Modified mean curvature flow of star-shaped hypersurfaces in hyperbolic space
aa r X i v : . [ m a t h . DG ] O c t MODIFIED MEAN CURVATURE FLOW OF STAR-SHAPEDHYPERSURFACES IN HYPERBOLIC SPACE
LONGZHI LIN AND LING XIAO
Abstract.
We define a new version of modified mean curvature flow (MMCF)in hyperbolic space H n +1 , which interestingly turns out to be the naturalnegative L -gradient flow of the energy functional defined by De Silva andSpruck in [DS09]. We show the existence, uniqueness and convergence of theMMCF of complete embedded star-shaped hypersurfaces with fixed prescribedasymptotic boundary at infinity. As an application, we recover the existenceand uniqueness of smooth complete hypersurfaces of constant mean curvaturein hyperbolic space with prescribed asymptotic boundary at infinity, whichwas first shown by Guan and Spruck in [GS00], see also [DS09]. Introduction
Let F ( z, t ) : S n + × [0 , ∞ ) → H n +1 be the complete embedded star-shaped hyper-surfaces (as complete radial graphs over S n + ) moving by the modified mean curvatureflow ( MMCF ) in hyperbolic space H n +1 , where S n + is the upper hemisphere of theunit sphere S n in R n +1 and the half-space model of H n +1 is used. That is, F ( · , t ) isa one-parameter family of smooth immersions with images Σ t = F ( S n + , t ) , satisfyingthe evolution equation(1.1) ∂∂t F ( z , t ) = ( H − σ ) ν H , ( z , t ) ∈ S n + × [0 , ∞ ) , F ( z ,
0) = Σ , z ∈ S n + , where H denotes the hyperbolic mean curvature of Σ t , σ ∈ ( − ,
1) is a constant, and ν H denotes the outward unit normal of Σ t with respect to the hyperbolic metric.By the half-space model of H n +1 , we mean H n +1 = { ( x ′ , x n +1 ) ∈ R n +1 : x n +1 > } equipped with the hyperbolic metric ds H = 1 x n +1 ds E , where ds E denotes the standard Euclidean metric on R n +1 . One identifies the hy-perplane { x n +1 = 0 } = R n × { } ⊂ R n +1 as the infinity of H n +1 , denoted by ∂ ∞ H n +1 .In this paper we consider the questions of the existence, uniqueness and con-vergence of the MMCF of complete embedded star-shaped hypersurfaces (as radial Mathematics Subject Classification.
Primary 53C44; Secondary 35K20, 58J35. graphs) in the hyperbolic space H n +1 with a fixed prescribed asymptotic boundaryat infinity, under some natural geometric conditions on the initial hypersurfaces.Namely, we consider the following Dirichlet problem of the MMCF:(1.2) ∂∂t F ( z , t ) = ( H − σ ) ν H , ( z , t ) ∈ S n + × [0 , ∞ ) , F ( z ,
0) = Σ , z ∈ S n + , F ( z , t ) = Γ( z ) , ( z , t ) ∈ ∂ S n + × [0 , ∞ ) , where σ ∈ ( − ,
1) and Γ = ∂ Σ is the boundary of a star-shaped C domainin { x n +1 = 0 } (the case of Γ being only continuous will also be discussed). Asan application, we shall also show that we can use the MMCF to deform a com-plete regular hypersurface to get one with constant hyperbolic mean curvature σ in hyperbolic space H n +1 .Mean curvature flow ( MCF ) was first studied by Brakke [B78] in the contextof geometric measure theory. Later, smooth compact surfaces evolved by MCF inEuclidean space were investigated by Huisken in [H84] and [H90], and on arbitraryambient manifolds in [H86]. The study of the evolution of complete graphs byMCF in R n +1 was also studied in [EH89], the result being improved in [EH91].See also [H89] for the nonparametric MCF with Dirichlet boundary condition. In[U03], Unterberger considered the MCF in hyperbolic space, namely, the case of σ = 0 in equation (1.1). And he obtained that if the initial surface Σ has boundedhyperbolic height over S n + then under the MCF, Σ t converges in C ∞ to S n + which hasconstant mean curvature 0. Note that no Dirichlet boundary data was imposed in[U03] . We shall remark that a similar MMCF (which is called the volume preservingMCF) was studied by Huisken in [H87] for closed, uniformly convex hypersurfacein R n +1 , where the constant σ in (1.1) was replaced by the average of the meancurvature of Σ t , see also [CM07] for this volume preserving MCF in the hyperbolicspace. With the average of the mean curvature of Σ t in the place of the constant σ , one cannot expect what the flow will converge to (if it converges), while we seedirectly that if the MMCF (1.1) converges then it converges to a hypersurface withconstant mean curvature σ . Namely, we can actually prescribe the constant meancurvature σ ∈ ( − ,
1) for the limiting hypersurface. This is the important featureand novelty of our version of MMCF defined in this work, which is also special forthe hyperbolic setting. Finally, we shall remark that it would be very interestingto see what the corresponding MMCF is in the Euclidean setting.The problem of finding smooth complete hypersurfaces of constant mean curva-ture in hyperbolic space with prescribed asymptotic boundary at infinity has alsobeen studied over the years, see [A82], [HL87], [Lin89], [T96] and [NS96]. In [GS00]Guan and Spruck proved the existence and uniqueness of smooth complete hyper-surfaces of constant mean curvature σ ∈ ( − ,
1) in hyperbolic space with prescribed
ODIFIED MEAN CURVATURE FLOW IN HYPERBOLIC SPACE 3 { x n +1 = ǫ } Γ ǫ Ω ǫ Σ ǫ O { x n +1 = 0 } Figure 1. asymptotic boundary at infinity. In [DS09], among other, De Silva and Spruck re-covered this result using the method of calculus of variations and representationtechniques. We remark that our paper can be thought of as a flow version of theirvariational method, see Section 2 . For the existence of hypersurfaces of constant(general) curvature in hyperbolic space H n +1 which have a prescribed asymptoticboundary at infinity, see [GSZ09] and [GS08] .Due to the degeneracy of the MMCF (1.2) for radial graphs at infinity (seeequation (2.8) below), we will begin with considering the approximate problem.For fixed ǫ > ǫ be the vertical translation of Γ to the plane { x n +1 = ǫ } and let Ω ǫ be the subdomain of S n + such that Γ ǫ is the radial graph over ∂ Ω ǫ (see Figure 1). We consider the following Dirichlet problem of the approximatemodified mean curvature flow ( AMMCF ):(1.3) ∂∂t F ( z , t ) = ( H − σ ) ν H , ( z , t ) ∈ Ω ǫ × [0 , ∞ ) , F ( z ,
0) = Σ ǫ , z ∈ Ω ǫ , F ( z , t ) = Γ ǫ ( z ) , for all ( z , t ) ∈ ∂ Ω ǫ × [0 , ∞ ) , where Σ ǫ = F (Ω ǫ , ∂ Σ ǫ = Γ ǫ and σ ∈ ( − ,
1) .For any ǫ ≥ P ∈ ∂ Σ ǫ = Γ ǫ (denoting Σ = Σ and Γ = Γ), the uniform star-shapedness of Γ ǫ implies there exist balls B R ( a, P )and B R ( b, P ) with radii R > R > a = ( a ′ , − σR ) and b = ( b ′ , σR ), respectively (see also “equidistance spheres” in Section 3.2 below),such that { x n +1 = ǫ } ∩ B R ( a, P ) is internally tangent to Γ ǫ at P and { x n +1 = ǫ } ∩ B R ( b, P ) is externally tangent to Γ ǫ at P . Note that in a small neighborhood B δ ( P ) around P for some δ >
0, both ∂B R ( a, P ) ∩ B δ ( P ) and ∂B R ( b, P ) ∩ B δ ( P )can be locally represented as radial graphs. To state our main results appropriately,we say that the initial hypersurfaces Σ ǫ ’s satisfy the uniform interior (resp. exterior) LONGZHI LIN AND LING XIAO { x n +1 = ǫ } Σ ǫ B R ( a, P ) B R ( b, P ) ( b ′ , σR )( a ′ , − σR ) Γ ǫ P Figure 2. local ball condition if for all ǫ ≥ P ∈ Γ ǫ , Σ ǫ ∩ B δ ( P ) ∩ B R ( a, P ) = { P } (resp. Σ ǫ ∩ B δ ( P ) ∩ B R ( b, P ) = { P } , see Figure 2), and thelocal radial graph ∂B R ( a, P ) ∩ B δ ( P ) (resp. ∂B R ( b, P ) ∩ B δ ( P )) has a uniformLipschitz bound depending only on the star-shapedness of Γ. If Σ ǫ ’s satisfy bothof the uniform interior and exterior local ball conditions, then we say Σ ǫ ’s satisfythe uniform local ball condition. The main results in this paper are the following.
Theorem 1.1.
Let Γ be the boundary of a star-shaped C domain in { x n +1 =0 } = ∂ ∞ H n +1 and Γ ǫ be its vertical lift to { x n +1 = ǫ } for ǫ > sufficiently small.Let Σ = lim ǫ → Σ ǫ be the limiting hypersurface of radial graphs Σ ǫ ∈ C (Ω ǫ ) with ∂ Σ ǫ = Γ ǫ . Suppose Σ ǫ ’s have a uniform Lipschitz bound and satisfy theuniform local ball condition. Then (i) there exists a unique solution F ( z , t ) ∈ C ∞ ( S n + × (0 , ∞ ) ∩ C , + ( S n + × (0 , ∞ )) ∩ C ( S n + × [0 , ∞ )) to the MMCF (1.2) ; (ii) there exist t i ր ∞ such that Σ t i = F ( S n + , t i ) converges to a unique sta-tionary smooth complete hypersurface Σ ∞ ∈ C ∞ ( S n + ) ∩ C ( S n + ) (as aradial graph over S n + ) which has constant hyperbolic mean curvature σ and ∂ Σ ∞ = Γ asymptotically. Also, each Σ t is a complete radial graph over S n + ; (iii) if additionally Σ ǫ has mean curvature H ǫ ≥ σ for all ǫ > sufficientlysmall, then Σ t converges uniformly to Σ ∞ for all t . In fact, if Σ ǫ has hyperbolic mean curvature H ǫ ≥ σ for all ǫ > ǫ ’s can be relaxed. Theorem 1.2.
Let Γ and Γ ǫ be as in Theorem 1.1 and Σ = lim ǫ → Σ ǫ be thelimiting hypersurface of radial graphs Σ ǫ ∈ C (Ω ǫ ) ∩ C (Ω ǫ ) with ∂ Σ ǫ = Γ ǫ . Such initial hypersurfaces exist and can be constructed explicitly since the balls B R ( a, P )and B R ( b, P ) can be constructed with uniform radii (see equation (8.5)) and the tangent planeto them at P can be computed explicitly as well (see equation (6.2)). ODIFIED MEAN CURVATURE FLOW IN HYPERBOLIC SPACE 5
Suppose Σ ǫ has mean curvature H ǫ ≥ σ for all ǫ > sufficiently small and Σ ǫ ’shave a uniform Lipschitz bound and satisfy the uniform exterior local ball condition.Then there exists a unique solution F ( z , t ) ∈ C ∞ ( S n + × (0 , ∞ ) ∩ C , ( S n + × (0 , ∞ )) ∩ C ( S n + × [0 , ∞ )) to the MMCF (1.2) . Moreover, Σ t = F ( S n + , t ) convergesuniformly for all t to a unique stationary smooth complete hypersurface Σ ∞ ∈ C ∞ ( S n + ) ∩ C ( S n + ) (as a radial graph over S n + ) which has constant hyperbolicmean curvature σ and ∂ Σ ∞ = Γ asymptotically. Also, each Σ t is a complete radialgraph over S n + . We will give an example of “good” initial hypersurfaces in Theorem 1.2 in Section8. As an immediately corollary of Theorem 1.1 or Theorem 1.2, we recover thefollowing existence and uniqueness results due to Guan and Spruck.
Corollary 1.3. [GS00]
Suppose Γ is the boundary of a star-shaped C domain in { x n +1 = 0 } and let | σ | < . Then there exists a unique smooth complete hypersur-face Σ of constant hyperbolic mean curvature σ in H n +1 with asymptotic boundary Γ . Moreover, Σ may be represented as a radial graph over S n + of a function in C ∞ ( S n + ) ∩ C ( S n + ) . With the aid of an a priori interior gradient estimate (see Section 9) and viaan approximation argument, the regularity of the boundary data Γ in Theorem 1.1and Theorem 1.2 could be further relaxed to be only continuous and a similar resultstill holds (see Theorem 9.2 below). As an application, we have
Corollary 1.4. [GS00] , [DS09] Suppose Γ is the boundary of a continuous star-shaped domain in { x n +1 = 0 } and let | σ | < . Then there exists a unique smoothcomplete hypersurface Σ of constant hyperbolic mean curvature σ in H n +1 withasymptotic boundary Γ . Moreover, Σ may be represented as a radial graph over S n + of a function in C ∞ ( S n + ) ∩ C ( S n + ) . The paper is organized as follows. In Section 2 we set up the problems, namely,the Dirichlet problems for the MMCF and AMMCF for radial graphs in hyperbolicspace. In Section 3 we state the short-time existence result for the AMMCF anddiscuss the equidistance spheres in H n +1 which will serve as good barriers in manysituations. We will prove Theorem 1.1 in sections 4 −
7. In Section 4 we prove aglobal gradient estimate for the solution to the AMMCF and therefore the long-time existence of the AMMCF. In Section 5 we prove the uniform gradient estimatefor the solutions to the AMMCF’s, which leads to the long-time existence of theMMCF, while in Section 7 we show the uniform convergence of the MMCF in thecase of H ǫ ≥ σ initially for all ǫ >
0. We show the boundary regularity of theMMCF in Section 6. In Section 8 we will prove Theorem 1.2 and give an exampleof “good” initial hypersurfaces in Theorem 1.2. In Section 9 we prove a version of a priori interior gradient estimate and therefore the existence result of the MMCFwith only continuous boundary data.
LONGZHI LIN AND LING XIAO MMCF and AMMCF for radial graphs in hyperbolic space
Let Ω ⊆ S n + , and suppose that Σ is a radial graph over Ω with position vector X in R n +1 . Then we can write X = e v ( z ) z , z ∈ Ω , for a function v defined over Ω. We call such function v the radial height of Σ.2.1. Gradient flow.
As in [DS09], one can define the energy functional I (Σ) as-sociated to Σ : I (Σ) = I Ω ( v ) = A Ω ( v ) + nσV Ω ( v )= Z Ω p |∇ v | y − n d z + nσ Z Ω v ( z ) y − ( n +1) d z , (2.1)where y = z n +1 and ∇ denotes the covariant derivative on the standard unit sphere.Note that in this energy functional I (Σ), the term A Ω corresponds to the area of Σ(under the hyperbolic metric) and the term V Ω corresponds to the radial volume ofthe cone region between Σ and the origin (up to a constant), see [DS09] for details .Then for a smooth solution F ( z , t ) to the MMCF (1.1), which can be representedas a complete radial graph over Ω = S n + , namely, F ( z , t ) = X ( z , t ) = e v ( z ,t ) z , ( z , t ) ∈ S n + × (0 , ∞ ) , we have ddt I (Σ t ) = − n Z Ω ( H − σ ) p |∇ v | y − n d z = − n Z Ω h ∂ F /∂t , ( H − σ ) ν H i H dA = − n Z Ω ( H − σ ) dA ≤ , (2.2)where in the first equality we used the Stokes’ theorem, equation (2.8) (see below)and the fact that (see equation (1.2) of [DS09])div z y − n ∇ v p |∇ v | ! = nHy − ( n +1) in Ω , and the second equality is just the first variation formula for I .From this point of view, one sees that the MMCF is the natural negative L -gradient flow of the energy functional I (Σ) . We have: Lemma 2.1.
Let F ( z , t ) = e v ( z ,t ) z be a smooth radial graph solution to the AMMCF (1.3) in Ω × [0 , T ] . Then for all t ∈ [0 , T ) we have (2.3) I (Σ ǫt ) + n Z t Z Ω ( H − σ ) dAdt = I (Σ ǫ ) . Remark . We point out that equation (2.2) is a natural analog of the well-knownformula for the classic MCF: ddt
Area(Σ t ) = − Z H dA ≤ . ODIFIED MEAN CURVATURE FLOW IN HYPERBOLIC SPACE 7
The hyperbolic mean curvature.
We will begin with fixing some nota-tions, and collecting some relevant facts about the hyperbolic space H n +1 . Wherenecessary, expressions in the Euclidean and hyperbolic spaces will, be denoted bythe subscript or superscript E and H , respectively. Let ∇ denote the covariantderivative on the standard unit sphere S n in R n +1 and y = e · z for z ∈ S n ⊂ R n +1 , where, throughout this paper, e is the unit vector in the positive x n +1 directionin R n +1 , and ‘ · ’ denotes the Euclidean inner product in R n +1 . Let τ , ..., τ n bea local frame of smooth vector fields on the upper hemisphere S n + . We denote by γ ij = τ i · τ j the standard metric of S n + and γ ij its inverse. For a function v on S n + ,we denote v i = ∇ i v = ∇ τ i v, v ij = ∇ j ∇ i v , etc.Suppose that locally Σ is a radial graph over Ω ⊆ S n + . Then the Euclideanoutward unit normal vector and mean curvature of Σ are respectively ν E = z − ∇ vw and H E = a ij v ij − nne v w , where a ij = γ ij − γ ik v k v j w , ≤ i, j ≤ n and w = (1 + |∇ v | ) / . The hyperbolic outward unit normal vector is ν H = u ν E , where u = e · X = e · e v z = y e v is called the height function. Moreover, using the relation between the hyperbolicand Euclidean principle curvatures κ Hi = e · ν E + u κ Ei , i = 1 , ..., n , we have (see equation (2.1) of [GS00], cf. equation (1.8) of [GS08])(2.4) H = e · ν E + u H E , which gives the hyperbolic mean curvature of Σ :(2.5) H = y e v H E + y − e · ∇ vw = y a ij v ij n w − e · ∇ vw , and therefore(2.6) a ij v ij = ny ( Hw + e · ∇ v ) . LONGZHI LIN AND LING XIAO
Degenerate parabolic equation.
The first equation of the MMCF (1.2)implies(2.7) (cid:28) ∂∂t F , ν H (cid:29) H = (cid:28) ∂∂t ( e v z ) , ν H (cid:29) H = e v uw ∂v∂t = 1 yw ∂v∂t = H − σ . Therefore by equation (2.5) we have(2.8) ∂v ( z , t ) ∂t = yw ( H − σ ) = y a ij v ij n − y e · ∇ v − σyw . Suppose Γ is the radial graph of a function e φ over ∂ S n + , i.e., Γ can be representedby X = e φ ( z ) z , z ∈ ∂ S n + . Then one observes that the Dirichlet problem for the MMCF (1.2) is equivalentto the following (degenerate parabolic) Dirichlet problem (the MMCF for radialgraphs):(2.9) ∂v ( z , t ) ∂t = y a ij v ij n − y e · ∇ v − σyw , ( z , t ) ∈ S n + × (0 , ∞ ) ,v ( z ,
0) = v ( z ) , z ∈ S n + ,v ( z , t ) = φ ( z ) , ( z , t ) ∈ ∂ S n + × [0 , ∞ ) , where we represent Σ as the radial graph of the function e v over S n + and v (cid:12)(cid:12) ∂ S n + = φ .2.4. Approximate problem.
Due to the degeneracy of equation (2.9) at infinity(i.e., y = 0), we consider the corresponding approximate problem for a fixed ǫ > ∂v ( z , t ) ∂t = y a ij v ij n − y e · ∇ v − σyw , ( z , t ) ∈ Ω ǫ × (0 , ∞ ) ,v ( z ,
0) = v ǫ ( z ) , z ∈ Ω ǫ ,v ( z , t ) = φ ǫ ( z ) , ( z , t ) ∈ ∂ Ω ǫ × [0 , ∞ ) , where we represent Σ ǫ as the radial graph of the function e v ǫ over Ω ǫ and v ǫ (cid:12)(cid:12) ∂ Ω ǫ = φ ǫ , and φ ǫ is a function defined on ∂ Ω ǫ ⊂ S n + such that Γ ǫ can be represented as aradial graph of e φ ǫ over ∂ Ω ǫ , i.e.,(2.11) X = e φ ǫ ( z ) z , z ∈ ∂ Ω ǫ . We denote the regular solution to (2.10) by v ǫ .3. The short-time existence and equidistance spheres
Short-time existence.
In the rest of the paper, we will focus on the caseof σ ∈ [0 ,
1) and the case of σ ∈ ( − ,
0) can be dealt with in the same way afterusing the hyperbolic reflection over S n + . The standard parabolic PDE theory withSchauder estimates guarantees the short-time existence of a regular solution (up to ODIFIED MEAN CURVATURE FLOW IN HYPERBOLIC SPACE 9 the parabolic boundary) to the AMMCF (2.10) with a C ∞ initial hypersurface andcompatible boundary data (i.e., H = σ on ∂ Σ ǫ ). And for a C ∞ initial hypersurfacewith incompatible boundary data, a solution exists at least for short time andbecomes regular immediately after t = 0 (cf. [Ha75]) . This is the statement of thenext lemma. Lemma 3.1.
There exists T ⋆ǫ > such that the AMMCF (2.10) with initial data v ǫ ∈ C ∞ (Ω ǫ ) has a solution v ǫ ∈ C ∞ (Ω ǫ × [0 , T ⋆ǫ )) except on the corner ∂ Ω ǫ × { t =0 } . For less regular (e.g. C ) initial and boundary data, the short-time existencelemma will remain true (see e.g. [L96, theorem 8.2] and [LSU68, theorem 4.2,P.559]) . Lemma 3.2.
There exists T ⋆ǫ > such that the AMMCF (2.10) with initial data v ǫ ∈ C (Ω ǫ ) has a solution v ǫ ∈ C ∞ (Ω ǫ × (0 , T ⋆ǫ )) ∩ C (Ω ǫ × [0 , T ⋆ǫ )) . Moreover, as we shall see, the passage to the limit of { v ǫ } as ǫ → ǫ .3.2. Equidistance spheres.
In the following, let T ǫ (possibly ∞ ) be the maximaltime up to which the AMMCF (1.3) for radial graphs or equivalently the solutionto (2.10) exists, and let V ǫ = ∪ ≤ t ≤ T ǫ Σ ǫt denote the flow region in H n +1 , whereΣ ǫt = F (Ω ǫ , t ) is the hypersurface moving by the AMMCF (1.3) at time t .Our estimates in the proof of the main theorems are all based on the followingfact: let B = B R ( a ) be a ball of radius R centered at a = ( a ′ , − σR ) ∈ R n +1 where a ′ ∈ R n and σ ∈ ( − , S = ∂B ∩ H n +1 has constant hyperbolic meancurvature σ with respect to its outward normal. Similarly, let B = B R ( b ) be a ballof radius R centered at b = ( b ′ , σR ) ∈ R n +1 , then S = ∂B ∩ H n +1 has constanthyperbolic mean curvature σ with respect to its inward normal. These so calledequidistance spheres will serve as good barriers in many situations (see Lemma 3.3below). Let D ⊂ { x n +1 = 0 } be the domain enclosed by Γ and D ǫ ⊂ { x n +1 = ǫ } be the domain enclosed by Γ ǫ . Lemma 3.3.
Let B and B be balls in R n +1 of radius R centered at a = ( a ′ , − σR ) and b = ( b ′ , σR ) , respectively. (i) If Σ ǫ ⊂ B , then V ǫ ⊂ B (see Figure 3) ; (ii) If B ∩ { x n +1 = ǫ } ⊂ D ǫ and B ∩ Σ ǫ = ∅ , then B ∩ V ǫ = ∅ ; (iii) If B ∩ D ǫ = ∅ and B ∩ Σ ǫ = ∅ , then B ∩ V ǫ = ∅ . Proof.
This lemma follows from the maximum principle by performing homotheticdilations (hyperbolic isometries) from ( a ′ ,
0) and ( b ′ , B continuously until it contains Σ ǫ ; for (ii) and (iii) we shrink B and { x n +1 = ǫ } B ( a ′ , − σR ) Σ ǫt Γ ǫ S Figure 3. B until they are respectively inside and outside Σ ǫ . We note that Σ ǫt satisfiesequation (2.8) as a radial graph and its mean curvature is calculated with respectto its outward normal direction. Also S , S have constant mean curvature σ withrespect to the outward and inward normal respectively, and locally as radial graphsthey both satisfy equation (2.8) (statically) too. Then from the maximum principlewe see that Σ ǫt cannot touch B or B when we reverse this process. (cid:3) Similarly, for the stationary case we have
Lemma 3.4. [GS00, lemma 3.1]
Let B and B be balls in R n +1 of radius R centered at a = ( a ′ , − σR ) and b = ( b ′ , σR ) , respectively. Suppose Σ has constanthyperbolic mean curvature σ . Then (i) If ∂ Σ ⊂ B , then Σ ⊂ B ;(ii) If B ∩ { x n +1 = ǫ } ⊂ D ǫ , then B ∩ Σ = ∅ ; (iii) If B ∩ D ǫ = ∅ , then B ∩ Σ = ∅ . Global gradient bounds and long time existence of the AMMCF
Before we begin our proof, we would like to collect some important formulasthat were first derived in [GS00]. From now on, we assume the local vector fields τ , ..., τ n to be orthonormal on S n + so that γ ij = δ ij and thus a ij = δ ij − v i v j w . Thecovariant derivatives of y are(4.1) y i = ∇ i y = ( e · z ) i = e · τ i ,y ij = ∇ i ∇ j y = e · ∇ i ∇ j z = e · ∇ i τ j = − yδ ij . Therefore e · ∇ y = X ( e · τ i ) = 1 − y , ∇ v · ∇ y = e · ∇ v and ∇ w · ∇ y = e · ∇ w . ODIFIED MEAN CURVATURE FLOW IN HYPERBOLIC SPACE 11
Note that we also have the identities a ij v i = v j w , a ij v i v j = 1 − w , X a ii = n − w . Moreover,(4.2) w i = v k v ki w , w ij = v k v kij w + 1 w a kl v ki v lj and ( ∇ k a ij ) v ij = − w a ij w i v kj . Straight forward calculations also show that( e · ∇ v ) i = ( e · τ k v k ) i = e · τ k v ki − yv i = y k v ki − yv i , ( e · ∇ v ) ij = e · τ k v kij − yv ij − e · τ j v i = y k v kij − yv ij − y j v i and(4.3) ∇ v · ∇ ( e · ∇ v ) = v i ( e · τ k v ki − yv i ) = w e · ∇ w − y ( w − . We also have the formula for commuting the covariant derivatives(4.4) v ijk = v kij + v j δ ik − v k δ ij . Now we are ready to state our first main technical lemma.
Lemma 4.1.
Let v ∈ C , (Ω × (0 , T )) be a function satisfying equation (2.8) forsome T > and Ω ⊆ S n + . Then (4.5) ( ∂∂t − L ) w ≤ − σ ( e · ∇ v ) + y ( w − nw − H w ≤ w in Ω × (0 , T ) , where L is the linear elliptic operator L ≡ y n (cid:18) a ij ∇ ij − w a ij w i ∇ j − nwy ( σ ∇ v + w e ) · ∇ (cid:19) . Proof.
By equation (2.8) we have ∂∂t w = 1 w ∇ v · ∇ ( v t ) = ∇ vw · ∇ ( yw ( H − σ ))= ∇ vw · ( ∇ yw ( H − σ ) + y ∇ w ( H − σ ) + yw ∇ H )= e · ∇ v ( H − σ ) + y ( H − σ ) w ∇ v · ∇ w + y ∇ v · ∇ H Differentiating both sides of the equation (2.6) with respect to τ k gives (using alsothe equation (4.2))( ∇ k a ij ) v ij + a ij v ijk = a ij v ijk − w a ij w i v kj = ny ( H k w + Hw k + ( e · ∇ v ) k ) − ny ( Hw + e · ∇ v ) y k . Therefore a ij v kij = ny ( H k w + Hw k + ( e · ∇ v ) k ) − ny ( Hw + e · ∇ v ) y k + 2 w a ij w i v kj − v k w + ( n − w ) v k (4.6) and a ij v k v ijk − w a ij w i v k v kj = ny ∇ v · ( ∇ Hw + H ∇ w + ∇ ( e ·∇ v )) − n e · ∇ vy ( Hw + e ·∇ v ) . Note that we also have a ij w ij = a ij ( v k v kij w + 1 w a kl v ki v lj )= 1 w ( v k a ij ( v ijk − v j δ ik + v k δ ij )) + 1 w a ij a kl v ki v lj . Now by the definition of the operator L , we have( ∂∂t − L ) w = e · ∇ v ( H − σ ) + y ( H − σ ) w ∇ v · ∇ w + y ∇ v · ∇ H − y n (cid:18) a ij w ij − w a ij w i w j − nwy ( σ ∇ v + w e ) · ∇ w (cid:19) = e · ∇ v ( H − σ ) + y ( H − σ ) w ∇ v · ∇ w + y ∇ v · ∇ H − y n (cid:20) nwy ∇ v · (cid:0) ∇ Hw + H ∇ w + ∇ ( e · ∇ v ) (cid:1) − n e · ∇ vwy ( Hw + e · ∇ v ) (cid:21) + y a ij v i v j nw − y ( w − nw ( n − w ) − y nw a ij a kl v ki v lj − y w n a ij w i v k v kj + 2 y wn a ij w i w j + yw ( σ ∇ v + w e ) · ∇ w = e · ∇ v (2 H − σ ) − yw ( ∇ v · ∇ ( e · ∇ v ) − w e · ∇ w ) + ( e · ∇ v ) w + y nw (1 − w ) − y ( w − w )(1 − n + 1 nw ) − y nw a ij a kl v ki v lj ≤ e · ∇ v (2 H − σ ) − yw ( − y ( w − e · ∇ v ) w + y nw (1 − w ) − y ( w − w )(1 − n + 1 nw ) − w ( Hw + e · ∇ v ) = − σ ( e · ∇ v ) + y n ( w − w ) − H w . Here we used the equations (4.3), (2.6) and (by Cauchy-Schwarz inequality) a ij a kl v ki v lj ≥ n ( a ij v ij ) = ny ( Hw + e · ∇ v ) . Hence we conclude that ( ∂∂t − L ) w ≤ w . (cid:3) For any ǫ ≥ z ∈ ∂ Ω ǫ corresponding to P = e φ ǫ ( z ) z ∈ Γ ǫ ,let B ǫ = B ǫR ( a ′ , − σR ) and B ǫ = B ǫR ( b ′ , σR ) be the (Euclidean) balls with radii R > R >
0, respectively, such that B ǫ and B ǫ are tangent at P , and B ǫ ∩ { x n +1 = ǫ } is internally tangent to Γ ǫ at P , and B ǫ ∩ { x n +1 = ǫ } is externally ODIFIED MEAN CURVATURE FLOW IN HYPERBOLIC SPACE 13 tangent to Γ ǫ at P . Recall that S ǫ = ∂B ǫ ∩ H n +1 has constant (hyperbolic) meancurvature σ with respect to its outward normal while S ǫ = ∂B ǫ ∩ H n +1 has constantmean curvature σ with respect to its inward normal. Moreover, we can represent S ǫ and S ǫ near P as radial graphs X i = e ϕ ǫi z , i = 1 , z ∈ Ω ǫ ∩ B ǫ ( z ) where ǫ depends only on the radii of B ǫi ’s and the uniformly star-shapedness of Γ. Thenthe uniform local ball condition implies(4.7) ϕ ǫ ( z ) ≤ v ǫ ≤ ϕ ǫ ( z ) , z ∈ Ω ǫ ∩ B ǫ ( z ) . From this point of view, one sees that S ǫ and S ǫ serve as good local barriers of Σ ǫ around P and |∇ v ǫ | ( P ) ≤ C , where C is independent of ǫ and P ∈ Γ ǫ . Moreover,note that S ǫ and S ǫ have constant hyperbolic mean curvature σ and they are staticunder the MMCF (2.8) as local radial graphs. Therefore by the maximum principle,they also serve as good local barriers of Σ ǫt around ( P , t ) for all t ∈ [0 , T ǫ ) and wehave(4.8) |∇ v ǫ | ( P , t ) ≤ C for all t ∈ [0 , T ǫ ), where C is independent of ǫ and P by the uniform local ballcondition. Lemma 4.2.
Locally S ǫ is interior to V ǫ and S ǫ is exterior to V ǫ .Proof. This follows from the maximum principle . (cid:3)
Let P Ω ǫ ( T ⋆ǫ ) = Ω ǫ ×{ }∪ ∂ Ω ǫ × [0 , T ⋆ǫ ) be the parabolic boundary of Ω ǫ × [0 , T ⋆ǫ ).Then Lemma 4.1, equation (4.8) and the Lipschitz bound on the initial radial graphΣ ǫ immediately yield (see e.g. [L96, thoerem 9.5])(4.9) w ǫ ( z , t ) ≤ e T ⋆ǫ max ( z ,t ) ∈ P Ω ǫ ( T ⋆ǫ ) w ǫ ( z , t ) ≤ C ( ǫ ) , ( z , t ) ∈ Ω ǫ × [0 , T ⋆ǫ ) . With this gradient estimate (and therefore the H¨older gradient estimate, see e.g.[L96, theorem 12.10]), for any fixed ǫ > v ǫ ∈ C ∞ (Ω ǫ × (0 , ∞ )) ∩ C , (Ω ǫ × (0 , ∞ )) ∩ C (Ω ǫ × [0 , ∞ )) by Schauder estimates. Therefore we have proved Theorem 4.3.
Let Γ , Γ ǫ and Σ ǫ ’s be as in Theorem 1.1. Then there exists a uniquesolution F ( z , t ) ∈ C ∞ (Ω ǫ × (0 , ∞ )) ∩ C , (Ω ǫ × (0 , ∞ )) ∩ C (Ω ǫ × [0 , ∞ )) tothe AMMCF (1.3) . Sharp gradient estimates
Since the earlier gradient estimate is too crude to prove the uniform convergenceof the AMMCF’s to the MMCF as ǫ →
0, we need a uniform sharp gradientestimate. To do this, we will need the next main technical result.
Theorem 5.1.
Let v ∈ C , (Ω × (0 , T )) be a function satisfying equation (2.8) forsome T > and Ω ⊆ S n + . Then (5.1) ( ∂∂t − L )( e v ( w + σ ( y + e · ∇ v ))) ≤ in Ω × (0 , T ) , where L is the linear elliptic operator from Lemma 4.1 .Proof. From the proof of Lemma 4.1 we know that(5.2) ( ∂∂t − L ) w ≤ − σ ( e · ∇ v ) + y n ( w − w ) − H w . We also have ( ∂∂t − L ) y = − L ( y )= − y n ( a ij y ij − w a ij w i y j − nwy ( σ ∇ v + w e ) · ∇ y )= − y n ( − y X a ii − w a ij w i y j − nwy ( σ ∇ v + w e ) · ∇ y )(5.3) = − y n ( − w a ij w i y j − nwy ( σ e · ∇ v + w ) + y − yw )= 2 y nw a ij w i y j + yw ( σ e · ∇ v + w ) − y n + y nw , and ( ∂∂t − L )( e · ∇ v ) = e · ∇ v t − L ( e · ∇ v )= e · ∇ ( yw ( H − σ )) − y n (cid:2) a ij ( e · ∇ v ) ij − w a ij w i ( e · ∇ v ) j − nwy ( σ ∇ v + w e ) · ∇ ( e · ∇ v ) (cid:3) = e · ( ∇ yw ( H − σ ) + y ∇ w ( H − σ ) + yw ∇ H ) − y n (cid:2) a ij ( y k v kij − yv ij − y j v i ) − w a ij w i ( y k v kj − yv j ) − nσwy ∇ v · ∇ ( e · ∇ v ) − ny e · ∇ ( e · ∇ v ) (cid:3) = (1 − y ) w ( H − σ ) + ∇ w · ∇ yy ( H − σ ) + yw e · ∇ H − y n (cid:2) y k (cid:0) ny ( H k w + Hw k + ( e · ∇ v ) k ) − ny ( Hw + e · ∇ v ) y k + 2 w a ij w i v kj − v k w + ( n − w ) v k (cid:1) − ∇ v · ∇ yw − n ( Hw + e · ∇ v ) − w a ij w i y k v kj + 2 yw a ij w i v j − nσwy ∇ v · ∇ ( e · ∇ v ) − ny e · ∇ ( e · ∇ v ) (cid:3) = 2 wH − σw (1 − y ) − σy ∇ w · ∇ y + (1 + y n + y nw ) e · ∇ v − y nw ∇ v · ∇ w + yσw ∇ v · ∇ ( e · ∇ v ) , ODIFIED MEAN CURVATURE FLOW IN HYPERBOLIC SPACE 15 where we used equations (2.6), (4.1)-(4.3) and (4.6) . Moreover,(5.4) ( ∂∂t − L ) v = yw ( H − σ ) − y n ( a ij v ij − w a ij w i v j − nwy ( σ ∇ v + w e ) · ∇ v )= yw ( H − σ ) − y n ( ny Hw − w ∇ v · ∇ w − nσwy + nσwy )= yw ( H − σ ) − yHw + 2 y nw ∇ v · ∇ w + yσw − yσw = 2 y nw ∇ v · ∇ w − yσw . Next, we note that for a function η defined on Ω × (0 , T ) ,(5.5) e − v ( ∂∂t − L )( e v η ) = η ( v t − Lv ) + ( η t − Lη ) − y n a ij v i v j η − y n a ij v i η j . In particular, e − v ( ∂∂t − L )( e v w ) ≤ w (cid:18) y nw ∇ v · ∇ w − yσw (cid:19) + (cid:20) − σ ( e · ∇ v ) + y n ( w − w ) − H w (cid:21) − y n a ij v i v j w − y n a ij v i w j = 2 y nw ∇ v · ∇ w − yσ − σ ( e · ∇ v ) + y n ( w − w )(5.6) − H w − y n ( w − w ) − y nw ∇ v · ∇ w = − yσ − σ ( e · ∇ v ) − H w , and e − v ( ∂∂t − L )( e v y ) = y (cid:18) y nw ∇ v · ∇ w − yσw (cid:19) + 2 y nw a ij w i y j + yw ( σ e · ∇ v + w ) − y n + y nw − y n a ij v i v j − y n a ij v i y j = 2 y nw ∇ v · ∇ w − y σw + 2 y nw ∇ y · ∇ w − y nw ( ∇ v · ∇ w )( ∇ y · ∇ v )+ σyw ( e · ∇ v ) + y − y n (1 − w ) − y nw ∇ v · ∇ y , and also e − v ( ∂∂t − L )( e v ( e · ∇ v ))= ( e · ∇ v )( 2 y nw ∇ v · ∇ w − yσw ) + 2 wH − σw (1 − y ) − σy ∇ w · ∇ y + ( e · ∇ v )(1 + y n + y nw ) − y nw ∇ v · ∇ w + yσw ∇ v · ∇ ( e · ∇ v ) − y n ( e · ∇ v )(1 − w ) − y n ∇ v · ∇ ( e · ∇ v ) w
26 LONGZHI LIN AND LING XIAO = 2 y nw ( ∇ v · ∇ w )( e · ∇ v ) − yσw ( e · ∇ v ) + 2 wH − σw (1 − y ) − σy ∇ w · ∇ y + ( e · ∇ v )(1 + 2 y nw ) − y nw ∇ v · ∇ w + ( yσw − y nw )( w e · ∇ w − y ( w − , . Therefore, combining the above two equations gives e − v ( ∂∂t − L )( e v ( y + ( e · ∇ v )))= − y σw + ( 2 y nw − σyw ) y ( w −
1) + y − y n (1 − w )(5.7) + 2 wH − σw (1 − y ) + e · ∇ v = y + 2 wH − σw + e · ∇ v . Finally, combining equations (5.6) and (5.7) implies(5.8) ( ∂∂t − L )( e v ( w + σ ( y + e · ∇ v ))) ≤ − e v ( H − σ ) w ≤ . (cid:3) Combing the uniform local ball condition (see equation (4.8)) and Theorem 5.1and appealing to the maximum principle, we conclude
Corollary 5.2.
Let v ǫ be the regular solution to the AMMCF (2.10) with initialhypersurface Σ ǫ as in Theorem 1.1. Then we have (5.9) |∇ v ǫ ( z , t ) | ≤ C , for all ( z , t ) ∈ Ω ǫ × [0 , ∞ ) , where C is a constant independent of ǫ . With the aid of Corollary 5.2 and the Arzel`a-Ascoli theorem, letting ǫ → { Σ ǫ i t } to the AMMCF (1.3),converging uniformly to Σ t ∈ C ∞ ( S n + × (0 , ∞ )) ∩ C , ( S n + × (0 , ∞ )) ∩ C ( S n + × [0 , ∞ )) which solves the MMCF (1.2) with initial hypersurface Σ = lim ǫ i → Σ ǫ i .6. The boundary regularity
In this section we show the boundary regularity of the MMCF (1.2) in Theorem1.1. The proof closely follows the idea in section 4.3 of [GS00], cf. [NS96]. Usingthe uniform local ball condition, we let P ∈ Γ and set ǫ = 0 in equation (4.7) anddenote ϕ = ϕ and ϕ = ϕ . For some ǫ > ϕ ( z ) ≤ v ( z , t ) ≤ ϕ ( z ) , ( z , t ) ∈ (cid:0) S n + ∩ B ǫ ( z ) (cid:1) × [0 , ∞ ) . Note that the tangent plane T to S at P is a radial graph T = e η z in S n + ∩{ z · ν > } with(6.2) η ( z ) = log P · e λy + z · e where λ = σ √ − σ and ν = σ e + √ − σ e is the unit normal vector to S at P . We also have(6.3) ϕ ( z ) ≤ η ( z ) ≤ ϕ ( z ) , z ∈ S n + ∩ B ǫ ( z ) . We will need the following more precise estimate on v . Lemma 6.1. v ( z , t ) = η ( z ) + O ( | z − z | ) in ( S n + ∩ B ǫ ( z )) × [0 , ∞ ) .Proof. This follows immediately from equation (6.1) and the estimates | ϕ i − η | ( z ) = O ( | z − z | ) , i = 1 , (cid:3) Now let p ∈ S n + and δ be the geodesic distance of p to ∂ S n + with δ < ǫ . Let q ∈ ∂ S n + be the closest point to p. Introduce normal coordinates x = ( x , . . . , x n )in T q S n + with x ( p ) = (0 , . . . , , δ ) . We observe that equation (2.8) may be writtenas ∂v∂t − y wn ∇ i (cid:18) ∇ i vw (cid:19) + y ∇ y · ∇ v + σyw = 0or in local coordinates (cf. equation (4.33) of [GS00]):(6.4) ∂v∂t − y wn √ γ ∂∂x i (cid:18) √ γγ ij w ∂v∂x j (cid:19) + yγ kl ∂y∂x k ∂v∂x l + σyw = 0 , where γ = det( γ ij ) and w = 1 + γ ij ∂v∂x i ∂v∂x j . One sees easily that both v and η satisfy equation (6.4) (note that the hyperplane T has constant hyperbolic meancurvature σ as well).Set ˜ v ( x ) = δ v ( δx ) and ˜ η ( x ) = δ η ( δx ) . Then (6.4) transforms to(6.5) ∂ ˜ v∂t − ˜ y ˜ wn √ ˜ γ ∂∂x i (cid:18) √ ˜ γ ˜ γ ij ˜ w ∂ ˜ v∂x j (cid:19) + ˜ y ˜ γ kl ∂ ˜ y∂x k ∂ ˜ v∂x l + σ ˜ y ˜ w = 0 , where ˜ y ( x ) = δ v ( δx ) , ˜ γ ij ( x ) = γ ij ( δx ) , ˜ γ = det(˜ γ ij ) and ˜ w = 1 + ˜ γ ij ∂ ˜ v∂x i ∂ ˜ vx j . Under this transformation we can move point p to the “interior” point ˜ p =(0 , ..., , T > B T = B (˜ p ) × (0 , T ), one observes that ˜ y = O (1).Also since sup |∇ ˜ v | = sup |∇ v | ≤ C and by [L96, theorem 12.10], ˜ v is uniformly C α, α . Moreover, since ˜ η satisfies the same equation (6.4), ˜ v − ˜ η satisfies alinear uniformly parabolic equation L (˜ v − ˜ η ) = 0 with uniformly H¨older continuouscoefficients. Then by the standard parabolic Schauder-type estimates and Lemma6.1 we get sup B T (cid:0) |∇ (˜ v − ˜ η ) | + |∇ (˜ v − ˜ η ) | (cid:1) ≤ C sup B T | ˜ v − ˜ η | ≤ Cδ .
Returning to the original variable we obtain(6.6) |∇ v | + |∇ v | ≤ C , where C is independent of δ . Now by equation (2.2) and Lemma 2.1, the energy functional I is non-increasing astime t increases and the MMCF subconverges to a smooth complete hypersurfaceΣ ∞ ∈ C ∞ ( S n + ) ∩ C ( S n + ) with constant hyperbolic mean curvature σ and ∂ Σ ∞ =Γ ⊂ ∂ ∞ H n +1 . Thus we have proved Theorem 6.2.
Let v ∈ C ∞ ( S n + × (0 , ∞ )) ∩ C , ( S n + × (0 , ∞ )) ∩ C ( S n + × [0 , ∞ )) be a solution to the MMCF (2.9) and φ ∈ C ( ∂ S n + ) . Then v ∈ C ∞ ( S n + × (0 , ∞ )) ∩ C , + ( S n + × (0 , ∞ )) ∩ C ( S n + × [0 , ∞ )) . Moreover, there exist t i ր ∞ suchthat Σ t i = F ( S n + , t i ) converges to a unique stationary smooth complete hypersurface Σ ∞ ∈ C ∞ ( S n + ) ∩ C ( S n + ) (as a radial graph over S n + ) which has constant hyperbolicmean curvature σ and ∂ Σ ∞ = Γ asymptotically. So now all that is left to prove of Theorem 1.1 is the uniform convergence of theMMCF in the case that Σ ǫ has mean curvature H ǫ ≥ σ for all ǫ > Uniform convergence
In this section we will show the uniform convergence of the regular solution tothe MMCF (1.2) as t → ∞ in the case of H ǫ ≥ σ initially for all ǫ >
0. To do this,we first show that for any fixed ǫ sufficiently small and for any z ∈ Ω ǫ , v ǫ ( z , t )is non-decreasing along the flow, where v ǫ is the regular solution to the AMMCF(2.10) for radial graphs . This is an immediate corollary of the following lemma. Lemma 7.1.
Let v ∈ C , (Ω × (0 , T )) be a function satisfying equation (2.8) forsome T > and Ω ⊆ S n + . Then (7.1) ( ∂∂t − e L )( yw ( H − σ )) = 0 in Ω × (0 , T ) , where e L is the linear elliptic operator e L ≡ y n a ij ∇ ij + (cid:20) y nw ( ∇ w · ∇ v ) ∇ v − y ∇ wnw − σyw ∇ v − y e (cid:21) · ∇ . Proof.
Let g = H − σ and h = ywg , we have(7.2) ∂v∂t = yw ( H − σ ) = ywg = h , (7.3) ∂w∂t = 1 w ∇ v · ∇ ( ywg ) = 1 w ∇ v · ∇ h , (7.4) ∂a ij ∂t = 2 v i v j ∇ v · ∇ hw − h i v j + h j v i w , and(7.5) ∂H∂t = ynw ( a ijt v ij + a ij ( v t ) ij ) − ya ij v ij w t nw − ( e · ∇ v ) t w + ( e · ∇ v ) w t w . ODIFIED MEAN CURVATURE FLOW IN HYPERBOLIC SPACE 19
Therefore by equations (7.3)-(7.5) and (2.6), we have ∂h∂t = yw t g + ywg t = yw t g + yw " ya ijt v ij + ya ij h ij nw − ya ij v ij w t nw − ( e · ∇ v ) t w + ( e · ∇ v ) w t w = yHw t − σyw t + y v ij n (cid:18) v i v j ∇ v · ∇ hw − h i v j + h j v i w (cid:19) + y n a ij h ij − y ( Hw + e · ∇ v ) w w t − y ( e · ∇ v ) t + yw ( e · ∇ v ) w t = yHw t − σyw ∇ v · ∇ h + 2 y nw ( ∇ w · ∇ v )( ∇ v · ∇ h ) − y ∇ w · ∇ hnw + y n a ij h ij − yHw t − y ( e · ∇ v ) t = y n a ij h ij + 2 y nw ( ∇ w · ∇ v )( ∇ v · ∇ h ) − y ∇ w · ∇ hnw − σyw ∇ v · ∇ h − y ( e · ∇ h ) . This completes the proof of the lemma using the definition of the operator e L . (cid:3) Corollary 7.2.
Suppose Σ ǫ has mean curvature H ǫ ≥ σ . Then ∂v ǫ ∂t = yw ǫ ( H ǫ − σ ) ≥ for all ( z , t ) ∈ Ω ǫ × [0 , ∞ ) .Proof. Since for any ǫ , v ǫ ( z , t ) ≡ φ ǫ ( z ) , z ∈ ∂ Ω ǫ , we have v t ≡ ∂ Ω ǫ × (0 , ∞ ).Then the condition H ǫ ≥ σ at t = 0, Lemma 7.1 and the maximum principle implythat ∂v ǫ ∂t = yw ǫ ( H ǫ − σ ) ≥ . (cid:3) Theorem 7.3.
Let Γ , Γ ǫ and Σ ǫ ’s be as in Theorem 1.1 and suppose Σ ǫ has meancurvature H ǫ ≥ σ for all ǫ > sufficiently small. Then Σ t converge uniformly for all t to a unique smooth complete star-shaped hypersurface Σ ∞ ∈ C ∞ ( S n + ) ∩ C ( S n + ) with constant hyperbolic mean curvature σ and boundary Γ .Proof. The subconvergence of the flow follows from Theorem 6.2. Corollary 7.2then yields ∂v∂t ≥
0, where v is the regular solution to the MMCF (2.9) for radialgraphs. This monotonicity of v implies that the regular solution Σ t to the MMCF(1.2) with initial hypersurface Σ converges uniformly for all t to Σ ∞ . (cid:3) This completes the proof of Theorem 1.1 .8.
Proof of Theorem 1.2 and “good” initial hypersurfaces
In this section we will prove Theorem 1.2 and give an example of “good” initialhypersurfaces for the Dirichlet problems (2.10) and (2.9).
Proof. (of Theorem 1.2) Note that since for any ǫ > H ǫ ≥ σ , Σ ǫ (as aradial graph of the function e v ǫ over Ω ǫ ) is a subsolution to the AMMCF (2.10).Therefore Σ ǫ serves as a natural lower barrier for the AMMCF. Combining thiswith the uniform exterior local ball condition yields the same proof as the one of Theorem 1.1 given in the previous sections, except the C boundary regularityof the flow. The C boundary regularity of the limiting hypersurface Σ ∞ followsfrom an elliptic version of the argument given in Section 6, see also section 4 . (cid:3) To find an example of “good” initial hypersurfaces in Theorem 1.2, namely, forany ǫ > C -) hyper-surface Σ ǫ = F (Ω ǫ ,
0) that can be represented as a radial graph of the function e v ǫ over Ω ǫ ⊂ S n + , having hyperbolic mean curvature H ǫ ≥ σ and Γ ǫ as its boundary.Moreover, Σ ǫ ’s satisfy the uniform exterior local ball condition and |∇ v ǫ | ( z ) ≤ C for all z ∈ Ω ǫ , where C is a constant independent of ǫ . For any ǫ > H n +1 of constant hyperbolic mean curvature close to 1 with boundaryΓ ǫ to serve as such “good” initial hypersurface Σ ǫ .From equations (2.5) and (2.11), one observes that if a smooth radial graph ofthe function e v over Ω ǫ has constant mean curvature σ with prescribed boundaryΓ ǫ , then v satisfies(8.1) a ij v ij = ny ( σw + e · ∇ v ) in Ω ǫ ,v = φ ǫ on ∂ Ω ǫ , where φ ǫ ∈ C ( ∂ Ω ǫ ) is assumed.It is clear that for σ = 1, the flat domain D ǫ ⊂ { x n +1 = ǫ } enclosed by Γ ǫ (known as “horosphere”) is the corresponding smooth radial graph satisfying (8.1).Therefore, there exists σ ∈ [0 , ∩ [ σ,
1) with σ being sufficiently close to 1 sothat the implicit function theorem applies to (8.1). In this way, we can obtain ahypersurface Σ ǫ = { e v ǫ z : z ∈ Ω ǫ } , where v ǫ ∈ C ∞ (Ω ǫ ) ∩ C (Ω ǫ ). Moreover Σ ǫ has hyperbolic mean curvature σ and ∂ Σ ǫ = Γ ǫ . By continuity, Σ ǫ is close to theflat domain D ǫ and for all ǫ ≥ ǫ ’s.With this specific construction of the initial hypersurface, we next give a prelim-inary C estimate for the solution to the AMMCF (1.3) . Lemma 8.1. On Σ ǫt there holds the height estimate (8.2) u ǫ ( z , t ) < d ( D )2 r − σ σ + ǫ , ( z , t ) ∈ Ω ǫ × [0 , T ǫ ) , where d ( D ) is the Euclidean diameter of D (the flat domain enclosed by Γ ) .Proof. Let B be a ball of radius R with center on the plane { x n +1 = − σR } suchthat the n -ball B ∩ { x n +1 = ǫ } has radius r = d ( D ) / D ǫ . Bycontinuity, we can choose σ so small that B contains Σ ǫ as well. By (i) of Lemma3.3, Σ ǫt is contained in B ∩ H n +1 for any t ∈ [0 , T ǫ ), and therefore u ǫ ( z , t ) < (1 − σ ) R , ( z , t ) ∈ Ω ǫ × [0 , T ǫ ) . ODIFIED MEAN CURVATURE FLOW IN HYPERBOLIC SPACE 21
Moreover, R = ( ǫ + σR ) + r , which implies(8.3) r √ − σ + σ − σ ǫ ≤ R ≤ r √ − σ + 1 + σ − σ ǫ . This completes the proof. (cid:3)
Remark . In particular, on Σ ǫ there holds the height estimate(8.4) u ǫ < d ( D )2 r − σ σ + ǫ . See lemma 3.2 of [GS00] .The only thing left to show is |∇ v ǫ | ( z ) ≤ C for all ǫ and z ∈ Ω ǫ . The first stepis to obtain a good barrier for ∇ v ǫ ( · , t ) at any point z ∈ ∂ Ω ǫ corresponding to P = e φ ǫ ( z ) z ∈ Γ ǫ . For convenience, we choose a coordinate system around P sothat the exterior normal to Γ ǫ at P is e ǫ . Let δ > δ ) be such thatfor each point P ∈ Γ ǫ , a ball of radius δ (respectively δ ) is internally (respectivelyexternally) tangent to Γ ǫ at P . Let B ǫi = B ǫi ( σ ), i = 1 , R i centered at C i = P + ( − i δ i e ǫ + ( a i − ǫ ) e , where(8.5) R i = − ( − i ǫσ + p ǫ + δ i (1 − σ )1 − σ and a i = ( − i R i σ . Recall that S ǫ ( σ ) = ∂B ǫ ∩ H n +1 has constant (hyperbolic) mean curvature σ with respect to its outward normal while S ǫ ( σ ) = ∂B ǫ ∩ H n +1 has constant meancurvature σ with respect to its inward normal. Moreover, by our construction, B ǫ and B ǫ are tangent at P , B ǫ ∩ { x n +1 = ǫ } is internally tangent to Γ ǫ at P , and B ǫ ∩ { x n +1 = ǫ } is externally tangent to Γ ǫ at P . Lemma 8.3.
Locally S ǫ ( σ ) is interior to Σ ǫ ( σ ) and S ǫ is exterior to Σ ǫ .Proof. This follows from the maximum principle for the equation (2.5) . (cid:3)
Similar to equation (4.7), we see that S ǫ ( σ ) and S ǫ ( σ ) serve as good localbarriers of Σ ǫ around P and we obtain that(8.6) |∇ v ǫ | ( P ) ≤ C , where C is independent of ǫ and P ∈ Γ ǫ .The next step is to obtain the uniform interior gradient bound for v ǫ and oneobserves that we only need to bound X ǫ · ν ǫE = e v ǫ p |∇ v ǫ | from below uniformly in ǫ . This can be done as follows. Firstly note that since D ǫ is a vertical graph over D and by continuity (induced from the implicit functiontheorem used in the construction of Σ ǫ ), Σ ǫ is a vertical graph of the function u ǫ over D as well. And similar to Lemma 8.1, we have another height estimate forvertical graphs. Lemma 8.4. [GS00, lemma 3.5] On Σ ǫ there holds (8.7) u ǫ ( x ′ ) ≥ d ( x ′ ) r − σ σ + σ ǫ σ , x ′ ∈ D where d ( x ′ ) is the distance from x ′ to ∂D . Moreover, there exists ǫ > σ ∈ [1 − ǫ , δ = δ ( ǫ ) so that in the δ -neighborhood of Γ ǫ in D ǫ one has |∇ v ǫ | ≤ C , where C is the uniform gradient bound of v ǫ on Γ ǫ as in equation (8.6). Away from the δ -neighborhood, by Lemma 8.4 X ǫ · ν ǫE = X ǫ · e − X ǫ · ( e − ν ǫE ) ≥ δ r − σ σ − e v ǫ vuut − q | e ∇ u ǫ | , where e ∇ is the Levi-Civita connection on R n +1 and we used that ν ǫE = ( − e ∇ u ǫ q | e ∇ u ǫ | , q | e ∇ u ǫ | )since Σ ǫ is a vertical graph.Now using the fact that H ǫE is subharmonic on the constant mean curvaturehypersurface Σ ǫ (see Theorem 2.2 of [GS00]), we have Lemma 8.5. [GS00, corollary 2.3]
For any λ ∈ (0 , , (8.8) q | e ∇ u ǫ | ≤ − λ ) σ in Ω λ , where Ω λ = n x ∈ D : u ǫ ≤ λσ sup Γ ǫ H ǫE o . To make use of Lemma 8.5, we also need the following estimate on the Euclideanmean curvature H ǫE of Σ ǫ on ∂ Σ ǫ = Γ ǫ . For x ∈ ∂D = Γ, denote by r ( x ) and r ( x ) the radius of the largest exterior and interior spheres to ∂D at x , respectively,and let r = min x ∈ ∂D r ( x ) , r = min x ∈ ∂D r ( x ). Then we have Lemma 8.6. [GS00, lemma 3.3]
For ǫ > sufficiently small, − p − σ r − ǫ (1 − σ ) r < σ − e · ν ǫE u = H ǫE < p − σ r + ǫ (1 + σ ) r on Γ ǫ . In particular, e · ν ǫE → σ on Γ ǫ as ǫ → , provided that ∂D is C . Combing the estimates in Remark 8.2 and Lemmas 8.5, 8.6, we can choose σ sufficiently close to 1 (for fixed ǫ ) such that X ǫ · ν ǫE ≥ min (cid:26) C , δ r − σ σ (cid:27) uniformly in ǫ . Now we can conclude
ODIFIED MEAN CURVATURE FLOW IN HYPERBOLIC SPACE 23
Theorem 8.7.
There exist constants ǫ > and σ ∈ (0 , ∩ [ σ, that is suffi-ciently close to such that for all ≤ ǫ ≤ ǫ , there exists a smooth hypersurface Σ ǫ with ∂ Σ ǫ = Γ ǫ ⊂ { x n +1 = ǫ } and whose hyperbolic mean curvature is σ . Ad-ditionally, Σ ǫ can be represented as a radial graph of a function e v ǫ over Ω ǫ ⊂ S n + and (8.9) |∇ v ǫ | ( z ) ≤ C , z ∈ Ω ǫ , where C is a constant independent of ǫ . Moreover, the Σ ǫ ’s satisfy the uniformexterior local ball condition. Interior gradient bounds and continuous boundary data
Interior gradient bounds.
We will next provide a version of a priori interiorgradient estimate for the regular solution to the MMCF (2.9), which is essentialfor the existence result of the MMCF with less regular (e.g. continuous) boundarydata.
Lemma 9.1.
Let v be a C , function satisfying equation (2.9) in B ρ ( P ) × (0 , T ) for some T > , where B ρ ( P ) ⊂ { y ≥ ε } . Then p |∇ v | ( P, T ) = w ( P, T ) ≤ C e C ρ , where C , C are non-negative constants depending only on n, σ, ε, T and k v k L ∞ .Proof. Define L = ∂∂t − L , where L is the linear elliptic operator from Lemma 4.1 . Without loss of generalitywe may assume (by adding a constant to v ) 1 ≤ v ≤ C . We will derive a maximumprinciple for the function h = η ( z , t, v ( z , t )) w by computing L h in B ρ ( P ) × (0 , T ),where η is non-negative, vanishes on the set { t ( ρ − ( d P ( z ) ) = 0 } , and is smoothwhere it is positive. Here d P ( z ) is the distance function (on the sphere) from P ,the center of the geodesic ball B ρ ( P ). Then h is non-negative and vanishes on theparabolic boundary of B ρ ( P ) × (0 , T ) .Choose η ≡ g ( ϕ ( z , t, v ( z , t ))) ; g ( ϕ ) = e Kϕ − , with the constant K > ϕ ( z , t, v ( z , t )) = " − v ( z , t )2 v ( P, T ) + tT − (cid:18) d P ( z ) ρ (cid:19) ! + . By Lemma 4.1 we have L h = η L w + w L η − y n a ij η i w j = η L w + w (cid:18) η t − y n M η (cid:19) ≤ w (cid:18) η + η t − y n M η (cid:19) , (9.1) where M = a ij ∇ ij − ny (cid:18) σ ∇ vw + e (cid:19) · ∇ . We will choose K so that 2 η + η t − y n M η ≤ h > w islarge.A straightforward computation gives that on the set where h > M η = g ′ ( ϕ ) (cid:18) a ij ∇ ij ϕ − ny (cid:18) σ ∇ vw + e (cid:19) · ∇ ϕ (cid:19) + g ′′ ( ϕ ) a ij ∇ i ϕ ∇ j ϕ = Ke Kϕ " − nv t y v ( P, T ) − nσ ywv ( P, T ) − tρ T (cid:0) a ij ∇ i d P ∇ j d P + d P a ij ∇ ij d P (cid:1) + 2 ntρ yT (cid:18) σ ∇ vw + e (cid:19) · d P ∇ d P + K e Kϕ a ij (cid:18) v i v ( P, T ) + 2 tρ T d P ∇ i d P (cid:19) (cid:18) v j v ( P, T ) + 2 tρ T d P ∇ j d P (cid:19) . Using the definition of a ij we find a ij (cid:18) v i v ( P, T ) + 2 tρ T d P ∇ i d P (cid:19) (cid:18) v j v ( P, T ) + 2 tρ T d P ∇ j d P (cid:19) = |∇ v | v ( P, T )) w + 2 td P T v ( P, T ) ρ w h∇ v, ∇ d P i + 4 t d P T ρ − (cid:28) ∇ vw , ∇ d P (cid:29) ! , where h , i denotes the inner product with respect to the induced Euclidean metricon Σ t . Therefore we have2 η + η t − y n M η = 2 η + Ke Kϕ − v t v ( P, T ) + 1 − (cid:16) d P ρ (cid:17) T − y n M η ≤ η + Ke Kϕ T − y n M η − Ke Kϕ v t v ( P, T ) ≤ − y n e Kϕ (cid:20) K (cid:18) |∇ v | w ( v ( P, T )) − w (cid:18) ρ + |∇ v | v ( P, T )) (cid:19)(cid:19) − CKρ − C (cid:21) ≤ − y n e Kϕ (cid:20) K − CKρ − C (cid:21) , whenever w > max {√ , C ρ } = C ρ so that |∇ v | w > and w ρ < C .Thus, the choice of K = 32 CC (cid:16) C ρ (cid:17) gives(9.2) L h ≤ w (cid:20) η + η t − y n M η (cid:21) < h > w > C ρ . Then by the maximum principle, (9.2) gives(9.3) h ( P, T ) = (cid:16) e K − (cid:17) w ( P, T ) ≤ max h ≤ (cid:0) e K − (cid:1) C ρ ODIFIED MEAN CURVATURE FLOW IN HYPERBOLIC SPACE 25 and hence w ( P, T ) ≤ C e CC ρ for a slightly larger constant C . This completes the proof. (cid:3) Continuous boundary data.
By the standard modulus of continuity esti-mates (see e.g. [L96, theorem 10.18]) and with the aid of the a priori interiorgradient estimate (see Lemma 9.1) proved in the previous section, one can furtherrelax the regularity of the boundary data to be only continuous via an approxima-tion argument. We have
Theorem 9.2.
Let Γ be the boundary of a continuous star-shaped domain in { x n +1 = 0 } and Σ = lim ǫ → Σ ǫ be as in Theorem 1.1 or Theorem 1.2. Thenthere exists a unique solution F ( z , t ) ∈ C ∞ ( S n + × (0 , ∞ ) ∩ C ( S n + × [0 , ∞ )) to theMMCF (1.2) . Moreover, there exist t i ր ∞ such that Σ t i = F ( S n + , t i ) converges toa unique stationary smooth complete hypersurface Σ ∞ ∈ C ∞ ( S n + ) ∩ C ( S n + ) (as a ra-dial graph over S n + ) which has constant hyperbolic mean curvature σ and ∂ Σ ∞ = Γ asymptotically. Acknowledgement
The authors would like to thank Professor Joel Spruck for the continued guid-ance.
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