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Conformal Killing graphs with prescribed mean curvature

We prove the existence and uniqueness of graphs with prescribed mean curvature function in a large class of Riemannian manifolds which comprises spaces endowed with a conformal Killing vector field.


Introduction
Our aim in this paper is to continue the research theme elaborated in [2] where the existence of a Killing vector field in a manifold permitted to formulate and solve Dirichlet problems associated to the mean curvature equation. Here, we expand our scope by dealing with conformal Killing vector fields and the corresponding notion of graph. Our main goal is to provide a significant improvement of the results in [1] for graphs of constant mean curvature and trivial initial condition. This is achieved by means of a completely different methodology that allows us to generalize, under an unifying perspective, several previous results.
The pure analytic method used here enable us to discard or weaken several assumptions in [1]. For instance, we allow prescribed mean curvature and nontrivial boundary data. Moreover, we remove the restriction on the conformal Killing field to be closed. We also remove the restriction on the Ricci curvature to be minimal in the direction of the Killing field and weaken other requirements on the Ricci tensor. Furthermore, we ruled out some restrictions on the metric of the ambient space so that, in particular, our result applies successfully to a class of product spaces.
To explain the framework of this paper we first fix some terminology. LetM n+1 denote a Riemannian manifold endowed with a conformal Killing vector field Y whose orthogonal distribution D is integrable. Thus, there exists ρ ∈ C ∞ (M ) such that £ Yḡ = 2ρḡ, whereḡ is the metric inM . It results that the integral leaves of D are totally umbilical hypersurfaces. If in addition Y is closed, then they are spherical, i.e., have constant mean curvature.
We denote by Φ : I × M n →M n+1 the flow generated by Y , where I = (−∞, a) is an interval with a > 0 and M n is an arbitrarily fixed integral leaf of D labeled as t = 0. It may happen that a = +∞, i.e., the vector field Y is complete. For instance, this occurs when the trajectories of Y are circles and we pass to the universal cover in an appropriate manner. Since Φ t = Φ(t, · ) is a conformal map for any fixed t ∈ I, there exists a positive function λ ∈ C ∞ (I) such that λ(0) = 1 and Φ * tḡ = λ 2 (t)ḡ. Given a bounded domain Ω in M, the conformal Killing graph Σ = Σ(z) of a function z onΩ is the hypersurface Proving the existence of a conformal Killing graph with prescribed mean curvature and boundary requires establishing apriori estimates. This is accomplished by the use of Killing cylinders as barriers. The Killing cylinder K over Γ = ∂Ω is the hypersurface ruled by the flow lines of Y through Γ, that is, Let Ω 0 denote the largest open subset of points of Ω that can be joined to Γ by a unique minimizing geodesic. At points of Ω 0 , we denote where RicM is the ambient Ricci tensor and η ∈ T x M is a unit vector tangent to the the unique minimizing geodesic from x ∈ Ω 0 to Γ. The following result assures the existence of conformal Killing graphs with prescribed mean curvature H and boundary data φ. Here, the functions H and φ are defined onΩ and Γ, respectively. Moreover, H K denotes the mean curvature of K when calculated pointing inwards.
Then, there exists a unique function z ∈ C 2,α (Ω) whose conformal Killing graph has mean curvature function H and boundary data φ.
Proposition 6 below implies that Theorem 1 holds under weaker but somewhat more technical assumptions on the ambient Ricci tensor. We also point out that we can prove with minor modifications an existence result for functions H depending also on t by imposing the condition (λH) t ≥ 0 instead of λ t ≥ 0. In the case comprised in Theorem 1 the condition λ t ≥ 0 says that the mean curvatures of the leaves and of the graph have opposite signs.
It is worth to mention thatM is conformal to a Riemannian product manifold I ×M whereM is conformal to M. A quite remarkable fact is that the mean curvature equation for a general conformal metric to a product metric like this does not satisfy, in general, the maximum principle. In fact, in the final remark of the paper we see that the class of metrics which we deal in this paper stands as a borderline for the validity of the elliptic techniques in the treatment of the mean curvature equation.
Assume now that Y is closed or just a Killing field. In each case, a short additional argument to the proof of Theorem 1 yields the same conclusion under weaker assumptions. For instance, in the case of a Killing field we prove a generalization of the main result in [2] under a weaker assumption on the Ricci curvature.
Then, there exists a unique function z ∈ C 2,α (Ω) whose conformal Killing graph has mean curvature function H and boundary data φ.
The case of closed conformal Killing fields encompasses a broad range of examples, namely, product and warped ambient spaces, that have been extensively considered in the recent pertinent literature. In this case, we can state our result in terms of the Ricci tensor of M n .
We finally point out that concerning constant mean curvature Euclidean radial graphs over spherical domains, the above result extends the theorems for minimal radial graphs due to Rado and Tausch as well as for constant mean curvature by Serrin and Lopez (see [5] and references therein).

Preliminaries
Let (M n+1 ,ḡ) be a Riemannian manifold endowed with a conformal Killing vector field Y whose orthogonal distribution D is integrable. Let∇ denote the Riemannian connection inM n+1 and X, Z =ḡ(X, Z).
It follows from (1) and (2) thatγ We have from (1) and the integrability of D that for any X, Z ∈ D. Thus, the leaves M n t = Φ t (M) are totally umbilical and the mean curvature k = k(t, u) of M t with respect to the unit normal vector field Y /|Y | is We assign coordinates x 0 = t, x 1 , . . . , x n to points inM of the formū = Φ(t, u) where x 1 , . . . , x n are local coordinates in M. Then, the coordinate vector fields are The components of the ambient line element ds 2 in terms of these coordinates arē where σ ij are the local components of the metric dσ 2 in M n . Setting, we have thatM n+1 is conformal to the Riemannian warped product manifold M n × ψ I with conformal factor λ, i.e., Finally, after the change of variable we have that (6) takes the form of a Riemannian twisted product where θ(r) = λ(t(r)).
The above change of variable is essential to avoid the coefficients of the terms of second order in the quasilinear elliptic mean curvature equation for Σ(z) to depend on the function z itself.
We conclude this sections with a few examples to illustrate this change of variable. Observe that all Riemannian manifolds in the examples below fit the assumptions in Theorem 1. Moreover, these examples comprise Euclidean and hyperbolic space forms what implies that Theorem 1 assures existence for radial graphs with prescribed mean curvature in these spaces. (a) By means of the change of variable e t = r, we have that (b) By means of the change of variable t = 1 − e −r , we have that In the particular case when Y is closed conformal Killing field, that is, when we have that γ is constant. Thus, in this case,M n+1 has a warped product structure and is conformal with conformal factor λ to a Riemannian product manifold I × M n . Observe that now the leaves of D are spherical, that is, totally umbilical with constant mean curvature k = k(t).

Killing cylinders
Let Ω ⊂ M n be a bounded domain with regular boundary Γ. The Killing cylinder K over Γ determined by the conformal Killing field Y is the hypersurface defined by Let t 1 , . . . , t n−1 be local coordinates for Γ. We denote by (τ ij ) the components of the metric in Γ with respect to these coordinates. It results that t, t 1 , . . . , t n−1 are local coordinates for K. Let η be the inward unit normal vector along Γ as a submanifold of M.
is an unit normal vector field to K. Thus, Hence ∂ t is a principal direction of K with corresponding principal curvature It follows from (1) and (3) that It was shown in [4] that the distance function d(u) = dist(u, Γ) in Ω 0 has the same regularity as Γ. Hence, local coordinates inM near K can be defined setting t 0 = t and t n = d. We denote by (t ij ) the components of the metric for these coordinates. Thus, Lemma 5. The mean curvature H K of the Killing cylinder K is given by Proof: We have that and the proof follows.
We denote by Γ ǫ and K ǫ the level sets d = ǫ in M andM , respectively. By H Kǫ we denote the mean curvature of the Killing cylinder K ǫ over Γ ǫ . Proposition 6. Assume that the Ricci curvature tensor ofM satisfies Proof: The coordinate d-curve in Φ(t, u) is the image by Φ t of the coordinate d-curve passing through u ∈ M. Thus, Extendη near K as the velocity vector field of the geodesics in M t departing orthogonally from K ∩ M t . We obtain for 1 ≤ i, j ≤ n − 1 that Using (4) and (5), we havē where A ǫ denotes the Weingarten map of K ǫ relative toη. For the remaining case i = j = 0, we havē However, Thus, Using (12) we have It follows thatη We also havē Taking traces with respect to the metric (t ij ) in K with t 00 =γ and t 0i = 0, and using (11) and (13) gives However, We conclude that Since nḢ Kǫ =∇ηtrA ǫ = tr∇ηA ǫ , at d = ǫ, it follows that Therefore, at t = 0 and by the assumption on Ric rad M there exist constants c > 0 and d 0 > 0 such thatḢ
We show next that the partial differential equation for a prescribed mean curvature function H inΩ is the quasilinear elliptic equation of divergence form div ∇z Recall that ρ = λ t /λ. Here, the gradient ∇ and the divergence div are differential operators in the leaf M n endowed with the metric dσ 2 . A sufficient condition to have a maximum principle for (15) (see Theorem 10.1 in [3]) is The graph Σ is parametrized in terms of local coordinates by X(u) ∈ Σ → z(x 1 , . . . , x n ), x 1 , . . . , x n .
Thus, the tangent space to Σ at X(u) is spanned by the vectors where z i = ∂z/∂x i . Hence, the metric induced on Σ is The inverse is where z i = σ ik z k and |∇z| 2 = z j z j as usual.
Fix the orientation on Σ given by the unit normal vector field where Notice that N, Y > 0. We compute the second fundamental form (17) and (18) we obtain The Levi-Civita connections in M and M t are determined by the same Christoffel symbols sinceσ ij |ū = λ 2 σ ij | u if u ∈ M andū = Φ t (u) ∈ M t . We have from (2) and (4) that The terms involving the flow lines acceleration are and Similarly, Since z i;j = z ij − ∇ ∂ i ∂ j , ∇z are the Hessian components, we have We easily obtain Taking traces after dividing both sides by λ 3 W 3 yields Equivalently, we have as we wished.

The proof of Theorem 1
Finding a conformal Killing graph Σ(z) with prescribed mean curvature function H and boundary data φ amounts to solve the Dirichlet problem Proof of Theorem 1: We apply the well-known continuity method to the family parametrized by τ ∈ [0, 1] of Dirichlet problems Let I be the subset of [0, 1] consisting of values of τ for which the Dirichlet problem (23) has a C 2,α solution. Then, the proof reduces to show that I = [0, 1]. First, observe that I is nonempty since z = 0 is a solution for τ = 0. Moreover, we have that I is open in view of (16). The difficult part is to show that I is closed. This follows from the a priori estimates given below in Propositions 8, 9 and 10 and standard theory of quasilinear elliptic partial differential equations [3].
Remark 7. We point out that our existence results still hold if φ is only assumed continuous. We may approximate φ uniformly by smooth boundary data and use the interior gradient estimate to obtain strong convergence on compact subsets of Ω. A local barrier argument shows that the limiting solutions achieves the given boundary data.

Height estimates
In this section, we obtain an a priori height estimate. Proof: In view of (16) we may apply the Comparison Principle (cf. Theorem 10.1 in [3]) to (23). From we conclude that z τ ≤ τ sup φ. Similarly, for a solution z of (22) we obtain from Next we construct barriers on Ω 0 which are subsolutions to (22) of the form where d(u) = dist(u, Γ) and the real function f will be chosen later. Hence, At points in Ω 0 , we have |∇d| = 1.
It follows that where H Γǫ denotes the mean curvature of Γ ǫ ⊂ Ω 0 with respect to η. Combining with (20) yields where Using (25) and (28) we obtain We chose in (24) the test function where B > diam(Ω) and D > 0 is a constant to be chosen later. Then, Using (14) and that ρ t ≥ 0 by assumption, we obtain nk 2 − √γ k t = (n − 1)k 2 +γρ t ≥ 0.

Boundary gradient estimates
In this section, we establish an a priori gradient estimate along the boundary of the domain. Proof: We argue for τ = 1. We use barriers of the form w + φ along a tubular neighborhood Ω ǫ of Γ where we extended φ to Ω ǫ by taking and µ > 0,μ > 0 are constants. Hence, We chooseμ = c/ ln(1 + µ) with c > 0 to be chosen later. Hence, A simple estimate using (29) gives Here Λ = γ/U 3 is the lowest eigenvalue of the matrix and Proof: The proof follows a similar guideline as in [2]. Consider on Σ(z) the function where v = |∇z| 2 = z i z i and K > 0 is a constant. We already have the desired bound if χ achieves its maximum on Γ. Thus, we assume that χ attains maximum value at an interior pointū ∈ Ω where |∇z| > 0. This assumption enables us to choose a local normal coordinate system x 1 , . . . , x n such that ∂ 1 |ū = ∇z/|∇z| and σ ij (ū) = δ ij . Hence, atū we have Since v j = 2z l z l;j , we also obtain atū from χ j = 2e 2Kz Kvz j + z l z l;j = 0 that z l z l;j = −Kvz j . Therefore, atū we obtain Moreover, we may assume after rotating the ∂ j that (z i;j ), 2 ≤ i, j ≤ n, is diagonal. We write (20) asã ij z i;j =b, Hence, atū we have Covariant differentiation of (32) followed by contraction with ∇z atū gives The third derivatives of z satisfy the Ricci identity where by R ikjm we denote the coefficients of the curvature tensor of M. Hence, To estimateã ij z l z l;ij we use the Hessian matrix of χ. Since is nonpositive atū, it results thatã ij χ i;j ≤ 0. Hence, Using (34) and (36) we obtaiñ a ij z l z l;ij ≤ K 2 γv 2 − Kbv.
We may assume from now on that γ(ū) ≤ v(ū). It follows that there exist constants A i = A i (γ, λ, ρ, H, ∇γ, ∇H), 1 ≤ i ≤ 4, such that We also estimate With the above estimates, it follows from (39) that Choosing K large enough so that the coefficient of v 3 is positive implies that v(ū) is bounded by a constant K 0 which depends only on Ω and H.

Proof of corollaries
In this section we prove Corollaries 2 and 3 in the Introduction.

Proof of Corollary 2:
We first observe that the proofs of Propositions 9 and 10 still work under the weaker assumption inf Γ H K ≥ H ≥ 0. Thus, it suffices to show that Proposition 8 still works if H * ≥ H when Y is a Killing field. In this situation, it is easy to see that (30) is equivalent to Clearly, the last inequality holds for D sufficiently large and the proof follows.

Proof of Corollary 3:
Being Y closed we may assume γ = 1. Thus (8) and (9) yield On the other hand, the relation between the Ricci tensors ofM n+1 and M n is RicM (X, Z) = Ric M (X, Z) − (nk 2 − k t ) X, Z for any X, Z ∈ T M. Thus (10) is equivalent to and the proof follows.
It is thus natural to consider the general situation of an ambient space R × M n endowed with the conformal metric g = λ 2 (t, u)g = e 2ϕ(t,u) g where g is the product metric in R × M n . In this case, the mean curvature equation for the graph X = (z(u), u) is div ∇z where z(t, u) = z(u) and we compute ∇ z,∇ϕ in the ambient space. We easily conclude that in order to have a maximum principle for the above equation we have to ask ϕ t to be independent of t, that is, the function λ has to separate variables. But this is precisely the case we studied in this paper.