Entire hypersurfaces of constant scalar curvature in Minkowski space

Pierre Bayard; Andrea Seppi

doi:10.1515/crelle-2025-0023

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Entire hypersurfaces of constant scalar curvature in Minkowski space

Pierre Bayard and Andrea Seppi

Published/Copyright: April 30, 2025

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2025 Issue 824

Abstract

We show that every regular domain 𝒟 in Minkowski space R n , 1 which is not a wedge admits an entire hypersurface whose domain of dependence is 𝒟 and whose scalar curvature is a prescribed constant (or function, under suitable hypotheses) in ( − ∞ , 0 ) . Under rather general assumptions, these hypersurfaces are unique and provide foliations of 𝒟. As an application, we show that every maximal globally hyperbolic Cauchy compact flat spacetime admits a foliation by hypersurfaces of constant scalar curvature, generalizing to any dimension previous results of Barbot–Béguin–Zeghib (for n = 2 ) and Smith (for n = 3 ).

1 Introduction

The study of spacelike hypersurfaces in Minkowski space R n , 1 , and more generally in locally Minkowski manifolds, is a largely explored subject that connects differential geometry, geometric analysis, geometric topology and mathematical physics. A very natural class is that of entire spacelike hypersurfaces, that is, those which are graphs over R n , or equivalently which are properly embedded.

While for maximal hypersurfaces (i.e. of vanishing mean curvature) a Bernstein-type theorem holds in any dimension, meaning that every entire maximal hypersurface is a spacelike hyperplane (as proved in the 1970s by Calabi [12] for n ≤ 4 and by Cheng and Yau [13] for any 𝑛), over the 1980s, considerable interest developed in entire constant mean curvature (CMC) hypersurfaces, for which there is an abundance of non-trivial examples. This interest was initially motivated by the relation of CMC hypersurfaces to harmonic maps, provided by the Gauss map. The classification of entire CMC hypersurfaces was completed in [11], building on the earlier works [31, 14].

In parallel, again in the 1980s, Hano and Nomizu [21] initiated the study in dimension 2 + 1 of entire surfaces of constant (negative) Gaussian curvature – that is, surfaces intrinsically locally isometric to hyperbolic space, up to a factor. Here progress was made in [24, 20, 9], and the full classification was completed in [10].

There are (at least) two possible ways to generalize these results about constant Gaussian curvature surfaces in R 2 , 1 to higher dimensions. One possibility is to consider hypersurfaces in R n , 1 of constant Gauss–Kronecker curvature, that is, such that the determinant of the shape operator is constant. Although there are in general serious regularity issues in higher dimensions, results in this direction have been obtained in [15, 24, 19, 20, 6, 16, 8, 27].

The second possibility, which we develop in this work, is the classification problem for hypersurfaces of constant (negative) scalar curvature in any dimension n + 1 (equivalently, those for which the second elementary polynomial of the principal curvatures is constant). Let us also briefly mention that other symmetric functions of the principal curvatures were recently studied in [33, 29, 28].

Asymptotic data

Before stating our results, we need to take a step back and explain how the classification problem is formulated. In all the aforementioned classification results, the classifying data encode the asymptotic behaviour of the hypersurfaces. The information on such asymptotic behaviour can be expressed in several ways, which we now briefly describe. It is worth observing immediately that these formulations are all completely equivalent, at least for convex hypersurfaces – and more generally for hypersurfaces which are trapped between two convex hypersurfaces with the same asymptotics, which will be the case here.

Let us start with the “geometric” approach. Given an entire spacelike hypersurface Σ in R n , 1 , its domain of dependence is the set of points 𝑝 such that every inextensible timelike curve through 𝑝 intersects Σ. When Σ has mean curvature bounded below by a positive constant, which will be the case, up to time reversal, for hypersurfaces of constant negative scalar curvature (see Section 3.3), the domain of dependence is a future regular domain (denoted 𝒟), that is, a convex domain obtained as a non-trivial intersection of future half-spaces bounded by lightlike hyperplanes.

Now, let us consider the normal vectors of the lightlike hyperplanes whose future half-spaces contain 𝒟 (or equivalently, contain Σ). These normal vectors are in turn lightlike, since they are normal to a lightlike hyperplane, and therefore are collinear with ( y , 1 ) for some y ∈ S n − 1 . However, not all points of S n − 1 are realized in this way: it might happen that, for some y ∈ S n − 1 , there is no lightlike hyperplane normal to ( y , 1 ) (regardless of the value of its intercept with the vertical axis) that avoids 𝒟. For this reason, it is natural to consider a subset 𝐹 of the sphere, consisting of those points 𝑦 that are actually realized, and a real-valued function 𝜑 defined on 𝐹, so that φ ⁢ ( y ) equals, up to a sign, the maximal height of a lightlike hyperplane normal to ( y , 1 ) that avoids 𝒟. To simplify the set-up, it is helpful to consider 𝜑 as a function defined on the whole S n − 1 , declaring it to be identically + ∞ outside of 𝐹. It turns out that such function 𝜑 is lower semi-continuous. We then denote 𝐹 by F φ : = { φ < + ∞ } .

To make an example, if Σ is the future unit hyperboloid in R n , 1 , then its domain of dependence 𝒟 is the future cone over the origin, that corresponds to φ ≡ 0 and F φ = S n − 1 . As a somehow antipodal example, if Σ is the product of a hyperbola in R 1 , 1 and a copy of R n − 1 , then its domain of dependence 𝒟 is a wedge, that is, a future regular domain obtained as the intersection of precisely two future half-spaces bounded by non-parallel lightlike hyperplanes. In this case, 𝜑 takes finite values on only two points 𝑦 (the only two unit vectors such that ( y , 1 ) lies in the chosen copy of R 1 , 1 ), i.e. the cardinality of F φ = { φ < + ∞ } is 2.

The future regular domain 𝒟 is, in general, easily recovered from the data of the function 𝜑, as an intersection of the corresponding lightlike hyperplanes. We can formally summarize the whole discussion by saying that any future regular domain can be written as the supergraph of the 1-Lipschitz convex function V φ : R n → R defined by

(1.1) V φ ( x ) : = sup y ∈ F φ ( ⟨ x , y ⟩ − φ ( y ) ) ,

where φ : S n − 1 → R ∪ { + ∞ } is a lower semi-continuous function and F φ : = { φ < + ∞ } . We denote by D φ the future regular domain obtained by this construction, namely the supergraph of V φ .

The “analytic” approach, which is adopted among others in [31, 14], consists in expressing an entire spacelike hypersurface Σ as the graph of a function u : R n → R with | D ⁢ u | < 1 , defining

L = { y ∈ S n − 1 | lim r → + ∞ u ⁢ ( r ⁢ y ) / r = 1 }

and considering the function f : L → R ∪ { + ∞ } defined by

f ⁢ ( y ) = lim r → + ∞ ( r − u ⁢ ( r ⁢ y ) ) .

When Σ has mean curvature bounded below by a positive constant, 𝐿 coincides with the closure of F φ as above, and φ | L = f , while φ ≡ + ∞ on S n − 1 ∖ L . Hence finding a hypersurface of constant scalar curvature S < 0 with domain of dependence D φ is completely equivalent to finding 𝑢 such that L = F ̄ φ , f = φ | L and the graph of 𝑢 has constant scalar curvature 𝑆. See [11, Section 1.4] for the details of this equivalence and the additional viewpoint of the Penrose boundary, which will not be treated here.

Geometric formulation

The first result of this paper concerns existence and uniqueness of entire spacelike hypersurfaces with constant negative scalar curvature and prescribed domain of dependence D φ . We state it here in the “geometric” version.

Theorem 1.1

Theorem 1.1 (Existence and uniqueness – geometric version)

Let D ⊂ R n , 1 be a future regular domain which is not a wedge. Then, for every S < 0 , there exists an entire spacelike hypersurface Σ of constant scalar curvature 𝑆 whose domain of dependence is 𝒟. Moreover, if D = D φ for φ : S n − 1 → R a continuous function, then Σ is unique.

We also study foliations of 𝒟. Recall that a time function is a real-valued function on a Lorentzian manifold that is increasing along future-directed causal curves.

Theorem 1.2

Theorem 1.2 (Foliation)

Let φ : S n − 1 → R be a continuous function. Then D φ is foliated by hypersurfaces of constant scalar curvature in ( − ∞ , 0 ) . Moreover, the function associating to p ∈ D φ the value of the scalar curvature of the unique leaf of the foliation containing 𝑝 is a time function.

Actually, taking products, the conclusion holds for φ : S n − 1 → R ∪ { + ∞ } any function which is continuous and real-valued on S n − 1 ∩ A , where 𝐴 is an affine subspace of R n of dimension 2 ≤ k ≤ n intersecting S n − 1 non-trivially, and is identically equal to + ∞ on the complement of 𝐴, that is, when the domain 𝒟 is isometric to the product of a regular domain in R k , 1 and of R n − k .

Theorems 1.1 and 1.2 were proved for n = 2 (in that case, the scalar curvature is equal to twice the Gaussian curvature) in [10].

Analytic formulation

Let us now introduce the PDE set-up for Theorem 1.1. As already explained, up to time reversal, a hypersurface of constant negative scalar curvature necessarily has positive mean curvature (Section 3.3). It is thus natural to look for a solution of the problem in the space of spacelike C 2 functions u : R n → R , that we call admissible, satisfying the property that H 1 ⁢ [ u ] > 0 and H 2 ⁢ [ u ] > 0 . Here, for each 𝑘, H k ⁢ [ u ] denotes the 𝑘-th elementary symmetric polynomial of the principal curvatures of 𝑢, normalized so as to be equal to 1 on the function 𝑢 whose graph is the future unit hyperboloid. In other words, H 1 is a positive multiple of the mean curvature, and H 2 is a negative multiple of the scalar curvature. Theorem 1.3 can be rewritten as follows (recall that F φ = { φ < + ∞ } ).

Theorem 1.3

Theorem 1.3 (Existence and uniqueness – analytic version)

If φ : S n − 1 → R ∪ { + ∞ } is a lower semi-continuous function such that card ⁡ ( F φ ) ≥ 3 , then there exists an admissible function u : R n → R such that H 2 ⁢ [ u ] ≡ 1 and whose graph has domain of dependence D φ . Moreover, if 𝜑 is a continuous function, then the admissible solution is unique.

The second result deals with the additional requirement of the prescription of the scalar curvature as a function on the domain of dependence D φ .

Theorem 1.4

Theorem 1.4 (Prescribed curvature function)

Let φ : S n − 1 → R ∪ { + ∞ } be a lower semi-continuous function such that card ⁡ ( F φ ) ≥ 3 and F φ is not included in any affine hyperplane of R n and let H : D φ ⊂ R n , 1 → R be a function of class C k , α , k ≥ 2 , α ∈ ( 0 , 1 ) , such that h 0 ≤ H ≤ h 1 for some positive constants h 0 and h 1 . Then there exists an admissible function u : R n → R belonging to C k + 2 , α such that H 2 ⁢ [ u ] = H ⁢ ( ⋅ , u ⁢ ( ⋅ ) ) on R n and whose graph has domain of dependence D φ . Moreover, if 𝜑 is a continuous function and ∂ x n + 1 H ≥ 0 , then the solution is unique.

Clearly, Theorem 1.4 could be also reformulated in geometric terms as we did for Theorem 1.1. We prefer to omit the precise statement, that can be easily deduced. We instead remark that some hypothesis on the function 𝐻 must be imposed (in the statement above, ∂ x n + 1 H ≥ 0 ) in order to ensure uniqueness. Indeed, if 𝐻 is the opposite of the time function given by Theorem 1.2, any hypersurface of the foliation is a solution of the problem H 2 ⁢ [ u ] = H ⁢ ( ⋅ , u ) .

About the hypotheses on 𝜑

Theorems 1.3 and 1.4 are improvements of previous results in the literature. In [4], the existence was proved under the hypothesis φ ∈ C 2 ⁢ ( S n − 1 ) , while in [5] for any lower semi-continuous function that only takes the values 0 or + ∞ . When n = 2 , Theorem 1.3 is the main result of [10].

The assumption that 𝜑 is a lower semi-continuous function is not restrictive in any way. Indeed, any regular domain can be written as the supergraph of V φ as in (1.1), for 𝜑 a lower semi-continuous function that takes finite values on at least two points. In order to achieve the sharpest possible existence result, it thus remains to address the question of whether the condition that card ⁡ ( F φ ) ≥ 3 is a necessary condition. For n = 2 , this was proved in [11, Corollary C]. We answer this question affirmatively for n = 3 .

Theorem 1.5

Let Σ be an entire spacelike hypersurface in R 3 , 1 with scalar curvature bounded above by a negative constant. Then, up to a time-reversing isometry, D ⁢ ( Σ ) = D φ , where card ⁡ ( F φ ) ≥ 3 .

The strategy to prove Theorem 1.5 is the following. By contradiction, we suppose that Σ is a hypersurface of scalar curvature bounded above by a negative constant, whose domain of dependence is a wedge 𝑊. Up to isometries, we suppose W = { x 4 > | x 1 | } . As said before, the hypothesis implies that the mean curvature is bounded below by some c > 0 , up to time reversal (see (3.2)). By the comparison principle for mean curvature proved in [11, Theorem 2.1] (see Theorem 3.3), Σ stays below the hypersurface of constant mean curvature 𝑐 in the wedge 𝑊, which is the product of a hyperbola in the timelike plane P = { x 2 = x 3 = 0 } and of P ⟂ . Hence the intersection of Σ with the timelike hyperplane { x 1 = 0 } , which is a copy of R 2 , 1 , is contained between two parallel spacelike planes, and we observe that it has positive mean curvature too. Then we show that there is no entire spacelike surface in R 2 , 1 of positive mean curvature contained between two parallel spacelike planes, thus giving a contradiction.

Unfortunately, this argument does not extend to higher dimensions. In fact, using radially symmetric functions on R n , we show that, for n ≥ 3 , one can find an entire spacelike hypersurface in R n , 1 of positive mean curvature contained between two parallel spacelike hyperplanes. For n ≥ 5 , it can be constructed in such a way that the scalar curvature is also negative, and completing with hyperbolas provides a hypersurface of negative scalar curvature whose domain of dependence is the wedge in R n + 1 , 1 . (See Proposition 6.3.) This shows that an extension of Theorem 1.5 to arbitrary dimension would require a substantially different strategy.

Also, an entire hypersurface with scalar curvature bounded above by a negative constant and with domain of dependence a wedge, if it exists, cannot be convex (Remark 6.4). We do not know if non-convex entire hypersurfaces of constant negative scalar curvature do exist. Let us briefly mention some results for n = 3 . A particular case of the main result of [28] shows that any entire hypersurface of constant scalar curvature and bounded principal curvatures is necessarily convex. As a consequence of [30], strict convexity of the constant scalar curvature hypersurfaces holds for those regular domains that admit an MGHC quotient (see the more detailed discussion below). Moreover, a Splitting Theorem (cf. Theorem 3.4 for CMC hypersurfaces) holds to classify the non-strictly convex solutions Σ under suitable assumptions; see [30, Theorem 2.7.2] and [28].

Strategy of proof of the existence

Let us now outline the strategy to prove Theorems 1.3 and 1.4. We use an exhaustion method, namely we solve a suitable sequence of finite Dirichlet problems over balls. Then we exploit the C 1 and C 2 estimates developed in [4] and [32] respectively, to show that these solutions converge smoothly to an entire hypersurface with the right asymptotic behaviour.

The essential element for implementing this strategy is the presence of (upper and lower) barriers – meaning that any solution of the finite Dirichlet problem with Dirichlet condition bounded between the barriers still stays between the barriers – having domain of dependence 𝒟. The precise properties of the barriers are listed in Definition 4.1. We would like to underline only two important aspects here.

First, the construction of the lower barrier is an essential novel ingredient in this article, and is the key element that permits to largely improve the previous existence results in the literature. Typically, barriers are taken to be smooth sub/super-solutions to the constant (or prescribed) scalar curvature problem. Here our lower barrier is constructed as the supremum of functions obtained from surfaces of constant Gaussian curvature in subspaces 𝑄 of signature ( 2 , 1 ) , determined by triples of points in F φ , and taking products with Q ⟂ . In other words, the lower barrier is a sub-solution only in the viscosity sense. Our proof thus relies crucially on the existence of a surface of constant Gaussian curvature asymptotic to the cone in R 2 , 1 obtained as the intersection of three pairwise non-parallel lightlike planes – this is proved in [10] and is used there as an ingredient to prove the statement of Theorem 1.1 for n = 2 .

Second, in order to apply Urbas’ C 2 estimates, one needs a strictly convex upper barrier. Concretely, we choose the upper barrier to be the CMC hypersurface whose regular domain is 𝒟, whose existence is proved in [11]. As a consequence of the Splitting Theorem (see Theorem 3.4 and Corollary 3.5), this CMC hypersurface is not strictly convex precisely when 𝒟 splits as the product of a regular domain in a copy of a lower-dimensional R k , 1 , and of R n − k . So the above strategy allows us to prove Theorem 1.3 only under the additional assumption that 𝒟 does not split as a product, and to prove Theorem 1.4. The proof of Theorem 1.3 in full generality – that is, for a regular domain 𝒟 that might also split as a product – then follows simply by taking products and an inductive argument.

This discussion also explains why, in Theorem 1.4, we need to assume that F φ is not included in some affine hyperplane of R n . If F φ is included in some affine subspace of R n of minimal dimension 𝑘, we can assume that F φ ⊂ R k × { 0 } ⊂ R n , so that D φ = D φ ′ × R n − k , where D φ ′ is a regular domain in R k , 1 . In this case, we can only solve the prescribed scalar curvature problem as in Theorem 1.4 for a function 𝐻 that does not depend on the second factor, i.e. is of the form H ⁢ ( x ′ , x ′′ ) = c k , n ⁢ H ′ ⁢ ( x ′ ) for H ′ : D φ ′ → R and c k , n : = k ( k − 1 ) / n ( n − 1 ) . Indeed, Theorem 1.4 yields an admissible solution of H 2 ⁢ [ u ′ ] = H ′ in D φ ′ and therefore an admissible solution u ⁢ ( x ′ , x ′′ ) = u ′ ⁢ ( x ′ ) of H 2 ⁢ [ u ] = H in D φ .

MGHC flat spacetimes

An application of our existence, uniqueness and foliation results concern the so-called maximal globally hyperbolic Cauchy compact (MGHC) flat spacetimes. These have been first studied in [1, 7], extending to arbitrary dimension the results of [25] in dimension 2 + 1 . In [1], a classification of MGHC flat spacetimes was provided, showing that (up to finite coverings) they are divided in three classes: translation spacetimes, Misner spacetimes and twisted products of Cauchy hyperbolic spacetimes (which have also been carefully studied in [7]) with Euclidean tori. The last class is by far the most interesting, and is obtained as the quotient of the product of a regular domain D φ , for 𝜑 a continuous function on S n − d − 1 , and R d for d ≥ 0 . Theorems 1.1 and 1.2 then imply the following result, for 𝑀 of any dimension n + 1 .

Theorem 1.6

Let 𝑀 be a maximal globally hyperbolic Cauchy compact flat spacetime. Then 𝑀 has a foliation by closed hypersurfaces of constant scalar curvature. Unless 𝑀 is finitely covered by a translation spacetime or a Misner spacetime, every closed spacelike hypersurface in 𝑀 of constant scalar curvature coincides with a leaf of the foliation, the scalar curvature of the leaves varies in ( − ∞ , 0 ) , and it defines a time function on 𝑀.

When 𝑀 is finitely covered by a translation or Misner spacetime, the foliation by hypersurfaces of constant scalar curvature is very easily described – actually, all leaves of the foliation have vanishing sectional curvature. (Uniqueness of the foliation, however, does not hold for these cases; see Remarks 7.2 and 7.4.)

The non-trivial part of Theorem 1.6 hence concerns the case of twisted products of Cauchy hyperbolic spacetimes with Euclidean tori. Here we apply Theorems 1.1 and 1.2. The uniqueness part of Theorem 1.1 can be applied since 𝜑 is continuous for the universal cover of Cauchy hyperbolic spacetimes, and is crucial to ensure that the hypersurfaces of constant scalar curvature induce closed hypersurfaces in the quotient.

Observe also that, in dimension 2 + 1 , foliations by constant Gaussian curvature of MGHC flat spacetimes were initially constructed in [2]; see also [10, Theorem D]. Theorem 1.6 was proved in [30] for MGHC flat spacetimes 𝑀 of dimension 3 + 1 , by completely different methods. More precisely, the approach of [30] consists in studying (in any dimension) the so-called special Lagrangian curvature, which in dimension 3 + 1 coincides with scalar curvature.

Finally, we remark that the existence of closed hypersurfaces of constant, or prescribed, scalar curvature in globally hyperbolic Cauchy compact Lorentzian manifolds was proved in [17] assuming the existence of barriers and of a strictly convex function defined between the barriers. As explained in the previous subsection, the existence of a (non-smooth) lower barrier is an essential ingredient of our proof of Theorem 1.1, and therefore of Theorem 1.6. In other words, Theorem 1.6 could not be proved by applying directly the results of [17], in the absence of a suitable lower barrier.

2 Spacelike hypersurfaces and domains of dependence

The ( n + 1 ) -dimensional Minkowski space R n , 1 is the vector space R n + 1 endowed with the flat Lorentzian metric

g = d ⁢ x 1 + ⋯ + d ⁢ x n 2 − d ⁢ x n + 1 2 .

Its isometry group is the group O ⁢ ( n , 1 ) ⋊ R n + 1 . A vector 𝑣 is spacelike (resp. lightlike, timelike) if g ⁢ ( v , v ) is positive (resp. null, negative). An immersed submanifold is spacelike if all its non-zero tangent vectors are spacelike; equivalently, its first fundamental form is a Riemannian metric.

2.1 Entire graphs

The main object of this article is the study of spacelike hypersurfaces having certain curvature properties. We will restrict to the natural class of entire spacelike hypersurfaces, which are defined by the equivalent conditions of the following proposition.

Proposition 2.1

Proposition 2.1 ([10, Proposition 1.10])

Let Σ be an immersed smooth spacelike hypersurface of R n , 1 . Then the following are equivalent:

Σ is properly immersed;
Σ is properly embedded;
Σ is the graph of a function u : R n → R .

Moreover, if the conditions hold, then 𝑢 is a smooth function satisfying | D ⁢ u ⁢ ( x ) | < 1 for all x ∈ R n .

We call a function 𝑢 as in the conclusion of Proposition 2.1 a spacelike function. As a consequence, we remark that the condition of being the graph of a spacelike function is invariant under isometries of R n , 1 . We call entire the submanifolds Σ satisfying the equivalent conditions of Proposition 2.1.

2.2 Domains of dependence and regular domains

A causal curve is a smooth curve 𝛾 such that its tangent vector is either timelike or lightlike at any point. It is locally the graph of a function f : ( a , b ) → R n satisfying | D ⁢ f ⁢ ( t ) | ≤ 1 for every t ∈ ( a , b ) . The causal curve 𝛾 is inextensible if it is the graph of a globally defined function ℝ to R n , that is, γ = { ( f ⁢ ( t ) , t ) ∣ t ∈ R } with | D ⁢ f ⁢ ( t ) | ≤ 1 for every t ∈ R .

We can now define the domain of dependence. This could be defined more generally for achronal sets, but for the purpose of this work, it is sufficient to restrict to entire spacelike hypersurfaces.

Definition 2.2

Let Σ be an entire spacelike hypersurface. The domain of dependence D ⁢ ( Σ ) of Σ is the set of points 𝑝 of R n , 1 such that every inextensible causal curve containing 𝑝 intersects Σ.

It can be shown that D ⁢ ( Σ ) is open, and is the intersection of all open half-spaces ℋ such that ∂ H is a lightlike hyperplane and ℋ contains Σ (see for example [7, Section 3] or [10, Lemma 1.4]).

We now introduce a special type of convex subsets of R n , 1 , that arise as domains of dependence of entire spacelike submanifolds. Given a lower semi-continuous function

φ : S n − 1 → R ∪ { + ∞ } ,

we define the set

F φ : = { y ∈ S n − 1 ∣ φ ( y ) < + ∞ }

and the function V φ : R n → R ,

V φ ( x ) : = sup y ∈ F φ ( ⟨ x , y ⟩ − φ ( y ) ) .

Observe that if card ⁡ ( F φ ) = 1 , i.e. if 𝜑 is a function taking a finite value at a single point y 0 and + ∞ elsewhere, then V φ is the function x ↦ ⟨ x , y 0 ⟩ − φ ⁢ ( y 0 ) , whose graph is a lightlike hyperplane. We call its open supergraph a future half-space, and its open subgraph a past half-space.

Definition 2.3

A future regular domain is a subset of R n , 1 of the form

(2.1) D φ : = { ( x , x n + 1 ) ∈ R n , 1 ∣ x n + 1 > V φ ( x ) } ,

for some lower semi-continuous function φ : S n − 1 → R ∪ { + ∞ } such that card ⁡ ( F φ ) ≥ 2 .

In the definition above, it is not restrictive to suppose 𝜑 lower semi-continuous because, for any function 𝜙, the set D ϕ formally defined as in (2.1) is equal to D φ , where 𝜑 is the lower semi-continuous envelope of 𝜙. One can define a past regular domain analogously, replacing lower semi-continuous functions by upper semi-continuous functions, changing minus by plus and replacing supremum by infimum in (1.1), and reversing the inequality in (2.1).

Now, observe that the intersection of a future half-space H 1 and a past half-space H 2 can contain no entire spacelike hypersurface unless ∂ H 1 and ∂ H 2 are parallel. In conclusion of the above discussion, the domain of dependence of an entire spacelike hypersurface can be

the whole R n , 1 ,
a future half-space,
a past half-space,
the intersection of a future and a past half-space with parallel boundaries,
a future regular domain, or
a past regular domain.

We will see in Corollary 3.6 that the domain of dependence of an entire spacelike hypersurface of scalar curvature bounded above by a negative constant is necessarily a (future or past) regular domain. Up to applying a time-reversing isometry, we will restrict ourselves to the case of future regular domains. In this setting, the function 𝜑 determining the domain of dependence of Σ is easily recovered via the following statement.

Proposition 2.4

Suppose that Σ = graph ⁡ ( u : R n → R ) is an entire spacelike hypersurface whose domain of dependence D ⁢ ( Σ ) is a future regular domain. Then D ⁢ ( Σ ) = D φ , where φ : S n − 1 → R ∪ { + ∞ } is the following function:

(2.2) φ ⁢ ( y ) = sup x ∈ R n ( ⟨ x , y ⟩ − u ⁢ ( x ) ) .

Proof

We have already mentioned that D ⁢ ( Σ ) is the intersection of all half-spaces containing Σ whose boundary is a lightlike hyperplane. Moreover, since D ⁢ ( Σ ) is a future regular domain, all such half-spaces are future. Now, for a given y ∈ S n − 1 , the future half-space whose boundary is the graph of x ↦ ⟨ x , y ⟩ − c contains Σ if and only if c > ⟨ x , y ⟩ − u ⁢ ( x ) for all x ∈ R n . Hence D ⁢ ( Σ ) is the intersection of the open future half-spaces whose boundaries are the lightlike hyperplanes x ↦ ⟨ x , y ⟩ − φ ⁢ ( y ) , where 𝜑 is as in (2.2), for φ ⁢ ( y ) < + ∞ . Using Definition 2.3, this concludes the proof. ∎

2.3 Products

A simple way of producing (future) regular domains is to start from a (future) regular domain contained in a lower-dimensional copy of R k , 1 , and take a product with its orthogonal complement. Since those regular domains obtained in this way play an important role for Theorem 1.3, we introduce a definition.

Definition 2.5

Let 𝒟 be a regular domain in R n , 1 . We say that 𝒟splits if there exists a subspace 𝑃 of signature ( k , 1 ) , for 1 ≤ k ≤ n − 1 , such that

(2.3) D = { x + v ∣ x ∈ D ′ , v ∈ P ⟂ } ,

where D ′ is a regular domain in P ≅ R k , 1 .

For brevity, when 𝒟 splits as above, we will write that D ≅ D ′ × R n − k . The next proposition characterizes future regular domains that split in terms of the function 𝜑.

Proposition 2.6

Let D φ be a future regular domain, for a lower semi-continuous function φ : S n − 1 → R ∪ { + ∞ } such that card ⁡ ( F φ ) ≥ 2 . Then D φ splits if and only if there exists an affine subspace 𝐴 of dimension k ≤ n − 1 of R n such that F φ ⊂ S n − 1 ∩ A .

Proof

First, observe that F φ ⊂ S n − 1 ∩ A if and only if all lightlike hyperplanes of the form

(2.4) x n + 1 = ⟨ x , y ⟩ − φ ⁢ ( y )

for y ∈ F φ are invariant under translations in P ⟂ , where

P : = { λ ⁢ ( a , 1 ) ∣ a ∈ A , λ ∈ R } ̄ ⊂ R n , 1 .

So if there exists 𝐴 such that F φ ⊂ S n − 1 ∩ A , then D φ , which is the intersection of the future half-spaces bounded by hyperplanes of form (2.4), is invariant under translations in P ⟂ , and hence a product of D ′ : = D φ ∩ P and P ⟂ as in (2.3).

Conversely, suppose that D φ is of form (2.3) for D ′ ⊂ P . Then D φ contains an affine subspace parallel to P ⟂ . This implies that if a future half-space bounded by a hyperplane of form (2.4) contains D φ , then ( y , 1 ) ∈ P , that is, y ∈ A : = P ∩ { x n + 1 = 1 } . This shows that F φ ⊂ A and concludes the proof. ∎

3 Scalar and mean curvature

3.1 Some general formulae

Let Ω be a convex domain in R n (most of the time, Ω = R n ), let u : Ω → R be a smooth spacelike function (i.e. | D ⁢ u ⁢ ( x ) | < 1 for all x ∈ Ω ), and let Σ be the graph of 𝑢. When Ω = R n , Σ is thus an entire spacelike hypersurface (see Proposition 2.1). By an elementary computation, the first fundamental form of Σ is

g i ⁢ j = δ i ⁢ j − ∂ i u ⁢ ∂ j u

and the second fundamental form, computed with respect to the future unit normal vector

N = 1 1 − | D ⁢ u | 2 ⁢ ( ∂ 1 u , … , ∂ n u , 1 ) ,

h i ⁢ j = 1 1 − | D ⁢ u | 2 ⁢ ∂ i ⁢ j 2 u .

Since the inverse of the metric is

g i ⁢ j = δ i ⁢ j + ∂ i u ⁢ ∂ j u 1 − | D ⁢ u | 2 ,

the shape operator of Σ is given by

(3.1) h j i = 1 1 − | D ⁢ u | 2 ⁢ ∑ k = 1 n ( δ i ⁢ k + ∂ i u ⁢ ∂ k u 1 − | D ⁢ u | 2 ) ⁢ ∂ k ⁢ j 2 u .

Let us denote by λ 1 , … , λ n the principal curvatures of Σ (the eigenvalues of the shape operator) and by H k ⁢ [ u ] the normalized 𝑘-th elementary symmetric function of λ 1 , … , λ n (the 𝑘-th curvature)

H k ⁢ [ u ] = k ! ⁢ ( n − k ) ! n ! ⁢ σ k ⁢ ( λ 1 , … , λ n ) .

We are interested in the scalar curvature S ⁢ [ u ] of Σ, which is linked to H 2 ⁢ [ u ] by the identity S ⁢ [ u ] = − n ⁢ ( n − 1 ) ⁢ H 2 ⁢ [ u ] , and in the mean curvature, which is simply equal to H 1 ⁢ [ u ] and is given by

H 1 ⁢ [ u ] = 1 n ⁢ ∑ i = 1 n h i i = 1 n ⁢ 1 − | D ⁢ u | 2 ⁢ ∑ 1 ≤ i , j ≤ n ( δ i ⁢ j + ∂ i u ⁢ ∂ j u 1 − | D ⁢ u | 2 ) ⁢ ∂ i ⁢ j 2 u .

3.2 CMC hypersurfaces

Let us start by recalling some results on entire hypersurfaces of constant mean curvature that will be used in the following. By the Lorentzian Bernstein theorem, the only entire hypersurfaces with vanishing mean curvature are spacelike hyperplanes, so we consider non-zero values 𝑐 of the (constant) mean curvature. Up to applying a time-reversing isometry, we can assume that the mean curvature is positive.

First of all, the following result shows, in particular, that the domain of dependence of an entire CMC hypersurface with positive 𝑐 is a future regular domain.

Theorem 3.1

Theorem 3.1 ([11, Corollary 2.4])

Let Σ be an entire spacelike hypersurface with mean curvature bounded below by a positive constant. Then D ⁢ ( Σ ) is a future regular domain.

By Theorem 3.1, in order to obtain a classification result for entire CMC hypersurfaces, it is sufficient to consider future regular domains. The classification result is the following.

Theorem 3.2

Theorem 3.2 ([11, Theorem A])

Given any future regular domain 𝒟 and any c > 0 , there exists a unique entire spacelike hypersurface with constant mean curvature 𝑐 such that D ⁢ ( Σ ) = D .

The uniqueness part of Theorem 3.2 follows from the following comparison principle for mean curvature, that we will apply in this more general form.

Theorem 3.3

Theorem 3.3 ([11, Theorem 2.1])

Let Σ − and Σ + be entire spacelike hypersurfaces, which are the graphs of u − and u + respectively. Suppose that Σ + has constant mean curvature H 1 ⁢ [ u + ] = c > 0 , and Σ − has (possibly non-constant) mean curvature H 1 ⁢ [ u − ] ≥ c , and that Σ + ⊆ D ⁢ ( Σ − ) . Then u + ⁢ ( x ) ≥ u − ⁢ ( x ) for every x ∈ R n .

Let us now briefly describe the relation between entire CMC hypersurfaces and product regular domains. The key result is the following Splitting Theorem.

Theorem 3.4

Theorem 3.4 ([14, Theorem 3.1])

Let Σ be an entire spacelike hypersurface in R n , 1 with constant mean curvature c > 0 . Then exactly one of the following holds:

Σ is strictly convex, or
there exists a subspace 𝑃 of signature ( k , 1 ) , for 1 ≤ k ≤ n − 1 , such that
Σ = { x + v ∣ x ∈ Σ ′ , v ∈ P ⟂ } ,
for Σ ′ ⊂ P ≅ R k , 1 an entire hypersurface of constant mean curvature n ⁢ c / k .

Clearly, the domain of dependence of a CMC hypersurface as in item 2 splits (as per Definition 2.5). Conversely, by the uniqueness part of Theorem 3.2, if D = D ⁢ ( Σ ) splits, then using the invariance of 𝒟 by translations in P ⟂ , one sees immediately that Σ is in the form of item 2. We summarize this discussion in the following corollary.

Corollary 3.5

Let Σ be an entire spacelike hypersurface in R n , 1 with constant mean curvature c > 0 . Then Σ is strictly convex if and only if D ⁢ ( Σ ) does not split.

3.3 Scalar curvature and admissible functions

Let us now focus on the study of the scalar curvature (or equivalently, on the H 2 ) of spacelike hypersurfaces. From the elementary inequality

(3.2) ( ∑ i = 1 n λ i ) 2 ≥ 2 ⁢ ∑ 1 ≤ i < j ≤ n λ i ⁢ λ j ,

it follows that if Σ is a (connected) hypersurface of scalar curvature bounded above by a negative constant, then its mean curvature is either bounded below by a positive constant, or bounded above by a negative constant. Up to applying a time-reversal isometry in the latter case, and using Theorem 3.1, we obtain the following corollary.

Corollary 3.6

Let Σ be an entire spacelike hypersurface whose scalar curvature is bounded above by a negative constant. Then D ⁢ ( Σ ) is either a future regular domain or a past regular domain. In the former case, the mean curvature of Σ is bounded below by a positive constant; in the latter, the mean curvature of Σ is bounded above by a negative constant.

In order to obtain existence and uniqueness results for entire spacelike hypersurfaces with a prescribed domain of dependence, we will thus focus (which is again not restrictive, up to a time-reversing isometry) on the first situation in Corollary 3.6. It is thus natural to introduce the following set

Definition 3.7

A C 2 function u : R n → R is called admissible if | D ⁢ u | < 1 , H 1 ⁢ [ u ] > 0 and H 2 ⁢ [ u ] > 0 on R n . We then denote by K 2 the set of admissible functions, that is,

K 2 : = { u : R n → R ∈ C 2 ( R n ) ∣ | D u | < 1 , H 1 [ u ] > 0 and H 2 [ u ] > 0 } .

On K 2 , the operator H 2 is elliptic, the operator H 2 1 2 is concave with respect to the second derivatives, and the following Maclaurin inequality (which is stronger than (3.2)) holds:

0 < H 2 ⁢ [ u ] 1 2 ≤ H 1 ⁢ [ u ] .

Details are given in Appendix A.

Let us state the standard comparison principles, and strong maximum principle, for the curvature operators H 1 and H 2 .

Theorem 3.8

Let Ω be a bounded open subset of R n and let u , v ∈ C 2 ⁢ ( Ω ) ∩ C 0 ⁢ ( Ω ̄ ) be spacelike functions such that H 1 ⁢ [ u ] ≥ H 1 ⁢ [ v ] in Ω and u ≤ v on ∂ Ω . Then u ≤ v on Ω ̄ . Furthermore, if Ω is connected, then either u < v or u ≡ v .

Theorem 3.9

Let Ω be a bounded open subset of R n and let u , v ∈ C 2 ⁢ ( Ω ) ∩ C 0 ⁢ ( Ω ̄ ) be spacelike functions such that H 2 ⁢ [ u ] ≥ H 2 ⁢ [ v ] in Ω and u ≤ v on ∂ Ω . Assume moreover that 𝑢 is admissible. Then u ≤ v on Ω ̄ . Furthermore, if Ω is connected, then either u < v or u ≡ v .

Remark 3.10

Theorems 3.8 and 3.9 are equivalent to the following statements, that we record here since they will be useful later. Let u , v ∈ C 2 ⁢ ( Ω ) , for Ω an open subset of R n , and let x 0 ∈ Ω . Suppose u ⁢ ( x 0 ) = v ⁢ ( x 0 ) and u ≤ v on Ω. Then H 1 ⁢ [ u ] ⁢ ( x 0 ) ≤ H 1 ⁢ [ v ] ⁢ ( x 0 ) . If moreover 𝑢 is admissible, then H 2 ⁢ [ u ] ⁢ ( x 0 ) ≤ H 2 ⁢ [ v ] ⁢ ( x 0 ) .

4 Existence of entire solutions

4.1 Outline of the construction

Let us provide an outline of the strategy to prove the existence parts of Theorem 1.3 and Theorem 1.4. Given a future regular domain D φ for which the cardinality of F φ is at least three, we will construct two barriers u ¯ , u ̄ : R n → R whose graphs have domain of dependence D φ . We will take for the upper barrier u ̄ a solution of the prescribed constant mean curvature equation obtained in [11], and for the lower barrier u ¯ a supremum of solutions of the prescribed constant scalar curvature equation: the latter solutions will be products with linear spaces of surfaces with constant Gauss curvature in R 2 , 1 and triangular Gauss map image obtained in [10]. We then solve a sequence of Dirichlet problems between the barriers, and extract a convergent sub-sequence using a priori estimates essentially obtained in [4, 32].

A technical, but important, point is the following. We will need to use that the upper barrier u ̄ is strictly convex, which by the Splitting Theorem (see Corollary 3.5) is the case if and only if the set F φ ⊂ S n − 1 does not belong to any affine hyperplane of R n (i.e. if and only if D φ does not split; see Proposition 2.6). While this is a hypothesis of Theorem 1.4, for Theorem 1.3, we will use the following remark: if D φ splits as D φ ′ × R n − k with D φ ′ ⊂ R k , 1 , a solution Σ ′ of constant scalar curvature in R k , 1 and domain of dependence D φ ′ yields a hypersurface Σ : = Σ ′ × R n − k in R n , 1 with constant scalar curvature and domain of dependence D φ , so in that case, the existence of a solution readily follows from the existence of a solution in a space of smaller dimension. The proof of Theorem 1.3 will be concluded by an inductive argument; see Section 4.7.

In this section, we construct entire admissible solutions of H 2 ⁢ [ u ] = H ⁢ ( ⋅ , u ) with domain of dependence D φ . For this purpose, we first construct the barriers u ¯ and u ̄ in Section 4.2, then we solve the Dirichlet problem between the barriers (see Appendix B) and we reduce the construction of entire solutions to the obtention of C 1 and C 2 interior estimates in Section 4.3. Then we obtain these estimates in Sections 4.5 and 4.6 and we conclude the proofs of the existence parts of Theorems 1.3 and 1.4 in Section 4.7.

4.2 The construction of the barriers

Recall, for the following definition, that V φ was defined in (1.1).

Definition 4.1

Let φ : S n − 1 → R ∪ { + ∞ } be a lower semi-continuous function such that card ⁡ ( F φ ) ≥ 3 , and let h 0 , h 1 be constants such that 0 < h 0 ≤ h 1 . We say that u ¯ , u ̄ : R n → R is a pair of ( D φ , h 0 , h 1 ) -barriers if

u ̄ is smooth, spacelike and strictly convex;
u ¯ is 1-Lipschitz;
the domain of dependence of the graphs of u ¯ and u ̄ is D φ and the following inequalities hold for some C 0 > 0 :
(4.1) V φ < u ¯ < u ̄ < V φ + C 0 ;
for every bounded domain Ω ⊂ R n , every admissible function u : Ω ̄ → R such that
h 0 ≤ H 2 ⁢ [ u ] ≤ h 1 and u ¯ | ∂ Ω ≤ u | ∂ Ω ≤ u ̄ | ∂ Ω
satisfies u ¯ ≤ u ≤ u ̄ on Ω.

Remark 4.2

In item 3, the hypothesis that the domain of dependence of the graph of u ̄ equals D φ , together with (4.1), automatically implies that the domain of dependence of the graph of u ¯ also equals D φ .

Remark 4.3

Classically, the upper and lower barriers are taken as two admissible functions satisfying H 2 ⁢ [ u ̄ ] ≤ h 0 and H 2 ⁢ [ u ¯ ] ≥ h 1 , so that item 4 is satisfied as a consequence of Theorem 3.9. In fact, in our Definition 4.1, we require that u ̄ is smooth, and under this assumption, condition 4 actually implies that H 2 ⁢ [ u ̄ ] ≤ h 0 . However, in this work, we will need to construct a function u ¯ which is possibly not smooth, obtained as a supremum of admissible functions satisfying a lower bound on H 2 .

Proposition 4.4

Let φ : S n − 1 → R ∪ { + ∞ } be a lower semi-continuous function such that card ⁡ ( F φ ) ≥ 3 and such that F φ is not contained in any affine hyperplane of R n . Then, for any constants 0 < h 0 ≤ h 1 , a pair of ( D φ , h 0 , h 1 ) -barriers exists.

Proof

Let α ≤ h 0 . (We can take α = h 0 for the moment, but if h 0 = h 1 , then we will have to replace 𝛼 by a smaller constant later on.)

To construct u ̄ as in item 1, let us take the function whose graph has constant mean curvature H 1 = α and whose domain of dependence is D φ , given in [11]. This upper barrier u ̄ is strictly convex by Proposition 2.6 and Corollary 3.5.

Let us construct u ¯ as in item 2. Consider a subset T = { y a , y b , y c } of F φ ⊂ S n − 1 ⊂ R n formed by three pairwise distinct points. The null vectors

0 ⁢ y a → = ( y a , 1 ) , 0 ⁢ y b → = ( y b , 1 ) , 0 ⁢ y c → = ( y c , 1 )

of R n , 1 span the 3-dimensional linear space

L T : = Span ( 0 ⁢ y a → , 0 ⁢ y b → , 0 ⁢ y c → )

and we have the decomposition R n , 1 = L T ⊕ L T ⟂ . The restriction to L T of the metric in R n , 1 is of signature ( 2 , 1 ) and the null planes

P a = 0 ⁢ y a → ⟂ − ( 0 , φ ⁢ ( a ) ) , P b = 0 ⁢ y b → ⟂ − ( 0 , φ ⁢ ( b ) ) , P c = 0 ⁢ y c → ⟂ − ( 0 , φ ⁢ ( c ) )

define a triangular regular domain

D T = I + ⁢ ( P a ) ∩ I + ⁢ ( P b ) ∩ I + ⁢ ( P c )

in L T ; here we use I + ⁢ ( P ) to denote the future half-space whose boundary is the null plane 𝑃. By [10, Theorem A], there exists a spacelike surface Σ T in L T with constant Gauss curvature K = − n ⁢ ( n − 1 ) 2 ⁢ h 1 and whose domain of dependence is D T . Let us note that Σ T is asymptotic at infinity to the boundary ∂ D T of its domain of dependence. The product Σ T × L T ⟂ then defines an admissible hypersurface of R n , 1 = L T ⊕ L T ⟂ with constant scalar curvature H 2 = h 1 . It is the graph of an entire function z T : R n → R . Denoting by 𝒯 the set of triples T = { y a , y b , y c } of pairwise distinct points in F φ , we finally set

(4.2) u ¯ = sup T ∈ T z T .

Let us now show item 3. The domain of dependence of the graph of u ̄ is D φ by construction. By Remark 4.2, the same holds true for u ¯ if (4.1) holds. We thus only have to show (4.1), that is, that V φ < u ¯ < u ̄ < V φ + C 0 .

We first show that V φ < u ¯ . Let us set, for each T ∈ T ,

V T ( x ) : = sup y ∈ T ( ⟨ x , y ⟩ − φ ( y ) ) .

Recalling that

V φ ⁢ ( x ) = sup y ∈ F φ ( ⟨ x , y ⟩ − φ ⁢ ( y ) )

and F φ = ⋃ T ∈ T T , we have

(4.3) V φ = sup T ∈ T V T .

Since z T > V T for all T ∈ T , from (4.2) and (4.3), we deduce that u ¯ ≥ V φ . The inequality is strict: let us fix x ∈ R n ; since 𝜑 is lower semi-continuous, there exists y 0 ∈ F φ such that

(4.4) V φ ⁢ ( x ) = sup y ∈ F φ ( ⟨ x , y ⟩ − φ ⁢ ( y ) ) = ⟨ x , y 0 ⟩ − φ ⁢ ( y 0 ) .

If T = { y 0 , y 1 , y 2 } is a triple containing y 0 , with arbitrary other points y 1 , y 2 ∈ F φ (that exist since card ⁡ ( F φ ) ≥ 3 ), we have, by definition of y 0 ,

V T ⁢ ( x ) = max ⁡ ( ⟨ x , y 0 ⟩ − φ ⁢ ( y 0 ) , ⟨ x , y 1 ⟩ − φ ⁢ ( y 1 ) , ⟨ x , y 2 ⟩ − φ ⁢ ( y 2 ) ) = ⟨ x , y 0 ⟩ − φ ⁢ ( y 0 ) .

Hence V φ ⁢ ( x ) = V T ⁢ ( x ) for all T ⊂ F φ containing the point y 0 defined by (4.4). We thus have

V φ ⁢ ( x ) = V T ⁢ ( x ) < z T ⁢ ( x ) ≤ u ¯ ⁢ ( x ) ,

and thus V φ < u ¯ .

Second, we show that u ¯ < u ̄ . By Theorem 3.3 applied to u − = z T and u + = u ̄ , since H 1 ⁢ [ u ̄ ] = α and H 1 ⁢ [ z T ] ≥ H 2 ⁢ [ z T ] 1 / 2 = h 1 1 / 2 ≥ α , we have that z T ≤ u ̄ for all T ∈ T . Taking the supremum, we deduce that u ¯ ≤ u ̄ on R n . We may moreover suppose that the strict inequality holds, up to replacing 𝛼 by a slightly smaller constant. Indeed, if v ̄ , constructed by Theorem 3.2, has constant mean curvature α ′ = H 1 ⁢ [ v ̄ ] < H 1 ⁢ [ u ̄ ] = α , then u ̄ ≤ v ̄ by Theorem 3.3 and moreover u ̄ < v ̄ by the strong maximum principle; replacing u ̄ by v ̄ if necessary, we may therefore suppose that u ¯ < u ̄ .

Third, we show that u ̄ < V φ + C 0 . For this, we use the notion of cosmological time (see [7, Section 4] for some foundational properties) in the regular domain D φ , which is the function T : D φ → ( 0 , + ∞ ) such that T ⁢ ( p ) is the supremum of the length of every past-directed causal curve γ : [ 0 , a ] → D φ with γ ⁢ ( 0 ) = p , where past-directed means that ⟨ γ ′ ⁢ ( t ) , ∂ x n + 1 ⟩ > 0 and the length of 𝛾 is

ℓ ⁢ ( γ ) = ∫ 0 a | ⟨ γ ′ ⁢ ( t ) , γ ′ ⁢ ( t ) ⟩ | ⁢ d t .

By [11, Lemma 2.3], the cosmological time 𝑇 restricted to the graph of u ̄ is bounded from above by C 0 = 1 / α . This implies the desired inequality. Indeed, the cosmological time of the points on the graph of the function V φ + C 0 is at least C 0 , since the vertical segment connecting the points ( x , V φ ⁢ ( x ) ) and ( x , V φ ⁢ ( x ) + C 0 ) is a timelike curve of length C 0 . This finishes the proof of item 3.

We finally prove item 4. First, since H 1 ⁢ [ u ] ≥ H 2 1 / 2 ⁢ [ u ] ≥ h 0 1 / 2 ≥ α = H 1 ⁢ [ u ̄ ] , we have u ≤ u ̄ by the comparison principle (Theorem 3.8) for the mean curvature on the bounded domain Ω. Second, we have H 2 ⁢ [ u ] ≤ h 1 = H 2 ⁢ [ z T ] for every function z T in the construction of u ¯ ; hence, by the comparison principle for H 2 , we obtain u ≥ z T (Theorem 3.9, since moreover u ≥ u ¯ ≥ z T on ∂ Ω by hypothesis). Taking the supremum over all 𝑇, we conclude that u ≥ u ¯ . ∎

4.3 Construction of a solution

In the following, we suppose that F φ is not included in any affine hyperplane of R n and fix h 0 , h 1 positive constants such that h 0 ≤ H ≤ h 1 . Moreover, u ¯ and u ̄ will denote a pair of ( D φ , h 0 , h 1 ) -barriers as in Definition 4.1, constructed in Proposition 4.4.

Consider the balls B i : = { x ∈ R n ∣ | x | < i } for i ∈ N ∗ . We have R n = ⋃ i ∈ N ∗ B i with B i ⊂ B i + 1 . Let u i ∈ C 4 , α ⁢ ( B i ̄ ) be a solution of the Dirichlet problem

H 2 ⁢ [ u i ] = H ⁢ ( ⋅ , u i ) ⁢ in ⁢ B i and u i = u ̄ ⁢ on ⁢ ∂ B i

such that u ¯ ≤ u i ≤ u ̄ . We have included an outline of the proof of the solution of the Dirichlet problem in Appendix B. The solution u i is obtained by applying Theorem B.1 to φ 2 = u ̄ and φ 1 some subsolution (see Remark B.2). Theorem B.1 provides a solution such that

φ 1 ≤ u i ≤ φ 2 = u ̄ ,

and we actually have u ¯ ≤ u i by item 4 of Definition 4.1.

To construct an entire solution, we need the following interior estimates: if K ⊂ R n is a compact subset, there exist I K ∈ N , θ K ∈ ( 0 , 1 ] and C K ≥ 0 such that, for all i ≥ I K ,

(4.5) sup K | D ⁢ u i | ≤ 1 − θ K ,

(4.6) sup K | D 2 ⁢ u i | ≤ C K .

With these estimates at hand, a C 2 , α estimate is obtained using the Evans–Krylov theory (note that the C 0 estimate is trivial since u ¯ ≤ u i ≤ u ̄ ), and an entire solution is then obtained using the Arzelà–Ascoli theorem together with a diagonal process.

The interior C 1 and C 2 estimates (4.5) and (4.6) were obtained in [4, 32] once two auxiliary functions are constructed, as explained in Sections 4.5 and 4.6 below.

4.4 A continuity lemma

Before moving on to the interior C 1 and C 2 estimates, we need to establish an elementary lemma that will be used several times in the rest of the article. First, let us introduce some notation.

Definition 4.5

Let Σ be an entire spacelike C 1 surface in R n , 1 . We denote by Ξ the subset of R n , 1 × S n − 1 consisting of those pairs ( p , v ) such that the null line t ↦ p + t ⁢ ( v , 1 ) (where 𝑡 varies in ℝ) intersects Σ.

Remark 4.6

Observe that, since Σ is spacelike, i.e. the graph of a strictly 1-Lipschitz function, the intersection point between the line p + t ⁢ ( v , 1 ) and Σ is unique. Moreover, by definition of domain of dependence, Ξ includes all pairs ( p , v ) , where 𝑝 is in D ⁢ ( Σ ) .

Lemma 4.7

The subset Ξ ⊂ R n , 1 × S n − 1 is open, and the map associating to ( p , v ) the unique point of intersection between the line t ↦ p + t ⁢ v and Σ is continuous.

Proof

We prove the statement by the implicit function theorem. Let u : R n → R be the spacelike function whose graph is Σ. Finding the intersection point between the line p + t ⁢ ( v , 1 ) and Σ amounts to finding t ∈ R such that ( x + t ⁢ v , y + t ) is in the graph of 𝑢, where p = ( x , y ) for x ∈ R n and y ∈ R . That is, ( x , y , v , t ) is a solution of the equation F ⁢ ( x , y , v , t ) = 0 , where

F ( x , y , v , t ) : = y + t − u ( x + t v ) = 0 .

Now, suppose ( x 0 , y 0 , v 0 , t 0 ) is a solution. We have

∂ t F ⁢ ( x 0 , y 0 , v 0 , t 0 ) = 1 − ⟨ D ⁢ u ⁢ ( x 0 + t 0 ⁢ v 0 ) , v 0 ⟩ .

Since 𝑢 is spacelike, | D ⁢ u | < 1 . Together with | v 0 | = 1 , we thus have ∂ t F ⁢ ( x 0 , y 0 , v 0 , t 0 ) ≠ 0 . This shows that, as ( x , y , v ) vary in a small neighbourhood of ( x 0 , y 0 , v 0 ) in R n × R × S n − 1 , all solutions can be expressed as ( x , y , v , t ⁢ ( x , y , v ) ) , where t = t ⁢ ( x , y , v ) is a continuous function. This concludes the proof. ∎

4.5 The interior C 1 estimate

The estimate relies on the following result.

Theorem 4.8

Theorem 4.8 ([4, Section 4])

Let 𝐾 be a compact subset of R n and R > 0 such that K ⊂ B R . If there exists a smooth spacelike function ψ : B ̄ R → R with the following property:

(⋆) ψ u ̄ ⁢ on ⁢ ∂ B R ,

then there exists θ ∈ ( 0 , 1 ] such that every solution u : B R → R of H 2 ⁢ [ u ] = H ⁢ ( ⋅ , u ) with u ¯ ≤ u ≤ u ̄ satisfies

sup K | D ⁢ u | ≤ 1 − θ .

The number 𝜃 depends on K , R , u ¯ , u ̄ , ψ and 𝐻 on { ( x , t ) ∣ x ∈ B R , ψ ⁢ ( x ) ≤ t ≤ u ̄ ⁢ ( x ) } .

We now construct the required auxiliary function 𝜓, given a pair of ( D φ , h 0 , h 1 ) -barriers. Observe that the hypothesis on the pair of barriers used in the proof are the fact that u ̄ is spacelike and the fact that u ¯ > V φ .

Lemma 4.9

If 𝐾 is a compact subset of R n , there exist R > 0 , with K ⊂ B R , and a smooth spacelike function ψ : B ̄ R → R such that condition (⋆) holds.

Proof

Let us fix R 0 such that K ⊂ B ̄ R 0 and let ψ 0 : B ̄ R 0 → R be a 1-Lipschitz function such that ψ 0 0 such that α 0 < inf B ̄ R 0 ( u ¯ − V φ ) ). We then consider ψ : R n → R such that ψ = ψ 0 on B ̄ R 0 and ψ ⁢ ( ξ + t ⁢ ξ / R 0 ) = ψ 0 ⁢ ( ξ ) + t for all ξ ∈ ∂ B ̄ R 0 and t ≥ 0 . By construction, for p = ( ξ , ψ 0 ⁢ ( ξ ) ) and v = ( ξ / R 0 , 1 ) , the null line p + t ⁢ v , t ≥ 0 , belongs to the graph of 𝜓. We claim that there exists a (maybe large) 𝑅 such that ψ > u ̄ on R n ∖ B R . To see this, we apply Lemma 4.7 to the subset

{ ( p = ( ξ , ψ 0 ⁢ ( ξ ) ) , v = ( ξ / R 0 , 1 ) ) ∣ ξ ∈ ∂ B R 0 } ,

which is in Ξ since ( ξ , ψ 0 ⁢ ( ξ ) ) ∈ D φ (Remark 4.6). Lemma 4.7 shows that the set { ψ = u ̄ } is the image of a continuous function from ∂ B R 0 to the graph of u ̄ , whose image is compact since ∂ B R 0 is compact. Hence { ψ 0 such that

δ < min ⁡ ( inf ∂ B R ( ψ − u ̄ ) , inf K ( u ¯ − ψ ) ) ,

we consider ρ ε ∈ C c ∞ ⁢ ( B ε ) such that ρ ε ≥ 0 and ∫ R n ρ ε ⁢ ( z ) ⁢ d z = 1 and choose 𝜀 small so that the function ψ ε = ( 1 − ε ) ⁢ ψ ∗ ρ ε satisfies sup B ̄ R | ψ ε − ψ | ≤ δ . The function ψ ε is smooth, and it is spacelike since

| ψ ε ⁢ ( x ) − ψ ε ⁢ ( y ) | = | ( 1 − ε ) ⁢ ∫ R n ( ψ ⁢ ( x − z ) − ψ ⁢ ( y − z ) ) ⁢ ρ ε ⁢ ( z ) ⁢ d z | ≤ ( 1 − ε ) ⁢ | x − y |

for all x , y ∈ R n (since 𝜓 is 1 − Lipschitz). It is moreover such that ψ ε u ̄ on ∂ B R . ∎

4.6 The interior C 2 estimate

The estimate is a consequence of the following C 2 estimate of Urbas.

Theorem 4.10

Theorem 4.10 ([32])

Let 𝐾 be a compact subset of R n and R > 0 such that K ⊂ B R . If there exists a smooth and strictly convex function ϕ : B ̄ R → R with the following property:

(⋆⋆) ϕ > u ̄ ⁢ on ⁢ K and ϕ < u ¯ ⁢ on ⁢ ∂ B R ,

then there exists C > 0 such that every solution u : B R → R of H 2 ⁢ [ u ] = H ⁢ ( ⋅ , u ) , u ¯ ≤ u ≤ u ̄ , satisfies

sup K | D 2 ⁢ u | ≤ C .

The constant 𝐶 depends on K , R , u ¯ , u ̄ , ϕ , H on { ( x , t ) ∣ x ∈ B R , u ¯ ⁢ ( x ) ≤ t ≤ ϕ ⁢ ( x ) } and on θ ∈ ( 0 , 1 ] such that sup B R | D ⁢ u | ≤ 1 − θ .

Note that the existence of a controlled constant θ ∈ ( 0 , 1 ] such that sup B R | D ⁢ u | ≤ 1 − θ is granted for every solution u : B R ′ → R between the barriers by the interior C 1 estimate obtained in the previous section, if R ′ > R is sufficiently large (Theorem 4.8 and Lemma 4.9).

We now construct the auxiliary function 𝜙. Here the key property of the barriers is the strict convexity of u ̄ (for which the hypothesis that F φ is not contained in any affine hyperplane of R n is essential) and the inequalities in (4.1).

Lemma 4.11

If 𝐾 is a compact subset of R n , there exist R > 0 , with K ⊂ B R , and a smooth and strictly convex function ϕ : B ̄ R → R such that condition (⋆⋆) holds.

Proof

Applying an isometry of R n , 1 , we can assume that u ̄ ⁢ ( 0 ) = 0 and D ⁢ u ̄ 0 = 0 , which, by strict convexity of u ̄ , implies

(4.7) lim | x | → + ∞ u ̄ ⁢ ( x ) = + ∞ .

The proof is then analogous to the proof of [5, Lemma 3.7]. We fix R ′ sufficiently large such that K ⊂ B ̄ R ′ . We set ϕ 0 : = sup B R ′ u ̄ + 1 . Recalling (4.1), we have u ¯ ≥ u ̄ − c on R n . We thus get from (4.7) the existence of R > R ′ such that

inf { x ∣ | x | ≥ R } u ¯ ⁢ ( x ) > ϕ 0 + 1 .

We set, for all x ∈ R n ,

ϕ ( x ) : = ϕ 0 + 1 R 2 | x | 2 .

The function 𝜙 is strictly convex, ϕ ≥ u ̄ + 1 on B R ′ and ϕ < u ¯ on ∂ B R . ∎

Remark 4.12

Up to taking a larger value of 𝑅, we could assume that the function 𝜙 constructed in the proof is of spacelike type on B R . However, this is not necessary for the C 2 estimates (see Theorem 4.10).

4.7 Conclusion of the proofs

Proof of existence part of Theorem 1.4

By Proposition 4.4, let u ¯ and u ̄ be a pair of ( D φ , h 0 , h 1 ) -barriers, according to Definition 4.1, which exist because we are assuming that F φ is not contained in any affine hyperplane of R n .

Let B i , i ∈ N ∗ , be an increasing sequence of balls (of radius 𝑖, say). From Theorem B.1, let u i ∈ C 4 , α ⁢ ( B i ̄ ) be an admissible solution of the Dirichlet problem

{ H 2 ⁢ [ u i ] = H ⁢ ( ⋅ , u i ) in ⁢ B i , u i = u ̄ on ⁢ ∂ B i

such that φ 1 ≤ u i ≤ φ 2 : = u ̄ , where φ 1 is any subsolution (see Remark B.2). We then have u ¯ ≤ u i ≤ u ̄ by item 4 of Definition 4.1.

We have obtained in Sections 4.5 and 4.6 the following local C 2 estimates: for every compact subset K ⊂ R n , there exist I K ∈ N , θ K ∈ ( 0 , 1 ] and C 2 , K ≥ 0 such that

sup K | D ⁢ u i | ≤ 1 − θ K and ∥ u i ∥ C 2 ⁢ ( K ) ≤ C 2 , K

for all i ≥ I K , where ∥ ⋅ ∥ C 2 ⁢ ( K ) stands for the usual C 2 norm on the compact 𝐾. These estimates control the ellipticity of the equation; see Proposition A.1. If 𝐾 is an arbitrary compact subset of R n , we may thus assume that the u i are solutions of a uniformly elliptic equation on 𝐾, and we deduce from the Evans–Krylov C 2 , α estimate the following local C 2 , α estimate (see e.g. [18, Theorem 17.14]): for every compact subset K ⊂ R n , there exist α K ∈ ( 0 , 1 ) , I K ∈ N and C 2 , K ′ ≥ 0 such that

∥ u i ∥ C 2 , α K ⁢ ( K ) ≤ C 2 , K ′

for all i ≥ I K , where ∥ ⋅ ∥ C 2 , α K ⁢ ( K ) denotes the usual C 2 , α K norm on 𝐾. Let us note that, in order to apply the Evans–Krylov C 2 , α estimate, we also use here that the operator H 2 1 / 2 is a concave function of the second derivatives on the range of an admissible function, which is property (A.2). The Ascoli–Arzelà theorem and a standard diagonal process then yield a subsequence, still denoted u i , which converges in the C 2 norm on compact subsets of R n . The limit u : R n → R is a solution of the equation H 2 ⁢ [ u ] = H ⁢ ( ⋅ , u ) , it is admissible since so are the functions u i and by (3.2) (if H 1 ⁢ [ u ] ≥ 0 and H 2 ⁢ [ u ] > 0 , then H 1 ⁢ [ u ] > 0 ), and finally 𝑢 is C k + 2 , α by the elliptic regularity theory since 𝐻 is assumed to be C k , α .

Let Σ be the graph of 𝑢. Since, for every 𝑖, u ¯ ≤ u i ≤ u ̄ , then for the limit 𝑢, we still have u ¯ ≤ u ≤ u ̄ . By definition of the barriers, the graphs Σ ¯ of u ¯ and Σ ̄ of u ̄ are entire hypersurfaces whose domain of dependence is D φ . Hence any inextensible causal curve in D φ intersects both Σ ¯ and Σ ̄ , and therefore also Σ. This shows that D φ ⊆ D ⁢ ( Σ ) . Conversely, if p ∈ R n , 1 does not belong to D φ , then an inextensible causal curve through 𝑝 does not intersect Σ ¯ , and therefore does not intersect Σ. This shows that D ⁢ ( Σ ) = D φ and concludes the proof. ∎

Proof of existence part of Theorem 1.3

We provide the proof by induction on the dimension. First, if n = 2 , the result has been proved in [10, Theorem A]. Suppose now that the result is true for n ≤ n 0 , and we shall prove it for n = n 0 + 1 . Let φ : S n 0 → R ∪ { + ∞ } be a lower semi-continuous function with card ⁡ ( F φ ) ≥ 3 , and let D φ be the associated regular domain in R n 0 + 1 , 1 . If F φ is contained in an affine hyperplane, i.e. if D φ splits (Definition 2.5 and Proposition 2.6), let 𝐴 be an affine subspace of minimal dimension k ≤ n 0 containing F φ , and let P ≅ R k , 1 be the corresponding linear subspace constructed as in the proof of Proposition 2.6. Then Σ = { x + v ∣ x ∈ Σ ′ , v ∈ P ⟂ } , where Σ ′ is a hypersurface with H 2 ≡ c in the regular domain D φ ∩ P , satisfies H 2 ≡ c ⁢ k ⁢ ( k − 1 ) / ( n 0 + 1 ) ⁢ n 0 and has domain of dependence D φ . If instead F φ is not contained in any affine subspace of dimension k ≤ n 0 , then the conclusion follows as a special case of Theorem 1.4, for 𝐻 the constant function. ∎

5 Uniqueness and foliation

5.1 Uniqueness

In this section, we will prove the following theorem, and we deduce the uniqueness parts of Theorems 1.3 and 1.4.

Theorem 5.1

Let Σ − and Σ + be entire spacelike hypersurfaces, which are the graphs of two functions u − and u + respectively, with u − admissible. Suppose that Σ + ⊆ D ⁢ ( Σ − ) and H 2 ⁢ [ u + ] ≤ H 2 ⁢ [ u − ] . Assume moreover that D ⁢ ( Σ + ) = D φ + for a continuous function φ + . Then u + ⁢ ( x ) ≥ u − ⁢ ( x ) for every x ∈ R n .

Proof

Let ϵ > 0 . We claim that u + ⁢ ( x ) + ϵ ≥ u − ⁢ ( x ) for all x ∈ R n . For this purpose, we only need to claim that the subset { x ∈ R n ∣ u + ⁢ ( x ) + ϵ ≤ u − ⁢ ( x ) } is compact, so that we can apply Theorem 3.9 to Ω : = { x ∈ R n ∣ u + ( x ) + ϵ < u − ( x ) } and to the functions u = u − and v = u + + ϵ , which agree on ∂ Ω , to deduce that Ω is actually empty.

To prove the claim, note first that, since u + > V φ + , the set { u + + ϵ ≤ u − } is contained in the set V : = { V φ + + ϵ ≤ u − } . Hence we will show that the latter is compact. For this, observe that every line (say, ℓ) of the form t ↦ ( t ⁢ y , t − φ + ⁢ ( y ) + ϵ ) , for y ∈ S n − 1 , intersects Σ − . Indeed, by Proposition 2.4, there exists x ∈ R n such that ⟨ x , y ⟩ − u + ⁢ ( x ) ≥ φ + ⁢ ( y ) − ϵ ; hence u + ⁢ ( x ) ≤ ⟨ x , y ⟩ − φ + ⁢ ( y ) + ϵ . This shows that the null hyperplane 𝑃 containing ℓ, that is P = { x n + 1 = ⟨ x , y ⟩ − φ + ⁢ ( y ) + ϵ } , intersects Σ + . Now, it was shown in [7, Proposition 3.6] or [11, Proposition 1.18] that if a null line ℓ does not intersect an entire spacelike hypersurface in R n , 1 , then the null hyperplane containing ℓ also avoids the hypersurface. So what was shown above implies that the null line ℓ intersects Σ + . Since Σ + ⊆ D ⁢ ( Σ − ) , this implies that the line ℓ intersects Σ − as well.

We now consider the function from S n − 1 to Σ − that associates to every y ∈ S n − 1 the unique intersection point of the line t ↦ ( 0 , − φ + ⁢ ( y ) + ϵ ) + t ⁢ ( y , 1 ) with Σ − . Applying Lemma 4.7, this function is continuous by continuity of φ + . Hence its image is compact in Σ − . By the definition (1.1) of V φ + , when x = t ⁢ y ,

V φ + ⁢ ( x ) + ϵ ≥ ⟨ x , y ⟩ − φ + ⁢ ( y ) + ϵ = t − φ + ⁢ ( y ) + ϵ ,

and the latter is the height function along the line ℓ. Hence V = { V φ + + ϵ < u − } is bounded and this concludes the claim.

Since 𝜖 was arbitrary, the conclusion follows by taking the limit as ϵ → 0 . ∎

We can now conclude the proofs of the uniqueness parts of Theorems 1.3 and 1.4.

Proof of the uniqueness part of Theorem 1.3

Let φ : S n − 1 → R be a continuous function, and let Σ 1 = graph ⁡ ( u 1 ) and Σ 2 = graph ⁡ ( u 2 ) be two entire spacelike hypersurfaces such that D ⁢ ( Σ 1 ) = D ⁢ ( Σ 2 ) = D φ and H 2 ⁢ [ u 1 ] = H 2 ⁢ [ u 2 ] = c . Then, applying Theorem 5.1 twice, u 1 = u 2 and thus Σ 1 = Σ 2 . ∎

Proof of the uniqueness part of Theorem 1.4

We reason as in the proof of Theorem 5.1. Let u 1 and u 2 be admissible functions such that H 2 ⁢ [ u i ] = H ⁢ ( ⋅ , u i ) on R n and having the same domain of dependence D φ , for 𝜑 continuous. Fixing ϵ > 0 , the proof of Theorem 5.1 shows that Ω : = { x ∈ R n ∣ u 2 ( x ) + ϵ < u 1 ( x ) } is bounded. Hence we can apply the comparison principle for scalar curvature (Theorem 3.9) to Ω and to the functions u = u 1 and v = u 2 + ϵ . Using that ∂ x n + 1 H ≥ 0 , we see that

H 2 ⁢ [ v ] = H 2 ⁢ [ u 2 + ϵ ] ≤ H 2 ⁢ [ u 1 ] = H 2 ⁢ [ u ] on ⁢ Ω .

This shows that Ω is in fact empty, and therefore u 2 + ϵ ≥ u 1 . Letting ϵ → 0 , we obtain that u 2 ≥ u 1 , and reversing the roles of u 1 and u 2 , we conclude that u 1 = u 2 . ∎

5.2 Foliation

The existence of a foliation in Theorem 1.2 will be a consequence of the following more general result.

Theorem 5.2

Let φ : S n − 1 → R ∪ { + ∞ } be a lower semi-continuous function such that, for all c > 0 , there exists a unique admissible function u c : R n → R such that H 2 ⁢ [ u c ] = c and with domain of dependence D φ . Then there exists a unique foliation of D φ by hypersurfaces of constant scalar curvature S ∈ ( − ∞ , 0 ) .

Proof

Uniqueness is obvious and we focus on the existence part of the statement. Suppose first that the domain D φ does not split. Under this assumption, let us note first that if c 1 < c 2 , then u c 1 > u c 2 . Indeed, recall from Section 4.3 that u c can be constructed as limits of solutions of Dirichlet problems on a ball B i with boundary values u ̄ c on ∂ B i . Moreover, from the proof of Proposition 4.4, we see that we can take u ̄ c to be the hypersurface of constant mean curvature c / 2 whose domain of dependence is D φ . With this choice, we have u ̄ c 1 > u ̄ c 2 by the comparison principle for the mean curvature (Theorem 3.3). Hence, by the comparison principle for H 2 (Theorem 3.9) and taking limits, u c 1 ≥ u c 2 . Together with the strong maximum principle, we conclude the strict inequality.

We also have the following properties. First,

lim c → + ∞ u ̄ c = V φ .

Indeed, it was proved in [11] that, for a given regular domain D φ , the graphs of u ̄ c are entire CMC hypersurfaces that foliate D φ , and u ̄ c 1 > u ̄ c 2 if c 1 < c 2 . This immediately implies

lim c → + ∞ u c = V φ .

Second,

lim c → 0 u ¯ c = + ∞ .

To show this, fix a triplet T ⊂ F . Now, recall that, by construction of u ¯ c , we have u ¯ c ≥ z T , c , where z T , c is an admissible function whose graph is Σ T , c × L T ⟂ , and Σ T , c is a surface of constant Gaussian curvature K = − n ⁢ ( n − 1 ) 2 ⁢ c in the vector subspace L T ≅ R 2 , 1 . By the results of [10], the surfaces Σ T , c provide a foliation of the regular domain D T , and therefore

lim c → 0 z T , c = + ∞ .

This implies lim c → 0 u ¯ c = + ∞ as claimed. It follows that

lim c → 0 u c = + ∞ .

We thus only have to prove that any point X = ( x , t ) ∈ D φ belongs to the graph of a solution u c . Setting

c + : = sup { c ∣ u c ( x ) > t } , c − : = inf { c ∣ u c ( x ) < t } ,

we have c + ≤ c − and we may consider a non-decreasing sequence c n + → n → + ∞ c + and a non-increasing sequence c n − → n → + ∞ c − such that

u c n + ⁢ ( x ) > t > u c n − ⁢ ( x ) .

Using the estimates obtained in Section 4, we may suppose that u c n + and u c n − converge to functions u + and u − . These functions satisfy u + ⁢ ( x ) ≥ t ≥ u − ⁢ ( x ) and they are solutions of H 2 ⁢ [ u + ] = c + and H 2 ⁢ [ u − ] = c − . If c + = c − , the uniqueness of a solution implies u + = u − and the result follows. Assuming c + < c − and fixing c 0 such that c + < c 0 < c − , we have u − t yields a contradiction with the definition of c + and the inequality u c 0 ⁢ ( x ) < t a contradiction with the definition of c − . We deduce that u c 0 ⁢ ( x ) = t , which finishes the proof in the case where D φ does not split.

If D φ ≅ D ′ × R n − k (see Section 2.3), then the uniqueness of the solutions of H 2 ⁢ [ u c ] = c in D φ for every c > 0 implies, taking products, the uniqueness of the solutions of the same problem in D ′ . The existence of the foliation in D φ then trivially follows from the first part of the proof applied to D ′ , by taking products again. ∎

Proof of Theorem 1.2

The uniqueness part of Theorem 1.1, together with Theorem 5.2, immediately implies the existence of a foliation by hypersurfaces of constant scalar curvature as in Theorem 1.2. Moreover, the construction shows that the function that sends 𝑝 to the value of 𝑆 such that 𝑝 is contained in the leaf of the foliation with scalar curvature 𝑆 is a time function. Indeed, the proof shows that if c 1 < c 2 , then u c 1 > u c 2 and the scalar curvature of u c equals − n ⁢ ( n − 1 ) ⁢ c . This concludes Theorem 1.2. ∎

Taking products, we obtain immediately that the conclusion of Theorem 1.2 also holds for regular domains that split as D φ × R n − k , for φ : S k − 1 → R continuous. More precisely, using Proposition 2.6, the most general statement is the following.

Corollary 5.3

Let φ : S n − 1 → R ∪ { + ∞ } be a function which is continuous and real-valued on S n − 1 ∩ A , where 𝐴 is an affine subspace of R n of dimension 3 ≤ d ≤ n intersecting S n − 1 , and identically equal to + ∞ on the complement of 𝐴. Then D φ is foliated by hypersurfaces of constant scalar curvature in ( − ∞ , 0 ) and the scalar curvature function associated to this foliation defines a time function.

6 Finiteness conditions

6.1 Dimension 3 + 1

We prove here that, in R 3 , 1 , the domain of dependence D φ of an entire admissible solution of H 2 ⁢ [ u ] = 1 necessarily has card ⁡ ( F φ ) ≥ 3 . This relies on the following result.

Proposition 6.1

There exists no entire spacelike function u : R 2 → R such that

H 1 ⁢ [ u ] > 0 ⁢ on ⁢ R 2 and sup R 2 u < + ∞ .

Proof

By contradiction, assuming that the function u : R 2 → R is such that H 1 ⁢ [ u ] > 0 and sup R 2 u < + ∞ , we consider the harmonic function v : R 2 ∖ { 0 } → R , x ↦ log ⁡ | x | and for ε > 0 the function u ε : = u − ε v , which is such that

lim | x | → + ∞ u ε ⁢ ( x ) = − ∞ .

Let us first show that

(6.1) u ε ⁢ ( x ) ≤ max | x | = 1 ⁡ u ε

for all 𝑥 such that | x | ≥ 1 . If not, u ε would reach its maximum on the set { | x | ≥ 1 } at a point x 0 such that | x 0 | > 1 , and we would have D ⁢ u ε = 0 and D 2 ⁢ u ε ≤ 0 at that point, i.e.

D ⁢ u = ε r ⁢ ∂ r and D 2 ⁢ u ≤ ε ⁢ D 2 ⁢ v .

Up to applying a rotation, we can assume x 0 = ( r , 0 ) . In the following,

u i = ⟨ D ⁢ u , e i ⟩ , u i ⁢ j = D 2 ⁢ u ⁢ ( e i , e j ) and v i ⁢ j = D 2 ⁢ v ⁢ ( e i , e j )

stand for the usual first and second partial derivatives of 𝑢 and 𝑣. Since D ⁢ u ∥ ∂ r , we thus have u 1 = | D ⁢ u | and u 2 = 0 at x 0 , which implies that

H 1 ⁢ [ u ] = 1 2 ⁢ 1 − | D ⁢ u | 2 ⁢ ∑ 1 ≤ i , j ≤ 2 ( δ i ⁢ j + u i ⁢ u j 1 − | D ⁢ u | 2 ) ⁢ u i ⁢ j = 1 2 ⁢ 1 − | D ⁢ u | 2 ⁢ ( 1 1 − | D ⁢ u | 2 ⁢ u 11 + u 22 ) ≤ ε 2 ⁢ 1 − | D ⁢ u | 2 ⁢ ( 1 1 − | D ⁢ u | 2 ⁢ v 11 + v 22 )

at x 0 . Since Δ ⁢ v = v 11 + v 22 = 0 , we have v 22 = − v 11 , and since v 11 = − 1 / r 2 < 0 , the right-hand side term is

ε 2 ⁢ 1 − | D ⁢ u | 2 ⁢ ( 1 1 − | D ⁢ u | 2 − 1 ) ⁢ v 11 < 0 ,

which is impossible since H 1 ⁢ [ u ] > 0 . So (6.1) holds, which yields, for all 𝑥 such that | x | ≥ 1 ,

u ⁢ ( x ) ≤ max | x | = 1 ⁡ u + ε ⁢ log ⁡ | x | .

Taking the limit as 𝜀 tends to 0, we deduce that, for all 𝑥 such that | x | ≥ 1 ,

u ⁢ ( x ) ≤ max | x | = 1 ⁡ u .

We then consider x 0 ∈ B ̄ ⁢ ( 0 , 1 ) such that u ⁢ ( x 0 ) = max B ̄ ⁢ ( 0 , 1 ) ⁡ u : it is a global maximum of 𝑢 in R 2 at which D ⁢ u = 0 and D 2 ⁢ u ≤ 0 , which is impossible since H 1 ⁢ [ u ] > 0 . ∎

Corollary 6.2

Let 𝑐 be a positive constant. Then there exists no admissible function u : R 3 → R such that

(6.2) | x 1 | ≤ u ⁢ ( x ) ≤ x 1 2 + c

for all x = ( x 1 , x 2 , x 3 ) ∈ R 3 .

Before the proof, we remark that the function on the right-hand side of (6.2) is the function whose graph is an entire CMC hypersurface Σ which is the product of a hyperbola in R 1 , 1 and of R 2 ≅ ( R 1 , 1 ) ⟂ . The left-hand side is the corresponding function V φ , whose graph is the boundary of the domain of dependence of this Σ, which is obtained as the intersection of the two future half-spaces x 4 > x 1 and x 4 > − x 1 . Hence 𝜑 is the function which takes the value 0 at the points ( ± 1 , 0 , 0 ) ∈ S 2 , and + ∞ elsewhere.

Proof of Corollary 6.2

By Corollary A.4, a section of an admissible graph has positive mean curvature. So the section ( x 2 , x 3 ) ↦ u ⁢ ( 0 , x 2 , x 3 ) , with values in { x 1 = 0 } ≅ R 2 , 1 , would have positive mean curvature and would be bounded (by (6.2)), which is not possible by Proposition 6.1. ∎

We are now ready to prove Theorem 1.5.

Proof of Theorem 1.5

By contradiction, suppose that 𝑢 is an admissible function satisfying H 2 ⁢ [ u ] ≥ h 0 > 0 with domain of dependence D φ ⊂ R 3 , 1 such that card ⁡ ( F φ ) = 2 . Up to an isometry, we may suppose that (6.2) holds for some coordinates x 1 , x 2 , x 3 and a constant 𝑐 conveniently chosen. Indeed, the domain of dependence of 𝑢 is a wedge, that we may assume to be { x ∈ R 3 ∣ | x 1 | < x 4 } . Since H 1 ⁢ [ u ] is bounded below by a positive constant 𝛼 (by the Maclaurin inequality for admissible functions), the graph of 𝑢 stays below every CMC hypersurface with H 1 ≤ α and belonging to the wedge (by the entire comparison principle for the mean curvature operator, Theorem 3.3). This contradicts Corollary 6.2. ∎

6.2 Higher dimension

We prove here that the results in the previous section do not hold in higher dimension: there exists a bounded entire spacelike function u : R n → R with positive mean curvature if n ≥ 3 , which is moreover admissible if n ≥ 5 , and there exists an entire admissible function whose domain of dependence is a wedge if n ≥ 6 . In other words, when n ≥ 6 , in contrast to the case n = 3 obtained in Corollary 6.2, card ⁡ ( F ) = 2 might not be an obstruction to the existence of an admissible function solving H 2 ⁢ [ u ] = 1 in R n , 1 (see also Remark 6.4 below).

Proposition 6.3

The following holds.

If n ≥ 3 , there exists an entire spacelike function u : R n → R such that H 1 ⁢ [ u ] > 0 on R n and sup R n u < + ∞ .
If n ≥ 5 , there exists an entire spacelike function u : R n → R which is admissible and bounded.
If n ≥ 6 , there exists an entire admissible function u : R n → R such that
(6.3) | x 1 | ≤ u ⁢ ( x ) ≤ x 1 2 + c
for all x = ( x 1 , x ′ ) ∈ R n .

Proof

(1) Consider a radial function u : x ∈ R n ↦ v ⁢ ( | x | ) ∈ R , where v : [ 0 , + ∞ ) → R is C 2 , spacelike and such that v ′ ⁢ ( 0 ) = 0 , and denote the mean curvature of its graph at x ∈ R n by h ⁢ ( r ) , where r = | x | . Since, from (3.1), the principal curvatures of its graph are

λ 1 = ⋯ = λ n − 1 = 1 1 − v ′ ⁣ 2 ⁢ v ′ r and λ n = v ′′ ( 1 − v ′ ⁣ 2 ) 3 2 ,

we have

(6.4) h ⁢ ( r ) = 1 n ⁢ 1 − v ′ ⁣ 2 ⁢ ( v ′′ 1 − v ′ ⁣ 2 + ( n − 1 ) ⁢ v ′ r ) ,

which reads

( r n − 1 ⁢ v ′ ( 1 − v ′ ⁣ 2 ) 1 2 ) ′ = n ⁢ r n − 1 ⁢ h ⁢ ( r ) .

Setting

(6.5) H ( r ) : = n r n − 1 ∫ 0 r s n − 1 h ( s ) d s

for r > 0 and since v ′ ⁢ ( 0 ) = 0 , an integration between 0 and 𝑟 yields

(6.6) v ′ ( 1 − v ′ ⁣ 2 ) 1 2 = H ⁢ ( r ) and v ′ = H ⁢ ( r ) 1 + H 2 ⁢ ( r ) .

Let us note that, by (6.4), h ⁢ ( r ) → r → 0 v ′′ ⁢ ( 0 ) / n and (6.5) implies that H ⁢ ( r ) → r → 0 0 , so (6.6) holds in fact on [ 0 , + ∞ ) . We deduce that, for all x ∈ R n ,

(6.7) u ⁢ ( x ) = u ⁢ ( 0 ) + ∫ 0 | x | H ⁢ ( r ) 1 + H 2 ⁢ ( r ) ⁢ d r .

Conversely, if h : [ 0 , + ∞ ) → ( 0 , + ∞ ) is a positive continuous function so that the function 𝐻 defined by (6.5) satisfies

(6.8) ∫ 0 + ∞ H ⁢ ( r ) 1 + H 2 ⁢ ( r ) ⁢ d r < + ∞ ,

then 𝑢 defined by (6.7) is a bounded entire spacelike function with mean curvature h ⁢ ( | x | ) at the point ( x , u ⁢ ( x ) ) .

Let us suppose that h ⁢ ( s ) ∼ + ∞ C ⁢ s γ , where C > 0 and γ ∈ R are constants, and determine for which values of 𝛾 the integral in (6.8) is finite. By using (6.5), we easily obtain that

H ⁢ ( r ) ∼ + ∞ { C ⁢ n γ + n ⁢ r γ + 1 if ⁢ γ > − n , C ⁢ n ⁢ log ⁡ r r n − 1 if ⁢ γ = − n , C ′ r n − 1 if ⁢ γ < − n ,

where in the last case C ′ is a finite positive constant, and deduce that (6.8) holds if and only if γ < − 2 in the case γ > − n , and n ≥ 3 in the case γ ≤ − n . We conclude that (6.8) holds if and only if n ≥ 3 and γ < − 2 , and in particular that (6.8) is possible if (and only if) n ≥ 3 .

Let us also note that, to verify that 𝑢 is C 2 on R n , especially at x = 0 , we need to verify that v ′ : [ 0 , + ∞ ) → R is C 1 and such that v ′ ⁢ ( 0 ) = 0 , which by (6.6) is equivalent to proving that H : [ 0 , + ∞ ) → R is C 1 and such that H ⁢ ( 0 ) = 0 . Since, by (6.5), we have H ⁢ ( r ) ∼ r → 0 h ⁢ ( 0 ) ⁢ r , the only critical point is to verify that H ′ ⁢ ( r ) has a limit as 𝑟 tends to 0: differentiating (6.5), we get

( n − 1 ) ⁢ H ⁢ ( r ) r + H ′ ⁢ ( r ) = n ⁢ h ⁢ ( r )

and deduce that H ′ ⁢ ( r ) tends to h ⁢ ( 0 ) as 𝑟 tends to 0, which completes the proof.

(2) We consider the radial function u ⁢ ( x ) = v ⁢ ( | x | ) constructed above. The second symmetric function σ 2 of the principal curvatures of the graph of 𝑢 is

σ 2 = n − 1 1 − v ′ ⁣ 2 ⁢ v ′ r ⁢ ( v ′′ 1 − v ′ ⁣ 2 + n − 2 2 ⁢ v ′ r ) ,

which also reads, using (6.4),

(6.9) σ 2 = n − 1 1 − v ′ ⁣ 2 ⁢ v ′ r ⁢ ( n ⁢ h ⁢ 1 − v ′ ⁣ 2 − n 2 ⁢ v ′ r ) ,

where h : [ 0 , + ∞ ) → ( 0 , + ∞ ) still denotes the mean curvature. Let us show that if n ≥ 5 , we may choose ℎ so that (6.8) holds and (6.9) is positive. Fix 𝛿 such that 0 < δ < n / 2 − 2 (we assume n ≥ 5 ), consider a smooth non-decreasing function k : [ 0 , + ∞ ) → [ 0 , + ∞ ) such that, for some constant c 0 > 0 ,

k ⁢ ( r ) ∼ r → 0 c 0 ⁢ r n / 2 and k ⁢ ( r ) ∼ r → + ∞ r δ

and set h ⁢ ( r ) = r − n / 2 ⁢ k ⁢ ( r ) . We have h ⁢ ( r ) ∼ + ∞ r γ with γ = − n / 2 + δ < − 2 and deduce that (6.8) holds (as we saw in the proof of the previous item). Using (6.6), equation (6.9) may be written

σ 2 = n ⁢ ( n − 1 ) 1 − v ′ ⁣ 2 ⁢ v ′ r n + 1 ⁢ ( r n ⁢ h ⁢ ( r ) − n 2 ⁢ ∫ 0 r s n − 1 ⁢ h ⁢ ( s ) ⁢ d s ) .

The right-hand side term is positive if 𝑟 is small, since h ⁢ ( r ) tends to c 0 as 𝑟 tends to 0. Moreover, the function r ↦ r n ⁢ h ⁢ ( r ) − n 2 ⁢ ∫ 0 r s n − 1 ⁢ h ⁢ ( s ) ⁢ d s is non-decreasing since its derivative is

n 2 ⁢ r n − 1 ⁢ h + r n ⁢ h ′ = r n / 2 ⁢ ( r n / 2 ⁢ h ) ′ = r n / 2 ⁢ k ′ ≥ 0 .

So σ 2 is positive on [ 0 , + ∞ ) and 𝑢 is admissible.

(3) We construct a solution in the form

u ⁢ ( x ) = x 1 2 + u ′ ⁢ ( x ′ ) 2 ,

where x ′ = ( x 2 , … , x n ) and u ′ : R n − 1 → R is admissible and such that

0 < u ′ < sup R n − 1 u ′ < + ∞ ,

constructed above for n − 1 ≥ 5 . As a consequence of the construction, (6.3) holds. Moreover, it is admissible since u ′ is admissible and, among the 𝑛 principal curvatures of 𝑢, n − 1 are equal to the principal curvatures of u ′ , while the last principal curvature is positive (it coincides with the curvature of a hyperbola). Hence, using that H 1 ⁢ [ u ′ ] > 0 and H 2 ⁢ [ u ′ ] > 0 , one immediately sees that H 1 ⁢ [ u ] > 0 and H 2 ⁢ [ u ] > 0 . ∎

Remark 6.4

The existence of an admissible solution of H 2 ⁢ [ u ] = 1 in R n , 1 with n ≥ 4 and card ⁡ ( F φ ) = 2 is still an open question. Note that such a solution cannot be convex: in convenient coordinates, it would satisfy | x 1 | ≤ u ⁢ ( x ) ≤ x 1 2 + c on R n , and fixing x 1 ∈ R , the map x ′ ∈ R n − 1 ↦ u ⁢ ( x 1 , x ′ ) ∈ R would be convex and bounded, and therefore constant; 𝑢 would thus only depend on x 1 , which is impossible: the graph of an admissible function has necessarily at least two principal curvatures which are positive. This follows from the inequalities

σ 1 , i : = ∑ j ≠ i λ j > 0 , i = 1 , … , n ,

expressing that H 2 is elliptic on admissible functions (see Section 3.3). Note also that it is not known if there exist entire admissible solutions of H 2 ⁢ [ u ] = 1 which are not convex.

7 MGHC spacetimes

The purpose of this section is to prove Theorem 1.6 about maximal globally hyperbolic Cauchy compact (in short, MGHC) flat spacetimes.

7.1 Flat MGHC spacetimes

Recall that a Lorentzian manifold 𝑀 is globally hyperbolic if it contains a Cauchy hypersurface, that is, a smooth spacelike hypersurface that intersects every inextensible causal curve exactly once. It is maximal if every isometric embedding of 𝑀 into a globally hyperbolic Lorentzian manifold that sends a Cauchy hypersurface to a Cauchy hypersurface is surjective. It is Cauchy compact if it has a compact Cauchy hypersurface. (It turns out that all Cauchy hypersurfaces are diffeomorphic, and that 𝑀 is diffeomorphic to the product of a Cauchy hypersurface and ℝ.)

Let us now outline the classification result that was proved in [1]: every maximally globally hyperbolic Cauchy compact flat spacetimes is, up to taking a finite quotient, a translation spacetime, a Misner spacetime, or a twisted product of a Cauchy hyperbolic spacetime by a Euclidean torus. In the rest of the section, we will describe each of these three situations, and prove that, in each case, there exists a foliation by hypersurfaces of constant scalar curvature, where the scalar curvature of the leaves is zero for translation spacetimes and Misner spacetimes, whereas it varies monotonically in ( − ∞ , 0 ) for twisted products of Cauchy hyperbolic spacetimes by Euclidean tori – the latter being the most interesting case.

7.2 Translation spacetimes

A translation spacetime is a quotient of R n , 1 by a lattice Λ ≅ Z n contained in R n = { x n + 1 = 0 } , hence consisting entirely of spacelike vectors. We start by treating this simple case, which is instructive for the case of Misner spacetimes and, most importantly, twisted products of Cauchy hyperbolic spacetimes by Euclidean tori.

Proposition 7.1

Let 𝑀 be an MGHC spacetime which is finitely covered by a translation spacetime. Then 𝑀 has a foliation by totally geodesic (hence intrinsically flat) hypersurfaces. Moreover, if Σ 0 is a closed spacelike hypersurface of constant scalar curvature 𝑆, then S = 0 .

Proof

Every MGH finite quotient 𝑀 of a translation spacetime is isometric to a quotient R n , 1 / Λ ̂ by a discrete group Λ ̂ of Euclidean isometries (a crystallographic group) acting freely on R n . Now, each leaf of the foliation { x n + 1 = c } of R n , 1 by horizontal hypersurfaces is preserved by any subgroup of Isom ⁢ ( R n ) < Isom ⁢ ( R n , 1 ) , and therefore induces a foliation by totally geodesic Cauchy hypersurfaces of the quotient manifold 𝑀.

We now prove the second statement. Let F : M → R be a function whose level sets are precisely the leaves of the above foliation and which is increasing in the future direction – for instance, 𝐹 is the function induced in the quotient by the x n + 1 coordinate function. Since Σ 0 is closed, F | Σ 0 has a maximum p max and a minimum p min .

First, Σ 0 is tangent to the totally geodesic hyperplane F − 1 ⁢ ( F ⁢ ( p min ) ) and contained in its future. Hence its principal curvatures (with respect to the future unit normal vector) at p min are all non-negative, and therefore the H 2 of Σ 0 , which is constant by hypothesis, is non-negative. It remains to show that the H 2 of Σ 0 cannot be positive. But if the H 2 were positive at every point, then the mean curvature would never vanish by (3.2); it would be positive at p min by the previous observation that all principal curvatures at p min are non-negative, and negative at p max by the same argument, thus contradicting the continuity of the mean curvature. ∎

Remark 7.2

The foliation of a translation spacetime M = R n + 1 / Λ by intrinsically flat (hence of vanishing scalar curvature) hypersurfaces is highly non-unique. For example, supposing that Λ is the standard lattice Z n in R n only to simplify the formula, for any f : R → R such that | D ⁢ f | < 1 and f ⁢ ( t + 1 ) = f ⁢ ( t ) , the graphs of the functions

F c ⁢ ( x 1 , … , x n + 1 ) = f ⁢ ( x 1 ) + c ( for ⁢ c ∈ R )

are flat hypersurfaces foliating 𝑀.

7.3 Misner spacetimes

A Misner spacetime is a quotient of a wedge

W : = { x ∈ R n , 1 ∣ x n + 1 > | x 1 | } .

To exploit this, observe that the metric of 𝑊 can be written as t 2 ⁢ d ⁢ s 2 + d ⁢ x 2 2 + ⋯ + d ⁢ x n 2 − d ⁢ t 2 , where we have performed a simple change of coordinates from ( x 1 , x n + 1 ) to ( t , s ) in the copy of R 1 , 1 given by x 2 = ⋯ = x n = 0 , t = x n + 1 2 − x 1 2 ∈ ( 0 , + ∞ ) is the timelike distance from the origin and s ∈ R is the arclength parameter of every hyperbola of the form x n + 1 2 − x 1 2 = t 2 . From this expression, one sees that every Minkowski isometry preserving 𝑊 acts by Euclidean isometries on the hypersurface { t = 1 } , which is intrinsically isometric to R n , and by the identity on the 𝑡 factor. The quotient of 𝑊 by a discrete subgroup Λ acting on { t = 1 } as a lattice is called a Misner spacetime.

Proposition 7.3

Let 𝑀 be an MGHC spacetime which is finitely covered by a Misner spacetime. Then 𝑀 has a foliation by intrinsically flat hypersurfaces. Moreover, if Σ 0 is a closed spacelike hypersurface of constant scalar curvature 𝑆, then S = 0 .

Proof

We argue as in the case of translation spacetimes. Since every group of isometries of 𝑊 preserves the 𝑡-coordinate, it preserves its level sets, which are intrinsically flat hypersurfaces. Hence the quotient 𝑀 inherits a foliation by intrinsically flat hypersurfaces.

For the second part, as before, let F : M → R be a function whose level sets are precisely the leaves of the above foliation and which is increasing in the future direction. For instance, 𝐹 is the function induced by the 𝑡 coordinate. As in the previous proof, we perform all arguments, which are local, in a small Minkowski chart.

Let p max and p min be the maximum and minimum of F | Σ 0 . Then Σ 0 is a hypersurface of constant scalar curvature, which is tangent to the leaf F − 1 ⁢ ( F ⁢ ( p min ) ) , and contained in its future. Since every leaf { F = c } has one positive principal curvature and n − 1 zero principal curvatures, by Weyl’s monotonicity theorem, the principal curvatures of Σ 0 at p min are all non-negative, and at least one is positive. Hence the (constant) H 2 of Σ 0 is non-negative, and moreover, the mean curvature is positive at p min . (Alternatively, we could have used Remark 3.10 to infer that H 2 is non-negative, except that the leaf { F = c } is the graph of a non-admissible function (say, 𝑢) since its H 2 vanishes, so one has to first perturb 𝑢 and 𝑣 by adding a function of the form ϵ ⁢ ∥ x ∥ 2 and then let ϵ → 0 .)

It only remains to exclude the possibility that the H 2 is positive. By contradiction, suppose that the H 2 of Σ 0 is positive. Then the mean curvature of Σ 0 never vanishes by (3.2), and is positive everywhere because it is positive at p min by the previous discussion on the principal curvatures. Thus Σ 0 is the graph of an admissible function and we can apply Remark 3.10 at p max . This implies that the H 2 of F − 1 ⁢ ( F ⁢ ( p max ) ) is positive at p max , which gives a contradiction. ∎

Remark 7.4

As in Remark 7.2, one easily sees that uniqueness of the hypersurfaces of constant scalar curvature does not hold in Misner spacetimes. Taking the standard Z n lattice acting on { t = 1 } for simplicity, one can see that the hypersurfaces { t = F ⁢ ( x 2 , … , x n , s ) } , where F ⁢ ( x 2 , … , x n , s ) = f ⁢ ( s ) and 𝑓 is such that f ⁢ ( s + 1 ) = f ⁢ ( s ) and defines a spacelike curve in R 1 , 1 , all have vanishing sectional curvature.

7.4 The interesting case

We now turn our attention to the most important part of Theorem 1.6, namely the existence of foliations by hypersurfaces of constant scalar curvature in MGHC flat spacetimes which are not (up to finite cover) translation spacetimes or Misner spacetimes. This will follow from Corollary 5.3. Let us first recall the definitions.

A Cauchy hyperbolic spacetime is obtained as a quotient of a regular domain D φ ⊂ R n , 1 by a group Γ of isometries of R n , 1 , acting freely and properly discontinuously on D φ . It turns out that the linear part of Γ, i.e. the projection to O ⁢ ( n , 1 ) , is contained in the identity component of O ⁢ ( n , 1 ) and acts freely and properly discontinuously on

H n = { x 1 2 + ⋯ + x n 2 − x n + 1 2 = − 1 , x n + 1 > 0 } ,

with quotient a closed hyperbolic manifold diffeomorphic to any Cauchy hypersurface of 𝑀.

Moreover, very importantly, φ : S n − 1 → R is continuous. This fact can be deduced from various references: it is a consequence of [24, Theorem 3.1] together with the existence of a uniformly convex Cauchy hypersurface [8, Theorem 1.6]; it is contained in [3, Section 4]; it is proved in parts (1) and (2) of [26, Theorem F], specialized to a closed manifold, i.e. the divisible case (the proof is done for n = 2 but extends immediately to any dimension).

Now, a twisted product of 𝑀 as above and a Euclidean torus is the quotient of D φ × T (endowed with the product metric), where 𝑇 is a Euclidean torus, by the action of Γ defined by γ ⁢ ( p , q ) = ( γ ⋅ p , ρ ⁢ ( γ ) ⋅ q ) , for a representation ρ : Γ → Isom ⁢ ( T ) . When 𝜌 is the trivial representation, 𝑀 is simply the (untwisted) product of D φ / Γ and 𝑇. Observe that we allow 𝑇 to have dimension 0 (i.e. 𝑇 is a point) so as to include Cauchy hyperbolic spacetimes in this definition.

Corollary 7.5

Let 𝑀 be an MGHC spacetime which is finitely covered by a twisted product of a Cauchy hyperbolic spacetime with a Euclidean torus. Then 𝑀 has a foliation by hypersurfaces of constant scalar curvature. Moreover, every spacelike hypersurface of constant scalar curvature in 𝑀 is a leaf of this foliation. Finally, the function sending 𝑝 to the value of the scalar curvature of the unique leaf through 𝑝 defines a time function on 𝑀.

Proof

By the above description, 𝑀 is isometric to a quotient of D φ ̂ by the action of a group Γ ̂ , where φ ̂ : S n − 1 → R ∪ { + ∞ } is a lower semi-continuous function which is continuous and real-valued when restricted to S n − d − 1 ⊂ S n − 1 (here 0 ≤ d = dim ( T ) ≤ n − 2 and S n − d − 1 is defined by the vanishing of the last 𝑑 coordinates) and is identically + ∞ on the complement of S n − d − 1 . To understand the action of Γ ̂ , first observe that, since D φ ̂ splits as D φ × R d , Γ ̂ is a subgroup of the product Isom ⁢ ( R n − d , 1 ) × Isom ⁢ ( R d ) . We have a short exact sequence

1 → Z d → Γ ̂ → Γ → 1 ,

where Z d is acting as a lattice on R d , with quotient isometric to 𝑇, whereas Γ ≅ Γ ̂ / Z d is acting on D φ × T ≅ D φ ̂ / Z d by the 𝜌-twisted action described above. As a consequence of this discussion, the projection of Γ ̂ to Isom ⁢ ( R n − d , 1 ) coincides with Γ seen as acting on R n − d , 1 (preserving D φ ).

By Corollary 5.3, D φ ̂ admits a foliation by constant scalar curvature hypersurfaces Σ ̂ S , where the scalar curvature 𝑆 varies in ( − ∞ , 0 ) . These are obtained as products of R d and of the hypersurfaces Σ S of constant scalar curvature whose domain of dependence is the regular domain D φ ⊂ R n − d , 1 , where φ = φ ̂ | S n − d − 1 . We have shown that the projection of Γ ̂ preserves D φ , and therefore preserves each Σ S by the uniqueness part of Theorem 1.1. Therefore, Γ ̂ preserves each Σ ̂ S . Since the action of Γ ̂ on D φ ̂ is free and properly discontinuous, so is the action on Σ ̂ S ; hence each Σ ̂ S induces a Cauchy hypersurface in the quotient manifold 𝑀. This shows the first part of the statement.

Now, the function sending 𝑝 to the value of the scalar curvature of the leaf through 𝑝 in D φ ̂ is increasing along future-directed timelike curves and thus induces a time function in 𝑀. This proves the last part of the statement.

It thus remains to show the “moreover” part. Consider, as in Sections 7.2 and 7.3, a function F : M → R whose level sets are the hypersurfaces of the foliation. Concretely, we take for 𝐹 the time function described in the previous paragraph, which takes values in ( − ∞ , 0 ) . Given any closed spacelike hypersurface Σ 0 in 𝑀 of constant scalar curvature, let p min and p max be the minimum and maximum points of F | Σ 0 , and let 𝑐 be the (constant) value of the H 2 of Σ 0 . At p min , Σ 0 is tangent to the leaf F − 1 ⁢ ( F ⁢ ( p min ) ) and is contained in its future. Since the latter has H 2 identically equal to n ⁢ ( n − 1 ) ⁢ | F ⁢ ( p min ) | and positive mean curvature, we can apply Remark 3.10 in a small Minkowski chart and infer that

(7.1) c ≥ n ⁢ ( n − 1 ) ⁢ | F ⁢ ( p min ) | .

Now, by (3.2), the mean curvature of Σ 0 never vanishes. But, by Remark 3.10 again, the mean curvature of Σ 0 at p min is greater than or equal to that of F − 1 ⁢ ( F ⁢ ( p min ) ) at p min , which is positive, so Σ 0 has positive mean curvature everywhere. In other words, Σ 0 is locally, in any Minkowski chart, the graph of an admissible function. Then we can apply Remark 3.10 at p max and conclude that

(7.2) c ≤ n ⁢ ( n − 1 ) ⁢ | F ⁢ ( p max ) | .

Funding source: European Research Council

Award Identifier / Grant number: 101124349

Funding statement: The second author is funded by the European Union (ERC, GENERATE, 101124349).

A Some algebraic properties of the curvature operators

Let us denote by S n ⁢ ( R ) the set of n × n symmetric matrices with real coefficients, and set, for p ∈ B ⁢ ( 0 , 1 ) ⊂ R n and q ∈ S n ⁢ ( R ) ,

h j i = 1 1 − | p | 2 ⁢ ∑ k = 1 n ( δ i ⁢ k + p i ⁢ p k 1 − | p | 2 ) ⁢ q k ⁢ j .

Let us denote by H k ⁢ ( p , q ) the normalized 𝑘-th elementary symmetric function of the eigenvalues λ 1 , … , λ n of ( h j i ) i , j ,

H k ⁢ ( p , q ) = k ! ⁢ ( n − k ) ! n ! ⁢ σ k ⁢ ( λ 1 , … , λ n ) ,

which reads

(A.1) H k ⁢ ( p , q ) = k ! ⁢ ( n − k ) ! n ! ⁢ 1 ( 1 − | p | 2 ) k 2 ⁢ ∑ I , J ( δ i ⁢ j + p i ⁢ p j 1 − | p | 2 ) I , J ⁢ q I , J ,

where the sum is over all the multi-indices I = i 1 < ⋯ < i k , J = j 1 < ⋯ < j k , and where, if 𝐴 is an n × n matrix, A I , J stands for the determinant of the k × k matrix formed by the lines of indices 𝐼 and the columns of indices 𝐽 of 𝐴. Let us set, for p ∈ B ⁢ ( 0 , 1 ) ⊂ R n , the positive cone associated to the operator H m ,

Γ m ⁢ ( p ) = { q ∈ S n ⁢ ( R ) ∣ H m ⁢ ( p , q + η ) > H m ⁢ ( p , q ) > 0 ⁢ for all ⁢ η ∈ S n ⁢ ( R ) + ∗ } = { q ∈ S n ⁢ ( R ) ∣ H k ⁢ ( p , q ) > 0 , k = 1 , … , m } ,

where S n ⁢ ( R ) + ∗ stands for the set of n × n symmetric matrices which are positive definite. Let us finally define the set of positivity of H m by

E : = { ( p , q ) ∈ B ( 0 , 1 ) × S n ( R ) ∣ q ∈ Γ m ( p ) }

and recall that the following properties hold on ℰ:

the operator H m is elliptic: for all ( p , q ) ∈ E and all ξ ∈ R n ∖ { 0 } ,
∑ i , j ∂ H m ∂ q i ⁢ j ⁢ ( p , q ) ⁢ ξ i ⁢ ξ j > 0 ;
the operator H m 1 / m is concave with respect to the second variable 𝑞: for all ( p , q ) ∈ E and all ( ξ i ⁢ j ) i ⁢ j ∈ S n ⁢ ( R ) ,
(A.2) ∑ i , j , k , l ∂ 2 H m 1 m ∂ q i ⁢ j ⁢ ∂ q k ⁢ l ⁢ ( p , q ) ⁢ ξ i ⁢ j ⁢ ξ k ⁢ l ≤ 0 ;
the Maclaurin inequalities hold: for all ( p , q ) ∈ E and all i ∈ { 1 , … , m − 1 } ,
( H i ⁢ ( p , q ) ) 1 i ≥ ( H i + 1 ⁢ ( p , q ) ) 1 i + 1 .

The following result shows that uniform ellipticity is granted once 𝑝 and 𝑞 are bounded.

Proposition A.1

Let δ > 0 , θ ∈ ( 0 , 1 ] and C ≥ 0 be given constants. There exist two positive constants 𝜆 and Λ depending only on 𝛿, 𝜃 and 𝐶 such that, for all ( p , q ) ∈ E satisfying | p | ≤ 1 − θ , | q | ≤ C and H m ⁢ ( p , q ) ≥ δ ,

λ ⁢ | ξ | 2 ≤ ∑ k , l ∂ H m ∂ q k ⁢ l ⁢ ( p , q ) ⁢ ξ k ⁢ ξ l ≤ Λ ⁢ | ξ | 2

for all ξ ∈ R n .

Proof

The upper bound is straightforward and the lower bound relies on the following inequality: for all ( p , q ) ∈ E ,

∑ k , l ∂ H m ∂ q k ⁢ l ⁢ ( p , q ) ⁢ ξ k ⁢ ξ l ≥ 1 n − m + 1 ⁢ ( 1 − | p | 2 ) ⁢ H m ⁢ ( p , q ) | q | 1 ⁢ | ξ | 2

for all ξ ∈ R n , with | q | 1 = ∑ i ⁢ j | q i ⁢ j | ; see [22, Lemma 6] for the key inequality in Γ m . ∎

We finish that section showing that vertical sections of admissible graphs are admissible. We begin with a useful formula.

Lemma A.2

For all p ∈ B ⁢ ( 0 , 1 ) ⊂ R n and all q ∈ S n ⁢ ( R ) , the formula

∂ H k ∂ q 11 ⁢ ( p , q ) = n k ⁢ ( 1 − | p ′ | 2 ) k + 1 2 ( 1 − | p | 2 ) k 2 + 1 ⁢ H k − 1 ⁢ ( p ′ , q ′ )

holds for k ≥ 2 , where p ′ = ( p i ) 2 ≤ i ≤ n and q ′ = ( q i ⁢ j ) 2 ≤ i , j ≤ n .

Proof

This is an elementary direct computation using (A.1). See [23] for a similar formula for the 𝑘-th curvature operator in Euclidean space. ∎

Corollary A.3

Keeping the notation introduced above, if 𝑞 belongs to Γ m ⁢ ( p ) then q ′ belongs to Γ m − 1 ⁢ ( p ′ ) .

Proof

This is a consequence of the lemma, since ∂ H k / ∂ q 11 ⁢ ( p , q ) > 0 by the ellipticity of H k on E = { ( p , q ) ∈ B ⁢ ( 0 , 1 ) × S n ⁢ ( R ) ∣ q ∈ Γ m ⁢ ( p ) } , for all k = 2 , … , m . ∎

A spacelike function u : R n → R of class C 2 is said to be 𝑚-admissible if D 2 ⁢ u ⁢ ( x ) belongs to Γ m ⁢ ( D ⁢ u ⁢ ( x ) ) for all x ∈ R n . We readily obtain as a corollary that vertical sections of 𝑚-admissible graphs are ( m − 1 ) -admissible.

Corollary A.4

If u : R n → R is an 𝑚-admissible function, then u ′ : R n − 1 → R defined by u ′ ( x 2 , … , x n ) : = u ( 0 , x 2 , … , x n ) is ( m − 1 ) -admissible.

B The Dirichlet problem between barriers

We solve here the Dirichlet problem for the prescribed scalar curvature equation between two barriers, with a boundary condition given by the upper barrier.

Theorem B.1

Let Ω be a uniformly convex domain in R n , where ∂ Ω is C 4 , α for some α ∈ ( 0 , 1 ) , and let H ∈ C 2 , α ⁢ ( Ω ̄ × R ) be a positive function. Let φ 1 ∈ C 2 ⁢ ( Ω ̄ ) be an admissible function and let φ 2 ∈ C 4 , α ⁢ ( Ω ̄ ) be strictly convex and spacelike such that

H 2 ⁢ [ φ 1 ] ≥ H ⁢ ( ⋅ , φ 1 ) , H 2 ⁢ [ φ 2 ] ≤ H ⁢ ( ⋅ , φ 2 ) in ⁢ Ω

and φ 1 < φ 2 in Ω ̄ . Then there exists a spacelike function 𝑢 belonging to C 4 , α ⁢ ( Ω ̄ ) such that

(B.1) { H 2 ⁢ [ u ] = H ⁢ ( ⋅ , u ) in ⁢ Ω , u = φ 2 on ⁢ ∂ Ω

and φ 1 ≤ u ≤ φ 2 . If ∂ x n + 1 H ≥ 0 the solution is unique.

Remark B.2

Let us note that if 𝐻 is bounded above and φ 2 is given, it is immediate to find φ 1 satisfying the other conditions in the theorem, so that (B.1) is solvable: we may take for φ 1 a function smaller than φ 2 whose graph is a hyperboloid with scalar curvature H 2 ⁢ [ φ 1 ] = sup Ω ̄ × R H .

Proof

A very similar result was proven in [4, Theorem 2.1], with a boundary condition given by the lower barrier φ 1 instead of the upper barrier φ 2 . We only point out the slight differences in the proof, and will refer to [4] for the other arguments. We consider the compact set K = { ( x , z ) ∣ x ∈ Ω ̄ , φ 1 ⁢ ( x ) ≤ z ≤ φ 2 ⁢ ( x ) } and the non-negative constant

k = max ⁡ ( sup K 1 H ⁢ ∂ H ∂ x n + 1 , 0 )

so that the function z ↦ H ⁢ ( x , z ) ⁢ e − k ⁢ z is decreasing on [ φ 1 ⁢ ( x ) , φ 2 ⁢ ( x ) ] for all x ∈ Ω ̄ . We suppose that φ 2 is not a solution. We consider the Banach space

E = { v ̄ ∈ C 2 , α ⁢ ( Ω ̄ ) ∣ v ̄ = 0 ⁢ on ⁢ ∂ Ω } ,

the convex open set of 𝐸,

W = { v ̄ ∈ E ∣ v ̄ > 0 ⁢ in ⁢ Ω , ∂ n v ̄ > 0 ⁢ on ⁢ ∂ Ω ⁢ and ⁢ v ̄ < φ 2 − φ 1 ⁢ on ⁢ Ω ̄ } ,

where ∂ n denotes the interior normal derivative at the boundary and the map

T : [ 0 , 1 ] × W → E , ( t , v ̄ ) ↦ u ̄ ,

where u ̄ ∈ E is such that u = φ 2 − u ̄ is the admissible solution of the Dirichlet problem

{ H 2 ⁢ [ u ] ⁢ e − k ⁢ u = t ⁢ H ⁢ ( ⋅ , v ) ⁢ e − k ⁢ v + ( 1 − t ) ⁢ H ⁢ ( ⋅ , φ 2 ) ⁢ e − k ⁢ φ 2 in ⁢ Ω , u = φ 2 on ⁢ ∂ Ω

(see [32, Theorem 1.1]). Here v = φ 2 − v ̄ . We may then follow the lines of [4, Section 2] without modification, and prove that the functions φ 1 and φ 2 are still sub- and super-solutions of that Dirichlet problem (using that the function z ↦ H ⁢ ( x , z ) ⁢ e − k ⁢ z is decreasing), 𝑇 takes values in 𝑊 and the fixed points of 𝑇 satisfy the estimate ∥ u ̄ ∥ 2 , α < C for a controlled positive constant 𝐶. Considering the convex set W c = { v ̄ ∈ W ∣ ∥ v ̄ ∥ 2 , α < C } and T : [ 0 , 1 ] × W c → E , we then show that 𝑇 is continuous and compact, T ⁢ ( 0 , ⋅ ) maps ∂ W c into W c ̄ and T ⁢ ( t , ⋅ ) does not have any fixed point on ∂ W c . The fixed point theorem of Browder–Potter finally implies that T ⁢ ( 1 , ⋅ ) has a fixed point, which proves the theorem. The details are carried out in [4]. ∎

Acknowledgements

The second author would like to thank Graham Smith for many ad hoc explanations and for pointing out relevant references, and to Thierry Barbot, Francesco Bonsante and Peter Smillie for several discussions related to this work. The authors are very grateful to the referee for the very careful reading of a previous version of this article, and for many suggestions that largely helped improving the exposition. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.

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Received: 2024-08-19

Revised: 2025-02-04

Published Online: 2025-04-30

Published in Print: 2025-07-01

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