Asymptotic Dirichlet problems in warped products

We study the asymptotic Dirichlet problem for Killing graphs with prescribed mean curvature $H$ in warped product manifolds $M\times_\varrho \mathbb{R}$. In the first part of the paper, we prove the existence of Killing graphs with prescribed boundary on geodesic balls under suitable assumptions on $H$ and the mean curvature of the Killing cylinders over geodesic spheres. In the process we obtain a uniform interior gradient estimate improving previous results by Dajczer and de Lira. In the second part we solve the asymptotic Dirichlet problem in a large class of manifolds whose sectional curvatures are allowed to go to $0$ or to $-\infty$ provided that $H$satisfies certain bounds with respect to the sectional curvatures of $M$ and the norm of the Killing vector field. Finally we obtain non-existence results if the prescribed mean curvature function $H$ grows too fast.


Introduction
Let N be a Riemannian manifold of the form N = M × R, where M is a complete n-dimensional Riemannian manifold and ∈ C ∞ (M ) is a smooth (warping) function. This means that the Riemannian metricḡ in N is of the form g = ( • π 1 ) 2 π * 2 dt 2 + π * 1 g, (1.1) where g denotes the Riemannian metric in M whereas t is the natural coordinate in R and π 1 : M × R → M and π 2 : M × R → R are the standard projections. It follows that the coordinate vector field X = ∂ t is a Killing field and that = |X| on M . Since the norm of X is preserved along its flow lines, we may extend to a smooth function = |X| ∈ C ∞ (N ). From now on, we suppose that > 0 on M . In this paper we study Killing graphs with prescribed mean curvature. Such graphs were introduced in [10], where the Dirichlet problem for prescribed mean curvature with C 2,α boundary values was solved in bounded domains Ω ⊂ M under hypothesis involving data on Ω and the Ricci curvature of the ambient space N . Recall that given a domain Ω ⊂ M , the Killing graph of a C 2 function u : Ω → R is the hypersurface given by In other words, where Ψ : Ω × R → N is the flow generated by X. In [7] the Dirichlet problem was solved with merely continuous boundary data. Furthermore, the authors proved the existence and uniqueness of so-called radial graphs in the hyperbolic space H n+1 with prescribed mean curvature and asymptotic boundary data at infinity thus solving the asymptotic Dirichlet problem in H n × cosh r R. One of our goals in the current paper is to solve the asymptotic Dirichlet problem with prescribed mean curvature in a large class of negatively curved manifolds. On the other hand, it is an interesting question under which conditions on a Riemannian manifold M every entire constant mean curvature graph over M is a slice, i.e. a graph of a constant function. The first such result is the celebrated theorem due to Bombieri, De Giorgi, and Miranda [3] that an entire minimal positive graph over R n is a totally geodesic slice. Their result was extended by Rosenberg, Schulze, and Spruck [16] to a complete Riemannian manifold M with nonnegative Ricci curvature and the sectional curvature bounded from below by a negative constant. Ding, Jost, and Xin considered in [11] complete, noncompact Riemannian manifolds with nonnegative Ricci curvature, Euclidean volume growth, and quadratic decay of the curvature tensor. They proved that an entire minimal graph over such a manifold M must be a slice if its height function has at most linear growth on one side unless M is isometric to Euclidean space. In the recent paper [4] Casteras, Heinonen, and Holopainen showed that a minimal positive graph over a complete Riemannian manifold with asymptotically nonnegative sectional curvature and only one end is a slice if its height function has at most linear growth. Entire Killing graphs in M × R with constant mean curvature were studied in [8] and [9]. In particular, it was shown in [8] that a bounded entire Killing graph of constant mean curvature must be a slice if Ric M ≥ 0, K M ≥ −K 0 for some K 0 ≥ 0, and if ≥ 0 > 0, with || || C 2 (M ) < ∞.
Our current paper is inspired by the above mentioned research [10], [7], [8], and [9] on Killing graphs with prescribed mean curvature as well as by the recent paper [5]. In the latter, the asymptotic Dirichlet problem for f -minimal graphs in Cartan-Hadamard manifolds M has been studied. Recall that f -minimal hypersurfaces are natural generalizations of self-shrinkers which play a crucial role in the study of mean curvature flow. Moreover, they are minimal hypersurfaces of weighted manifolds M f = M, g, e −f d vol M , where (M, g) is a complete Riemannian manifold with the Riemannian volume element d vol M .
Returning to the Killing graph Σ u of a function u, we note that the induced metric in Σ u has components where g ij are local components of the metric g. The induced volume element in Σ u (or equivalently, on the domain Ω ⊂ M ) is given by We consider the constrained area functional and H is a smooth function on Ω. Given an arbitrary compactly supported function v ∈ C ∞ 0 (Ω) we have the first variation formula where W = −2 + |∇u| 2 and the differential operators ∇ and div are taken with respect to the metric g in M . Then the Euler-Lagrange equation of this functional is div ∇u W + ∇ log , ∇u W = nH (1. 3) and H(x) is the mean curvature of the graph Σ u ⊂ M × R at (x, u(x)). The equation (1.3) can be rewritten as where the weighted divergence operator corresponding to a smooth density function Note that this is the divergence-form operator that fits well with the weighted measure dM in the sense that a suitable version of the divergence theorem is still valid in this context. Reasoning another way around, since Σ is oriented by the normal vector field where∇ is the Riemannian connection in N , we can interpret as a weighted mean curvature of the submanifold Σ u in the Riemannian product M × R in the sense that the Euler-Lagrange PDE may be rewritten as More generally, if f is an arbitrary density in M we consider a weighted area functional of the form In this case, the Euler-Lagrange equation is As before, this equation may be rewritten either in terms of a modified weighted divergence or as a prescribed weighted mean curvature equation For the time being, we restrict ourselves to the case where f = 0. Intrinsically, given a hypersurface Σ ⊂ N and denoting u = t| Σ , the parametric counterpart of where T denotes tangential projection onto T Σ. Hence we obtain from where the formula (1.5) above follows.
In particular, minimal graphs in N = M × R have height function that satisfies the weighted harmonic equation This may be considered as a PDE in Ω if we replace the metric g by the induced metric with components given by (1.2). Denoting (1.7)

Main results
The existence of Killing graphs with prescribed mean curvature H over bounded domains Ω ⊂ M with continuous boundary data on ∂Ω was established in [7, Theorem 2] under suitable conditions on the Ricci curvature on Ω, the mean curvature function H, and on the mean curvature of the Killing cylinder over ∂Ω; see also [10].
In this paper we mainly focus on the setting where M is a Cartan-Hadamard manifold with sectional curvatures controlled from above and below by some radial functions. We prove quantitative a priori height and gradient estimates for solutions of (1.3) on geodesic balls Ω = B(o, k) ⊂ M under natural conditions on the prescribed mean curvature function in terms of sectional curvatures K M and the warping function . These estimates allow us to use the continuity method (the Leray-Schauder method) and hence are enough to guarantee the existence of solutions to the following Dirichlet problem where ϕ ∈ C(∂Ω). We formulate the (local) existence result in geodesic balls on Cartan-Hadamard manifolds.
Theorem 2.1. Let M be a Cartan-Hadamard manifold, Ω = B(o, k) ⊂ M , and ϕ ∈ C(∂Ω). Suppose that the prescribed mean curvature function and H k−d is the mean curvature of the Killing cylinder C k−d over the geodesic sphere ∂B(o, k − d). Then there exists a unique solution u ∈ C 2,α (Ω) ∩ C(Ω) to (2.1).
Above and in what follows we denote by r(x) = d(x, o) the distance from x to a fixed point o ∈ M . We notice that the mean curvature of the Killing cylinder C r over a geodesic sphere ∂B(o, r) is given by and therefore can be estimated from below in terms of a suitable model manifold is a rotationally symmetric Cartan-Hadamard manifold with radial sectional curvatures equal to −a 2 (r) and + : M → (0, ∞) is a positive rotationally symmetric C 1 function such that 1 ∇ , ∇r = ∂ r ≥ ∂ r + + .
As mentioned above the proofs of Theorem 2.1 and Corollary 2.2 for boundary data ϕ ∈ C 2,α (∂Ω) follow from the well-known continuity method once the a priori height and gradient estimates are at our disposal. The case of a continuous boundary values ϕ ∈ C(∂Ω) can be treated as in [7]; see also [5].
Our main object in this paper is the asymptotic Dirichlet problem for Killing graphs with prescribed mean curvature and behaviour at infinity. To solve the problem, we extend the given boundary value function ϕ ∈ C(∂ ∞ M ) to a continuous function ϕ ∈ C(M ); see Section 5 for the notation. Then we apply Corollary 2.2 for an exhaustion Ω k = B(o, k), k ∈ N, of M to obtain a sequence of solutions u k with boundary values u k |∂Ω k = ϕ. Under a suitable bound on |H| in terms of a comparison manifold M −a 2 (r) × + R we obtain a global height estimate and, consequently together with Schauder estimates, the sequence is uniformly bounded in the C 2,α -norm. Hence there exists a subsequence that converges in the C 2,α -norm to a global solution u to the equation Finally, under suitable curvature upper and lower bounds as well as conditions on |H| we are able to construct (local) barriers at infinity and prove that the solution u extends continuously to ∂ ∞ M and attains the given boundary values ϕ there.
The following two solvability theorems will be proven in Section 6.

A priori height and gradient estimates
Throughout this section we denote by Ω k = B(o, k) the geodesic ball centered at a given point o ∈ M with radius k ∈ N, and by d(·) = dist(·, ∂Ω k ) the distance function to the boundary of Ω k .
3.1. Height estimate. Fix k ∈ N and suppose that u k ∈ C 2 (Ω k ) is a solution of the Dirichlet problem (2.1). We aim to show that the function where h will be determined later, is an upper barrier for the solution u k . It suffices to show (see [17, p. 795] or [10, pp. 239-240]) that v k is a barrier in an open neighbourhood of ∂Ω k in which the points can be joined to ∂Ω k by unique geodesics. In this neighbourhood the distance function d has the same regularity as ∂Ω k and therefore the derivatives of d in the following computations are well-defined. Since X is Killing field, we have where κ is the principal curvature of the Killing cylinder C k−d over the geodesic sphere ∂B(o, k − d). This implies that where ∂ d denotes the derivative to the direction ∇d. However, where H k−d is the mean curvature of the cylinder C k−d , and we have Hence it follows that Suppose that the principal curvature of the Killing cylinder C k−d satisfies where 0 is a smooth positive increasing function on [0, ∞). We note already at this point that, in the case of Cartan-Hadamard manifolds, ∇d = −∇r and this agrees with the assumption (4.12). Then define the function h as for some constant C > 0 to be fixed later. Now, since h > 0, we have Assuming that |H| < H k−d inΩ k and choosing the constant C as and hence v k is an upper barrier for u k .
Similarly we see that the function is a lower barrier for u k and together these barriers give the following height estimate. inΩ k and that u k is a solution to the Dirichlet problem (2.1). Then there exists a 3.2. Boundary gradient estimate. For given ε > 0 we define an annulus In order to obtain a boundary gradient estimate, we aim to show that a function of the form is an upper barrier in the set U k (ε) for a fixed ε ∈ (0, 1/2) chosen so that d is smooth in U k (ε). Here we denote by ψ the extension of the boundary data that is constant along geodesics issuing perpendicularly from ∂Ω k , i.e. ψ(exp y t∇d(y)) = ϕ(y), where y ∈ ∂Ω k and ∇d(y) is the unit inward normal to ∂Ω k at y. From (1.7) we have that with ∇d, ∇ψ = 0, it follows that ∆w = g ∆d + g + ∆ψ and ∇ ∇w ∇w, ∇w = g 2 g − g ∇ ∇ψ ∇d, ∇ψ + ∇ ∇ψ ∇ψ, ∇ψ .
Using the expression (3.5) we obtain that and combining with the previous reasoning, this results to We note that and, on the other hand, Moreover, the matrix (σ ij ) has eigenvalues 1/( 2 W 3 ) and 1/W which can be estimated as When ≥ 1, this is trivial, and when < 1 we can choose the constant K in the definition (3.7) of g such that this holds. Therefore we are able to estimate where (3.8) Note that this choice yields g ≥ u k on the "inner" boundary {x ∈ Ω k : d(x) = ε} of U k (ε). Then, for K large, we have We also have which implies that Hence we obtain However, Combining these with the fact that W ≥ K 2 , we obtain the estimate Finally observe that (3.10) Taking (3.6), (3.8), (3.9) and (3.10) into account, we can choose . This suffices for the following boundary gradient estimate.

3.3.
Interior gradient estimate. In this subsection we prove a quantitative interior gradient estimate that is interesting on its own. The proof is based on the technique due to Korevaar and Simon [13], and further developed by Wang [18]. We will perform the computations in a coordinate free way. Let u be a (C 3 -smooth) positive solution of the equation (1.3) in a ball B(p, R) ⊂ M. Suppose that sectional curvatures in B(p, R) are bounded from below by −K 2 0 for some constant K 0 = K 0 (p, R) ≥ 0. We consider a nonnegative and smooth function η with η = 0 in M \ B(p, R) and define a function χ in B(p, R) of the form where the functions η, γ and ψ will be specified later. Suppose that χ attains its maximum at x 0 ∈ B(p, R), and without loss of generality, that η(x 0 ) = 0. Then at x 0 and therefore is non-positive at x 0 . Applying the Ricci identities for the Hessian of u we have and this yields On the other hand, denoting and differentiating both sides in (1.7) we have (3.14) Contracting (3.14) with u k , we get Using the previous identity, (3.13) and noticing that Notice that (3.13) yields to Plugging this into the previous estimate, we get Suppose that |∇u|(x 0 ) > 1. Otherwise we are done. Hence, following [18], we set By modifying the argument in [16, Proof of Theorem 4.1, Case 2] we may assume that the maximum point x 0 is not in the cut-locus C(p) of p. Then we choose η as As in [18], we set where M > 0 is a constant to be fixed later. Then γ = 0 and hence

14JEAN-BAPTISTE CASTERAS, ESKO HEINONEN, ILKKA HOLOPAINEN, AND JORGE LIRA
Let L = L(p, R) ≥ 0 be chosen in such a way that in B(p, R). Then we obtain Set M = maxB (p,R) u. We consider first the case Then we have On the other hand, when which implies that Hence at x 0 (3.22) Since η(p) = 1 and γ(p) ≥ 1 we conclude that We have proven the following quantitative gradient estimate. Here we denote by R B the Riemannian curvature tensor in a set B. If the gradient of u is continuous up to the boundary of Ω and Ω is bounded, we obtain the following quantitative global estimate. such that |∇u(p)| ≤ C for every p ∈Ω.
Proof. Let p ∈ Ω and R = diam(Ω). Define inΩ ∩ B(p, R) a function where η, γ, and ψ are as in the previous proof. If χ attains its maximum in an interior point x 0 ∈ B(p, R) ∩ Ω, the proof of Lemma 3.3 applies and we have a desired upper bound. Otherwise, χ attains its maximum at x 0 ∈ ∂Ω, but then |∇u(x 0 )| ≤ max ∂Ω |∇u| and again we are done.
We remark that a global gradient estimate for bounded Killing graphs follows immediately from (3.23), (3.

Global barriers
In this section we present two methods to obtain global (upper and lower) barriers for solutions to (2.1).
In the case when H is constant along flow lines of X, that is, when H is a function in M , there is a conservation law (a flux formula) corresponding to the invariance of A H with respect to the flow generated by X. This flux formula for graphs is stated as where Γ = ∂Ω and ν is the outward unit normal vector field along Γ ⊂ M . Suppose for a while that M is a model manifold with respect to a fixed pole o ∈ M and that = |X| is a radial function. In terms of polar coordinates (r, ϑ) ∈ R + × S n−1 centered at o the metric in M is of the form where dϑ 2 stands for the canonical metric in S n−1 . Suppose that H and u are also radial functions. Applying (4.1) to Ω = B(o, r), the geodesic ball centered at o with radius r, we obtain u (r) −2 (r) + u 2 (r) (r)ξ n−1 (r) = This is a first integral of (1.3) in this rotationally invariant setting. Indeed, taking derivatives on both sides of (4.2) with respect to r we get On the other hand in this particular setting (1.3) becomes It is convenient to write (4.2) in a "quadrature" form as follows where For instance, in the case when H is constant we have to impose a condition such as This discussion motivates us to define in the general case a function of the form for some nonnegative functions + (r(x)), ξ + (r(x)) and H(r(x)) to be chosen later.
Plugging u + (x) = u + (r(x)) into the differential operator Hence we obtain In order to prove that u + is indeed an upper barrier we next check that (4.9) Note that u + ≤ 0. We observe that ∂ ∂r if and only if But now integrating (4.8) we get we see that (4.10) holds. Therefore we are left to show that The conditions (4.4) and (4.5) in our mind, we choose H as n H(r) = (1 − ε) + (r) + (r) with some ε ∈ (0, 1). Note that then and we see that with this choice the denominator in the definition of u + stays bounded from 0. Moreover, we have and therefore u + is well defined, positive and decreasing function if Now we can compute −2 , and for example, taking ε = 1 − √ 2/2 we have −2 .
For the prescribed mean curvature we obtain the bound All together, we have obtained the following.
In a rotationally symmetric case if = + (r) (and (4.11) holds), we see that the bound for the mean curvature is

Example: hyperbolic space.
We consider the warped model of H n+1 given by H n × cosh r R, where r is a radial coordinate in H n defined with respect to a fixed reference point o ∈ H n . Then the hyperbolic metric is expressed as where dϑ 2 stands for the standard metric in S n−1 ⊂ T o H n . The flow of the Killing field X = ∂ t is given by the hyperbolic translations generated by a geodesic γ orthogonal to H n through o. Since (r) = cosh r and ξ(r) = sinh r in this case, we obtain lim r→∞ (r)ξ n−1 (r) r 0 (τ )ξ n−1 (τ ) dτ = lim r→∞ sinh n r + (n − 1) cosh 2 r sinh n−2 r cosh r sinh n−1 r = lim r→∞ sinh r cosh r + (n − 1) cosh r sinh r ≥ n.
Therefore a natural bound to the mean curvature function according (4.4) is that is, below the mean curvature of horospheres. We also have for |H| < 1 Therefore we have u 2 (r) ≤ 1.

4.2.
Global barrier V . In this subsection we construct a global barrier using an idea of Mastrolia, Monticelli, and Punzo [14]; see also [5]. Recall that + : [0, ∞) → (0, ∞) is an increasing smooth function satisfying + (0) = (o) and for all x ∈ M . Then we have an estimate for the weighted Laplacian of the distance function r. Let a 0 be a positive function such that We define where D is the constant given by (4.22). Denoting V (r) = V (r(x)), we observe that .
Since V (r) < 0, the limit exists. Furthermore, D ≤ 0 (see [14, (4.5)]) and finite by (4.20) and therefore V is well defined. Next we write and aim to prove that Q[V ] ≤ 0. First we estimate the weighted Laplacian of V by using (4.19) and thus the first term of (4.23) can be estimated as Then, for the last term of (4.23) we have .
Finally, if the prescribed mean curvature function satisfies Hence we have proved the following uniform height estimate.
with some positive functions + and a 0 satifying (4.18) and (4.20), respectively. Then Next we discuss possible choices of the functions + and a 0 and their influence on the bound of |H|. Notice that the right hand side of (4.24) can be written as −2 Hence if we can choose the comparison manifold M −a 2 (r) × + R and a 0 such that V (r) → −∞ and −2 asymptotically as r → ∞.

Barrier at infinity
In this section we assume that M is a Cartan-Hadamard manifold of dimension n ≥ 2, ∂ ∞ M is the asymptotic boundary of M , andM = M ∪ ∂ ∞ M the compactification of M in the cone topology. Recall that the asymptotic boundary is defined as the set of all equivalence classes of unit speed geodesic rays in M ; two such rays γ 1 and γ 2 are equivalent if sup t≥0 d γ 1 (t), γ 2 (t) < ∞. The equivalence class of γ is denoted by γ(∞). Throughout this section, we assume that the sectional curvatures of M are bounded from below and above by are assumed to be smooth such that a(t) = 0 and b(t) is constant for t ∈ [0, T 0 ] for some T 0 > 0, and that assumptions (A1)-(A7) hold. These curvature bounds are needed to control the first two derivatives of "barrier" functions that we will construct in the next subsection. We assume that function b in (5.1) is monotonic and that there exist positive constants T 1 ≥ T 0 , C 1 , C 2 , C 3 , and Q ∈ (0, 1) such that for all t ≥ T 1 and for all t ≥ 0. In addition, we assume that and that there exists a constant C 4 > 0 such that 3) for the definition of f a . We recall from [12] the following two examples of functions a and b.
Example 5.2. Let k > 0 and ε > 0 be constants and define a(t) = k for all t ≥ 0. Define b(t) = t −1−ε/2 e kt for t ≥ R 0 = r 0 + 1, where r 0 > 0 is so large that t → t −1−ε/2 e kt is increasing and greater than k for all t ≥ r 0 . Extend b to an increasing smooth function b : [0, ∞) → [k, ∞) that is constant in some neighborhood of 0. We can choose C 1 > 0 in (A1) as large as we wish. Then a and b satisfy (A1)-(A7) with constants C 1 , T 1 = C 1 /k, some C 2 > 0, some C 3 > 0, Q = 1/2, and any C 4 ∈ (0, ε/2). Thus P(ϕ) is an integral average of ϕ with respect to χ similar to that in [1, p. 436] except that here the function b is taken into account explicitly. If ϕ ∈ C(M ), we extend P(ϕ) : M → R to a functionM → R by setting P(ϕ)(x) = ϕ(x) whenever x ∈ M (∞). Then the extended function P(ϕ) is C ∞ -smooth in M and continuous inM ; see [12,Lemma 3.13]. In particular, applying P to the functionh yields an appropriate smooth extension Furthermore, we denote by Hess x u the norm of the Hessian of a smooth function u at x, that is |Hess u(X, X)|.
The following lemma gives the desired estimates for derivatives of h. We refer to [12] for the proofs of these estimates; see also [6]. ,

5)
for all x ∈ 3Ω \ B(o, R 1 ). In addition, Let A > 0 be a fixed constant, and R 3 > 0 and δ > 0 constants that will be determined later, and h the function defined in (5.4). We will show that a function is a supersolution In the proof we shall use the following estimates obtained in [12]: Lemma 3.17] There exist constants R 2 = R 2 (C) and c 2 = c 2 (C) with the following property. If δ ∈ (0, 1), then Let us denote where C 1 and C 4 are constants defined in (A1) and (A7), respectively.

Solving the asymptotic Dirichlet problem
In this section we solve the asymptotic Dirichlet problem (6.1) on a Cartan-Hadamard manifold M with given boundary data ϕ ∈ C(∂ ∞ M ). If the ambient manifold N = M × R is a Cartan-Hadamard manifold, too, we will interpret the graph S = {(x, u(x)) : x ∈ M } of the solution u as a Killing graph with prescribed mean curvature H and continuous boundary values at infinity. We recall from [2, 7.7] that N is a Cartan-Hadamard manifold if and only if the warping function is convex. In that case we may consider ∂ ∞ M as a subset of ∂ ∞ N in the sense that a representative γ of a boundary point x 0 ∈ ∂ ∞ M is also a representative of a point x 0 ∈ ∂ ∞ N since M is a totally geodesic submanifold of N . Given ϕ ∈ C(∂ ∞ M ) we define its Killing graph on ∂ ∞ N as follows. For x ∈ ∂ ∞ M , take the (totally geodesic) leaf where Ψ is the flow generated by X. Let γ x be any geodesic on M representing x. Thenγ x : t → Ψ(γ x (t), ϕ(x)) is a geodesic on M ϕ(x) and also on N since Ψ(·, ϕ(x)) is an isometry. Henceγ x defines a point in ∂ ∞ N which we, by abusing the notation, denote by (x, ϕ(x)). Using this notation, we call the set the Killing graph of ϕ. Note that, in general, ∂ ∞ N has no canonical smooth structure.
We formulate our global existence results in the following two theorems depending on the assumption on the prescribed mean curvature function H. Proof. The proofs of Theorems 6.2 and 6.3 are similar. The only difference is to use the global barrier u + in Lemma 4.1 for 6.2 relative to V in Lemma 4.3 for 6.3. Extend the boundary data function ϕ ∈ C(∂ ∞ M ) to a function ϕ ∈ C(M ) and let B k = B(o, k), k ∈ N be an exhaustion of M . Then by Corollary 2.2 there exist solutions u k ∈ C 2,α (B k ) ∩ C(B k ) to the Dirichlet problem By Lemma 4.1, we see that the sequence (u k ) is uniformly bounded. Applying the gradient estimates in compact domains and then the diagonal argument, we obtain a subsequence converging locally uniformly with respect to C 2 -norm to a solution u. Next we show that u extends continuously to the boundary ∂ ∞ M with u|∂ ∞ M = ϕ. Let x 0 ∈ ∂ ∞ M and ε > 0 be fixed. By the continuity of the function ϕ we find a constant L ∈ (8/π, ∞) so that |ϕ(y) − ϕ(x 0 )| < ε/2 whenever y ∈ C(v 0 , 4/L) ∩ ∂ ∞ M , where v 0 =γ o,x0 0 is the initial direction of the geodesic ray representing x 0 . Taking (4.15) into account, we can choose R 3 in Lemma 5.5 so big that u + (r) ≤ ||ϕ|| ∞ + ε/2 when r ≥ R 3 .
The point x 0 ∈ ∂ ∞ M and constant ε > 0 were arbitrary so this shows that u extends continuously to C(M ) and u|∂ ∞ M = ϕ. Finally, the uniqueness follows from the comparison principle.

Non-existence result
In the following, we state a non-existence result for the prescribed weighted mean curvature graph equation by adapting the approach of Pigola, Rigoli and Setti in [15]. We denote by A(r) the area of the geodesic sphere ∂B(o, r) centred at a fixed point o ∈ M . Then, if q ≡ 0, there are no solutions to (7.3).
Proof. The proof is very similar to that in [15], the only differences being our use of the divergence operator with respect to the weighted volume form dM and a suitable form of the Mikljukov-Hwang-Collin-Krust inequality which in our setting reads as follows Together these result in the extra factors of 0 in (7.1), (7.2), and on the right hand side of (7.3). Taking into account these differences the proof in [15] applies almost verbatim.
As direct corollaries of the previous theorem, we have