The prescribed curvature problem for entire hypersurfaces in Minkowski space

We prove three results in this paper. First, we prove for a wide class of functions $\varphi\in C^2(\mathbb{S}^{n-1})$ and $\psi(X, \nu)\in C^2(\mathbb{R}^{n+1}\times\mathbb{H}^n),$ there exists a unique, entire, strictly convex, spacelike hypersurface $M_u$ satisfying $\sigma_k(\kappa[M_u])=\psi(X, \nu)$ and $u(x)\rightarrow |x|+\varphi\left(\frac{x}{|x|}\right)$ as $|x|\rightarrow\infty.$ Second, when $k=n-1, n-2,$ we show the existence and uniqueness of entire, $k$-convex, spacelike hypersurface $M_u$ satisfying $\sigma_k(\kappa[M_u])=\psi(x, u(x))$ and $u(x)\rightarrow |x|+\varphi\left(\frac{x}{|x|}\right)$ as $|x|\rightarrow\infty.$ Last, we obtain the existence and uniqueness of entire, strictly convex, downward translating solitons $M_u$ with prescribed asymptotic behavior at infinity for $\sigma_k$ curvature flow equations. Moreover, we prove that the downward translating solitons $M_u$ have bounded principal curvatures.


Introduction
Let R n,1 be the Minkowski space with the Lorentzian metric In this paper, we will devote ourselves to the study of spacelike hypersurfaces with prescribed σ k curvature in Minkowski space R n,1 .Here, σ k is the k-th elementary symmetric polynomial, i.e., σ k (κ) = Any such hypersurface M can be written locally as a graph of a function x n+1 = u(x), x ∈ R n , satisfying the spacelike condition More precisely, we will focus on the following equation: where X = (x, u(x)) is the position vector of M u = {(x, u(x))|x ∈ R n }, ν = (κ 1 , • • • , κ n ) are the principal curvatures of M u .Thus equation (1.2) can be rewritten as σ k (κ[M u ]) = ψ(x, u(x), Du).(1.3) Notice that the right hand side functions ψ of (1.2) and (1.3) are different.Slightly extending the notation, we use the same symbol here.
The classical Minkowski problem asks for the construction of a strictly convex compact surface Σ whose Gaussian curvature is a given positive function f (ν(X)), where ν(X) denotes the normal to Σ at X.This problem has been discussed by Nirenberg [24], Pogorelov [27], and Cheng-Yau [11].The general problem of finding strictly convex hypersurfaces with prescribed surface area measures is called the Christoffel-Minkowski problem.This type of problems can be deduced to a fully nonlinear equation of the form (1.2).It may be traced back to Alexandrov [1] who established the problem of prescribing zeroth curvature measure.Later on, the prescribed curvature measure problem in convex geometry has been extensively studied by Alexandrov [2], Pogorelov [26], Guan-Lin-Ma [18], and Guan-Li-Li [17].A more general form of the prescribed curvature measure problem can be expressed as (1.3).In particular, Guan-Ren-Wang [19] solved this problem in Euclidean space for convex hypersurfaces.Other related studies and references may be found in [3,9,10,15,25,33].
Our goal here is to construct entire, spacelike hypersurfaces satisfy equation (1.2) in Minkowski space.The main results of this paper are the following.
The first result is to construct entire, strictly convex, spacelike hypersurfaces satisfying equation (1.2).
Remark 2. Indeed, from the proof of the C 2 global estimate Lemma 9 we can see that, the assumption ψ(X, ν) dose not depend on X can be replaced by a weaker assumption, that is, ψ −1/k (X, ν) is convex with respect to X and the corresponding form ψ(x, u, Du) dose not depend on |x|.
Remark 3. In the proof, we only can see that the hypersurface M u we constructed is convex.In order to say its strictly convex, we need to apply the Constant Rank Theorem (see Theorem 1.2 in [16] and Theorem 27 in [35]) and the Splitting Theorem (see Theorem 28 in [35]) to obtain that if M u has a degenerate point in the interior, then M u = M l ×R n−l , where M l ⊂ R l,1 is a strictly convex, space like hypersurface.This contradicts (1.4).
Before stating our second result, we need the following definition: Definition 4. A C 2 regular hypersurface M ⊂ R n,1 is k-convex, if the principal curvatures of M at X ∈ M satisfy κ[X] ∈ Γ k for all X ∈ M, where Γ k is the Gårding cone Using the newly developed methods in [28] and [29], we are able to generalize results in [5].We prove Theorem 5. Suppose ϕ is some C 2 function defined on S n−1 and ψ(x, u(x)) ∈ C 2 (R n+1 ) is a positive function satisfying c 1 ψ(x, u(x)) c 2 for c 1 , c 2 > 0. We further assume that k = n − 1, n − 2, and ψ u 0. Then there exits a unique, k-convex, spacelike hypersurface M u = {(x, u(x))|x ∈ R n } satisfying Moreover, as |x| → ∞, Now, let's consider the σ k curvature flow with forcing term in Minkowski space: where κ[M u ] ∈ Γ k .This can be rewritten as the equation for the height function u, The downward translating soliton to (1.8) is of the form u(x, t) = u(x) − t, (1.9) where u(x) satisfies The above equation (1.10) can be viewed as the "degenerate" type of (1.2).In this case, we prove the following theorem: Theorem 6. Suppose ϕ is a C 2 function defined on S n−1 2 and C > 1 is a constant.There exists a unique, strictly convex solution u : R n → R of (1.10) such that as |x| → ∞, Moreover, M u = {(x, u(x))|x ∈ R n } has bounded principal curvatures.
Remark 7.Under our assumptions on ψ, we can see that the linearized operators of equations (1.2), (1.5), and (1.10) satisfy the maximum principle.Therefore, the uniqueness properties in Theorem 1, 5, and 6 follow from the maximum principle directly.
The rest of this paper is organized as follows.In Section 2, we introduce some basic formulas and notations.The solvability of equations (1.2) and (1.5) on bounded domain (Dirichlet problem) is discussed in Section 3. We prove the local C 1 and C 2 estimates for solutions of equations (1.2) and (1.5) in Section 4. This leads to the completion of the proof of our first two main results, Theorem 1 and Theorem 5, in Section 5. Section 6 and Section 7 are devoted to Theorem 6.In particular, in Section 6, we study the radially symmetric solution to equation (1.10), this solution will be used to construct barrier functions in Section 7. We finish the proof of Theorem 6 in Section 7.

Preliminaries
In this paper, we will follow notations in [35].For readers convenience, we will include some basic notations and formulas in this section.Readers who are already familiar with calculations in Minkowski space can skip this section.
We first recall that the Minkowski space R n,1 is R n+1 endowed with the Lorentzian metric 2.1.Vertical graphs in R n,1 .A spacelike hypersurface M in R n,1 is a codimension one submanifold whose induced metric is Riemannian.Locally M can be written as a graph M u = {X = (x, u(x))|x ∈ R n } satisfying the spacelike condition (1.1).Let E = (0, • • • , 0, 1), then the height function of M is u(x) = − X, E .It's easy to see that the induced metric and second fundamental form of M are given by while the timelike unit normal vector field to M is where Du = (u x 1 , • • • , u xn ) and D 2 u = u x i x j denote the ordinary gradient and Hessian of u, respectively.By a straightforward calculation, we have the principle curvatures of M are eigenvalues of the symmetric matrix A = (a ij ) : where 1+w , which is the square root of (g ij ).Let S be the vector of n × n symmetric matrices and Throughout this paper we denote be a local orthonormal frame on T M. We will use ∇ to denote the induced Levi-Civita connection on M. For a function v on M, we denote Using normal coordinates, we also need the following well known fundamental equations for a hypersurface M in R n,1 : (2.2) and the Ricci identity, (2.3) 2.2.The Gauss map.Let M be an entire, strictly convex, spacelike hypersurface, ν(X) be the timelike unit normal vector to M at X. It's well known that the hyperbolic space H n (−1) is canonically embedded in R n,1 as the hypersurface By parallel translating to the origin we can regard ν(X) as a point in H n (−1).In this way, we define the Gauss map: Next, let's consider the support function of M. We denote Let {e 1 , • • • , e n } be an orthonormal frame on H n .We will also denote {e * 1 , • • • , e * n } the pull-back of e i by the Gauss map G. Similar to the convex geometry case, we denote Λ ij = v ij − vδ ij the hyperbolic Hessian.Here v ij denote the covariant derivatives with respect to the hyperbolic metric.
Let ∇ be the connection of the ambient space.Then, we have Note also that, This implies that the eigenvalues of the hyperbolic Hessian are the curvature radius of M. Therefore, equation (1.2) can be written as where where x k is the coordinate function.
2.3.Legendre transform.Suppose M is an entire, stictly convex, spacelike hypersurface.Then M is the graph of a convex function where E = (0, • • • , 0, 1).Introduce the Legendre transform Next, we calculate the first and the second fundamental forms in terms of ξ i .Since it is well known that, We have, the first and the second fundamental forms can be rewritten as: where u * ij denotes the inverse matrix of (u * ij ) and |ξ| 2 = i ξ 2 i .Now, let W be the Weingarten matrix of M, then Here, 1+w * is the square root of the matrix g ij .

The Dirichlet problem
We will divide this section into two subsections.In the first subsection, we only consider the convex solution to (1.2).In the second subsection, we restrict ourselves to the case when k = n − 1 (n 3), n − 2 (n 5), and we will consider the k-convex, spacelike solution to (1.5).When k = 2, this problem has been studied by [4] and [34].
3.1.Dirichilet problem for 1 k n.Recall that in [35] we proved the following Lemma.
Lemma 8. Let F ⊂ S n−1 , F = Conv(F), and u * be a solution of ).Then, the Legendre transform of u * denoted by u satisfies, when Notice that the proof of the above Lemma is independent of the equation that the function u * satisfies.Therefore, adapting the above Lemma to the settings in this paper, this Lemma tells us that if a strictly convex function u * : B 1 → R satisfies u * (ξ) = −ϕ(ξ) for ξ ∈ ∂B 1 , then the Legendre transform of u * denoted by u, satisfies u(x) → |x| + ϕ x |x| as |x| → ∞.Moreover, by Theorem 4 in [35], there exists two solutions ū, ū such that and as |x| → ∞ Here, the constants c 1 , c 2 are the same as the ones in Theorem 1.Throughout this paper, we will denote the Legendre transforms of ū, ū by ū * , ū * respectively.It's easy to see that ū * and ū * are the super-and sub-solutions of (2.9).Combining the discussions above with Section 2, we conclude that in order to find an entire, strictly convex solution u of (1.3), we only need to solve the following equation: where Note that by our assumption in Theorem 1 we have, Thus, equation (3.3) possesses the maximum principle.
Notice that equation (3.3) is degenerate on ∂B 1 .Therefore, we will consider the approximate equation: where 0 < r < 1.
By continuity method we know that, if we can obtain a prior estimates up to the second order, then we can show (3.5) has a unique, strictly convex solution u r * .In view of the super-and sub-solutions ū * , ū * , the C 0 estimates are easy to obtain.The C 1 estimates can be derived by following the argument in Subsection 9.2 of [30].The C 2 estimate on the boundary can be derived from Lemma 27 in [30] and the argument of Bo Guan [14].In the following, we only need to consider the global C 2 estimate.
Let M u = {(x, u(x))|x ∈ R n } be a strictly convex, spacelike hypersurface, v = X, ν be the support function of M u , and u * be the Legendre transform of u.From Subsection 2.2 and 2.3, we know that λ . Therefore, to study the global C 2 estimate of (3.5) is equivalent to study the global C 2 estimate of (2.6).
For our convenience, we will consider the equation , and v ij is the covariant derivatives with respect to the hyperbolic metric.We will use λ to denote the eigenvalues of the matrix Λ.We define the Riemann curvature tensor: Let {e 1 , e 2 , • • • , e n } be an orthonormal frame on H n , we use the notation Then the commutation formulae are Note that in hyperbolic space we have, Therefore, given an orthonormal frame on H n , we obtain the following geometric formulae: We will prove Lemma 9. Let v be the solution of (3.6) in a bounded domain U ⊂ H n .Denote the eigenvalues of where where x n+1 is the coordinate function.Without loss of generality, we assume M is achieved at an interior point P 0 ∈ U for some direction ξ 0 .Chose an orthonormal frame {e 1 , • • • , e n } around P 0 such that e 1 (P 0 ) = ξ 0 and Λ ij (P 0 ) = λ i δ ij .Now, let's consider the test function At its maximum point P 0 , we have In view of (3.7), we get This yields, By the concavity of (σ n /σ n−k ) 1/k we can see that the first term on the right hand side is nonnegative.Combining (3.10)-(3.12)we have, ( We need an explicit expression of F ii .A straightforward calculation gives where for 1 l n, σ ii l = ∂σ l ∂λ i .Since Here and in the following, σ l (λ|a) and σ l (λ|ab) are the l-th elementary symmetric polynomials of λ 1 , • • • , λ n with λ a = 0 and λ a = λ b = 0, respectively.It follows When i 2, we can see that Plugging (3.17) into (3.13),we obtain Here, in the last equality, we have used (3.8).Now, let's calculate ψ11 .We denote the connection of the ambient space by ∇, and {e * 1 , e * 2 , • • • , e * n } denotes the pull back of {e 1 , e 2 , • • • , e n } via the Gauss map.Differentiating ψ with respect to e 1 twice we get, where the first inequality comes from the locally strict convexity assumption on ψ −1/k , i.e., for any spacelike vector Here c 0 > 0 is some constant depending on the defining domain, and the Euclidean norm and Minkowski norm respectively.At the point P 0 , in view of (3.8) and the assumption that ψ x n+1 0 , we derive Here, in the last inequality we have assumed λ 1 = λ 1 (|ψ| C 2 ) > 0 is large at P 0 .On the other hand, note that the functional F is concave and homogenous of degree one.Therefore, Let N, λ 1 be sufficiently large, then we obtain a contradiction.This completes the proof of Lemma 9.
Notice that this is the only place we need to use the locally strict convexity assumption of ψ −1/k in Theorem 1.It's also clear that the above proof can be easily modified to the case when ψ −1/k is convex with respect to X and the corresponding ψ(x, u(x), Du) does not depend on |x| (see the second inequality in (3.21)), as stated in the Remark 2. Therefore, (3.5) is solvable when either ψ −1/k is locally strictly convex with respect to X or ψ −1/k is convex with respect to X and ψ(x, u(x), Du(x)) does not depend on |x|.

Dirichilet problem for
n}, we will consider the following Dirichlet problem: Note that since ū is strictly convex, Ω n is strictly convex.It's easy to see that if u is a solution of (3.23), then ū u ū.Therefore, in order to find a k-convex solution u for (3.23), we only need to study the C 1 and C 2 estimates of u.

C 1 estimate for equation (3.23).
Lemma 10.Let u be a solution of (3.23), then |Du| < C < 1.Here C is a constant depending on |D ū|Ω n and ψ.
and consider the test function φ = ln V + Ku, where K > 0 to be determined.If φ achieves its maximum at an interior point P 0 ∈ M u , then at this point, we may choose a normal coordinate This leads to a contradiction.3.2.2.C 2 boundary estimates for equation (3.23).Now, we will establish the C 2 boundary estimate.For our convenience, we will consider the solvability of the following Dirichlet problem: where Ω is strictly convex.We will follow the idea of [10].
Differentiating equation (3.25) with respect to t, then evaluate it at t = 1 we obtain Infinitesmal rotation in Minkowski space.Keeping the coordinates To the first order in θ the image of (x, u(x)) under such rotation is Therefore, to the first order in θ the image of Denote this image as a graph function then we have Notice that dv dθ θ=0 = x n − u n u, we obtain Thus, we conclude that Lemma 11.Let u be a solution of (3.24), then |D 2 u| < C on ∂Ω.Here C is a constant depending on Ω and ψ.
Proof.For any p ∈ ∂Ω, we suppose p is the origin and that the x n − axis is the interior normal of ∂Ω at p. We may also assume the boundary near the origin p is represented by where λ α > 0, 1 α n − 1 are the principal curvatures of ∂Ω at the origin.Let where a > 0 depends on the principal curvatures of ∂Ω.Notice that u is a spacelike function, we suppose |Du| θ 0 in Ω for some θ 0 ∈ (0, 1).Then we have 0 −u θ 0 β in Ω β .Therefore, on {x n = β} we obtain (3.31) with C being independent of β and δ.Moreover, (3.32) where δ and β are small positive constants.Now choose A = A(δ) > 0 large such that and LAh > |LT α u| in Ω β .By the maximum principle we conclude that On the other hand we have h(0) = T α u(0) = 0. Therefore, Since p ∈ ∂Ω is arbitrary, we get Applying Lemma 1.2 in [8] we obtain This completes the proof of this Lemma.
Lemma 12. Let u be a solution of (3.24) with ψ = ψ(X, ν), then Here C is a constant depending on |Du| Ω and ψ.
Proof.we consider the following test function whose form first appeared in [19], Here, the function P is defined by P = l e κ l and N is a sufficiently large constant to be determined later.We may assume that the maximum of φ is achieved at some point P 0 ∈ M u , where u is the solution of (3.24).Suppose Differentiating the function φ twice at P 0 , we have (3.33)φ i = P i P log P + N h ii u i = 0, and At P 0 , differentiating the equation (1.2) twice yields, where C is some uniform constant only depending on ψ.Note that Inserting (3.36) and (3.37) into (3.34),we obtain By (3.33) and (3.35), we have 1 Now, for any constant K > 1, we denote Combinning with (3.38), we get Claim 1.For any given 0 < ε < 1 2 , we let α = 1−2ε 1+ε .There exists a positive constant δ < 1 2 such that, for any |κ i | δκ 1 , 1 i n, if the constant K and the maximum principal curvature κ 1 both are sufficiently large, we have Applying Lemma 6 in [28], we can see that when K is chosen to be sufficiently large, then A i 0. By the Cauchy-Schwarz inequality, we have Thus, Let ε be equal to the ε T in Lemma 12 of [28].Then we know there exists a positive constant δ < ε such that, when (3.42) On the other hand, we have A straightforward calculation shows that when κ 1 is very large the following inequalities hold: Moreover, it is easy to see that By the Taylor expansion, we have Combining the previous four formulae with (3.44), we obtain when κ 1 is sufficiently large and |κ i | < δκ 1 , Therefore, Claim 1 is proved.Now, recall Section 4 of [28] and the proof of Theorem 14 in [29], we know the following claim is true.
).For any index 1 i n, if the positive constant K and the maximum principal curvature κ 1 both are sufficiently large, we have By Claim 1 and Claim 2, (3.39) becomes (3.47) 0 Here, the constant δ is the constant chosen in Claim 1. Choose N > 0 such that σ 11 k κ 2 1 (−N ν, E − 1) − Cκ 1 > 0, we get a contradiction.Therefore, our desired estimate follows immediately.By Lemma 10, Lemma 11, and Lemma 12, we conclude that when k = n−1, n−2, the Dirichlet problem (3.23) admits a k-convex solution.

The local estimates
We will devote this section to establishing the local C 1 and C 2 estimates for the solution u of (1.3).

4.1.
Local C 1 estimates.In this subsection, we will prove the local C 1 estimate.We will split it into two cases.In the first case, we will assume u is a convex solution of (1.2); in the second case, we will assume u is a k-convex solution of (1.5).Note that in both cases our results hold for 1 k n.
For strictly convex, spacelike hypersurfaces, Bayard-Schnürer [7] proved the following local gradient estimate lemma.Lemma 13. (Lemma 5.1 in [7]) Let Ω ⊂ R n be a bounded open set.Let u, ū, Ψ : Ω → R n be strictly spacelike.Assume that u is strictly convex and u < ū in Ω.Also assume that near ∂Ω, we have Ψ > ū.Consider the set, where u > Ψ.For every x in this set, we have the following gradient estimate for u : 1 For k-convex, spacelike hypersurfaces, Bayard [5] proved a similar result when k = 2.In the following, we will extend it to all k.Our argument is a modification of Bayards' in [5].We would also like to mention that the basic idea of this argument had appeared in Chow-Wang [12].
and u ū in Ω.Also assume that near ∂Ω, we have Ψ > ū.Consider the set, where u > Ψ.For every x in this set, we have the following gradient estimate for u : Here, N = N (n, k) is a uniform constant only depending on n, k, and Proof.Consider the test function: where N is a large undetermined constant.Assume the function φ achieves its maximum at P. We may choose a local normal coordinate {τ 1 , • • • , τ n } such that at P, h ij = κ i δ ij .Differentiating φ twice at P, we have, Contracting with σ ii k , we get Without loss of generality, we may assume that at P where ∇ is the Levi-Civita connection on M u .By (4.1), we have We may also assume |∇u(P )| is so large that | Ψ 1 u 1 | < 1 2 .Then at P we can see, Thus, if N is sufficiently large, κ 1 is negative and its norm is large.Using the inequality (26) in Lin-Trudinger [23], we obtain where η is a uniform constant only depending on n, k.Therefore, By (4.3), we get Inserting (1.2) and (4.4) into (4.2) yields, Notice that Combing (4.5) with (4.6), we get Notice that when κ 1 < 0, we have Moreover, − ν, E = 1 + |∇u| 2 .Let N be sufficiently large in (4.7), we obtain the desired estimate.
4.2.The Pogorelov type local C 2 estimates.Recall that in [35] (see Lemma 24), we proved the Pogorelove type local C 2 estimate for strictly convex, spacelike, constant σ k curvature hypersurfaces.With small modifications, we can show Lemma 15.Let u r * be the solution of (3.5) and u r be the Legendre transform of u r * .For any given s > 2C 0 +1, where C 0 > min ū is an arbitrary constant, let r s > 0 be a positive number such that when r > r s , u r | ∂Ωr > s, where Ω r = Du r * (B r ).Let κ max (x) be the largest principal curvature of M u r at x, where Then, for r > r s we have Here, C depends on the local C 1 estimates of u r and s.
In the rest of this subsection, we will establish the Pogorelov type local C 2 estimates for the k-convex solution of equation (1.2), where k = n−1 (n 3), n−2 (n 5).
Lemma 16.Let u n be the k-convex solution of (3.23) with ψ = ψ(X, ν), where Here, C depends on the local C 1 estimates of u m and s.
Proof.In this proof, for our convenience when there is no confusion, we will drop the superscript on u m .Now, on Ω m , we consider the following test function whose form first appeared in [19], Here the function P is defined by and β, N are constants to be determined later.
Let U s = {x ∈ R n |u(x) < s}, we may assume that the maximum of φ is achieved at Differentiating the function φ twice at P 0 , we get (4.9) At P 0 , differentiating the equation (1.2) twice yields, where C is some uniform constant.Note that Inserting (4.12) and (4.13) into (4.10),we obtain From (4.9) and (4.11), we deduce For any constant K > 1, denote Note that Therefore, (4.14) becomes Following the same argument as the one in the proof of Lemma 12, from (4.15) we obtain, (4.16) 0 Here, the constant δ is the same constant as the one chosen in Claim 1 of Lemma 12.Moreover, by (4.9), we have Choose β > 0 such that αβ > 2, then (4.16) implies , we let i 0 denote the index of the maximum value element of the set Then, we obtain which also implies our desired estimate.

The prescribed curvature problem
We will prove Theorem 1 and 5 in this section.Let's consider the proof of Theorem 1 first.Recall that in Subsection 3.1, we have solved the approximate Dirichlet problem (3.5) on B r , for r < 1.We will denote the strictly convex solution of (3.5) by u r * .We further denote the Legendre transform of (B r , u r * ) to be (Ω r , u r ), where Ω r = Du r * (B r ) is the domain of u r .By Lemma 19 and 20 in [35] we have ū u r ū, (5.1) in Ω r .
In the following, we will denote Ωr = D ū * (B r ) to be the domain of ūr := ū| Ωr .It is not difficult to see that these domains are increasing, namely, Ωr ⊂ Ωs , for r < s.
Moreover, by the choice of ū in Subsection 3.1, we have ū| ∂ Ωr → +∞, as r → 1.Thus, by the comparison principle, we have From this we can see that, as r → 1, u r | ∂Ωr → +∞.This in turn implies, for any compact set K ⊂ R n , there exists a constant c K = c(K) < 1 such that when r > c K , Ω r ⊃ K. Therefore, for any compact set K ⊂ R n , we can apply Lemma 13 and Lemma 15 to obtain uniform C 1 and C 2 bounds for u r in K.
More precisely, in order to obtain the local C 1 estimate, we introduce a new subsolution ū1 of (1.2), where ū1 satisfies By the strong maximum principle we have, when x ∈ R n ū1 (x) < ū(x).Thus, for any compact convex domain K, let 2δ = min We define a strict spacelike function Ψ = ū1 + δ.Denote K ′ = {x ∈ R n ; Ψ ū}.Since as |x| → ∞, ū1 − ū → 0, we know that K ′ is a compact set only depending on K. Applying Lemma 13, for any (Ω r , u r ), if K ′ ⊂ Ω r , we have the gradient estimate: Next, we want to show that for any given compact set K ⊂ R n , {|D 2 u r |} is uniformly bounded in K. Without loss of generality, let's consider any B R ⊂ R n .Let C 0 = max B R ū and s = 2C 0 + 1 in Lemma 15.Denote U s = {x ∈ R n ; ū(x) < s}, then by earlier discussion, it's easy to see that there exists r s > 0 such that when r > r s , Ω r ⊃ U s .Applying Lemma 15 we obtain when r > r Here C depends on the upper bound of on Ūs , which is independent of r.Using the classical regularity theorem and convergence theorem, we conclude that (Ω r , u r ) converges locally smoothly to an entire, smooth convex function u satisfying (1.2).In view of (5.1) and the asymptotic behavior of ū, ū, we know that as |x| → ∞, u → |x| + ϕ x |x| .Moreover, by Remark 2 we also know that u is strictly convex.Therefore, its Gauss map image is B 1 , i.e., Du(R n ) = B 1 .
Theorem 5 follows by replacing Lemma 13 and Lemma 15 in the proof of Theorem 1 with Lemma 14 and Lemma 16.

The radial downward translating soliton
In this section, we will study the radially symmetric downward translating soliton.Recall that we say M u is a downward translating soliton when its principal curvatures satisfy (6.1) where C > 1 is a constant.We want to point out that in this section and the next section, C is the fixed constant in (6.1).We also denote C = 1 − 1 C 2 as in Theorem 6.The following theorem is a generalization of Theorem 1 in [6].
Moreover, as |x| → ∞, u(x) has the following asymptotic expansion for some constant c 0 ∈ R. In particular, the radial solution u is unique up to the addition of a constant.
For radial solutions, we will reduce the equation (6.1) to an ODE.Let u = u(r) and y = ∂u ∂r , then a straightforward calculation yields, and (6.1) becomes By a small modification of the proof of Proposition 2.1 in [6], we obtain Proposition 18.Under the hypotheses of Theorem 17, there exists a solution y of (6.3), which is defined on [0, +∞) and smooth on (0, +∞), such that Moreover, as r → 0+, we have Since the proof is a small modification of the proof of Proposition 2.1 in [6], we skip it here.Now, let's study the asymptotic behavior of y.
Proposition 19.Let y be the solution of (6.3).Then as r → ∞, y has the following asymptotic expansion Proof.By Proposition 18 we may assume (6.4) Then we have, Differentiating (6.4), then substituting it into (6.3),we get By (6.5), (6.6) can be simplified as Thus, we obtain Applying Proposition 18 we can see that Here, we have used lim r→∞ z r = 0, which is a direct consequence of Proposition 18. Next Lemma is a generalization of Proposition A.2 in [6].
Lemma 20.Assume z : (0, +∞) → R is a positive solution of the equation where A, B : (0, ∞) → R are continuous functions such that Proof.In order to prove this Lemma, we only need to prove Claim 3. Assume z : (0, +∞) → R is a positive solution of the equation where Notice that lim r→+∞ w r = 0 and D(r) has a uniform positive lower bound.In the following, we want to find a positive upper bound for F (r).Using the expressions (6.8) of B(r), C(r), we obtain .
Therefore, we only need to show r(y − A(r)/C) is bounded as r → ∞.By (6.5), we have Combining (6.14) with the expression of y and (6.5), we can derive (6.15) From (6.14), (6.15), and Lemma 20 we conclude that r(y − A(r)/C) is uniformly bounded from above.Thus, F (r) has an uniform upper bound.Applying Proposition A.3 in [6], we obtain a uniform upper bound for w.This completes the proof.
It's not hard to see that Theorem 17 follows from Proposition 18 and Proposition 19.

The existence results
In this section we will prove Theorem 6.First, we want to prove the following existence Theorem.
7.1.Constructing barriers.We first construct the barrier functions of equation (1.10).Following the ideas of [31,32], we denote the radial solution of (1.10) by z k 0 (|x|), whose asymptotic expansion satisfies (6.2) with c 0 = 0. Let p i ( Cy) = Dϕ( Cy) + (−1) i+1 2M Cy, i = 1, 2 for any y ∈ S n−1 .Set, is a subsolution of (1.10) and is a supersolution of (1.10).Moreover, q k 1 (x) q k 2 (x) and when |x| → +∞, we have 7.2.The Dirichlet problem.First, let's solve equation (1.10) for the case when k = n.For any t > min R n q n 2 , we let ∂Ω t = {x ∈ R n |q n 1 (x) < t < q n 2 (x)}, and Ω t be a smooth, strictly convex domain in R n .Consider the following Dirichlet problem: By a small modification of [13], we know that there exists a unique solution u t of (7.2).Then, applying the local C 1 , C 2 estimates obtained in [7] we conclude that, there exists a subsequence {u t i } ∞ i=1 (t i → ∞ as i → ∞), that converges to an entire, strictly convex solution u of (1.10) for k = n.Moreover, it's easy to see that u(x) satisfies the desired asymptotic behavior as |x| → ∞.From now on, we will denote this solution by u n .We will also denote the Legendre transform of u n by u n * .
Next, we consider the case when k < n.We denote the legendre transform of z k 0 by (z k 0 ) * , that is, Using the asymptotic expansion of z 0 derived in Section 6, we know We denote its principal part: Since the proof is the same as the proof of Lemma 5.1 in [7], we skip it here.We now construct Ψ.Following the argument in Section 4 of [6], let It is clear that when |x| sufficiently large we have Ψ(x) > q 2 (x).On the other hand, for any compact set K ⊂ R n , we can always choose A 0 sufficiently large such that Ψ(x) < q 1 (x) in K. Applying Lemma 23 we obtain that for any K ⊂ R n and any strictly convex function q 1 (x) < u(x) < q 2 (x) satisfying (1.10), whose domain of definition contains K, there exists a local C 1 bound C K for u(x) in K that is only depending on K.
Using the idea of [35], we can prove the following Pogorelov type local C 2 estimate for translating solitons.
Lemma 24.Let u be the solution of (1.10) defined on Ω.For any given s > min R n u(x) + 1, suppose u| ∂Ω > s.Let κ max (x) be the largest principal curvature of M u = {(x, u(x))|x ∈ Ω} at x.Then, we have Here, C 1 only depends on the local C 1 estimate of u.More specifically, C 1 depends on the lower bound of C + ν, E .
Following the argument in Section 5, we complete the proof of Proposition 21. 7.4.Proof of Theorem 6.In this subsection, we will prove that the hypersurface M u constructed in Proposition 21 has bounded principal curvatures.This completes the proof of Theorem 6.For our convenience, in the following, we will drop the superscript k, and the updated configuration z k 0 now becomes z 0 .Suppose u is a strictly convex solution of (1.10) and u * is the Legendre transform of u.Then u * satisfies (7.5) We also denote the Legendre transform of z 0 by z * 0 , that is, , where τ = ∂z 0 ∂r .
Using the asymptotic expansion of z 0 derived in Section 6, we know We denote its principal part as Lemma 25.Let u * and z * 0 be defined as above.Then we have, lim Proof.We will use the auxiliary functions z i (x, y), i = 1, 2, constructed in Subsection 7.1.It's easy to see that z 1 (x, y) < u(x) < z 2 (x, y), for any x ∈ R n , y ∈ S n−1 .

Now we let
∂ξ i be the angular derivative.Similar to Section 10 in [30], we obtain following Lemmas.Proof.Notice that ∂|ξ| 2 = 0, we have the angular derivative of the right hand side of equation (7.5) is zero.Therefore, following the proof of Lemma 29 and 30 in [30], we have In view of (7.6) and the maximum principle, we obtain the desired estimates.
We further have Lemma 27.Let u * be the solution of equation (7.5).There is a positive constant b such that , the rest of the proof is same as the one of Lemma 5.3 in [21].
Proof.We modify the proof of Li [21].We first consider the lower bound.For any ξ ∈ S n−1 (r), using Lemma 28, we have u * ( ξ) = ū * 0 ( ξ), and u * (ξ) < ū * 0 (ξ) for ξ ∈ S n−1 (r) \ { ξ}.Thus, using ū * 0 is a supersolution, we get u * (ξ) < ū * 0 (ξ) in B r .Therefore, at ξ, we get u Using the asymptotic behavior of z 0 , we have We denote Therefore, by (7.9), we obtain for r being sufficiently close to C, which we may assume r > a 0 .For r < a 0 , without loss of generality, we can assume u 1. Therefore Thus, we obtain the uniform lower bound.For the upper bound.Applying a similar argument, for r being sufficiently close to C , which we will still assume r a 0 , we have We obtain a uniform upper bound.
Finally, we are ready to adapt the ideas in [30,21] to estimate the principal curvatures of M u .Proposition 31.Let u be the solution of equation (1.10).Then the hypersurface M u = {(x, u(x))| x ∈ R n } has bounded principal curvatures.
Proof.We will establish a Pogorelov type interior estimate.For any s > 0, consider where P m = j κ m j and m, N > 0 are constants to be determined later.Without loss of generality, we also assume u 1 in R n .It's easy to see that φ achieves its local maximum at an interior point of U s = {x ∈ R n | u(x) < s}, we will assume this point is x 0 .We can choose a local normal coordinate {τ Differentiating log φ at x 0 we get, (7.10)where K 0 = K 0 (n, k, C) > 0 is a constant depending on n, k and C. Recall that in Minkowski space we have Thus, (7.13) Combining (7.13) with (7.11) we obtain (7. Here σ pp,qq k h ppi h qqi , for some constant K > 1, and By Lemma 8, Lemma 9, and Corollary 10 in [22] we can assume the following claim holds.
Claim 4.There exists two small positive constants δ and η < 1.If κ k δκ 1 , we have where m > 0 is sufficiently large.
Since x is arbitrary, we finish proving Proposition 31.
Theorem 6 follows from Proposition 21 and Proposition 31 immediately.

√ 1 −
|Du| 2 is the upward unit normal lying on the hyperboloid H n , and κ[M u ] = Research of the first author is supported by NSFC Grant No. 11871243 and the second author is supported by NSFC Grant No.11871161 and 11771103.

Lemma 26 .
Let u * be the solution of equation (7.5).Then, |∂u * | are bounded above by a constant depends on |ϕ| C 1 and ∂ 2 u * are bounded above by a constant depends on |ϕ| C 2 .
a are constants depending on ξ, and a > 0, a C2 − | ξ| 2 < C 1 , where C 1 is a positive constant only depending on |ϕ| C 2 .Using Lemma 28 and Lemma 29 we can show Lemma 30.Let u be the solution of equation (1.10) and u * be the Legendre transform of u.There are positive constants d 2 > d 1 such that (7.8) 0 < d 1 u( C2 − |Du| 2 ) d 2 .
and C is a positive constant only depending on U and ψ.
E .