Non-parametric mean curvature flow with prescribed contact angle in Riemannian products

Assuming that there exists a translating soliton $u_\infty$ with speed $C$ in a domain $\Omega$ and with prescribed contact angle on $\partial\Omega$, we prove that a graphical solution to the mean curvature flow with the same prescribed contact angle converges to $u_\infty +Ct$ as $t\to\infty$. We also generalize the recent existence result of Gao, Ma, Wang and Weng to non-Euclidean settings under suitable bounds on convexity of $\Omega$ and Ricci curvature in $\Omega$.


INTRODUCTION
We study a non-parametric mean curvature flow in a Riemannian product N × R represented by graphs with prescribed contact angle with the cylinder ∂Ω × R. We assume that N is a Riemannian manifold and Ω N is a relatively compact domain with smooth boundary ∂Ω. We denote by γ the inward pointing unit normal vector field to ∂Ω. The boundary condition is determined by a given smooth function φ ∈ C ∞ (∂Ω), with |φ| ≤ φ 0 < 1, and the initial condition by a smooth function u 0 ∈ C ∞ (Ω).
The function u above in (1.1) is a solution to the following evolution equation u(·, 0) = u 0 inΩ, (1.2) where W = 1 + |∇u| 2 and ∇u denotes the gradient of u with respect to the Riemannian metric on N at x ∈Ω. The boundary condition above can be written as ν, γ = φ, (1.3) where ν is the downward pointing unit normal to the graph of u, i.e.
The longtime existence of the solution u t := u(·, t) to (1.2) and convergence as t → ∞ have been studied under various conditions on Ω and φ. Huisken [5] proved the existence of a smooth solution in a C 2,α -smooth bounded domain Ω ⊂ R n for u 0 ∈ C 2,α (Ω) and φ ≡ 0. Moreover, he showed that u t converges to a constant function as t → ∞. In [1] Altschuler and Wu complemented Huisken's results for prescribed contact angle in case Ω is a smooth bounded strictly convex domain in R 2 . Guan [4] proved a priori gradient estimates and established longtime existence of solutions in case Ω ⊂ R n is a smooth bounded domain. Recently, Zhou [8] studied mean curvature type flows in a Riemannian product M × R and proved the longtime existence of the solution for relatively compact smooth domains Ω ⊂ M. Furthermore, he extended the convergence result of Altschuler and Wu to the case M is a Riemannian surface with nonnegative curvature and Ω ⊂ M is a smooth bounded strictly convex domain; see [8,Theorem 1.4].
The key ingredient, and at the same time the main obstacle, for proving the uniform convergence of u t has been a difficulty to obtain a timeindependent gradient estimate. We circumvent this obstacle by modifying the method of Korevaar [6], Guan [4] and Zhou [8] and obtain a uniform gradient estimate in an arbitrary relatively compact smooth domain Ω ⊂ N provided there exists a translating soliton with speed C and with the prescribed contact angle condition (1.3).
Towards this end, let d be a smooth bounded function defined in some neighborhood ofΩ such that d(x) = min y∈∂Ω dist(x, y), the distance to the boundary ∂Ω, for points x ∈ Ω sufficiently close to ∂Ω. Thus γ = ∇d on ∂Ω. We assume that 0 ≤ d ≤ 1, |∇d| ≤ 1 and | Hess d| ≤ C d inΩ. We also assume that the function φ ∈ C ∞ (∂Ω) is extended as a smooth function to the wholeΩ, satisfying the condition |φ| ≤ φ 0 < 1.
Our main theorem is the following: Theorem 1.1. Suppose that there exists a solution u ∞ to the translating soliton equation where C ∞ is given by (1.5) Then the equation (1.2) has a smooth solution u ∈ C ∞ (Ω, [0, ∞)) with W ≤ C 1 , where C 1 is a constant depending on φ, u 0 , C d , and the Ricci curvature of Ω. Moreover, u(x, t) converges uniformly to u ∞ (x) + C ∞ t as t → ∞.
Notice that the existence of a solution u ∈ C ∞ Ω × [0, ∞) to (1.2) is given by [ More precisely, let Ω N be a relatively compact, strictly convex domain with smooth boundary admitting a smooth defining function h such that h < 0 in Ω, h = 0 on ∂Ω, for some constant k 1 > 0 and sup Ω |∇h| ≤ 1, h γ = −1 and |∇h| = 1 on ∂Ω. Furthermore, by strict convexity of Ω, the second fundamental form of ∂Ω satisfies where κ 0 > 0 is the minimal principal curvature of ∂Ω. In the Euclidean case, N = R n , such functions h are constructed in [2]. We give some simple examples at the end of Section 3.
We will sketch the proof of Theorem 1.3 in Section 3.

PROOF OF THEOREM 1.1
Let u be a solution to (1.2) inΩ × R. Given a constant C ∞ ∈ R we define, following the ideas of Korevaar [6], Guan [4] and Zhou [8], a function η :Ω × R → (0, ∞) by setting where K and S are positive constants to be determined later. We start with a gradient estimate.
there exists a constant C 0 only depending on C d , φ, C ∞ , and the lower bound for the Ricci curvature in Ω such that W(x 0 , t 0 ) ≤ C 0 .
Proof. Let g = g ij dx i dx j be the Riemannian metric of N. We denote by (g ij ) the inverse of (g ij ), u j = ∂u/∂x j , and u i;j = u ij − Γ k ij u k . We set and define an operator L by Lu = a ij u i;j − ∂ t u. Observe that (1.2) can be rewritten as Lu = 0. In all the following, computations will be done at the maximum point (x 0 , t 0 ) of ηW. We first consider the case where x 0 ∈ ∂Ω. We choose normal coordinates at x 0 such that g ij = g ij = δ ij at x 0 , ∂ n = γ, This implies that We have Using our coordinate system, we get for some constant C depending only on C d and φ. So choosing S ≥ C + 1, we get that Next we assume that x 0 ∈ Ω and that S ≥ C + 1, where C is as above. Let us recall from [8, Lemma 3.5] that where ν N = ∇u/W and |A| 2 = a ij a k u i;k u j; /W 2 is the squared norm of the second fundamental form of the graph M t . Since 0 = W i η + Wη i , for every i = 1, . . . , n, we deduce that This yields to 1 To simplify the notation, we set So we have 1 We can compute Lh as and, by Young's inequality for matrices, we get the estimate by using the assumption that Ric is bounded. Next we turn our attention to the other terms in (2.5). We have Then we note that by the assumptions, we clearly have and we are left to consider Plugging the estimates (2.6), (2.7), (2.8), and (2.9) into (2.5) and using (2.4) with the Ricci lower bound we obtain Then collecting the terms including |A| 2 and noticing that Now choosing K large enough, we obtain W(x 0 , t 0 ) ≤ C 0 , where C 0 depends only on C ∞ , d, φ, the lower bound of the Ricci curvature in Ω, and the dimension of N. We notice that the constant C 0 is independent of T. Since We observe that the function u ∞ (x) + Ct solves the equation (1.2) with the initial condition u 0 = u ∞ if u ∞ is a solution to the elliptic equation (1.4) and C is given by (1.5 for some constant c 2 only depending on u 0 , φ, and Ω.
whereã ij ,c ij are positive definite matrices and b i ∈ R. Then the proof of the lemma follows by applying the maximum principle.
In view of Lemma 2.2, taking C ∞ = C, and observing that the constant C 0 is independent of T, we get from (2.10) a uniform gradient bound. (1.4) admits a solution u ∞ with the unique constant C given by (1.5). Let u be a solution to (1.2). Then W(x, t) ≤ C 1 for all (x, t) ∈ Ω × [0, ∞) with a constant C 1 depending only on φ 0 , u 0 , and Ω.

Lemma 2.3. Suppose that
Having a uniform gradient bound in our disposal, applying once more the strong maximum principle for linear uniformly parabolic equations, we obtain: Theorem 2.4. Suppose that (1.4) admits a solution u ∞ with the unique constant C given by (1.5). Let u 1 and u 2 be two solutions of (1.2) with the same prescribed contact angle as u ∞ . Let u = u 1 − u 2 . Then u converges to a constant function as t → ∞. In particular, if C is given by (1.5), then u 1 (x, t) − u ∞ (x) − Ct converges uniformly to a constant as t → ∞.
Proof. The proof is given in [1, p. 109]. We reproduce it for the reader's convenience. One can check that u satisfies whereã ij ,c ij are positive definite matrices and b i ∈ R. By the strong maximum principle, we get that the function F u (t) = max u(·, t) − min u(·, t) ≥ 0 is either strictly decreasing or u is constant. Assuming on the contrary that lim t→∞ u is not a constant function, setting u n (·, t) = u(·, t − t n ) for some sequence t n → ∞, we would get a non-constant solution, say v, defined on Ω × (−∞, +∞) for which F v would be constant. We get a contradiction with the maximum principle. The only extra ingredient we must take into account in our non-flat case is the following Ricci identity for the Hessian ϕ i;j of a smooth function ϕ For the convenience of the reader, we mostly use the same notations as in [3]. Thus let h be a smooth defining function of Ω such that h < 0 in Ω, h = 0 on ∂Ω, (h i;j ) ≥ k 1 (δ ij ) for some constant k 1 > 0 and sup Ω |∇h| ≤ 1, h γ = −1 and |∇h| = 1 on ∂Ω. Furthermore, by strict convexity of Ω, the second fundamental form of ∂Ω satisfies where κ 0 > 0 is the minimal principal curvature of ∂Ω.
We consider the equation for small ε > 0. Writing φ = − cos θ, v = 1 + |∇u| 2 and where w(x) = v − u h cos θ and α > 0 is a constant to be determined, we assume that the maximum of Φ is attained in a point x 0 ∈Ω. If x 0 ∈ ∂Ω, we can proceed as in [3, pp. 34-36]. Thus choosing 0 < α < κ 0 and 0 < ε 0 ≤ ε α < 1 such that where M 1 = supΩ |∇ 2 h|, yields an upper bound for the tangential component of ∇u on ∂Ω. Combining this with the boundary condition u γ = −v cos θ gives an upper bound for |∇u(x 0 )| and hence for Φ(x 0 ). The only difference to the Euclidean case occurs when x 0 ∈ Ω, i.e. is an interior point of Ω. At this point we have, using the same notations as in [3, p. 42], and 0 ≥ a ij Φ i;j (x 0 ) = a ij w i;j w − α 2 a ij h i h j + αa ij h i;j =: I + I I + I I I.
In some special cases we get sharper estimates than those above.

Conflict of interest:
Authors state no conflict of interest.