Spacelike Graphs of Prescribed Mean Curvature in the Steady State Space

1 Introduction and Statement of Results

The steady state space ℋn+1 is a model for the universe proposed by Bondi and Gold [4], and Hoyle [12] which is homogeneous and isotropic, that is, it looks the same not only at all points and in all directions, but also at all times [10, Section 5.2]. If ℝ1n+2=(ℝn+2, 〈,〉=dx12+⋯+dxn+12-dxn+22) denotes the n+2-dimensional Minkowski space, the steady state space ℋn+1 is the half of the de Sitter space 𝕊1n+1={p∈ℝ1n+2:〈p,p〉=1}, given by ℋn+1={p∈𝕊1n+1:〈p,a〉>0}, where a∈ℝ1n+1 is a non-zero null vector in the past half of the null cone. This space is a non-complete manifold foliated by a uniparametric family of totally umbilical hypersurfaces {Lτ:τ∈(0,∞)}, called slices, where Lτ={p∈ℋn+1:〈p,a〉=τ}, and all have constant mean curvature H=-1. The boundary of ℋn+1 is the null hypersurface L0={p∈𝕊1n+1:〈p,a〉=0} that represents the past infinity and L∞={p∈𝕊1n+1:〈p,a〉=∞} is the limit boundary that stands as the future infinity.

An equivalent model of ℋn+1 is (ℝ+n+1,g), where ℝ+n+1=ℝn×ℝ+, is the upper half-space of vector space ℝn+1 endowed with the metric

where x=(x1,…,xn+1) are the canonical coordinates of ℝn+1. By the expression of the metric g in (1.1), the steady state space ℋn+1 is the Lorentzian analogue to the upper half-space model of the hyperbolic space viewed as a subset of the (n+1)-dimensional Lorentz–Minkowski space ℝ1n+1. In this model a slice is a horizontal hyperplane L⁢(h)={x∈ℝn+1:xn+1=h} for h>0.

In this paper we study the Dirichlet problem for the prescribed mean curvature equation on a domain of a slice: If Ω⊂ℝn is a smooth bounded domain, given φ∈C0⁢(∂⁡Ω) and H∈ℝ, find a solution u∈C2⁢(Ω)∩C0⁢(Ω¯) of

Here div and D are the Euclidean divergence and gradient operators of ℝn, respectively. If u is a solution of (1.2), its graph Σu={(x1,…,xn,u⁢(x1,…,xn)):(x1,…,xn)∈Ω} is a spacelike hypersurface in ℋn+1 of constant mean curvature H with respect to the upwards orientation, and its boundary is the graph of φ. Spacelike hypersurfaces with constant mean curvature are of interest in a spacetime, as for example ℋn+1, because they are used as convenient initial hypersurfaces for the Cauchy problem corresponding to the Einstein equations [15]. Also it is interesting to have foliations of the spacetime by hypersurfaces with constant mean curvature because all points of each leaf of the foliation are instantaneous observers (or normal observers) and the (timelike) unit normal vector of the hypersurfaces measures how the observers get away with respect to the next ones. Thus, the graph of a solution of (1.2) can be viewed as a local result (on the domain Ω) of prescribing the behavior of normal observers. In the simplest spacetime, the Lorentz–Minkowski space, the first remarkable result on the Dirichlet problem is due to Bartnik and Simon [3] who proved the solvability for almost any domain Ω and boundary values φ .

In the literature, the first existence result of the Dirichlet problem (1.2) appears in [16]. More recently, the existence of radially symmetric spacelike hypersurfaces in ℋn+1 with constant mean curvature was shown in [7]. The methods employed therein were the classical Schauder fixed point theorem, and the results were extended to a wider class of spacetimes, namely, the Robertson–Walker space. However, our approach for solving (1.2) is the method of continuity following [3, 16]. Proving the existence of graphs with prescribed mean curvature and boundary requires establishing a priori C1 estimates, and these estimates are accomplished by the use of hyperbolic planes as barriers. For the question of uniqueness of solutions of (1.2), standard arguments of elliptic PDEs do not provide a complete answer (see Remark 2.2). On the other hand, the Dirichlet problem of the Gauss curvature equation was studied in [17]. Recently, the author has proved that planar discs and hyperbolic caps are the only compact spacelike surfaces with constant mean curvature in ℋ3 spanning a circle [13].

The solvability of the Dirichlet problem (1.2) depends on whether H is less than or greater than -1, which is just the value of the mean curvature of slices. For example, for φ=h, the graph Σu lies in one side of the slice L⁢(h), namely, Σu is above L⁢(h) (or u>h) if H<-1 and below L⁢(h) (or u<h) if H>-1. Notice that the space is not symmetric with respect to L⁢(h) and one can not expect similar results if H<-1 or H>-1; see also the difference about the uniqueness in Remark 2.2. In the mentioned paper [16], Montiel showed the existence of solutions of (1.2) for φ=h and H<-1, assuming that Ω is strictly mean convex. This result allowed to prove the existence of non-compact spacelike hypersurfaces with boundary in the infinity. By the maximum principle, the value of H for these hypersurfaces must be less than -1 and, after letting h→0 together with a suitable control of the C1 estimates, he established successfully the existence of complete spacelike hypersurfaces with boundary at the past infinity. The goal of the present paper is to consider the Dirichlet problem for H>-1, where getting a priori estimates is not a consequence of the case H<-1, and the technical difficulties are hard. Indeed, the existence result that we will prove holds when -1≤H<0 and Ω satisfies stronger convexity assumptions. Here we recall that if κ>0, a domain Ω⊂L⁢(h) is said to be κ-convex if the principal curvatures κ1,…,κn-1 of ∂⁡Ω with respect to the inward normal vector satisfy κi≥κ. The main result that we prove is the following.

Let us point out that some kind of smallness assumption on the domain Ω is needed because the following result was proved by the author in [13].

The proof of this result uses an argument of comparison of the graph of a prospective solution with a uniparametric family of hyperbolic planes that have the role of barrier hypersurfaces. The value of r0 in (1.4) implies that r0>1 and this is the reason that we suppose, in Theorem 1.1, that the domain Ω is contained in a ball of radius 1. On the other hand, as L⁢(h) is isometric to the Euclidean hyperplane of equation xn=h, then the κ-convexity of Ω means that Ω is also κ/h-convex as a submanifold of the Euclidean hyperplane xn=h.

This paper is organized as follows. In Section 2 we recall some basics of the steady state space and we obtain the prescribed mean curvature equation. Section 3 is devoted to giving the notion of a graph on a slice. The height and gradient estimates needed in the continuity method are obtained in Section 4. Finally, in Section 5 we prove Theorem 1.1. As a consequence of the methods employed in the previous sections, in Section 6 we obtain a priori gradient estimates for a solution of the Dirichlet problem for other values of H and boundary conditions φ.

2 Preliminaries

Let Σ be a connected hypersurface. A smooth immersion ψ:Σ→ℋn+1 is said to be a spacelike hypersurface if the induced metric via ψ is a Riemannian metric on Σ. A spacelike hypersurface is always orientable because the causal character of the ambient space allows to choose a unique unit timelike normal vector field N, globally defined on Σ, which is future-directed. As the metric of ℋn+1 is conformal to the Minkowski metric, the causal character of ℋn+1 is the same to that of the upper half-space viewed as an open set of the Minkowski space ℝ1n+1. Recall that there exists an explicit isometry Φ:ℋn+1⊂𝕊1n+1→ℋn+1⊂ℝ+n+1 between both models which inverts the orientation [16]. We will also use the terminology horizontal and vertical in the Euclidean sense considering the target ambient space ℝ+n+1. In this model of ℋn+1, the isometries are the conformal transformations of the Minkowski space ℝ1n+1 that preserve the upper half-space ℝ+n+1. For example, the rotations about a vertical straight line are isometries of ℋn+1 as well as the horizontal translations or homotheties from any point of ℝn-1×{0}.

If ∇ stands for the Levi-Civita connection on Σ, the mean curvature H is defined as

where W is the shape operator and κi are the principal curvatures of ψ. When H is constant we say that Σ is a CMC hypersurface or a H-hypersurface if we want to emphasize the value of the mean curvature. We point out that the choice of the minus sign in (2.1) follows [1, 5] and it is of opposite sign to the one adopted in [16]. According to (2.1), a slice Lτ has constant mean curvature H=-1 with respect to the future unit normal vector.

In view of the metric (1.1), we can consider on Σ the Minkowski metric 〈,〉 and the metric g. If N is the Gauss map of ψ:Σ→ℋn+1, then N′=N/xn is the Gauss map of the immersion ψ:Σ→ℝ1n+1. Since g and 〈,〉 are conformal metrics by (1.1), we have κi=xn⁢κi′+(xn∘N′), where κi and κi′ are the principal curvatures of (Σ,g) and (Σ,〈,〉), respectively. From the definition of H in (2.1), we conclude

where H and H′ are the mean curvatures of Σ⊂ℋn+1 and Σ⊂ℝ1n+1, respectively.

3 Graphs in the Steady State Space

The notion of a graph for the Dirichlet problem (1.2) coincides with the graph on a domain of a slice of ℋn+1. We need to link this concept in both models to obtain, in Section 4, the height and gradient estimates needed for solving (1.2). Let h>0. Given a domain Ω⊂ℝn, take Ω×{h}⊂L⁢(h). If f is a function on Ω, we associate to each q∈Ω×{h} the point of the geodesic of ℋn+1 passing by q and orthogonal to L⁢(h) at distance f⁢(q) from L⁢(h). In the upper half-space model, this type of geodesic is a vertical straight line, so a graph on Ω×{h} is an Euclidean graph xn+1=u⁢(x1,…,xn) on Ω, where u=h⁢ef. In the model of ℋn+1 as a subset of 𝕊1n+1, the orthogonal geodesic to Ω⊂Lτ at q∈Ω is

If f is a function on Ω, the graph Σf of f is

Notice that

The tangent space of Σ at X⁢(q) is

where ∇ is the gradient operator in Lτ. The unit normal vector field N pointing to the future is

A computation on 𝕊1n+1 then gives

By the isometry Φ, let Σu denote the corresponding graph on the domain Φ⁢(Ω)≡Ω⊂L⁢(h), where we relate the functions f and u. If we let p stand for a point in 𝕊1n+1 and Φ⁢(p)=x∈ℝ+n+1, then from (3.1) it follows that

so h=1/τ. We also express N, defined in Σ⊂𝕊1n+1, with the Gauss map N′ given in (2.4). By the isometry Φ between both models, we find 〈N,a〉=(xn+1∘N′)/xn+1, and thus

From the expression of N′ in (2.4), we have

Two examples of graphs are slices and hyperbolic planes. For a slice L⁢(h), we have u=h, N′=𝐞n+1 and 〈N′,𝐞n+1〉=-1. For a hyperbolic plane ℍn⁢(r;t), t∈ℝ, we have

and

The control of the functions 〈p,a〉 and 〈N,a〉 given in (3.2) and (3.3), respectively, will be decisive in the proof of Theorem 1.1 when we establish a priori C1 estimates of a solution u. These estimates will be based on the expressions of the Laplacians of 〈p,a〉 and 〈N,a〉. For the function 〈p,a〉, we are identifying p with its image ψ⁢(p) by the immersion ψ:Σ→𝕊1n+1. With respect to the induced metric on Σ via ψ, the Laplacian of 〈p,a〉 (or 〈ψ⁢(p),a〉) is

If we now suppose that the mean curvature is constant, then the Laplacian of 〈N,a〉 is

where σ is the second fundamental form of ψ, see [16].

4 The Method of Continuity

The technique used in the proof of Theorem 1.1 is the method of continuity [9], in the context of the prescribed mean curvature equation, see, e.g., [2, 6, 14]. After an isometry of ℋn+1, we assume that h=1, that is, Ω is included in the slice L⁢(1). In this slice, the curvature ∂⁡Ω coincides with the Euclidean one as subset of ℝn. The method of continuity considers the family of Dirichlet problems

where H⁢(t)=t⁢(1+H)-1 and t∈[0,1]. We show that the subset of [0,1] defined by

is non-empty, closed and open in [0,1]. In such case, 1∈A, proving the existence of a solution u∈C2,α⁢(Ω¯). As H is constant and Ω is smooth, any C2,α solution will be smooth on Ω¯ (see [9, Theorem 6.17]), proving Theorem 1.1.

Let us observe that if H=-1 then the solution of (1.2) for φ=1 is the constant function u=1. Moreover, this solution is unique as a consequence of the tangency principle comparing Σu with the slices L⁢(c) for c>0.

The proof that A is closed follows once we establish a priori C1 estimates of the prospective solutions of (1.2), that is, height and gradient estimates for every solution u of (1.2), see [9]. The convexity condition (1.3) is required first to obtain an estimate of the C0 norm of the solution comparing the graph with hyperbolic planes of type ℍn⁢(r;c) and, in addition, these hyperbolic planes will provide the boundary gradient estimates. Establishing gradient estimates, |D⁢u| goes through to prevent that |D⁢u|→1 at a point of Ω¯ or, in the terminology of Marsden and Tipler, that the hypersurface cannot ‘go null’ [15, p. 124]. This is the hard part of the proof of Theorem 1.1, and it will be proved in the next subsections.

4.1 Height Estimates

We introduce the next notation. For each t≥0, the light cone with apex ξ=(0,…,0,t) is defined by 𝒞t={x∈ℝ+n+1:〈x-ξ,x-ξ〉=0}. The apex ξ separates the cone in two half-cones. Let 𝒞t+ denote the upper half-cone and let int⁡(𝒞t+) be the convex domain of ℝ+n+1 bounded by 𝒞t+.

Let -1<H<0 and let u be a solution of (1.2) for u=1 along ∂⁡Ω. By the tangency principle, comparing with slices L⁢(h) for values of h going from h=∞ to h=1, we have u<1 in Ω. For any t∈[0,1), let 𝒞t+ be the upper light cone. Denote by Ωρ⊂ℝn the ball of radius ρ>0 centered at the origin O and suppose Ω⊂Ω1. Since the inclusion Ω⊂Ω1 is strict, let ρ0<1 be a number such that Ω¯⊂Ωρ0. As u=1 on ∂⁡Ω, the spacelike condition |D⁢u|<1 and the convexity of Ω imply that |u⁢(q)-u⁢(q′)|<|q-q′| whenever q,q′∈Ω for q≠q′. This inequality together with the fact that ∂⁡Σu⊂Ωρ0×{1} imply Σu⊂int⁡(𝒞1-ρ0+), see Figure 1. Because the apex of 𝒞1-ρ0+ is (1-ρ0)⁢p0, where p0=(0,0,1), we find

(4.2) 1 - ρ 0 < x n + 1 ⁢ ( p ) ≤ 1

for all p∈Σu, obtaining the uniform height estimate for u.

Figure 1

Proof of the boundary gradient estimates in Theorem 1.1.

5 Proof of Theorem 1.1

Now that the height and gradient estimates are obtained, for a solution of the Dirichlet problem, we prove definitively Theorem 1.1. Again suppose h=1 on the boundary values in (1.2). If H=-1, then u=1 is a solution of (1.2). We suppose -1<H<0 and use the notation of Section 4. In the continuity method, let ut be a solution of the Dirichlet problem (4.1). First, we note that A is non-empty because u=1 is a solution, so 0∈A.

In order to prove that A is a closed subset of [0,1], we have to find C1 estimates of the solutions ut for t∈A. The height estimates (4.2) hold for all values of the mean curvature between -1 and 0. From the global gradient estimate (4.8), we have

The proof that A is an open subset of [0,1] is a consequence of the implicit function theorem. The linearization of the mean curvature is the Jacobi operator, corresponding to the second variation of the area functional, and it is given by L=Δ-|σ|2+n, see [16]. In this context, if we show that the kernel of L is trivial, then L is a self-adjoint Fredholm operator of index 0, hence L is invertible. From the implicit function theorem for Banach spaces, it follows that if the Dirichlet problem (1.2) can be solved for the value H=H0, then it can also be solved in an open interval around H0. However, we point out that we cannot apply the standard theory [9, Theorem 10.1].

We prove the openness of A by considering the following eigenvalue problem for L:

Associated to L, there is a quadratic form Q acting on the subspace C0∞⁢(Σ) of smooth functions on Σ satisfying the condition f=0 on ∂⁡Σ. Here Q is

We say that Σ is strongly stable if Q⁢(f)≥0 for all f∈C0∞⁢(Σ) which, in terms of the spectrum of L, is equivalent to say that the first eigenvalue λ1⁢(L) of L is non-negative, so the kernel of L is zero. The next result closely follows [8, Theorem 1].