Translating solitons over Cartan-Hadamard manifolds

We prove existence results for entire graphical translators of the mean curvature flow (the so-called bowl solitons) on Cartan-Hadamard manifolds. We show that the asymptotic behaviour of entire solitons depends heavily on the curvature of the manifold, and that there exist also bounded solutions if the curvature goes to minus infinity fast enough. Moreover, it is even possible to solve the asymptotic Dirichlet problem under certain conditions.

In this paper we study the existence of translating solitons in Riemannian products N × R, where N is an n-dimensional Cartan-Hadamard manifold, i.e. a complete, simply connected Riemannian manifold with non-positive sectional curvature. A submanifold M of N × R is a translating soliton of the mean curvature flow if where H is the mean curvature vector field, X = ∂ t is the standard coordinate vector field of R, and c ∈ R is a constant that indicates the velocity of the flow. Recall from [5,Prop. 6] that, given a domain Ω ⊂ N and a C 2 function u : Ω → R, the graph is a translating soliton if and only if u satisfies the quasilinear partial differential equation with some constant c ∈ R.
In [1] Altschuler and Wu studied surfaces over a convex domain in R 2 that are evolving by the mean curvature flow and have prescribed contact angle with the boundary cylinder. They also proved the existence of a convex rotationally symmetric translating soliton over the entire plane. Clutterbuck, Schnürer, and Schulze [3] constructed entire, rotationally symmetric, strictly convex graphical translating solitons in R n+1 , n ≥ 2, known as bowl solitons. They also classified all translating solitons of revolution giving a one-parameter family of rotationally symmetric "winglike" solitons M ε , where ε represents the neck-size of the winglike soliton M ε and the limit as ε → 0 consists of a dounle copy of the bowl soliton. We recall that Wang [14] characterized the bowl soliton as the only convex translating solitons in R n+1 that is an entire graph. Spruck and Xiao [13] proved that a translating soliton which is graph over the whole R 2 must be convex and hence the bowl soliton. In recent years several families of new translating solitons in the Euclidean space have been constructed by using different techniques, see [4], [7], [8], [10], [11], [12]. For instance, in [7] Hoffman, Ilmanen, Martín and White gave a full classification of complete translating graphs in R 3 and constructed (n − 1)-parameter families of new examples of translating graphs in R n+1 .
In [5] de Lira and Martín extended the constructions of Clutterbuck, Schnürer, and Schulze to rotationally symmetric Cartan-Hadamard manifolds N ξ whose metric can be written as where r = r(x) = d(o, x) is the distance to a pole o ∈ N ξ and dϑ 2 is the metric on the unit sphere S n−1 ⊂ T o N ξ . They proved the existence of a oneparameter family of rotationally symmetric translating solitons M ε , ε ∈ [0, +∞), embedded into the Riemannian product N ξ × R. The translating soliton M 0 , again called the bowl soliton, is the graph of an entire solution to (1.1), whereas each M ε , ε > 0, is a bi-graph over the exterior of the geodesic ball B(o, ε) ⊂ N ξ and is called a winglike soliton. Notice that a radial function u = u(r) is a solution to (1.1) if and only if it satisfies the ODE where the prime denotes derivatives with respect to the radial coordinate r. De Lira and Martín also constructed entire grim reaper graphs on a class of complete, not necessarily rotationally symmetric, Riemannian manifolds by using Fermi coordinates attached to a geodesic.
Our main motivation in this paper is to prove existence results for graphical translating solitons on Cartan-Hadamard manifolds that need not be rotationally symmetric. We assume that the radial sectional curvatures of a Cartan-Hadamard manifold N satisfy where r is the distance function to a fixed point o ∈ N, a, b : [0, ∞) → [0, ∞) are smooth functions, and P x is any 2-plane in the tangent space T x N containing (∂ r ) x = ∇r(x). We call a and b the radial curvature functions of N. Notice that this is not a restriction at all since there are such functions for any Cartan-Hadamard manifold. We will denote by N a and N b the rotationally symmetric model manifolds with radial sectional curvatures −a(r) 2 and −b(r) 2 , respectively. Note that the Riemannian metric g a on N a (similarly on N b ) can be written as where f a is the solution for the 1-dimensional Jacobi equation (1.4) Next we describe the structure of our paper and state some of our results. The main theme in Section 2 is the search of global super-and subsolutions to (1.1) that stay of bounded distance from each other and use them as barriers in construction of entire solutions. First we implement the rotationally symmetric bowl solitons from the models N a and N b obtained by de Lira and Martín to the actual Cartan-Hadamard manifold N and obtain solutions to (1.1) on geodesic balls B(o, R) ⊂ N with constant boundary values on ∂B(o, R). In Subsection 2.1 we first improve the estimate of the asymptotic behavior of a rotationally symmetric solutions obtained in [3] and [5]. Then we consider asymptotically rotationally symmetric manifolds and utilize the bowl solitons in rotationally symmetric models as global upper and lower barriers. If the sectional curvature upper bound goes to minus infinity fast enough, there can be bounded entire solutions to (1.1). Indeed, we have the following: where the curvature upper bound goes to −∞ fast enough so that Then there exists a translating soliton in N × R that is the graph of an entire bounded solution u : N → R to the equation (1.1).
Furthermore, under suitable curvature bounds, it is even possible to solve the asymptotic Dirichlet problem for (1.1). For instance, if the radial sectional curvatures are bounded as −2 cosh cosh r(x) ≤ K(P x ) ≤ − cosh 2 r(x) − sinh r(x) coth sinh r(x) and we are given a continuous function ϕ ∈ C(∂ ∞ N) on the sphere at infinity, there exists a unique entire solution u ∈ C 2 (N) ∩ C(N) to (1.1) with boundary values u|∂ ∞ N = ϕ at infinity. See Theorem 2.14 in Subsection 2.3 for the general result. In Subsection 2.4 we construct global super-and subsolutions whose difference goes to zero at infinity under certain (implicit) assumptions on the Riemannian metric that are more general than those in 2.1. Then we apply these barriers in construction of entire solutions. The following two corollaries provide examples of such suitable metrics. Corollary 1.2. Let (N 2 , g) be a 2-dimensional Cartan-Hadamard manifold with the Riemannian metric g = dr 2 + h(r, θ) 2 dθ 2 , where h(r, θ) = 1 a cos 2 θ + b sin 2 θ sinh(ar cos 2 θ + br sin 2 θ).
Then there exists a bowl soliton on (N 2 , g).
Then there exists a bowl soliton on (N 2 , g).
Acknowledgements. We would like to thank Francisco Martín for several useful discussions during the preparation of the paper.

CONSTRUCTION OF ENTIRE SOLUTIONS
In this section we search conditions on N that guarantee the existence of an entire solution to (1.1). We start with the following application of the Laplace comparison theorem and the existence result of de Lira and Martín for rotationally symmetric models. Proof. We prove the claim for subsolutions in the case c > 0, the other cases are similar. Let u a : N a → R be the radial solution on N a to (1.1) given by [5,Theorem 7]. Using u a we define a radial function, also denoted by u a , on N by setting u a (x) = u a (r(x)), where r(x) is the distance to the fixed point o ∈ N. Hence ∇u a = u a ∇r. Applying the Laplace comparison ∆r ≥ ∆ a r, where ∆ a denotes the Laplace-Beltrami operator on N a , and denoting Above we used the fact u a ≥ 0 that follows from the maximum principle since the radial solution u a on N a can not have interior maxima. Similarly, the entire radial solution u b on N b yields an entire supersolution on N. In fact, any constant function on N is a supersolution for c ≥ 0.
From now on we assume, without loss of generality, that the constant c in (1.1) is nonnegative. The sub-and supersolutions u a and, respectively, u b can be uses as barriers in order to obtain solutions in geodesic balls with constant boundary values.  Proof. Let u a and u b be the radial sub-and supersolutions given by Lemma 2.1. By adding suitable constants we may assume that u a = u b = m on ∂B.
Then u a is a lower barrier and u b is an upper barrier in B, and therefore we obtain a priori boundary gradient estimate for the Dirichlet problem (2.1). By [2, Lemma 2.3] we have a priori interior gradient estimate, hence the existence of a solution to (2.1) follows from the Leray-Schauder method [6, Theorem 13.8].

Remark 2.3.
Although u a and u b are global sub-and supersolutions it seems difficult to use them as global barriers. The difficulty being that the difference |u b − u a | remains bounded only in a very special case of asymptotically rotationally symmetric manifolds; see Lemma 2.8 and Remark 2.9.
2.1. Asymptotically rotationally symmetric case. Next we will prove the existence of entire solutions of (1.1) under assumptions on the asymptotic behaviour of the radial sectional curvatures of M. For this we first slightly improve the estimate about the asymptotic behaviour of the rotationally symmetric solutions obtained in [5]; see also [3].
Proposition 2.4. Let N be a complete rotationally symmetric Riemannian manifold whose radial sectional curvatures satisfy Suppose, furthermore, that ξ ξ as r → ∞ for some smooth positive h. Then the rotationally symmetric translating solitons M ε , ε ∈ [0, +∞), are described, outside a cylinder over a geodesic ball B R (o) ⊂ N, as graphs or bi-graphs of functions with the following asymptotic behavior For > 0 we denote We claim that for every given > 0 and r 0 > R there exists r 1 > r 0 such that ζ(r 1 ) < ϕ(r 1 ). Indeed, if this were not the case, there would exist > 0 and r 0 > R such that for all r > r * > r 0 . This contradicts the fact that the solution is entire and hence r → +∞.
Next we claim that ξ (r) that holds true for all sufficiently large r > R by the assumption (i). Then adjusting r 1 > r 0 to be sufficiently large we conclude from a standard comparison argument for nonlinear ODEs that ζ(r) < ϕ(r) for every r > r 1 , with r 1 > r 0 > R sufficiently large. We conclude that for every given > 0 and r 0 > R there exists r 1 > r 0 such that Similarly, given > 0 we denote Next we prove that for all sufficiently large r, or equivalently that ξ (r) that holds true for all sufficiently large r again by the assumption (i). Therefore we can again conclude that for every given > 0 and r 0 > R there exists r 1 > r 0 such that We set and claim that lim r→+∞ ψ(r) = 0.
If this were not the case, then Note that there are arbitrary large r * such that ψ(r * ) < −δ. For every such r * ≥ r 2 we have, by the assumption (i), that is finite, hence ψ(r 4 ) = −δ and ψ (r 4 ) ≥δ. This leads to a contradiction since ψ(t) > −δ for all t ∈ (r 3 , r 4 ).
Given any δ > 0, with there are arbitrary large r * such that ψ(r * ) > δ. Again, for every such r * ≥ r 2 we have ψ (r * ) < −δ whenever r 2 is large enough. As above, this leads to a contradiction.
Next we study the rate of the convergence ψ → 0. For this, we denote and show that λ(r) → 0 as r → ∞. Since Assuming that λ(r) ≤ −δ < 0 for some arbitrarily large r implies that for such r by assumptions (ii) and (iii). By a similar argument as before this implies that lim inf On the other hand, if λ(r) ≥ δ > 0 for some arbitrarily large r, then λ (r) ≤ −δ < 0 for such r, and again we can conclude that lim sup r→+∞ λ(r) ≤ 0.
In fact, also h(r) = sinh r will do as can be seen in the estimation of λ (r). (c) If ξ(r) = sinh(sinh r), we can choose h(r) = e r / log r.
Next, let us discuss the condition (i) of Proposition 2.4. Before doing so, we need the following lemma which provides criteria to compare the sectional curvature K(P x ) with (ξ /ξ) r(x) when r(x) is very large. To simplify notation, we set −a 2 = K and f a = ξ.
(ii) If (2.4) does not hold, we assume that a (t) ≤ −c a 2 (t) < 0, with some constant c > 0, for all sufficiently large every t and, furthermore, either Proof. The proof of (i) can be found in [9, Lemma 2.3]. Let us prove next (ii). Its proof follows closely [9, Lemma 2.3]. For k > 0 we define g k (t) = exp k t 0 a(s)ds and notice that g k (t) = k a(t)g k (t) and g k (t) = k a (t) + k 2 a 2 (t) g k (t).
Since a (t) ≤ −c a 2 (t) < 0 for all t ≥ t 1 , we obtain Therefore Hence and consequently This proves (2.5).
Notice that Next, let us assume that a satisfies the assumptions (ii) in Lemma 2.6. First, integrating the inequality a (t) ≤ −ca 2 (t), we deduce that there exists a constant C such that a(t) ≤ C/t =ã(t). Therefore, using Example 2.1 of [9], there exist two positive constants α > 1 and c such that lim t→∞ fã(t) t α = c and lim t→∞ f ã (t) t α−1 = αc. This implies that lim t→∞ fã f ã (t) = ∞. Concerning conditions (ii) and (iii) of Proposition 2.4, we see that if lim t→∞ a(t) = ∞ then we can take h = f a / f a . Indeed, (ii) is a direct consequence of the fact that ( f a / f a ) (t) → 0 as t → ∞. Concerning (iii), we notice that Using that we deduce that h = o(h 2 ) which is equivalent to (iii). We do not comment on these conditions in the case lim t→∞ a(t) = 0 since in any case, we will need a refined asymptotic expansion.
Proposition 2.7. Let N be a complete rotationally symmetric Riemannian manifold whose radial sectional curvatures satisfy Suppose, furthermore, that and ξ(r) ξ (r) → ∞ as r → ∞. Then the rotationally symmetric translating solitons M ε , ε ∈ [0, +∞), are described, outside a cylinder over a geodesic ball B R (o) ⊂ N, as graphs or bi-graphs of functions with the following asymptotic behavior (2.10) as r → +∞.
Proof. By the proof of 2.4 where ψ(r) → 0 as r → ∞. We write and claim that (2.9) holds for λ. First we compute If, for fixed δ > 0 for some arbitrary large r, then λ (r) >δ for such r. Hence (2.9) holds. Next we write from which we deduce as earlier that (2.10) holds.
We claim that the assumptions of the previous lemma hold under the hypothesis of Lemma 2.6 (ii) with K = −a 2 . By Lemma 2.6, (2.6) and (2.7), we see that f a f a (r) ≥ C and So the condition .
Since a f a / f a ≤ 1 and a goes to 0, this holds true. This proves the claim.
and that

12)
for some smooth, increasing, positive function h such that 1 h ∈ L 1 (+∞) (2.13) and h h (r) → 0 (2.14) as r → ∞. Then there exist entire radial sub-and supersolutions, u a and u b , of (1.1) such that the difference |u b (x) − u a (x)| remains bounded as r(x) → ∞. Proof. Since by [9, Lemma 2.2], we see that the assumptions in Proposition 2.4 hold with a function h satisfying (2.12) and (2.14). Let u a and u b be the entire radial sub-and, respectively, supersolution on N obtained from the model manifolds N a and N b as in Lemma 2.1. By considering u i − u i (o), i = a, b, we can assume that u i (o) = 0 and hence, using (2.2), (2.11) and (2.13), the difference of the these functions can be estimated by  Proof. Let u a and u b be the entire radial sub-and, respectively, supersolution on N obtained from the model manifolds N a and N b as in Lemma 2.1. By Lemma 2.2 there exists, for each k ∈ N, a function On the other hand, by Lemma 2.8, there exists a constant C > 0 such that u b + C ≥ u a on N. Furthermore, u b + C is a (global) supersolution. Therefore the sequence (u k ) is locally uniformly bounded and hence there exists a subsequence converging locally uniformly with respect to C 2 -norm to an entire solution u.
As in [5] we call the graph M = {(x, u(x)) ∈ N × R : x ∈ N} a bowl soliton.
We want to give some examples about metrics that satisfy the assumptions (2.11) -(2.14).

Example 2.11.
1. We may choose f a (r) = sinh r and f b (r) = g(r) sinh r, where g is a smooth positive function such that g(r) = c e −1/r for all large r with a suitable positive constant c and that 2g (r) coth r g(r) + g (r) g(r) is nonnegative and bounded and, furthermore, that g(0) = 1 and g (odd) (0) = 0. Then and f a and f b satisfy the assumptions (2.11)-(2.14) with h(r) = r 2 . 2. Another example is given by f a (r) = 1 2 sinh r + 1 2 sinh 2r and f b (r) = 1 2 sinh 2r. Then a 2 (r) = f a (r) f a (r) = sinh r + 2 sinh(2r) sinh r + 1 2 sinh 2r and, furthermore, f a and f b satisfy (2.11)-(2.14) with h(r) = e r . In fact, any choice f a (r) = s sinh r + (1 − s) 1 b sinh(bt) and f b (t) = 1 b sinh bt, with 0 < s < 1 and b > 1, will do.

Bounded solutions.
Next we show that if the sectional curvatures of N are negative enough near the infinity, it is possible to have entire bounded solutions of (1.1). Recall that, in the radially symmetric case, the soliton equation (1.1) can be written as where the curvature upper bound goes to −∞ fast enough so that Then there exists a translating soliton in N × R that is the graph of an entire bounded solution u : N → R to the equation (1.1).
Proof. We will prove that the bounded function is a subsolution to (1.1) if the constant c 1 is large enough. It follows from the Laplace comparison where∇ f is the gradient of a smooth function f : N × R → R with respect to the product Riemannian metric of N × R and ν denotes the downward unit normal to the graph of u, i.e.
where ∂ t denotes the standard coordinate vector field on R. Indeed, we obtain (1.1) in the case f (x, t) = −ct. Asymptotic Dirichlet problem for (2.16) was solved in [2] under assumptions on f and the radial curvature functions a and b that are not directly applicable in the setting of the current paper. In [2] the authors applied the assumptions and results from [9] to construct local barriers at the sphere at infinity ∂ ∞ N. These barriers consist of an angular part and of a radially decaying part. The assumptions on a and b are needed to control effectively first and second order derivatives of the barriers. We refer to [2] and [9] for the definition of the sphere at infinity and other relevant notions concerning the asymptotic Dirichlet problem.
Since we are looking for, first of all, bounded solutions, it is natural to assume that f a / f a is integrable; see (2.2) and Theorem 2.12. In the light of [2, Lemma 4.3] we assume that lim t→∞ f a (t)t 1+ε f a (t) = 0 (2.17) for some ε > 0. Scrutinizing the reasoning in [9] we see that the assumptions on a and b can be weakened to the setting of the current paper. We assume that the radial curvature functions a and b (≥ a) are increasing, Furthermore, we assume that there exists a constant κ > 0 such that The assumption (2.19) will be used instead of [9, Lemma 3.10] that was one of the important tools in [9] then the assumptions (2.17)-(2.21) hold. In this case we have radial curvature bounds −2 cosh cosh r(x) ≤ K(P x ) ≤ − cosh 2 r(x) − sinh r(x) coth sinh r(x) .

Global barriers and applications to the construction of bowl solitons.
In this section, we are going to construct global barriers for our problem under certain assumptions on the Riemannian metric that are more general than those in Subsection 2.1. To do so, we first focus on the asymptotics at infinity of solutions to the soliton equation (1.1). Our idea is to make (implicit) assumptions on the metric in order to mimic the asymptotic behavior of radial solutions. In particular, we want to neglect all the non-radial terms in the equation. We will see in the following that we are able to construct suitable metrics with pinched sectional curvature. Thanks to the knowledge of this asymptotic behavior, we are able to construct suband supersolutions to our equation at infinity in such a way that the difference between them goes to zero. To extend these barriers to the whole manifold, we only match these roughly using cut-off function with radial sub-and supersolutions on compact sets. We start with noticing that the Riemannian metric on N can be written as where r is the distance to a fixed point o ∈ N and dϑ 2 is the (induced) Riemannian metric on the geodesic sphere S(r) := S(o, r). Then ∆u = u rr + u r ∆r + ∆ S(r) u, where u rr = ∂ r (∂ r u) and ∆ S(r) is the Laplacian on the Riemannian submanifold S(r). It follows that the soliton equation (1.1) is equivalent to u rr + u r ∆r + ∆ S(r) u − Hess u(∇u, ∇u) where We also set E(u) = u rr + ∆ S(r) u − H(u), For the next proposition we define a sequence of functions v i : where γ is the unique unit speed geodesic joining o = γ(0) and x = γ r(x) . In what follows the notation f = o R (g) means that lim R→∞ f /g = 0.
Proposition 2.15. Suppose that for some ε > 0, and (ii) there exists i 0 ∈ N such that v i 0 (x) → 0 as r(x) → ∞, and that (iii) for some very large R 1 , E(v i 0 ) has a constant sign on N \ B(o, R 1 ).
Then there exists R > 0 such that Moreover, letting g = u 1 − u 2 , we have that |g(x)| → 0 as r(x) → ∞. Furthermore, if for some ε > 0, there exist radial functions F − , F + such that Then, there exist U 1 and U 2 such that Proof. We proceed by induction. Notice that, for any functions v and w, we have Next we substitute v by ∑ i−1 j=0 v j and w by (1 ± ε)v i , for ε > 0. We get Since by assumption, one can show that ∑ i−1 j=0 v j + (1 ± ε)v i is a sub-or supersolution depending on the sign of M ∑ i−1 j=0 v j . We stop as soon as v i (x) → 0 as r(x) → ∞. Next we construct a global sub-and supersolutions. To do so, let χ(x) be a radial function such that for some A, B to be determined later. We set By assumption, we have that Thanks to this choice, we get By assumptions, we have We proceed in the same way to prove that U 1 = χF + + (1 − χ)u 1 is a global supersolution. Theorem 2.16. Let (N, g) be a Cartan-Hadamard manifold whose Riemannian metric g admits functions v i , i ≥ 0, F − , and F + satisfying the assumptions in Proposition 2.15. Then there exists a global bowl soliton on (N, g).
Proof. Let U 1 be an entire supersolution and U 2 an entire subsolution, with U 1 (x) − U 2 (x) → 0 as r(x) → ∞, provided by Proposition 2.15. It follows from the comparison principle that U 1 (x) ≥ U 2 (x) for every x ∈ N. Indeed, if U 1 (y 0 ) < U 2 (y 0 ) for some y 0 ∈ N, let D be the y 0 -component of the set {x ∈ N : U 1 (x) < U 2 (x) − δ/2}, with δ = U 2 (y 0 ) − U 1 (y 0 ) > 0. Then D is an open relatively compact subset of N and U 1 (x) = U 2 (x) − δ/2 on ∂D. Hence U 1 (x) = U 2 (x) − δ/2 in D by the comparison principle leading to a contradiction. Now the sets Ω j = {x ∈ N : U 1 (x) > U 2 (x) + 1/j}, for all sufficiently large j, exhaust N. We can use U 1 and U 2 + 1/j as upper and lower barriers in Ω j in a similar fashion than u b and u a in Theorem 2.10 to obtain a global solution.
Next, we use the previous results to prove the existence of bowl solitons on two manifolds with pinched sectional curvatures. The sectional curvatures of the first one are pinched between two arbitrary negative constants (which are attained for some planes) while the sectional curvatures of the second one go quadratically to 0 at infinity and take the values α 2 /r 2 and β 2 /r 2 , β > α > 4 √ 5, for some planes.
This proves that (i) of Proposition 2.15 holds for i = 0. Thanks to this estimate, we get that max{e −aγ(t) , e −bγ(t) } c dt.
Therefore, v 1 (x) → 0 as r(x) → ∞ which is (ii) of Proposition 2.15. Finally, notice that v 1 and none of the terms of E(v 1 ) oscillates at infinity so clearly E(v 1 (x)) has a sign provided that r(x) is large enough. Thus (iii) of Proposition 2.15 holds. To conclude the proof, we are left with the existence of F ± . Recall from de Lira and Martin [5], that if P is a rotationally symmetric manifold with constant sectional curvature, there exists a radially symmetric function w P such that M(w P ) = 0, for any c P ∈ R and (w P ) rr (x) goes to 0 when r(x) goes to infinity. Since the sectional curvatures of our manifold are pinched between −a 2 and −b 2 , F ± can be obtained by taking w P for some appropriate manifolds P and some constant c P by standard comparison theorems. This proves the corollary.
We deduce from this that By construction, we have that (i) of Proposition 2.15 holds true. Observe that v 2 (x) → 0 as r(x) → ∞ therefore (ii) of Proposition 2.15 holds true forĩ = 2. As in the previous corollary, none of the terms of E(v 2 ) oscillate so it has a sign at infinity i.e. (iii) holds. To prove the existence of F ± , we proceed as in the previous corollary. Let us point out that the condition F ± → 0 of Proposition 2.15 can be replaced by F ± / 1 + |∇U 2 | 2 → 0 which is satisfied in our current situation since |∇U 2 | → ∞.