Translating Solutions of the Nonparametric Mean Curvature Flow with Nonzero Neumann Boundary Data in Product Manifold Mn×R?

Ya GAO Yi-Juan GONG Jing MAO

Abstract In this paper,the authors can prove the existence of translating solutions to the nonparametric mean curvature flow with nonzero Neumann boundary data in a prescribed product manifold Mn ×R, where Mn is an n-dimensional (n ≥2) complete Riemannian manifold with nonnegative Ricci curvature, and R is the Euclidean 1-space.

Keywords Translating solutions, Singularity, Nonparametric mean curvature flow,Convexity, Ricci curvature.

1 Introduction

The mean curvature flow (MCF for short) is one of the most important extrinsic curvature flows and has many nice applications. For instance, by using the curve shortening flow(i.e.,the lower-dimensional case of MCF), Topping [21] successfully gave an isoperimetric inequality on simply connected surfaces with Gaussian curvature satisfying some integral precondition. This result extends those isoperimetric inequalities (introduced in detail in, e.g., [6, 19]) obtained separately by Alexandrov, Fiala-Huber, Bol, and Bernstein-Schmidt. Applying the long-time existence and convergence conclusions of graphic MCF of any codimension in prescribed product manifolds(see[23]), Wang[24]showed that for a bounded C2convex domain D(with diameter δ and boundary ?D) in the Euclidean n-space Rnand φ : ?D →Rma continuous map, there exists a map ψ : D →Rm, with ψ|?D= φ and with the graph of ψ a minimal submanifold in Rn+m, provided ψ|?Dis a smooth map andThis conclusion provides classical solutions to the Dirichlet problem for minimal surface systems in arbitrary codimensions for a class of boundary maps. Specially, when m = 1, the existence of ψ was obtained by Jenkins and Serrin [14] already. Inspired by Wang’s work mentioned above, by applying the spacelike MCF in the Minkowski space Rn+m,n, Mao [16] can successfully get the existence of ψ for maximal spacelike submanifolds (with index n) in Rn+m,n.

Problem 1When does the MCF exist for all the time?

That is to say, under specified setting, there is no singularity formed during the evolution of MCF.

If there exists a constant vector V such that

then the evolving submanifold Xt: Mn→Rn+mis called a translating soliton of the MCF equation (1.1). Here (·)⊥denotes the normal projection of a prescribed vector to the normal bundle ofn Rm+n. It is easy to see that the translating soliton gives an eternal solution Xt= X0+tV to (1.1), which is called the translating solution. Translating solitons play an important role in the study of type-II singularities of the MCF. For instance, Angenent and Vel′azquez [4–5] gave some examples of convergence which implies that type-II singularities of the MCF are modeled by translating surfaces. Clearly, the existence of translation solutions to(1.1) can give a positive answer to Problem 1.

Huisken[13]considered the evolution of graphic hypersurfaces over a bounded domain(with smooth boundary)in Rnunder the MCF with a vanishing Neumann boundary condition(NBC for short), and proved that the flow exists for all the time and evolving graphic hypersurfaces in Rn+1converge to the graph of a constant function as t →∞. The vanishing NBC here has strong geometric meaning, that is, the evolving graphic hypersurface is perpendicular with the parabolic boundary during the evolution (or the contact angle between the evolving graphic hypersurface and parabolic boundary is. Is the vanishing NBC necessary? What about the non-vanishing case? There are many literatures working on this direction and we would like to mention some of them. When the dimension n satisfies n =1 or n =2, Altschuler and Wu[2–3] gave a positive answer to these questions. In fact, they proved:

?When n=1,a graphic curve defined over an open bounded interval evolves along the flow given by a class of quasilinear parabolic equations (of course, including the MCF as a special case), with arbitrary contact angle (i.e., with nonzero NBC), would exist for all the time, and the evolving curves converge as t →∞to a solution moving by translation with speed uniquely determined by the boundary data.

?When n=2,a graphic surface defined over a compact strictly convex domain(with smooth boundary)in R2evolves along the MCF,with arbitrary contact angle(i.e., with nonzero NBC),would exist for all the time,and the evolving surfaces converge as t →∞to a surface(unique up to translation) which moves at a constant speed (uniquely determined by the boundary data).

For the higher dimensional case, Guan [10] has given a partial answer. In fact, he can get the long-time existence of the evolution of graphic hypersurfaces, defined over a bounded domain (with smooth boundary)in Rn, under a nonparametric mean curvature type flow (i.e.,the MCF with a forcing term given by an admissible function defined therein) with nonzero NBC.However,the asymptotic behavior of the flow cannot be obtained in his setting. Zhou[25]extended Altschuler-Wu’s conclusion(see[3])to the situation that graphic surfaces were defined over a compact strictly convex domain (with smooth boundary) in 2-dimensional Riemannian surfaces M2with nonnegative Ricci curvature, and extended Guan’s conclusion (see [10]) to the situation that graphic hypersurfaces were defined over a bounded domain (with smooth boundary) in n-dimensional (n ≥2) Riemannian manifolds Mn. However, similar to Guan’s work (see [10]), Zhou [25] also cannot give the asymptotic behavior of the MCF with a forcing term (given by an admissible function) and with nonzero NBC in product manifolds Mn×R.Recently, Ma, Wang and Wei [15] improved Huisken’s work (see [13]) to a more general setting that the vanishing NBC therein can be replaced by a nonzero NBC of specialized type.

Our purpose here is trying to extend the main conclusion in [15] to a more general case –the ambient space Rn+1will be replaced by product manifolds of type Mn×R, where Mnis a complete Riemannian manifold of nonnegative Ricci curvature.

and the corresponding upward unit normal vector is given by

Moreover,the scalar mean curvature2In fact, the mean curvature H is computed as with the second fundamental form hij given However, if one uses another definition for hij, that is, (equivalently, choosing an opposite orientation for the unit normal vector), then and consequently, the evolution equation in (?) does not change. Obviously, there is no essential difference between these two settings.of G is

Hence, in our situation here, the evolution of G under the MCF with nonzero NBC in Mn×R with the metriccan be reduced to solvability of the following initial-boundary value problem(IBVP for short)

Here (1.3) is called compatibility condition of system (?), and a comma “,” in the subscript means doing covariant derivative with respect to a prescribed tensor. This convention will also be used in the sequel. For the IBVP (?), we can prove the following theorem.

Theorem 1.1If the Ricci curvature of Mnis nonnegative, ? ?Mnis a compact strictly convex domain with smooth boundary ??, then for the IBVP (?), we have

(1) the IBVP (?) has a smooth solution u(x,t) on×[0,∞);

(2) the smooth solution u(x,t) converges as t →∞to λt+w(x), i.e.,

where λ ∈R and w ∈C2,α((unique up to a constant) solving the following boundary value problem (BVP for short)

Here 0<α<1 and λ is called the additive eigenvalue of the BVP (?).

Remark 1.1(I) By (1.2), it is easy to know that

which, substituting into the first equation of (?), implies

where u = w(x) is the solution to the BVP (?). Integrating the above equality and using the divergence theorem, one can get

Clearly, if φ(x) ≡0, then λ = 0. Moreover, in this setting, for the IBVP (?), as t →∞, its smooth solution u(x,t) would converge to a constant function defined over ? ?Mn.

(II) We would like to mention one thing, that is, if Mn=Rnand φ(x) ≡0, then Theorem 1.1 here degenerates into Huisken’s main conclusion in [13]; if Mn= Rn, our main conclusion here becomes exactly [15, Theorems 1.1–1.2].

(III) Recent years, the study of submanifolds of constant curvature in product manifolds attracts many geometers’ attention. For instance, Hopf in 1955 discovered that the complexification of the traceless part of the second fundamental form of an immersed surface Σ2, with constant mean curvature (CMC for short) H, in R3is a holomorphic quadratic differential Q on Σ2, and then he used this observation to get his well-known conclusion that any immersed CMC sphere S2R3is a standard distance sphere with radiusBy introducing a generalized quadratic differentialfor immersed surfaces Σ2in product spaces S2×R and H2×R, with S2, H2the 2-dimensional sphere and hyperbolic surface, respectively, Abresch and Rosenberg[1] can extend Hopf’s result to CMC spheres in these target spaces. Meeks and Rosenberg[18] successfully classified stable properly embedded orientable minimal surfaces in the product space M×R,where M is a closed orientable Riemannian surface. In fact,they proved that such a surface must be a product of a stable embedded geodesic on M with R, a minimal graph over a region of M bounded by stable geodesics, M ×{t} for some t ∈R, or is in a moduli space of periodic multigraphs parameterized by P ×R+, where P is the set of primitive (non-multiple)homology classes in H1(M). Mazet, Rodr′?guez and Rosenberg [17] analyzed properties of periodic minimal or constant mean curvature surfaces in the product manifold H2×R, and they also constructed examples of periodic minimal surfaces in H2×R. In [20], Rosenberg, Schulze and Spruck showed that a properly immersed minimal hypersurface in M×R+equals some slice M ×{c} when M is a complete, recurrent n-dimensional Riemannian manifold with bounded curvature. Of course, for more information, readers can check references therein of these papers. Hence, it is interesting and important to consider submanifolds of constant curvature in the product manifold of type Mn×R. Based on this reason, in our setting here, it should be interesting and important to consider the following CMC equation with nonzero NBC:

Of course, all the symbols in the above system have the same meaning as those in (?). The existence and uniqueness of solution to the BVP (?) have been obtained recently (see [9] for details).

(IV) The evolution of space-like surfaces in the Lorentz 3-manifold M2× R under the MCF with arbitrary contact angle (of course, in this situation, the NBC is nonzero) has been investigated in [7], and the long-time existence and the existence of translating solutions to the flow have been obtained.

(V) As we know, if the warping function was chosen to be a constant function, then warped product manifolds would degenerate into product manifolds. Hence, one might ask “whether one could expect to get a similar conclusion to Theorem 1.1 in warped product manifolds or not?”. By constructing an interesting graphic hypersurface example in a prescribed warped product (see [25, Appendix A]), Zhou gave a negative answer to this question. Speaking in other words, he showed that the MCF with nonzero NBC in warped product manifolds would form singularities within finite time.

(VI) In fact, Huisken [13] considered the following IBVP:

which,as mentioned before,describes the evolution of graphic hypersurfaces over ? ?Rnunder the MCF with a zero NBC, and obtained the long-time existence, i.e., T = ∞. The vanishing NBC here means that for any q >0, and similar conclusion to Theorem 1.1 could be derived.

This paper is organized as follows. In Section 2, the time-derivative estimate, the gradient estimate, and the estimate for higher-order derivatives of u will be shown in detail, which naturally lead to the long-time existence of the IBVP (?). In Section 3, by using an approximating approach, the solvability of the BVP (?) can be given first, which will be used later to get the asymptotic behavior of solutions u to the IBVP (?).

2 The Long-Time Existence

For convenience, we use several notations as follows:

For vectors V, W or matrices A, B, we shall use the shorthand as follows:

First, by applying a similar method to that in the proof of[3, Lemma 2.2], we would like to show the time-derivative estimate for u.

Lemma 2.1For the IBVP (?), we have

That is to say, there exists some positive constant c0= c0(u0)∈R+such that for any (x,t) ∈×[0,T], we have

ProofWe first show that the maximum of |ut| must occur on (??×[0,T])∪?0. By a direct computation, we have

which implies

by directly applying the weak maximum principle.

Next, we expel the possibility that the maximum occurs at ??×[0,T]. Assume that

Lemma 2.2Assume that u(x,t)∈C3,2(?×[0,T)) is a solution to the IBVP (?), and the Ricci curvature of Mnis nonnegative. Then there exists a constant c1:=c1(n,?,u0,φ(x)) such that

ProofTo reach the conclusion of this lemma, we only need to prove that for 0

Let

where

and ζ is a positive constant which will be determined later. For convenience, denote by G =?φ(x)β.

We firstly show that the maximum of Φ(x)on×[0,T′]cannot be achieved at the boundary??×[0,T′].

Choose a suitable local coordinate around a point x0∈such that τnis the inward unit normal vector of ??, and τi, i = 1,2,··· ,n ?1 are the unit smooth tangent vectors of ??.Denote by Dτiu := ui, Dτju := uj, DτiDτju := uijfor 1 ≤i,j ≤n.3Covariant derivatives of other tensors can be simplified similarly. For instance, one has ωi = Dτiω, ωij =DτiDτj ω.By the boundary condition, one has

If Φ(x,t) attains its maximum at (x0,t0)∈??×[0,T′], then at (x0,t0), we have

Hence, by taking 0 < ζ < 2κ1, the maximum of Φ can only be achieved in ?×[0,T′]. By the way, there is one thing we would like to mention here, that is, in (2.1), the relation

holds. Here we have used the convention in Riemannian geometry to deal with the subscripts and superscripts, and this convention will also be used in the sequel.

Assume that Φ(x,t) attains its maximum at (x0,t0)∈?×[0,T′]. By direct calculation, we have

and

where

and

At (x0,t0), one can make a suitable change4This change can always be found. In fact, one can firstly rotate τi, i = 1,2,··· ,n, such that the gradient vector Du lies in the same or the opposite direction with τ1. Denote the hyperplane, which is orthogonal with τ1, by Π. Then rotate τ2,τ3,··· ,τn in Π, corresponding to an orthogonal matrix, such that the real symmetric matrices (uij)2≤i,j≤n, (σij)2≤i,j≤n change into diagonal matrices.to the coordinate vector fields {τ1,τ2,··· ,τn}such that |Du| = u1, (uij)2≤i,j≤nis diagonal, and (σij)2≤i,j≤nis diagonal. Clearly, in this setting, σ11=1. Besides, we have

Assume that u1is big enough such that u1, ω1, ω1, |Dω| and v are equivalent with each other at (x0,t0). Otherwise, the conclusion of Lemma 2.2 is proved. It is also noticeable that|ωi| ≤c2, i = 2,··· ,n for some nonnegative constant c2. Here, in the proof, c2is denoted to be a nonnegative constant which may change in different places but has nothing to do with T′.Since (βij)n×n≥k0(δij)n×n, one can easily get

Next we deal with J2,J3. In fact, using the nonnegativity of the Ricci curvature on Mn, we have

and

By (2.2)–(2.3), for 2 ≤i ≤n, we have

and

By (2.8), for 2 ≤i ≤n, it follows that

By (2.9), we have

By (2.10)–(2.11), we have

So

Now, we deal with J31,J32,J33,J34, respectively. It is obvious that

For the term J32,

Besides, we have

By (2.13)–(2.15), it follows that

Then, for J4, we have

By (2.6)–(2.7), (2.16) and (2.18), we write all the terms containing uiiin J as below

where the inequality holds since ax2+bx ≥?for a>0. Therefore, we can obtain

Hence,

By (2.4)–(2.5) and (2.19), at the maximum point (x0,t0), we can get

Let λ=min(σii), Λ=max(σii), i ≥2. Taking 0<ζ

where c3is independent of T′. Then the conclusion of Lemma 2.2 follows immediately.

By Lemmas 2.1–2.2, together with the Schauder estimate for parabolic PDEs (i.e., one can control C2,αby Cα,and then,by iterating,the regularity can be improved),we can get uniform estimates in any Ck-norm for the derivatives of u, and locally (in time) uniform bounds for the C0-norm,which leads to the long-time existence, with uniform bounds on all higher derivatives of u, to the IBVP (?). This finishes the proof of (1) of Theorem 1.1.

3 Asymptotic Behavior

In order to study the asymptotic behavior of the solution to the IVBP (?), we need the following two conclusions.

Lemma 3.1Let ? be a strictly convex bounded domain in Mn, n ≥2, and ?? ∈C3.Assume that ε > 0, the Ricci curvature of Mnis nonnegative, φ is a function defined onand there exists a positive constant L>0 such that

Let u ∈C2(∩C3(?) be a solution to the following BVP:

Then there exists a constant c4:=c4(n,?,L)>0 such that

ProofLet Φ(x) = log|Dω|2+ζβ, where ω = u+φ(x)β, and ζ will be determined later.Denote by G=?φ(x)β.

If one chooses 0 < ζ < 2κ1, using an almost same procedure as that in (2.1), it is easy to show that the maximum of Φ can only be achieved in the interior of ?.

Assume that Φ(x) attains its maximum at x0∈?, then we have at this point that

and

It follows that

where

and As in Lemma 2.2, one can choose suitable local coordinates around x0such that |Du| = u1,(uij)2≤i,j≤nis diagonal, and (σij)2≤i,j≤nis diagonal. Similarly, for the term I2, at x0, we have

Without loss of generality, one may assume that u1is big enough, then

Otherwise, the conclusion of Lemma 3.1 follows.

As in Lemma 2.2, the other terms J1, J2, J3, J5can be controlled similarly. In(3.2), taking 0<ζ

for some positive constant c5:= c5(n,?,L), which is independent of ε. Then the assertion of Lemma 3.1 follows.

Theorem 3.1Let ? be a strictly convex bounded domain in Mnwith C3boundary ??,n ≥2. Assume that the Ricci curvature of Mnis nonnegative. For φ(x) ∈C3(there exists a unique λ ∈R and ω ∈C2,α(solving the BVP (?). Moreover, the solution ω is unique up to a constant.

ProofWe use a similar method to that of the proof of [15, Theorem 1.2].

For each fixed ε>0, we firstly show the existence of the solution to the BVP (3.1). Based on the C1-estimate (see Lemma 3.1), the only obstacle is to derive a priori C0-estimate for the solution uε(x) to the BVP (3.1).

Together with the fact f(x)?uε(x)≥f(ρ)?uε(ρ) for x ∈?, we have

Similarly, one can get a lower bound for εuε(x). Therefore,holds for some nonnegative constant c7. By the standard theory of second-order elliptic PDEs,5In fact, here we can use two steps based on the standard theory of second-order elliptic PDEs to get the existence.Step 1: Denote by (?)i a family of boundary value problems indexed by a parameter i in a bounded closed interval, say [0,1]. Moreover, when i = 1, (?)1 is exactly the target equation (i.e., BVP (3.1)), and when i = 0,(?)0 is a BVP which can be solved.Step 2: Show that the set A of all i ∈[0,1]for which BVPs(?)i can be solved is not only open but also closed,which means A should be the whole segment 0 ≤i ≤1. Therefore, the target equation can be solved.one can get the existence of the solution to the BVP (3.1).

By

Let

where(λ,ω)is the solution to the BVP(?). It is easy to check that ~ω solves the following IBVP

As mentioned at the end of Section 2, by Lemmas 2.1–2.2, the Schauder theory for parabolic PDEs, one can obtain the long-time existence for the IBVP (?), i.e., T =∞.

Corollary 3.1For a solution u = u(x,t) to the IBVP (?), there exists a positive constant c10, independent of t, such that

ProofSet z(x,t):=u(x,t)?(x,t). By the linearization process, it is easy to check that z(x,t) satisfies

which implies the conclusion of Corollary 3.1.

By the strong maximum principle of second-order parabolic PDEs and Hopf’s lemma,one knows that osc(u)(t) is a strictly decreasing function unless u is a constant.

Now, we claim that

and

The second equality in (3.7) holds since u1,n(·,t) and u1,n(·,t) are uniformly convergent.

Besides, it is easy to check that u?satisfies

Clearly, by Corollary 3.1 and Lemma 3.2, we know that the limit of any solution to the IBVP (?) is= w+λt up to a constant, where (λ,ω) is the solution to the BVP (?). This completes the proof of (2) of Theorem 1.1.