Mean curvature flow of graphs in warped products

Let $M$ be a complete Riemannian manifold which either is compact or has a pole, and let $\varphi$ be a positive smooth function on $M$. In the warped product $M\times_\varphi\mathbb R$, we study the flow by the mean curvature of a locally Lipschitz continuous graph on $M$ and prove that the flow exists for all time and that the evolving hypersurface is $C^\infty$ for $t>0$ and is a graph for all $t$. Moreover, under certain conditions, the flow has a well defined limit.


Introduction
Let M be a n-dimensional manifold, (M , g) a n + 1 dimensional Riemannian manifold. where H(·, t) is the mean curvature vector of the immersion F t . We shall use the following convention signs for the mean curvature H, the Weingarten map A and the second fundamental form (h for the scalar valued version and α for its vector valued version). For a chosen unit normal vector N , they are: AX = −∇ X N , α(X, Y ) = ∇ X Y, N N = AX, Y N , h(X, Y ) = α(XY ), N and H = trA = n i=1 h(E i , E i ), for a local orthonormal frame E 1 , ..., E n of the submanfold, and H = n i=1 α(E i , E i ) = H N . Along the rest of the paper, by M t we shall denote both the immersion F t : M −→ M and the image F t (M ), as well as the Riemannian manifold (M, g t ) with the metric g t induced by the immersion. Analogous notation will be used when we have a single immersion F : M −→ M . Notice that, since in this paper we shall deal with graphs, all the immersions that will appear evolving by mean curvature flow will be, in fact, embeddings.
In two fundamental papers [3] and [4], Ecker and Huisken studied the evolution of noncompact hypersurfaces in the Euclidean Space. In [3] they studied the evolution of a graph in R n+1 and showed that (A) If F 0 a "locally Lipschitz" continuous graph and has linear growth rate for its height, then (1.1) with initial condition F 0 has a smooth solution for all time.
(B) If, moreover, F 0 is "stright" at the infinity, F t asymptotically approaches a selfsimilar solution of (1.1). They give an example showing that the condition cannot be weakened.
In [4], Ecker and Huisken obtained some interior estimates and applied them to prove that the hypothesis of linear growth in (A) is not necessary, that is: (A') If F 0 a "locally Lipschitz" continuous graph, then (1.1) with initial condition F 0 has a smooth solution for all time.
In [12] and [13], Unterberger extended result (A') to the Hyperbolic Space H n+1 and gave a result of type (B). In this space, the first problem to face with is the choosing of the right concept of "graph". A natural one is to say that a hypersurface M of H n+1 is a graph over a totally geodesic hypersurface H n if all the geodesics orthogonal to H n cut M once and transversally (we shall call it a geodesic graph). But, in his thesis [12], Unterberger found an example of hypersurface which is a geodesic graph but loses this property when it evolves under (1.1). Then he considered another concept of graph. Let p ∈ H n be a fixed point, Γ a geodesic through p orthogonal to H n . We shall call equidistant curves all the curves which are at constant distance from Γ. Then we say that M is an equidistant graph over H n if it cuts once and transversally all the equidistant curves. Unterberger proved the exact analog of (A') for equidistant graphs. As a result of type (B), he proved that if F 0 : M −→ H n+1 is a "locally Lipschitz" equidistant graph and is at bounded distance from H n , then it converges asymptotically to H n .
After H n+1 , other ambient spaces natural for trying to extend the results of Ecker and Huisken are products M × R or, more generally, warped products R × ϕ M or M × ϕ R. As usual, by a warped product M × ϕ N of two Riemannian manifolds (M, g) and (N , h) we understand the riemannian manifold (M×N , g +ϕ 2 h), being ϕ : M −→ R a positive smooth map. For these spaces it is natural to say that a hypersurface of M × ϕ R (or R × ϕ M ) is a graph if it is a graph of a function u : M −→ R and u(M ) cuts transversally the curves s → (x, s). The interest in the last years for studying minimal and constant mean curvature surfaces in these ambient spaces produces also a natural interest on the study of MCF on them.
When ϕ(s) = cosh s and M = H n , R × ϕ M is H n+1 and the concept of being a graph coincides with that of geodesic graph in H n+1 . Then the counterexample of Unterberger gives us few hope of getting general results. Computations of the evolution of the gradient of u (see the appendix) give some analytic reasons of the failing of the preservation of geodesic graphs under MCF. Also the last paragraph in Remark 4 and the pictures in the appendix can help to get some gometric insight on this fact.
When M = H n , x 0 ∈ M and ϕ(x) = cosh(dist(x 0 , x)), M × ϕ R is H n+1 and the concept of being a graph coincides with that of equidistant graph in H n+1 . Then, general M × ϕ R seem to be good general ambient spaces where to extend results of type (A') and (B). In this paper we show that this is in fact the case, proving an extension of (A') for general M × ϕ R (theorems 9, 11 and 13), where the only conditions to be satisfied by M and ϕ are: the quotients | ∇ m ϕ| ϕ are bounded, the curvature of M and all its covariant derivatives are bounded and, when M is non-compact, M has a pole (that is, a point with empty cut-locus). May be the last condition can be weakened with some stronger analytic tools. On the other hand, like in [4], the only condition for M 0 is to be a "locally Lipschitz" graph. We also get a result of type (B) (theorems 15 and 16) imposing on M 0 the condition that its distance to M is bounded on it and on M a pinching condition on its sectional curvature related to the pinching of the hessian of ϕ.
We use mainly the methods in [4] and [13], where we have to introduce necessary technical tricks to choose the right functions to get estimates having into account the complications introduced by the terms containing ϕ and the curvature of M . We also needed to use the comparison theory of Riemannian Geometry to bound some functions of the distance to a point or to a hypersurface. Moreover we cannot use barriers as it is done in [13] because we are not in a model space where we know the evolution of some hypersurfaces and, at some points, we need to substitute barriers arguments for others (see the end of the proofs of Theorem 10 and Lemma 14).
Somewhat surprising for us has been the fact that the qualitative results of type (A') (not the estimates) do not depend on the curvature of M .
In contrast, the curvature of M plays an essential role in the theorem of convergence. The paper is organized as follows: in section 2 we state the notation and recall some lemmas that will be used later. In section 3 we collect the properties of the ambient spaces and their hypersurfaces that we shall need. Section 4 is a short comment about short time existence when M is compact and also a useful description of the evolution of some geometric quantities under an equivalent flow in the direction of the last coordinate u in M × ϕ R. In section 5 we give the gradient etimate, which, when M is compact, is enough to conclude the preservation of the property of being a graph under MCF . In section 6 we obtain the higher order estimates which give rise to the long time existence (and conclude -module the existence theorems below-with the main theorems of the paper when the initial condition is smooth). In section 7 we complete the discussion of the existence when M is non-compact, and, in section 8 we discuss the existence theorems for Lipschitz initial conditions. It is more usual to give the theorems for smooth and Lipschitz initial conditions simultaneously but, for the sake of non expert readers, we preferred to do it separately. In this way appears more clearly why in the Lipschitz case we cannot use the initial conditions to bound the second derivatives and beyond, which forces us to introduce bounds depending on t although these are becoming worse when t goes to 0. Finally, in section 9 we give a case where the flow has a limit and determine the limit.
Acknowledgments: This work was done while the first author was Visiting Professor at the University of Valencia in 2008, supported by a "ayuda del Ministerio de Educación y Ciencia SAB2006-0073." He wants to thank that university and its Department of Geometry and Topology by the facilities they gave him.
Second author was partially supported by DGI(Spain) and FEDER Project MTM2007-65852.

Preliminaries
In this paper we shall consider a Riemannian manifold (M, g) and an immersion F : M −→ M into another Riemannian manifold (M , g), and we shall denote by g the metric induced on M by the immersion F . We shall use the notation |X| or X, Y for indicating the g-norm, g-norm or g-norm of X or the g-product, g-product or g-product of X and Y if X and Y are, respectively, tangent to M , M or F (M ).
For any vector X ∈ T F (x) M , we shall denote by X the component of X tangent to F (M ).
We shall use ∇, ∇ and ∇ to denote the covariant derivative and the gradient in (M , g), (M, g) and (F (M ), g) respectively. By ∆, ∆ and ∆ we shall denote the corresponding laplacians. The convention sign for the laplacians wiil be: ∆f = tr∇ 2 f.
For any λ ∈ R, we shall use the notation: These functions satisfy the following computational rules: When λ < 0 one has s λ (t) = sinh( |λ| t) |λ| and c λ (t) = cosh( |λ| t). Given a point x 0 in M , we shall denote by r the g-distance to x 0 in M . We shall denote by The comparison theorem for the hessian of the distance function r says Lemma 1 (cfr [6] or [11]) If Sec ≥ µ, then, at points of M between x 0 and its cutlocus, one has This will be essential for the gradient estimates we shall obtain in Theorem 7. The next comparison theorem will be used for proving the convergence of the solution of (1.1) in certain cases.
Lemma 2 (cf. [10]) Let M be a complete riemannian manifold, with sectional curvature Sec ≤ k < 0, let M be a complete totally geodesic hypersurface of M . Let be the gdistance to M , ∂ := ∇ . Let us denote by S the Weingarten map (associated to ∂ )nof the hypersurface τ M at distance of M (that is, S X = −∇ X ∂ for every X ∈ T τ M ). At every point between M and the cut locus of M in M one has S X, X ≤ k s k c k |X| 2 for every X ∈ T τ M.
Also in the proof of convergence we shall use the following maximum principle for noncompact manifolds due to Ecker and Huisken: Let M be a manifold with a family of Riemannian metrics g t satisfying, for some 3 The geometric setting

The ambient space
Given (M, g) a n-dimensional riemannian manifold and ϕ : M −→ R + a C ∞ f unction, our ambient space will be M = (M × R, g) with g = g + ϕ(x) 2 du 2 (usually denoted as M × ϕ R).
If X (resp. θ) is a vector field (resp. a differential form) on M , we shall denote by the same letter the vector fields (resp. forms) on M induced by the family of embeddings In a g-orthonormasl local frame e 1 , ..., e n of (M, g), if θ 1 , ..., θ n is the dual frame, g = θ 1 2 + ... + θ n 2 . In the extension of this frame to M completed with du, g is written as and its dual frame e 0 , e 1 , ..., e n in M is a g-orthonormal frame related with the e i by e i = e i , and e 0 = 1 ϕ From now on i, j = 1, ..., n and a, b = 0, 1, ..., n. In this subsection we shall use he Einstein convention of summing repeated indices when one is a subindex and the other a super-index.

The Levi-Civita connection of M
In the frames given above, we compute the Cartan connection forms ω b a defined by Differentiating the formulae for θ a and comparing with (3.1), we have and the solution of these two equations is, denoting ϕ i := e i (ϕ), The relation between ω and ∇ is then we have Differentiating in (3.2) and applying equations (3.4), where Ω i j are the curvature 2-forms of (M, g). From the expressions for the curvature 2-forms we obtain for the components of the curvature tensor: The general setting in this paper is that the geometry of M is bounded, that is, |R|,  Now we use the expression of the curvature components to obtain the sectional curvatures:

7)
Remark 3 When M is the simply connected space of constant sectional curvature λ, M n λ and ϕ(x) = c λ ( r(x)) ( r : for some x 0 fixed), then M must be M n+1 λ ), the simply connected space of constant sectional curvature λ, as can be checked from the above formulae for curvatures. The frame e i of M induces a frame (on the submanifold F (M )) e i = F * e i = e i + u i ∂ u = e i + u i ϕe 0 , where u i = e i (u). In this frame, the matrix of the metric g of the submanifold and its inverse, and the dual frame θ i are given by:

The submanifold
A unit normal vector to F (M ) can be found using ξ = a ∇u + b∂ u , imposing the condition ξ, e i = 0 and dividing by the g-norm. We choose Then N, ∂ u = ϕ ϕ 2 | ∇u| 2 + 1 and N, e 0 = 1 The gradient of u in F (M ) can be computed using 3.9) or ∇u = g ij u i e j . In both cases we obtain: From (3.9) or (3.10) one gets and If we define σ = N, e 0 and v = 1 σ , the above formulae read From the above formulae we also get

Relations of ∇v and ∇ 2 u with the second fundamental form
For computations, on F (M ) we shall consider another local frame E i orthonormal and satisfying (∇ E i E j ) p = 0 at the point p where we are doing the computations. Using it we compute the hessian of u. First: and its norm where e 0 := e 0 − e 0 , N N = ϕ∇u. For ∇v we have, for every X tangent to F (M ), 4 Evolution of u up to tangential diffeomorphisms and short time existence when M is compact.
As it is well known, the equation (1.1) is not parabolic, but it is also known (see [5], or also [1])) that a solution F (p, t) of (1.1) can be written (as far as it remains a graph over M ) as the composition F (p, t) = F (Φ(p, t), t), where Φ(·, t) is a family of diffeomorphismd depending on t and F satisfies ∂F ∂t , N = H. and can be written, for each t, as a graph F (x, t) = (x, u(x, t)) over M . Using this parametrization of F the equation (4.1) becomes In order to apply the standard theory of P.D.E., we compute H in terms of u : M −→ R and ∇. We shall use the vectors e i and the expression for N given in section 3.2. The second fundamental form can be computed using the expressions (3.3) for ∇, which gives:

From (4.3) and (3.8),
By substitution of this expression in (4.2), which is a parabolic equation whereas v remains bounded. Then, when M is compact, the existence of solution of (4.5) for a small interval of time and smooth initial condition follows directly from the theory of parabolic equations. When M is non-compact, we postpone the discussion to section 7.
For sections 7 and 8 it will be useful to know the evolution under (4.1) of N , v and H. We get them now.
Using the parametrization F (x, t) = (x, u(x, t)) and (3.3), From the above evolution equation and the definition of v, From (4.4) we get the following evolution for H The following observation will also be useful Using the above expressions for X and ∇ f and the formulae (3.3) for ∇, It follows from (4.8) that | ∇ f | is bounded if |∇f | and | ∇u| are bounded, and from (4.9) that

Preserving the property of being a graph
The condition "cuts transversaly the curves s → (x, s)" in the definition of a graph means that σ := N, e 0 > 0, which is equivalent to say 1 < v = 1 σ < ∞. Therefore, our first goal is to obtain an upper bound for v. To achieve this, we need the evolution equation for v under (1.1).
where " 1" is the unit vector in the direction ∇u.
Proof First we compute ∆σ. To do so, we shall use an orthonormal frame of M of the form E 1 , ..., E n as was introduced before.
Now, let us compute From the expressions (3.6) of the components of R, it follows that Ric(e 0 , ∇u) = 0 (5.8) Joining the expressions for ∆v, ∆σ and ∂v ∂t , we obtain and sustitution in the evolution equation for v gives Notation 1 A general hypothesis in this work is that M has bounded geometry, in particular that | ∇ϕ| ϕ , | ∇ 2 ϕ| ϕ and | R| are bounded. Then there are constants η, µ 1 , µ 2 and µ satisfying and we define the constant ν by the formula Proof Using the notation (5.18) and (5.19), we have From (5.1) and forgetting about the negative summands, we reach the inequality: The hypothesis M non compact also imposes µ ≤ 0 (cf. (5.18)). On the other hand, if Sec ≥ 0, it is also true that Sec > µ for every µ < 0, then we can suppose, without loose of generality, that µ < 0.
Proof Let us recall that r denotes the g-distance from x 0 . As in the compact case, we have the inequality (5.20). An standard procedure to apply the maximum principle to v in the non-compact case is to look at the evolution of the product φv of v times some cut-function φ suitably chosen. Inspired in [5] and [13], we define first a function where the function f and the constant β will be defined later, and where, again, the constant α > 0 will be chosen later. Let us observe that: In order to compute the evolution of φv we need first to compute the evolution of r.
In order that f > 0 and the first summand in the last expression be zero, we take f = c µ −µ and the above expression gives where we have used By substitution of this value of β in (5.28), using c µ ≥ |µ|s µ , we obtain and, finaly, the evolution of φv where we have used ν ≥ 0 and |∇ϕ| ϕ ≤ | ∇ϕ| ϕ ≤ η, for the last inequality. If ν < 0, we can forget the last summand in (5.31). To consider both cases in the future we define Now, from the obvious inequality (|∇φ| − 2φ 2 ) 2 ≥ 0, we obtain Then, if we take α = 1 + η 4 , we obtain Let us consider the sets (where ")" means "]" if τ < T and "[" if τ = T ). Observe that φv vanishes on the boundary of S ρ,t and that By the maximum principle applied to the function φv, on S ρ,T , φv(F (x, t)) is bounded from above by the solution of the equation y = (2η 2 + (n − 1) ν)y with the initial conditions which, for every F (x, t) in the slice M t , taking ρ big enough, gives an upper bound of v(F (x, t)) (depending on t and ρ), which finishes the proof of the claim.
Remark 4 Let us observe that, unlike in the compact case, the conclusion of Theorem 7 is just a bound on v, with no conclusion about the preservation of the property of being a graph, which will be a consequence of the proof of the existence theorem in section 7 and the prolongation theorem 10. The reason of this is that, when M t is not compact, it is not clear that v bounded at each point implies M t is a graph. However, if M t is complete and has |A| bounded by an universal constant, it is true that v bounded at each point implies that M t is a graph over M . In fact, let us suppose that M t is a graph over a proper subset of M and has |A| bounded, let x ∈ M such that the line L = {x} × R does not cut M t . The distance d to L is a C ∞ function on M t , and its infimum is the distance δ between M t and L. Then, by Omori's Lemma (cf [9]), there is a sequence of points p n ∈ M t such that d(p n ) − δ < 1 n and |∇d|(p n ) < 1 n . For each p n , let q n ∈ L such that dist(q n , p n ) = d(p n ). Let γ n (t) be the geodesic from q n to p n realizing d(p n ). It is orthogonal to L at q n and, since the hypersurfaces u =constant are totally geodesic (cf. Remark 2), γ n is contained in such a hypersurface and orthogonal to the vector field e 0 . Moreover, it is an integral curve of ∇d which is then, orthogonal to e 0 . On the other hand ∇d = ∇d − ∇d, N N , then |∇d|(p n ) < 1 n implies ∇d(p n ) approach N (p n ) as n → ∞, then, as n → ∞, e 0 , N pn approach 0 and v goes to ∞. But, taking ρ > dist(x, x 0 ), this is in contradiction with (5.35). Then, on [0,T[, M t remains to be a graphic over all M .
The property above gives a geometric difference between the concepts of graph in M × ϕ R and R × ϕ M . In fact, for graphs in R × ϕ M and ϕ not constant, it is no longer true that the hypersurfaces u =constant be totally geodesic, then ∇d is not orthogonal to e 0 and the argument fails. It is not dificult, using this idea, to construct a hypersurface in the hyperbolic space H n+1 which is complete, a geodesic graph on an open set U of H n (U = H n ) and with v bounded. For instance, in the Poincaré's ball model, take a disc with boundary at the infinite and parallel to the equator.
Remark 5 For use in the long time existence theorem, it is convenient to give a more explicit (although less precise) bound for v than that obtained in (5.35). For this, first we consider a smaller set S ρ,t,γ than S ρ,t , defined, for any positive γ < 1, by From (5.35) and some obvious inequlities one has, for every F (x, t) ∈ S R,T,γ , Since this formula is true for any γ between 0 and 1, we have Notation 2 For the following sections, it will be convenient to introduce the following notation: given any ρ > 0 and 0 < γ < 1, we define ρ i , i = 1, 2, 3, ... by: It is simportant to reamrk that S ρ i ,τ,γ = S ρ i+1 ,τ ⊂ S ρ i ,τ and also that, given any ρ > 0, i ∈ N and 0 < γ < 1, there is a ρ such that ρ = ρ i .

Long time existence
First remember the evolution of |A| 2 , which we take from [2] (erasing the terms with H) because the notation used there is more similar to that of this paper.
And the formula of the theorem follows putting all this together and using the relation between R and R computed in section 3.1.2. Proof From (6.5) and (6.6) we obtain the following inequalities and define the constants K and C by From these definitions of K and C and Lemma 4 it follows We remark that K and C depend on T only through δ, δ depends on sup v and, according to Theorem 6, when M is compact and ν ≤ 0, sup v does not increase with time.
As we remarked in section 4, when M is compact, there is a solution of (1.1) for a maximal interval [0, T [. Let t 1 ∈ [0, T [. Let us suppose that (x 0 , t 0 ) is the point where g attains its maximum for t ≤ t 1 and 0 < t 0 < t 1 .
If g 0 := g(x 0 , t 0 ) > 1, then it has to satisfy 0 ≤ −2 δ(t 0 )g 2 Since ψ and g are bounded, |A| 2 is bounded on [0, T [ with a bound which does not depends on T when ν ≤ 0, because in this case neither K, C and δ depend on t.
Once we achieve the upper bound for |A| 2 , it follows, like in [7] and [8], that |∇ j A| is bounded for every j ≥ 1. If T < ∞, these bounds imply (cf. [7] pages 257, ff.) that X t converges (as t → T , in the C ∞ -topology) to a unique smooth limit X T . Now we can apply the short time existence theorem to continue the solution after T , contradicting the maximality of [0, T [. The next is a theorem modulo the existence theorem that will be proved in section 7. For this reason, in its hypotheses appears the existence of a short time existence theorem. Proof Let η, ν and ν be as defined by (5.18), (5.19) and (5.32). For the terms B and C in Lemma 8 we have the same bounds that in Theorem 9, so we define K and C again by (6.15) and (6.16) respectively. Now we shall use the functions φ and ζ defined in Theorem 7 and compute the evolution of φg for points in S ρ 1 ,T .
By substitution in the evolution of φg, we obtain Using ∇ψ = ψ ∇v and the expression (3.17) of ∇v, ∇ϕ ϕ , N e 0 + Ae 0 , then (6.21) On the other hand s µ 2 e 2βnt ≤ c µ e βnt 2 ≤ c µ (ρ 1 ) 2 on S ρ 1 ,T . (6.23) Pluging these inequalitues in (6.20), At a point where a maximum for φg is attained it follows from the above inequality that, if this happens when t = 0, and, multiplying by having into account that Having into account that s µ (ρ 1 ) ≤ c µ (ρ 1 ) and dividing by c µ (ρ 1 ) 3 , we get As the coefficient of is positive, the above inequality implies that where D is the biggest solution of the third order polynomial equation given by (6.27) when we change the inequality by an equality. D depends on the coefficients of the equation, then it only depends on the geometry of M and δ. From the above inequality we have that, every then, remembering that √ g = √ ψ |A| and the last inequality (6.26), This shows that, on S ρ 2 ,T = S ρ 1 ,T,γ , |A| is bounded by a bound C(0, ρ, γ, T ) depending T , γ and ρ (only through δ) and on M 0 . From here, a variant of the procedure used by Ecker, Huisken and Unterberger works. Let us give some details of these computations. Let us suppose, by induction, that, on S ρ 2+k ,T , where C(k, ρ, γ, T ) depends on ρ, γ and k (through ρ 2+k ), the bounds of |∇ i A| for 0 ≤ i ≤ k − 1 and the geometry of M (the bounds on |∇ j R|, j = 0, ..., k).
Following the same procedure of [7] and [8], we can find a constant D 1 depending on m, n, the geometry of M (that is, the bounds on |∇ k R|, k = 0, ..., m) and the bounds of |∇ k A|, Next, we define f := |∇ m A| 2 + ξ|∇ m−1 A| 2 , (6.32) being ξ a constant to be specified later. For the time derivative of f , (6.31) yields Observe that the last addend on the right hand side of (6.33) can be estimated using again (6.31). In fact, being D 2 a constant depending on C(m − 1, ρ, γ, T ). Substituting this in (6.33), we get If we choose ξ ≥ D 1 and D 3 := D 1 + ξD 2 , then Considering the same function φ as before, the same computational rules and the evolution of φ, we get The same computations than in the proof of Theorem 10 give now At a point of S ρ m+1 ,T where a maximum for φf is attained it follows from the above inequality that then, at any point in S ρ 2+m ,T = S ρ m+1 ,T,γ , since, on these points, φ ≥ α If we take ξ = max{D 1 , 7µ 2 (1 − γ) 2 }, we have, for points in S ρ 1 ,T,γ , Finally, by (6.32), |∇ m A| 2 ≤ f ≤ C(m, ρ, γ, T ) on S ρ 2+m ,T . Now the final argument cannot be exactly like in the compact case because there, to bound dist(F (p, t), x 0 ), one uses integration along t, but here it could happen that F (p, t) ∈ S ρ 2+m ,T but F (p, s) / ∈ S ρ 2+m ,T for s < t, then we do not know anything about the bounds of |A| in F (p, t). To avoid this problem, we have to use the parametrization F (·, t) of M t and the equation (4.1) or its equivalent (4.5). From the bounds on v obtained in the section 5 and (3.13) we get that the gradient of u is bounded on S ρ 1 ,T . From this, formula (4.3) and the above bounds on |∇ m A| we get that the higher order derivatives of u are bounded. And also from the bounds on v and |A| and formula (4.2) it follows that |u(t)| ≤ |u(0)|+ T 0 v ϕ n|A| is bounded on S ρ 2 ,T . Once we have these bounds, we have, on each )e βnt ≥ 0 a well defined limit of M t when t → T , which allows to continue the flow after T . Then T = ∞.

Existence of solution when M is non-compact
When M is complete non-compact and posses a pole x 0 , the existence of solution follows from the parabolic theory using the estimates of section 5 and 6 in the following way.
Given ρ 0 > 0, let us define ρ = ρ(ρ 0 ) by the expression For any ρ ≥ ρ, by the theory of parabolic equations, there is a unique solution u ρ of (4.5) on B(x 0 , ρ ) × [0, T ] satisfying the conditions: t)) ∈ S ρ 2+m ,T and, from the estimates (5.37), (6.29) and (6.38) we get that there are constants c(m, ρ, γ, T ) such that | ∇ m u R | ≤ c(m, ρ, γ, T ) and, since ρ is determined from ρ 0 and γ by (7.1), we can say that | ∇ m u R | are bounded by constants C(m, ρ 0 , γ, T ). From these bounds one gets also an estimate for |u ρ | in the same way that was done at the end of the proof of Theorem 10.
As a consequence, for every ρ 0 > 0, the family of C ∞ functions {u ρ } ρ ≥ρ converges to a smooth function u ρ 0 on B(x 0 , ρ 0 ) which is at least C 1 on t and a solution of (4.5) on B(x 0 , ρ 0 ) × [0, T ] (then, by parabolic theory, it is also C ∞ on t). Given a sequence ρ 1 0 < ρ 2 0 < · · · → ∞, for j > i the families {u ρ } ρ ≥ρ(ρ i 0 ) and {u ρ } ρ ≥ρ(ρ j 0 ) , coincide for ρ ≥ ρ(ρ j 0 ), then their limits u ρ i satisfy the property that u ρ j | b B(x 0 ,ρ i ) = u ρ i , then they define a smooth function u on M which is the C ∞ limit on the compacts of the family u ρ when ρ → ∞, and it is a solution of (4.5). Then, joining these arguments with those of the proof of Theorem 10 we have Theorem 11 Let M be complete non compact with a pole x 0 . If M = M × ϕ R and M 0 is a C ∞ graph over M , then there is a solution M t of (1.1) with initial condition M 0 which is a graph over M and is defined on [0, ∞[.

Existence of solution for a Lipschitz initial condition
The existence of solution of (1.1) when M 0 is a graph over M given by a Lipschitz continuous function follows by approximating M 0 by a sequence of smooth graphs M n and applying the existence theorems 9 and 11 to these approximations. But, in order to show that the solutions of (1.1) with initial conditions M n converge to a smooth solution, we need to get bounds of |∇ m A| that do not depend on the bounds on the initial condition, because these could go to ∞ as n → ∞ because the limit M 0 of M n is only Lipschitz. In this section we shall obtain these estimates.
Moreover, the constant α m depends on m, the geometry of M and T in the first case, and also on ρ and γ in the second.
Proof First, let us consider the case M compact. For obtaining the bound when m = 0 we start using the inequality (6.17) to obtain the following inequation for the evolution of tg Given t ∈]0, T [, let t 0 be the time when tg attains it maximum value t 0 g 0 in [0, t] × M (then g 0 = max x∈Mt 0 g(x)). By the maximum principle, from (8.2) we get If g 0 ≤ 1, then, by definition of maximum, t g ≤ t 0 g 0 ≤ t 0 ≤ t, then g ≤ 1 on M t , and, by the definition of g and (6.5), If g 0 ≥ 1, then t 0 √ g 0 ≤ t 0 g 0 and it follow from (8.3) and the definition of maximum that then, using again (6.5), with α 0 = (1 − δ)(max{K + C, 1})/(2δ). Now, let us suppose that (8.1) holds for values of m between 0 and m − 1. Let us show that it is true also for m. We start with the well known formulae (cf. [8]): where D is a constant which depends only on m and n. Also, by repetitive use of the Gauss formula for a submanifold, for the restriction of any tensor field B on M to M t one has where " * " has the same meaning that in [8].
If, for every k, |∇ k B| is bounded in M by some constant b k , from (8.7) and the induction hypothesis we obtain, renaming the constants each time we need, where b k = max{b k , c(n, k, M , T )}. Like in [3], we consider the function f m = t m+1 |∇ m A| 2 + ξt m |∇ m−1 A| 2 , where ξ is some constant that will be defined later, and estimate its evolution with time under (1.1). Using (8.6), we get From the induction hypothesis, and because 0 For the second summand in (8.13), denoting by r j the analog of b j in (8.8) when B is R, using (8.8) and the induction hypothesis Analogously, denoting byr j the analog of b j in (8.8) when B is ∇R, By substitution of all these inequalitues in (8.13) we get Using again the induction hypothesis, grouping terms and renaming constants, With this choosing of ξ, the coefficient of t m+1 |∇ m A| 2 in (8.21) is bounded from above by This shows that, on S ρ 1 ,T,γ , t|A| is bounded by a bound depending T and ρ 1 (only through δ) and on M 0 (only through the maximum of v in S ρ 1 ,T ). Renaming the constants, and, since 1 ≤ 1 + t, t|A| 2 ≤ α 0 (1 + t) on S ρ 1 ,T,γ = S ρ 2 ,T .
This proves (8.1) for M non-compact and m = 0. To prove it for every m we consider the evolution of φf m . From (5.30) and (8.21), computing like we did for getting (6.19) and renaming some constants, From here, reasoning like we did from equations (6.34) to (6.38), but taking, at the end,

Proof
We shall write the details for the case M non-compact. When M is compact the arguments are similar, simplified by the fact that we can take always M or M t instead of B(x 0 , ρ 0 ) or S ρ k ,t,γ respectively.
Let M k be a sequence of smooth manifolds given by smooth graphs u k over M and converging to M 0 . For each M k , let M kt be the smooth solution of (4.1) which has M k as initial condition, which exists and is defined for t ∈ [0, ∞[ by theorems 9 and 11. Each M kt is represented by the graph of a function u k (·, t) which is a solution of (4.5) with the initial condition u k (·, 0).
Given ρ 0 > 0 and T > 0, let us define ρ by (7.1). It follows from (5.37) and (3.13) that where c is a constant which depends only on ρ 0 (through ρ), γ, T , an upper bound of v on S k ρ,0 ⊂ M k and the bounds of ϕ. Since M 0 is Lipschitz, v is bounded on the corresponding S ρ,0 ⊂ M 0 . Since the M k converge to M 0 , the maxima of v on S k ρ,0 ⊂ M k will be near the maximum of v on S ρ,0 ⊂ M 0 for k big enough. Then we can take the constant c in (8.31) independent of k. From the definition of ρ, if (x, t) ∈ B(x 0 , ρ 0 ) × [0, T ] then (x, u k (x, t)) ∈ S ρ,T,γ and (8.31) holds on B(x 0 , R) × [0, T ].
The same argument as given at the end of last paragraph shows that, by Lemma 12, where the constants α m depend only on m, ρ 0 , γ and T . Now, let us consider the functions t). Through the relation between u and A, the bounds (8.31) and (8.32) give that for every t ∈]0, T ], there are constants β(m, t, ρ 0 , T, γ) depending only on m, t, ρ 0 , T and γ such that | ∇ m u kρ 0 (·, t)| ≤ β(m, t, ρ 0 , T, γ). (8.33) then, by the Ascoli-Arzela lemma, for each t and each ρ 0 , there is a subsequence u kρ 0 (·, t) which converges to a C ∞ function u 0ρ 0 (·, t).
If we consider the u kρ 0 (·, t) and u 0ρ 0 (·, t) as functions of t, the same arguments used in section 7 (now with bound (8.1)) show that ∂u ∂t and ∂ 2 u ∂t 2 are bounded on B(x 0 , ρ 0 ))×[t 0 , T ] by a constant depending only on t 0 > 0, ρ 0 , T and γ, but not on k. Once we know that, we can conclude that the limit u = u 0ρ 0 is, at least, of class C 1 on t and, since all u kρ 0 satisfy equation (4.2) on B(x 0 , ρ 0 )) × [t 0 , T ], also does u 0ρ 0 . Taking ρ 0 and T bigger, we have sequences of functions satisfying similar conditions and which coincide with the older ones for the older values of ρ 0 and T , then for these new values of ρ 0 and T we have a new limit function which coincides withe the older limit when restricted to the older values of ρ 0 and T . Letting ρ 0 an T go to infinity and t 0 to 0, this gives a function u 0 which is a solution of (4.2), satisfies the initial condition u 0 (·, 0) = the function defining M 0 as a graph, and, for every t ∈]0, ∞[, u 0 (·, t) is C ∞ as a function of M .

Some results on convergence
Lemma 14 Let M be a complete (may be compact) riemannian manifold, with bounded sectional curvature k 0 ≤ Sec ≤ k < 0 and having a complete totally geodesic hypersurface M . Let M t be the evolution of a complete hypersurface M 0 at time t by (1.1). Let us suppose that, for every x ∈ M 0 , the distance (x) from x to M is bounded from above by some constant 0 . If M is non-compact, we add the hypothesis that the norm |A t | of the Weingarten map A t of the hypersurface M t is bounded by a constant (depending on t). Then, one has s k ( (F (x, t))) ≤ s k ( 0 )e knt (9.1) Proof A calculation similar to that done before for u, but now for is and, using (2.3), one gets If M is compact, then the maximum principle states that is bounded by the solution of the ODE s k ( ) = n k s k ( ) with the initial condition (0) = 0 , which is s k ( ) = s k ( 0 )e nkt , from which the statement of the theorem follows.
If M is non-compact, let us consider first the evolution of , which follows from (9.3) as above we observe first that the added hypothesis implies that we can apply the maximum principle for noncompact manifolds given in Lemma 3 to f = − 0 . In fact, our hypothesis, together with the hypothesis k 0 ≤ Sec and the Gauss formulae give that the sectional curvature of each M t is bounded from below, which shows that the hypothesis on the growing of volume of geodesic balls of that Lemma is satisfied. Conditions (i) and (ii) are obvious. Condition iv) follows from the evolution equation ∂ ∂t g = −2Hα and the added hypothesis. Condition (iii) follows from |∇ | ≤ 1 and the fact that dµ t ≤ s λ n (r) r n dµ e , where λ is determined by µ 0 and the upper bounds of |A t | and dµ e is the euclidean measure. Then we can conclude, from the quoted theorem, that − 0 ≤ 0 for all time.
Once we know that is bounded, we can study the evolution of u = s k ( ) − s k ( 0 )e nkt , which, from (9.4) is ∂u ∂t = ∆u + n k u, (9.6) and, again, this equation satisfies the conditions of Theorem 4.3 in [4], the only condition that has to be checked now is (iii), which follows because |∇u| = c k ( )|∇ | ≤ c k ( 0 ) and now we know that ≤ 0 . Then, as before, we can conclude that u ≤ 0 and the theorem is proved. Proof Notice that the hypothesis on the lower bound of Sec is equivalent to ν ≤ 0, then, from theorems 6 and 9 it follows that both v and |∇ i A|, i = 0, 1, ... are bounded by some constants not depending on time, then, by Ascoli-Arzel, the M t converge to some limit M ∞ . The hypotheses −µ 1 > Sec implies −µ 1 > Sec, then we can apply Lemma 14 to conclude that M ∞ must be at distance 0 from M , then it must be M . Proof Looking at (5.38) and at the proof of Theorem 10, one observes that, when v and |∇ i A|, i = 0, 1, ..., are bounded on M 0 , then both v and |∇ i A| are bounded by some constants depending on time, but not on R. Then, we can apply Lemma 3 to the evolution equation (5.1) satisfied by v when ν ≤ 0 to conclude that v is bounded by a bound not depending on time. Using this fact in the proof of Theorem 10, we see that the bounds of |∇ i A| do not depend on time, then, as above, we can apply Ascoli-Arzela to conclude that M t has a limit M ∞ , and, using Lemma 14 as before, we conclude that M ∞ must be M . where Ric b 1b 1 is the Ricci curvature in the direction ∇u. The term − ϕ ϕ 2Hv 2 in this equation do not allow to apply the maximum principle as we did with (5.1) and, in fact, can be considered as the analytical reason why the property of being a graph is not preserved in general under the MCF in R × ϕ M . Looking again at (10.1) it is possible to think that we can get interesting results for hypersurfaces satisfying that the sign of H and the sign of ϕ is the same. But this condition has many drawbacks : 1. A computation shows that if M 0 is the graph of a function u, Then, when M is compact, it follows from this formula and the maximum principle that H has the same sign that ϕ if and only if u = 0.
2. When M is non-compact, the known proof (for the euclidean space) given in [4] that the sign of H is preserved uses the bounds of |A|, and, for them, bounds of v are also used. But in our case, are just the bounds of v which we do not know.
Another idea of why it is easy that the property of being a graph is preserved for M × ϕ R and not for R × ϕ M is given by the following pictures. In both M = H n and ϕ(u) = cosh u for the first and ϕ(x) = cosh(dist(x 0 , x)) for the second. Then M = H n+1 in both, but a graph for the first one corresponds to the sense of graph in this appendix (a graph for geodesics) and the second one corresponds to a graph in the sense o the previous sections of this paper, that is a graph for "equidistant" curves. a graphic for geodesics a graphic for equidistants which is not a graphic for geodesics