Émery-Ricci tensor of Hnf satisfies Ricf(n1). Thus, the weighted Lorentzian product space R1×Hnf is such that its base Hnf has nonnegative Bakry-Émery-Ricci tensor.

    So, from Theorem 1 and Example 2 we also get the following result.

    Corollary 2. Let R1×Hnf be the weighted Lorentzian product space with weight function f defined by (3.12). Let ψ:ΣnR1×Hnf be a complete spacelike translating soliton of the mean curvature flow with respect to t, having soliton constant c and constant future f-mean curvature. If either H2 attains its maximum on Σn or HLqf(Σ), for some q>2, then Σn is maximal.

    In [45], Yau established the following version of Stokes' Theorem on an n-dimensional complete noncompact Riemannian manifold Σn: if ωΩn1(Σn) is an integrable (n1)-differential form on Σn, then there exists a sequence Bi of domains on Σn such that

    BiBi+1,Σn=i1Bi,andlimiBidω=0.

    Later on, supposing that Σn is oriented by the volume element dΣ and denoting by ω=ιXdΣ the contraction of dΣ in the direction of a smooth vector field X on Σn, Caminha extended this Yau's result showing that if the divergence of X, divΣX, does not change sign and that |X| is Lebesgue integrable on Σn, then divΣX must be identically zero on Σn; see [12,Proposition 2.1].

    Taking into account (2.7), from [12,Proposition 2.1] above mentioned we obtain our next auxiliary lemma.

    Lemma 3. Let u be a smooth function on a complete Riemannian manifold Σn endowed with a weight function f:ΣnR, such that Δfu does not change sign on Σn. If |u|L1f(Σ), then Δfu vanishes identically on Σn.

    We will use Lemma 3 to establish our next uniqueness result.

    Theorem 3. Let R1×Pnf be a weighted Lorentzian product space such that its Riemannian base Pn is complete, with nonnegative Bakry-Émery-Ricci tensor. Let ψ:ΣnR1×Pnf be a complete spacelike translating soliton of the mean curvature flow with respect to t, having soliton constant c and constant future f-mean curvature. If H is bounded and |H|L1f(Σ), then Σn is maximal. Moreover, if in addition Pn has nonnegative sectional curvature and Σn lies in vertical half-space of R1×Pnf, then Σn must be a slice Pnt0 for some t0R.

    Proof. Since we are assuming that H is bounded and |H|L1f(Σ), we obtain

    |(H2)|=2|H||H|L1f(Σ).

    Therefore, we can reason as in the proof of Theorem 1 applying Lemma 3 instead of Lemma 1 to obtain the proof of Theorem 3.

    From Theorem 3 we obtain the following consequence.

    Corollary 3. Let R1×Hnf be the weighted Lorentzian product space with weight function f defined by (3.12). Let ψ:ΣnR1×Hnf be a complete spacelike translating soliton of the mean curvature flow with respect to t, having soliton constant c and constant future f-mean curvature. If H is bounded and |H|L1f(Σ), then Σn is maximal.

    Example 3. Considering the same context of Example 2 and fixing a constant cR with 0<|c|<1, from [15,Example 4.4] we have that

    Σn={(clnxn,x1,...,xn):xn>0}R1×Hn

    is a complete spacelike translating soliton of the mean curvature flow with respect to t, having soliton constant c and constant future mean curvature

    H=c1c2=cΘ.

    Consequently, we have that the assumption that Hf being constant in Theorem 3 is a necessary hypothesis to conclude that the spacelike translating soliton is maximal.

    We recall that (according to the classical terminology in linear potential theory) a weighted manifold Σn 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 Σn. In this context, from [4,Corollary 2] we have the following f-parabolicity criterion.

    Lemma 4. Let R1×Pnf be a weighted Lorentzian product space whose Riemannian base Pn is complete with f-parabolic universal Riemannian covering and let ψ:ΣnR1×Pnf be a complete spacelike hypersurface. If the hyperbolic angle function Θ is bounded, then Σn is f-parabolic.

    Using this previous parabolicity criterion, we obtain the following result.

    Theorem 4. Let R1×Pnf be a weighted Lorentzian product space such that its Riemannian base Pn is complete, with nonnegative Bakry-Émery-Ricci tensor and f-parabolic universal Riemannian covering. Let ψ:ΣnR1×Pnf be a complete spacelike translating soliton of the mean curvature flow with respect to t, having soliton constant c and constant future f-mean curvature. If H is bounded, then Σn is maximal. Moreover, if in addition Pn has nonnegative sectional curvature and Σn lies in vertical half-space of R1×Pnf, then Σn must be a slice Pnt0 for some t0R.

    Proof. We argue as the proof of Theorem 1. If c=0, the there is nothing to do. So assume that c0. Thus, since we are assuming that H is bounded, we define the smooth function φ on Σn by

    φ:=12{(supΣH2)H2}.

    Since ~Ricf is supposed to be nonnegative, from (3.9) we get

    Δfφ(~Ricf(N,N)+|A|2)H20.

    Consequently, φ is a nonnegative, f-superharmonic function defined on Σn.

    On the other hand, from (3.1) we have that the boundedness of H also implies the boundedness of the hyperbolic angle function Θ. So, Lemma 4 guarantees that Σn is f-parabolic.

    Hence, we conclude that H must be constant on Σn. At this point, we can reason as in the proof of Theorem 1 to conclude our result.

    From [25,Corollary 3] we have that the Gaussian space Gn has finite f-volume, where f is the Gaussian probability measure defined in (3.11). Consequently, taking into account [28,Remark 3.8], we conclude that Gn is f-parabolic. This fact enable us to state the following application of Theorem 4.

    Corollary 4. The only complete spacelike translating solitons ψ:ΣnR1×Gn of the mean curvature flow with respect to t, having constant future f-mean curvature (where f is the Gaussian probability measure defined in (3.11)), lying in a vertical half-space of R1×Gn and such that H is bounded, are the slices {t}×Gn.

    For our next results we need the following definition, which first appeared with Omori [38] for the Hessian and, afterwards, with Yau [44] for the Laplacian. For the drift Laplacian, we can find some general versions in [6].

    Definition 1. Let (Σn,,) be a Riemannian manifold and let f:ΣnR be a smooth function. We say that the drift Laplacian Δf satisfies the Omori-Yau's maximum principle on Σ if, for every uC2 with u=supΣu<, there exists a sequence {xk}Σ such that

    u(xk)>u1k,|u(xk)|<1k,andΔfu(xk)<1k,

    for every kN.

    The proposition below gives sufficient conditions to guarantee that the drift Laplacian on a spacelike translating soliton immersed in a Lorentzian product space satisfies the Omori-Yau maximum principle for a weight function f whose Hessian is assumed to be bounded from below.

    Proposition 1. Let ¯Mn+1=R1×Pnf be a weighted Lorentzian product space, whose Riemannian base Pn is complete with sectional curvature KPn such that KPnκ for some positive constant κ. Let ψ:Σn¯Mn+1 be a complete spacelike translating soliton with soliton constant c0, bounded mean curvature and the Hessian of f+ch along Σn is bounded from below. Then, the drift Laplacian Δf on Σn satisfies the Omori-Yau's maximum principle.

    Proof. We recall that the curvature tensor R of a spacelike hypersurface ψ:ΣnR1×Pn can be described in terms of the corresponding shape operator A and the curvature tensor ¯R of R1×Pn by the so-called Gauss equation given by

    R(X,Y)Z=(¯R(X,Y)Z)AX,ZAY+AY,ZAX, (3.13)

    for every tangent vector fields X,Y,ZX(Σ). Here, as in [39], the curvature tensor R is given by

    R(X,Y)Z=[X,Y]Z[X,Y]Z,

    where [,] denotes the Lie bracket and X,Y,ZX(Σ).

    Let us consider XX(Σ) and a local orthonormal frame {E1,...,En} of X(Σ). Then, it follows from the Gauss equation (3.13) that

    Ric(X,X)=ni=1¯R(X,Ei)X,EiHAX,X+|AX|2. (3.14)

    Moreover, we have that

    ¯R(X,Ei)X,Ei=R(X,Ei)X,EiPn=KPn(X,Ei)(X,XPnEi,EiPnX,Ei2Pn). (3.15)

    On the other hand, since X=X+X,tt, Ei=Ei+Ei,tt and h=t, with a straightforward computation we see that

    X,XPnEi,EiPn=(1+Ei,h2)(|X|2+X,h2) (3.16)

    and

    X,Ei2Pn=X,Ei2+2X,hEi,hX,Ei+X,h2Ei,h2. (3.17)

    Then, since we are supposing that KPnκ for some positive constant κ, inserting (3.16) and (3.17) into (3.15) we obtain

    ni=1¯R(X,Ei)X,Eiκ((n1)|X|2+(n2)X,h2+|X|2|h|2). (3.18)

    Consequently, from (2.3) and (3.18) we get

    ni=1¯R(X,Ei)X,Ei(n1)κΘ2|X|2. (3.19)

    Thus, from (3.14), (3.19), and taking into account that |H|n|A|, we obtain

    Ric(X,X)(n1)κΘ2|X|2HAX,X+|AX|2{(n1)κΘ2}|X|2+Hess(ch)(X,X). (3.20)

    Hence, since Hess(f+ch)β, H is bounded and using (2.6), (3.1) and (3.20) we get

    Ricf(X,X)((n1)κsupΣΘ2+β)|X|2.

    Therefore, we can apply [14,Theorem 1] to conclude our result.

    As a direct consequence of the previous result we have the following corollary.

    Corollary 5. Let ¯Mn+1=R1×Pnf be a weighted Lorentzian product space, whose Riemannian base Pn is complete with sectional curvature KPn such that KPnκ for some positive constant κ, and Hessian of f bounded from below. Let ψ:Σn¯Mn+1 be a complete spacelike translating soliton with soliton constant c0, Hf constant and bounded second fundamental form. Then, the drift Laplacian Δf on Σn satisfies the Omori-Yau's maximum principle.

    Proof. We notice that ¯Hessf(X,Y)=Hessf(X,Y)+fNAX,Y, for all X,YX(Σ), where fN stands for N,¯f. Thus, since Hf=HfN is a constant e and Hess(ch)=HA, we deduce that:

    ¯Hessf=Hess(f+ch)eA.

    From this expression and our hypothesis, we obtain that

    Hess(f+ch)βandHisbounded,

    and the result follows from Proposition 1.

    Next, we apply Proposition 1 to establish the following nonexistence result.

    Theorem 5. Let R1×Pnf be a weighted Lorentzian product space such that its Riemannian base Pn is complete, with sectional curvature KPn such that KPnκ for some positive constant κ, and nonnegative Bakry-Émery-Ricci tensor. There does not exist a complete spacelike translating soliton ψ:ΣnR1×Pnf of the mean curvature flow with respect to t, having soliton constant c0, bounded mean curvature and Hessian of f+ch is bounded from below.

    Proof. Let us suppose by contradiction the existence of such a spacelike translating soliton Σn. Since H2n|A|2, from our assumptions jointly with inequality (3.9), we can apply Proposition 1 obtaining a sequence of points {pk} in Σn such that

    012lim supkΔfH2(pk)lim supk(~Ricf(N,N)+|A|2)(pk)supΣH20. (3.21)

    Therefore, since our hypothesis that c0 implies that supΣH2>0 and using once more that H2n|A|2, from (3.21) we get

    0<supΣH2nlimk|A|2(pk)=0,

    reaching at a contradiction.

    As a consequence of the proof of Theorem 5 and using Corollary 5 we obtain the following consequence.

    Corollary 6. There does not exist a complete spacelike translating soliton immersed in either R1×Gn or R1×Hnf (where the weight function f is defined by (3.12)) of the mean curvature flow with respect to t, having soliton constant c0, constant future f-mean curvature and bounded second fundamental form.

    We recall that an entire graph over the Riemannian base Pn is determined by a smooth function uC(Pn) and it is given by

    Σ(u)={(u(x),x);xPn}R1×Pnf.

    The metric induced on Pn from the Lorentzian metric on the ambient space via Σ(u) is

    ,=du2+,Pn. (4.1)

    It can be easily seen from (4.1) that a graph Σ(u) is a spacelike hypersurface if, and only if, |Du|2Pn<1, Du being the gradient of u in Pn and |Du|Pn its norm, both with respect to the metric of Pn. It is well known that in the case where Pn is a simply connected manifold, every complete spacelike hypersurface Σn immersed in R1×Pn is an entire spacelike graph over Pn; see, for instance, [3,Lemma 3.1]. It is interesting to observe that, according to the examples of non-complete entire maximal graphs in R1×H2 due to Albujer in [2,Section 3], an entire spacelike graph Σ(u) in R1×Pn is not necessarily complete, in the sense that the induced Riemannian metric is not necessarily complete on Pn. However, it was proven in the beginning of [16,Corollary 1] that if Pn is complete and |Du|Pnα for certain positive constant α<1, then Σ(u) must be complete.

    The future-pointing Gauss map of an entire spacelike graph Σ(u) constructed over the Riemannian fiber Pn is given by the vector field

    N(x)=11|Du(x)|2Pn(t|(u(x),x)+Du(x)),xPn. (4.2)

    Moreover, the second fundamental form A of Σ(u) with respect to its orientation (4.2) is given by

    AX=11|Du|2PnDXDuDXDu,DuPn(1|Du|2Pn)3/2Du, (4.3)

    for any smooth vector field X tangent to Pn. Consequently, if Σ(u) is a spacelike entire graph over the Riemannian fiber Pn of a weighted Lorentzian product space R1×Pn, it is not difficult to verify from (2.9) and (4.3) that the future f-mean curvature function Hf(u) of Σ(u) is given by

    Hf(u)=divf(Du1|Du|2Pn).

    When an entire spacelike graph Σ(u)R1×Pnf is a spacelike translating soliton of the mean curvature flow with respect to t, it is called an entire spacelike translating graph. In this context, it is not difficult to verify that from Theorem 4 we obtain the following nonexistence result.

    Theorem 6. Let R1×Pnf be a weighted Lorentzian product space such that its Riemannian base Pn is complete, with nonnegative sectional curvature, nonnegative Bakry-Émery-Ricci tensor and f-parabolic universal Riemannian covering. For any constants C0 and 0<α<1, there does not exist an entire spacelike translating graph Σ(u)R1×Pnf such that the corresponding smooth function uC(Pn) is a semi-bounded solution of the system

    {divf(Du1|Du|2Pn)=C|Du|Pnα.

    Remark 2. Taking into account Theorem 2 and since (4.2) implies that the mean curvature H of an entire spacelike translating graph Σ(u)R1×Pnf is such that

    H2=c21|Du(x)|2Pn,

    we can replace the assumption that the Riemannian base Pn has f-parabolic universal Riemannian covering in Theorem 6, by the hypothesis that |Du|Pn attains its maximum on Σ(u). $

    When the ambient space is R1×Gn, Theorem 6 reads as follows.

    Corollary 7. For any constants C0 and 0<α<1, there does not exist an entire spacelike translating graph Σ(u)R1×Gn such that the corresponding smooth function uC(Rn) is a semi-bounded solution of the system

    {divf(Du1|Du|2Rn)=C|Du|Rnα,

    where f is the Gaussian probability measure defined in (3.11).

    We say that uC(Pn) has finite C2 norm when

    uC2(Pn):=sup|γ|2|Dγu|L(Pn)<.

    In this context, observing that (4.3) guarantees that a spacelike translating graph Σ(u)R1×Pnf whose corresponding smooth function uC(Pn) has finite C2 norm has bounded second fundamental form, from the proof of Theorem 5 and Corollary 5 we obtain the following nonexistence result.

    Theorem 7. Let R1×Pnf be a weighted Lorentzian product space such that its Riemannian base Pn is complete, with sectional curvature KPn such that KPnκ for some positive constant κ, and the Hessian of the weight function f bounded from below, and nonnegative Bakry-Émery-Ricci tensor. For any constants C0 and 0<α<1, there does not exist an entire spacelike translating graph Σ(u)R1×Pnf such that the corresponding smooth function uC(Pn) has finite C2 norm and it is a solution of the system

    {divf(Du1|Du|2Pn)=C|Du|Pnα.

    We close our paper with the following consequences of Theorem 7.

    Corollary 8. For any constants C0 and 0<α<1, there does not exist an entire spacelike translating graph Σ(u)R1×Gn such that the corresponding smooth function uC(Rn) has finite C2 norm and it is a solution of the system

    {divf(Du1|Du|2Rn)=C|Du|Rnα,

    where f is the Gaussian probability measure defined in (3.11).

    Corollary 9. For any constants C0 and 0<α<1, there does not exist an entire spacelike translating graph Σ(u)R1×Hnf such that the corresponding smooth function uC(Hn) has finite C2 norm and it is a solution of the system

    {divf(Du1|Du|2Hn)=C|Du|Hnα,

    where f is the weight function defined in (3.12).

    The authors would like to thank the referee for his/her careful reading and many useful comments which improved the presentation of this paper. The first and third authors are partially supported by the Brazilian National Council for Scientific and Technological Development [grants: 308440/2021-8 and 405468/2021-0 to M.B., 301970/2019-0 to H.dL.]. The second author is partially supported by INdAM-GNAMPA Research Project 2020 titled Equazioni alle derivate parziali: problemi e modelli.

    The authors declare no conflict of interest.



    [1] M. A. S. Aarons, Mean curvature flow with a forcing term in Minkowski space, Calc. Var., 25 (2005), 205–246. http://doi.org/10.1007/S00526-005-0351-8 doi: 10.1007/S00526-005-0351-8
    [2] A. L. Albujer, New examples of entire maximal graphs in H2×R1, Differ. Geom. Appl., 26 (2008), 456–462. https://doi.org/10.1016/j.difgeo.2007.11.035 doi: 10.1016/j.difgeo.2007.11.035
    [3] A. L. Albujer, L. J. Alías, Calabi-Bernstein results for maximal surfaces in Lorentzian product spaces, J. Geom. Phys., 59 (2009), 620–631. https://doi.org/10.1016/j.geomphys.2009.01.008 doi: 10.1016/j.geomphys.2009.01.008
    [4] A. L. Albujer, H. F. de Lima, A. M. Oliveira, M. A. L. Velásquez, ϕ-Parabolicity and the uniqueness of spacelike hypersurfaces immersed in a spatially weighted GRW spacetime, Mediterr. J. Math., 15 (2018), 84. https://doi.org/10.1007/s00009-018-1134-8 doi: 10.1007/s00009-018-1134-8
    [5] L. J. Alías, A. G. Colares, Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson-Walker spacetimes, Math. Proc. Cambridge, 143 (2007), 703–729. http://doi.org/10.1017/S0305004107000576 doi: 10.1017/S0305004107000576
    [6] L. J. Alías, P. Mastrolia, M. Rigoli, Maximum principles and geometric applications, Cham: Springer, 2016. https://doi.org/10.1007/978-3-319-24337-5
    [7] H. V. Q. An, D. V. Cuong, N. T. M. Duyen, D. T. Hieu, T. L. Nam, On entire f-maximal graphs in the Lorentzian product Gn×R1, J. Geom. Phys., 114 (2017), 587–592. http://doi.org/10.1016/j.geomphys.2016.12.023 doi: 10.1016/j.geomphys.2016.12.023
    [8] C. P. Aquino, H. I. Baltazar, H. F. de Lima, A new Calabi-Bernstein type result in spatially closed generalized Robertson-Walker spacetimes, Milan J. Math., 85 (2017), 235–245. https://doi.org/10.1007/s00032-017-0271-z doi: 10.1007/s00032-017-0271-z
    [9] M. Batista, H. F. de Lima, Spacelike translating solitons in Lorentzian product spaces: Nonexistence, Calabi-Bernstein type results and examples, Commun. Contemp. Math., 24 (2022), 2150034. http://doi.org/10.1142/S0219199721500346 doi: 10.1142/S0219199721500346
    [10] D. Bakry, M. Émery, Diffusions hypercontractives, In: Séminaire de Probabilités XIX 1983/84, Berlin: Springer, 1985,177–206. https://doi.org/10.1007/BFb0075847
    [11] S. Bernstein, Sur les surfaces d'efinies au moyen de leur courboure moyenne ou totale, Annales scientifiques de l'École Normale Supérieure, Serie 3, 27 (1910), 233–256. https://doi.org/10.24033/asens.621 doi: 10.24033/asens.621
    [12] A. Caminha, The geometry of closed conformal vector fields on Riemannian spaces, Bull. Braz. Math. Soc., 42 (2011), 277–300. https://doi.org/10.1007/s00574-011-0015-6 doi: 10.1007/s00574-011-0015-6
    [13] J. S. Case, Singularity theorems and the Lorentzian splitting theorem for the Bakry-Émery-Ricci tensor, J. Geom. Phys., 60 (2010), 477–490. https://doi.org/10.1016/j.geomphys.2009.11.001 doi: 10.1016/j.geomphys.2009.11.001
    [14] Q. Chen, H. Qiu, Rigidity of self-shrinkers and translating solitons of mean curvature flows, Adv. Math., 294 (2016), 517–531. https://doi.org/10.1016/j.aim.2016.03.004 doi: 10.1016/j.aim.2016.03.004
    [15] H. F. de Lima, E. A. Lima Jr, Generalized maximum principles and the unicity of complete spacelike hypersurfaces immersed in a Lorentzian product space, Beitr. Algebra Geom., 55 (2013), 59–75. http://doi.org/10.1007/s13366-013-0137-7 doi: 10.1007/s13366-013-0137-7
    [16] H. F. de Lima, A. M. Oliveira, M. S. Santos, Rigidity of complete spacelike hypersurfaces with constant weighted mean curvature, Beitr. Algebra Geom., 57 (2016), 623–635. http://doi.org/10.1007/s13366-015-0253-7 doi: 10.1007/s13366-015-0253-7
    [17] J. H. S. de Lira, F. Martín, Translating solitons in Riemannian products, J. Differ. Equations, 266 (2019), 7780–7812. https://doi.org/10.1016/j.jde.2018.12.015 doi: 10.1016/j.jde.2018.12.015
    [18] K. Ecker, On mean curvature flow of spacelike hypersurfaces in asymptotically flat spacetime, J. Aust. Math. Soc., 55 (1993), 41–59. https://doi.org/10.1017/S1446788700031918 doi: 10.1017/S1446788700031918
    [19] K. Ecker, Interior estimates and longtime solutions for mean curvature flow of noncompact spacelike hypersurfaces in Minkowski space, J. Differ. Geom., 46 (1997), 481–498. http://doi.org/10.4310/jdg/1214459975 doi: 10.4310/jdg/1214459975
    [20] K. Ecker, Mean curvature flow of spacelike hypersurfaces near null initial data, Commun. Anal. Geom., 11 (2003), 181–205. https://doi.org/10.4310/CAG.2003.v11.n2.a1 doi: 10.4310/CAG.2003.v11.n2.a1
    [21] K. Ecker, G. Huisken, Parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes, Commun. Math. Phys., 135 (1991), 595–613. http://doi.org/https://doi.org/10.1007/BF02104123 doi: 10.1007/BF02104123
    [22] G. J. Galloway, E. Woolgar, Cosmological singularities in Bakry-Émery spacetimes, J. Geom. Phys., 86 (2014), 359–369. https://doi.org/10.1016/j.geomphys.2014.08.016 doi: 10.1016/j.geomphys.2014.08.016
    [23] S. Gao, G. Li, C. Wu, Translating spacelike graphs by mean curvature flow with prescribed contact angle, Arch. Math., 103 (2014), 499–508. https://doi.org/10.1007/s00013-014-0699-0 doi: 10.1007/s00013-014-0699-0
    [24] M. Gromov, Isoperimetry of waists and concentration of maps, Geom. Funct. Anal., 13 (2003), 178–215. http://doi.org/10.1007/s000390300004 doi: 10.1007/s000390300004
    [25] D. T. Hieu, T. L. Nam, Bernstein type theorem for entire weighted minimal graphs in Gn×R, J. Geom. Phys., 81 (2014), 87–91. http://doi.org/10.1016/j.geomphys.2014.03.011 doi: 10.1016/j.geomphys.2014.03.011
    [26] E. Hopf, Elementare Bemerkungen über die Lösungen partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus, Akad. Wiss. Berlin: S.-B. Preuss, 1927: 147–152.
    [27] G. Huisken, S. T. Yau, Definition of center of mass for isolated physical system and unique foliations by stable spheres with constant curvature, Invent. Math., 124 (1996), 281–311. http://doi.org/10.1007/s002220050054 doi: 10.1007/s002220050054
    [28] D. Impera, M. Rimoldi, Stability properties and topology at infinity of f-minimal hypersurfaces, Geom. Dedicata, 178 (2015), 21–47. http://doi.org/10.1007/s10711-014-9999-6 doi: 10.1007/s10711-014-9999-6
    [29] H. Jian, Translating solitons of mean curvature flow of noncompact spacelike hypersurfaces in Minkowski space, J. Differ. Equations, 220 (2006), 147–162. https://doi.org/10.1016/j.jde.2005.08.005 doi: 10.1016/j.jde.2005.08.005
    [30] H. Ju, J. Lu, H. Jian, Translating solutions to mean curvature flow with a forcing term in Minkowski space, Commun. Pure Appl. Anal., 9 (2010), 963–973. https://doi.org/10.3934/cpaa.2010.9.963 doi: 10.3934/cpaa.2010.9.963
    [31] B. Lambert, A note on the oblique derivative problem for graphical mean curvature flow in Minkowski space, Abh. Math. Semin. Univ. Hambg., 82 (2012), 115–120. https://doi.org/10.1007/s12188-012-0066-7 doi: 10.1007/s12188-012-0066-7
    [32] B. Lambert, The perpendicular Neumann problem for mean curvature flow with a timelike cone boundary condition, Trans. Amer. Math. Soc., 366 (2014), 3373–3388. https://www.jstor.org/stable/23813865
    [33] B. Lambert, J. D. Lotay, Spacelike mean curvature flow, J. Geom. Anal., 31 (2021), 1291–1359. https://doi.org/10.1007/s12220-019-00266-4 doi: 10.1007/s12220-019-00266-4
    [34] G. Li, I. Salavessa, Mean curvature flow of spacelike graphs, Math. Z., 269 (2011), 697–719. https://doi.org/10.1007/s00209-010-0768-4 doi: 10.1007/s00209-010-0768-4
    [35] A. Lichnerowicz, Variétés Riemanniennes à tenseur C non négatif, C. R. Acad. Sci. Paris Sér. A, 271 (1970), 650–653.
    [36] A. Lichnerowicz, Variétés Kählériennes à première classe de Chern non negative et variétés Riemanniennes à courbure de Ricci généralisée non negative, J. Differ. Geom., 6 (1971), 47–94. https://doi.org/10.4310/jdg/1214428089 doi: 10.4310/jdg/1214428089
    [37] F. Morgan, Manifolds with density, Notices of the American Mathematical Society, 52 (2005), 853–858.
    [38] H. Omori, Isometric immersions of Riemannian manifolds, J. Math. Soc. Japan, 19 (1967), 205–214. https://doi.org/10.2969/jmsj/01920205 doi: 10.2969/jmsj/01920205
    [39] B. O'Neill, Semi-Riemannian geometry with applications to relativity, London: Academic Press, 1983.
    [40] S. Pigola, M. Rigoli, A. G. Setti, Vanishing theorems on Riemannian manifolds, and geometric applications, J. Funct. Anal., 229 (2005), 424–461. https://doi.org/10.1016/j.jfa.2005.05.007 doi: 10.1016/j.jfa.2005.05.007
    [41] P. Pucci, J. Serrin, The strong maximum principle revisited, J. Differ. Equations, 196 (2004), 1–66. https://doi.org/10.1016/j.jde.2003.05.001 doi: 10.1016/j.jde.2003.05.001
    [42] G. Wei, W. Willie, Comparison geometry for the Bakry-Émery Ricci tensor, J. Differ. Geom., 83 (2009), 377–405. https://doi.org/10.4310/jdg/1261495336 doi: 10.4310/jdg/1261495336
    [43] E. Woolgar, Scalar-tensor gravitation and the Bakry-Émery-Ricci tensor, Class. Quantum Grav., 30 (2013), 085007. https://doi.org/10.1088/0264-9381/30/8/085007 doi: 10.1088/0264-9381/30/8/085007
    [44] S. T. Yau, Harmonic functions on complete Riemannian manifolds, Commun. Pure Appl. Math., 28 (1975), 201–228. https://doi.org/10.1002/cpa.3160280203 doi: 10.1002/cpa.3160280203
    [45] S. T. Yau, Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry, Indiana Univ. Math. J., 25 (1976), 659–670. https://doi.org/10.1512/iumj.1976.25.25051 doi: 10.1512/iumj.1976.25.25051
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