Mean curvature flow of Killing graphs

We study a Neumann problem related to the evolution of graphs under mean curvature flow in Riemannian manifolds endowed with a Killing vector field. We prove that in a particular case these graphs converge to a bounded minimal graph which contacts the cylinder over the domain orthogonally along its boundary.


Introduction
Let M be a (n + 1)-dimensional Riemannian manifold endowed with a Killing vector field Y . Suppose that the distribution orthogonal to Y is of constant rank and integrable. Given an integral leaf P of that distribution, let Ω ⊂ P be a bounded domain with regular boundary Γ = ∂Ω. Let ϑ : I ×Ω → M be the flow generated by Y with initial values in M , where I is a maximal interval of definition. In geometric terms, the ambient manifold is a warped product M = P × 1/ √ γ I where γ = 1/|Y | 2 .
Notice that this definition could be slightly more general if we suppose that the coordinates of x ∈Ω change with the parameter t ∈ [0, T ). To abolish this possibility is equivalent to rule out tangential diffeomorphisms of Ω.
The Killing cylinder K over Γ is by its turn defined by Let N be a unit normal vector field along Σ t . In what follows, we denote by H the mean curvature of Σ t with respect to the orientation given by N . We are then concerned with establishing conditions for longtime existence of a prescribed mean curvature flow of the form X(0, ·) = ϑ(u 0 (·), ·), for given functions u 0 :Ω → R and H :Ω → R. In order to define boundary conditions for the evolution problem (2) we consider a function φ ∈ C ∞ (Γ) such that |φ| ≤ φ 0 < 1 for some positive constant φ 0 . Let ν be the inward unit normal vector field along K. We 1 impose the following Neumann condition associated to (2) where ·, · denotes the Riemannian metric in M . The main result in this paper may be stated as follows Theorem 1. There exists a unique solution u :Ω × [0, ∞) → I to the problem (2)- (4). Moreover, if φ = 0 and H = 0 the graphs Σ t converge to a minimal graph which contacts the cylinder K orthogonally along its boundary.
Theorem 1 extends Theorem 1.1 in [3] as well as Theorem 2.4 in [2] and Theorem 2.4 in [1] in a twofold way. The corresponding theorems in [3] and [2] concern evolution of graphs in Euclidean space whereas [1] deals with the case of graphs in Riemannian product spaces of the form P × R. Moreover those earlier results hold only for the case when the prescribed mean curvature is H = 0. Some related results may be also found in [7] and [8].
The paper is organized as follows. Section 2 describes the evolution problem in nonparametric terms. Height and boundary gradient a priori estimates for (2)-(4) are presented respectively in sections 3 and 4. Interior gradient estimates are obtained in Section 5. Some technical computations needed in the body of the proofs are collected in an appendix.
In Section 6 we prove the following result about the asymptotic behavior of the mean curvature flow (2) for general φ and H.
Theorem 2. Suppose that there exists a solution v ∈ C ∞ (Ω) of the elliptic Neumann problem Then the mean curvature flow (2)-(4) converges to a graph with prescribed mean curvature H and prescribed contact angle φ.
In a forthcoming paper the authors prove an existence result for (5)- (6). We also consider there another stationary regimen of the mean curvature flow (2)-(4), namely translating solitons.

Fundamental equations
Since we will consider the mean curvature flow in nonparametric terms it seems adequate to describe all geometric invariants as well as their evolution equations in terms of graphical coordinates.
Let x 1 , . . . , x n be local coordinates in P . This system is augmented to be a coordinate system in M by setting x 0 = s, the flow parameter of Y . The tangent space of Σ t at a point X(t, x), x ∈Ω, is spanned by the coordinate vector fields In terms of these coordinates the induced metric in Σ t is expressed in local components by (8) g where γ = 1 |Y | 2 and σ ij are the local components of the metric in P . In order to compute the mean curvature of Σ t , we fix N as the vector field where ∇u is the gradient of u in P and (10) The second fundamental form of Σ t calculated with respect to this choice of normal vector field has local components where∇ denotes the covariant derivative in M . We then compute Hence using the fact that the maps x → ϑ(s, x) are isometries and that the hypersurfaces defined by {ϑ(s, x) : x ∈ P }, s ∈ I, are totally geodesic one concludes that It follows from Killing's equation that It turns out that a ij could be also expressed by Taking traces with respect to the induced metric one obtains the following expression for the mean curvature H of the hypersurface Σ t Alternatively one has At this point we recall that Using this one easily verifies that (14) may be written in divergence form as ∇γ, ∇u = nH.
In fact we have We conclude that (2) may be written nonparametrically as Indeed it holds that Using (14) one verifies that (21) is equivalent to We conclude that the Neumann problem (2)-(4) has the following nonparametric form in Ω × {0} (24) with boundary condition This boundary value problem describes the evolution of the Killing graph of the function u(·, t) by its mean curvature in the direction of the unit normal N with prescribed contact angle at the boundary. The standard theory for quasilinear parabolic equations [5] guarantees that the problem of solving (2)-(4) is reduced to obtaning a priori height and gradient estimates for solutions to (23)-(25).

Height estimates
From now on, we consider the parabolic linear operator given by where v ∈ C ∞ (Ω × [0, T )).
Then it follows that max |u| ≤ CT * for a given constant C > 0 which depends on T * .
Proof. First of all we verify that u t is a solution for a linear parabolic equation. Indeed one has However since γ = γ(x) in (22) and x is independent of t it follows that In the same way we have We conclude that Now using the fact that σ ij ;t = 0 and γ t = 0 we have Hence it follows that Hence we choose a coordinate system adapted to the boundary Γ in such a way that ∂ ∂x n = ν at x 0 . Then, at the point (x 0 , t 0 ) we have where we used (25) and (27). On the other hand, (25) implies that at (x 0 , t 0 ). We conclude that (1 − φ 2 (x 0 ))u n;t = 0. However since | φ |< 1, it follows that u t;n = 0 what contradicts the parabolic Hopf Lemma [5].
From this contradiction we conclude that t 0 = 0. Since T * is arbitrary, the conclusion follows.

Boundary gradient estimates
Now we will prove a gradient bound for a solution of (23)-(25) by applying a modification of the Korevaar's technique [4] which appeared formerly in [2].
From now on, we consider a non-negative extension d :Ω → R of the distance function dist P (·, Γ) satisfying |∇d| ≤ 1 inΩ. In the same way, we consider a C ∞ extension of the boundary data φ to the domainΩ which we denote also by φ. Then we define where K and α are positive numbers to be fixed later.
Proposition 2. For α > 0 sufficiently large independent of K and t, if for some t ≥ 0 fixed, ηW (·, t) attains a local maximum value at a point x 0 ∈ ∂Ω, then W (x 0 , t) ≤ K.
for a point x 0 ∈ Γ. Hence we choose a coordinate system adapted to Γ such that ∂ ∂x n = ν at x 0 and (32) We have at x 0 from what follows that On the other hand at x 0 we have Since (ηW ) n ≤ 0 at x 0 it holds that On the other hand Therefore since for a given constant C depending solely on γ and φ. It follows that W (x 0 , t) ≤ K if α is chosen large enough and independent of K and t.

Interior gradient estimates
In this section we deduce a global gradient bound using the techniques in [1] and [2]. However the more general context of warped product gives rise to a long list of additional terms which require a careful tracking along the calculations.
In the sequel, we consider the parabolic linear operator given by Proof. We can assume x 0 ∈ Ω and t 0 > 0. At a point (x 0 , t 0 ) where ηW attains maximum value we have We conclude that Thus the expression for Lh in Appendix allows us to conclude that On the other hand Lemma 3 yields Now we use the fact that x 0 is a critical point to ηW . We have However We then conclude that Then we have We conclude that However using some standard inequalites we obtain Using that W 2 ≥ γ and choosing α sufficiently large and depending only on n, γ, φ and κ we have where C is a constant depending on n, γ, φ, d, κ and H.
Hence we obtain

It follows that
Otherwise we are done. In this case we have W ≤ W 2 and absorbing the terms with W into that one with W 2 transforms the inequality above into Then for α > 1 d 0 max{1, 2C/γ} we have It follows that for K > . This finishes the proof of the proposition. Proof. Propositions 1, 2 and 3 yield the following global gradient bound for (x, t) ∈Ω × [0, T * ], where C 1 and C 2 are positive constants and It results that (23) is uniformly parabolic and then the standard theory of quasilinear parabolic PDEs may be applied for assuring the existence of a unique smooth solution to (23)-(25).

Asymptotic behavior
In the particular case when u( Conversely, notice that if v(x) is a solution of (41)-(42) then u = v + Ct is a solution of (23) which is translating along the flow lines of Y with speed C.
Therefore it follows from divergence theorem that φ.
Since the integrands do not depend on s we have from what results that Since W ≤ C 1 e C 2 M T and |φ| < 1 we conclude that Comparing an arbitrary solution of the mean curvature flow with translating graphs yields Proposition 4. Suppose that there exists a solution of (41) for a given C. Then given a solution u(x, t) of (23) there exists a constant M such that Proof. Consider a solution v of (41). Then consider the functions v which are also solutions of (41). By definition we have v 1 ≤ u 0 ≤ v 2 .
Hence the parabolic maximum principle implies that for t ∈ [0, T ) from what we obtain (47).
Theorem 4. Suppose that there exists a solution of (41) for C = 0. Then lim t→∞ u t = 0.
In particular the mean curvature flow converges to a graph with prescribed mean curvature H and prescribed contact angle φ.
Proof. Since C = 0 Proposition (4) implies that It follows that

Appendix
In what follows, II and A denote respectively the second fundamental form and the Weingarten map of Σ t . Their components are given by Some lemmata will be needed in the sequel. Their content could be also of independent interest for other applications. Lemma 1. Denote θ = ∇d, N . The differentials of the functions θ and h have components given by where we used the fact that [ ∂ ∂x 0 ,∇d] = 0 and that P is totally geodesic. Thus we conclude that ∂θ However This finishes the proof of the proposition.
We denote the components of the tensor X * II in P by Notice that the covariant derivatives of X * II and II are related by However since X * Hence using (12), (16) and (17) we obtain Hence it follows that We conclude that that is, Now we use (55) for computing the Hessian of the function θ.
Lemma 2. The trace of the Hessian of θ in Ω calculated with respect to the metric in Σ is given by Proof. Notice that we may write (53) as Hence we have However Hence using Codazzi's equation we obtain Using that g jl u l = γ W 2 u j we conclude that However we have from what follows that Now using the fact that g ij u j = γ W 2 u i and therefore g ij N j = γ W 2 N i we obtain Similarly we have Replacing this above we obtain However Hence we have This finishes the proof of the Lemma.
Using Lemma 2 we will obtain an expression for Lh. Notice that Moreover it holds that We conclude that Now we compute the derivatives with respect to t. We have Hence we have Moreover we have Therefore We also compute ∇γ, ∇h = α ∇d, ∇γ + φ A∇ Σ d, X * ∇γ − ∇φ, ∇γ θ − φd i;j γ i N j + κφ N, ∇γ .

Now we obtain
Moreover we have Therefore grouping and rearranging these expressions we obtain Lemma 3. We have Proof. Notice that Hence it follows that Hence we obtain Now we compute However we have Moreover we compute We also have 2W g ij W j AY T , X * ∂ ∂x i = 2W 2 AY T , ∇ Σ γ 2γ − W 3 ∇γ γ , N AY T , Y T + 2W 3 AY T , AY T .

Now we compute
g ij W N l b il;j = W N l g ij ∇ Σ j a il + 1 γ W 2 g ij a ij a m l N l u m + 1 γ W 2 g ij a lj N l a m i u m +W g ij u i u j 2γ 2 a lm N l γ m + W g ij N l u l u j 2γ 2 a im γ m .
Hence we have We conclude that