Guan-Li type mean curvature flow for free boundary hypersurfaces in a ball
aa r X i v : . [ m a t h . DG ] J un GUAN-LI TYPE MEAN CURVATURE FLOW FOR FREEBOUNDARY HYPERSURFACES IN A BALL
GUOFANG WANG AND CHAO XIA
Abstract.
In this paper we introduce a Guan-Li type volume preserving meancurvature flow for free boundary hypersurfaces in a ball. We give a concept ofstar-shaped free boundary hypersurfaces in a ball and show that the Guan-Li typemean curvature flow has long time existence and converges to a free boundaryspherical cap, provided the initial data is star-shaped. Introduction
Let B n +1 ⊂ R n +1 be the open unit Euclidean ball centered at the origin and S n = ∂ B n +1 ⊂ R n +1 the unit sphere. In this paper, we shall consider a meancurvature type flow for compact hypersurfaces in B n +1 with free boundary on S n .Let Σ ⊂ ¯ B n +1 be a properly embedded compact hypersurface with boundary, whichis given by x : M → ¯ B n +1 , where M is a compact Riemannian manifold with boundary ∂M . Here properlyembedded means thatint(Σ) = x (int( M )) ⊂ B n +1 and ∂ Σ = x ( ∂M ) ⊂ ∂ B n +1 . We further assume that Σ has free boundary, in the sense that Σ intersects ∂ B n +1 = S n orthogonally, that is, h ν, µ ◦ x i = 0 on ∂M, where ν is a unit normal vector field of x , which will be specified later, and µ is theoutward unit normal vector field of S n , i.e., µ ◦ x = x along ∂M .Let e ∈ S n ⊂ R n +1 be a fixed unit vector field. Consider a family of properly em-bedded compact hypersurfaces { Σ t } t ∈ [0 ,T ) with free boundary, given by embeddings x : M × [0 , T ) → ¯ B n +1 , satisfying(1) (cid:26) ∂ t x = ( n h x, e i − H h X e , ν i ) ν in M × [0 , T ) , h ν, µ ◦ x i = 0 on ∂M × [0 , T ) . Mathematics Subject Classification. with an initial surface x ( · ,
0) = x . Here ν and H are a unit normal vector field andthe mean curvature of x ( · , t ) respectively, X e is a fixed vector field in R n +1 given by X e = X e ( x ) = h x, e i x −
12 ( | x | + 1) e, for a fixed unit vector e . This vector field plays an important role in our recent paper[11]. We choose ν in the following way. Let Ω t be the component of the encloseddomain by Σ t and S n which contains e in its interior. Then ν is chosen to be theoutward normal of Σ t with respect to Ω t . Also, throughout this paper, we make theconvention that the enclosed domain Ω t of Σ t and S n is the one e in its interior. Thevolume of the enclosed domain Ω t of Σ is called the enclosed volume of Σ t .The flow is designed in this way so that the enclosed volume of Σ t is preservedalong the flow (1). We will discuss it later. Such kinds of flow was first consideredby Guan-Li [5] in the setting of closed hypersurfaces in space forms and by Guan-Li-Wang [6] in the setting of closed hypersurfaces in warped product spaces.The main objective of this paper is to study the existence and the convergence ofthe flow (1). For this aim we introduce a concept of star-shaped hypersurfaces withfree boundary in ¯ B n +1 . To arrive at this, we should first make some comments onthe vector field X e above. X e is a conformal Killing vector field with h X e ( x ) , x i = 0 , ∀ x ∈ ∂ B n +1 . More precisely, denoting the Euclidean metric by δ , we have L X e δ = h x, e i δ. Let φ t : ¯ B n +1 → ¯ B n +1 be the one-parameter family of conformal transformationsgenerated by X e . Let Π e be the hyperplane which passes through the origin andis orthogonal to e . For each point p ∈ Π e , there exists a unique planar circlepassing through p and ± e . One can check that the integral curves of X e are givenby the intersection of all such planar circles with B n +1 . We introduce star-shapedhypersurfaces with free boundary in ¯ B n +1 . Definition 1.1. Σ ⊂ ¯ B n +1 is called star-shaped(with respect to e ) if Σ intersects each integral curve of X e exactly once.2). A proper embedded hypersurface Σ ⊂ ¯ B n +1 is called strictly star-shaped (withrespect to e ) if (2) h X e , ν i > . For our purpose we will consider strictly star-shaped hypersurfaces in ¯ B n +1 inthis paper. This condition is slightly stronger than the condition of star-shapedness,but clearly much weaker than the convexity. For the simplicity in this paper we callhypersurfaces satisfying (2) star-shaped hypersurfaces.From now on we consider star-shaped hypersurfaces. Being such a hypersurface, itis necessary that M is of ball type. Therefore we use M = ¯ S n + , the closed hemisphere.Our main result is the following Theorem 1.1.
Let Σ ⊂ ¯ B n +1 ( n ≥ be a properly embedded compact hypersurfacewith free boundary, given by x : ¯ S n + → ¯ B n +1 , which is star-shaped with respect to e .Then there exists a unique solution x : ¯ S n + × [0 , ∞ ) → ¯ B n +1 to (1) . Moreover, x ( · , t ) converges smoothly to a spherical cap or the totally geodesic n -ball, whose encloseddomain has the same volume as Σ . When n ≥ , or n = 2 and the enclosed volumeof x is not that of a half ball, x ( · , t ) converges exponentially fast. UAN-LI TYPE MEAN CURVATURE FLOW 3
The family of spherical caps is given by C ± r ( e ) = { x ∈ ¯ B n +1 : | x ± p r + 1 e | = r } , r > n -ball is given by C ∞ ( e ) = { x ∈ ¯ B n +1 : h x, e i = 0 } . It is clear that either each spherical cap C ± r ( e ) or the totally geodesic n -ball C ∞ ( e )has free boundary, that is, it intersects the support S n orthogonally.As a direct consequence, we give a flow proof of the isoperimetric problem for freeboundary hypersurfaces in B n +1 . Corollary 1.1.
Among star-shaped free boundary hypersurfaces with fixed enclosedvolume, the totally geodesic n -ball or the spherical caps have minimal area. For general hypersurfaces it is a classical result proved by Burago-Mazaya [3],Bokowsky-Sperner [2] and Almgren [1], by using the method of symmetrization.The introduction of flow (1) is motivated by the paper of Guan-Li [5], in whichthey used at the first time the Minkowski formula for closed hypersurfaces to definea geometric flow for isoperimetric problems. In the same spirit, the flow (1) isbased on the following two Minkowski formulas obtained in [11] for free boundaryhypersurfaces n Z Σ h x, e i = Z Σ h X e , ν i H, (3) Z Σ h x, e i H = 2 n − Z h X e , ν i σ ( κ ) . (4)Here κ = ( κ , κ , · · · , κ n ) are principal curvatures of Σ and σ ( κ ) is the 2nd ordermean curvature. From these formulas, one can show that flow (1) preserves thevolume of Ω t and decreases the area of Σ t . See Proposition 4.1. These are crucialproperties of this flow.To prove Theorem 1.1, we first transform the flow equation to a scalar flow (19) on S n + by using star-shapedness. By using the M¨obius transformation between the halfspace ¯ R n +1+ and the unit ball ¯ B n +1 , a star-shaped hypersurface in ¯ B n +1 is equivalentto a classical star-shaped hypersurface in ¯ R n +1+ with a conformal flat metric. Weremark that a different reparametrization based on M¨obius transformation betweenround cylinder and B n +1 was used by Lambert-Scheuer [7]. For the scalar flow(19), the C estimate follows directly from the barrier argument. We then show thegradient estimate for (19).Finally we mention some previous results on curvature flows with free boundaryin B n +1 . The classical mean curvature flow was considered by Stahl [9, 10], whereit was shown that strictly convex initial data are driven to a round point in a finitetime. The classical inverse mean curvature flow was treated by Lambert-Scheuer [7],where it was shown that strictly convex initial data are driven to a flat perpendicular n -ball in a finite time. Following a similar idea of this paper, a fully nonlinear inversecurvature type flow was considered by Scheuer and the authors [8] to show a classof new Alexandrov-Fenchel’s inequalities for convex free boundary hypersurfaces in B n +1 .The rest of this paper is organized as follows. In Section 2 we introduce theM¨obius transformation between ¯ R n + and ¯ B n , and reduce flow (1) to a scalar flow GUOFANG WANG AND CHAO XIA (19), provided that all evolving hypersurfaces are star-shaped. In Section 3, weshow that C and C estimates of (1). As consequence, we prove in Section 4 thatthe global convergence of (1), Theorem 1.1 and its consequence, Corollary 1.1.2. A scalar flow
In this section we reduce (1) to a scalar flow, provided that all evolving hypersur-faces are star-shaped.Without loss of generality, from now on, we assume e = E n +1 , the ( n + 1)-coordinate vector. Let R n +1+ = { z = ( z , · · · , z n +1 ) ∈ R n +1 : z n +1 > } be the half space. Define f : ¯ R n +1+ → ¯ B n +1 , (5) ( z ′ , z n +1 ) (cid:18) z ′ | z ′ | + (1 + z n +1 ) , | z | − | z ′ | + (1 + z n +1 ) (cid:19) . (6)Here z ′ = ( z , · · · , z n ) ∈ R n . f is bijective and f ( R n +1+ ) = B n +1 , (7) f ( ∂ R n +1+ ) = ∂ B n +1 , (8) f ( {| z | = 1 } ) = { x n +1 = 0 } . (9)Moreover, f is a conformal diffeomorphism between ( ¯ R n +1+ , δ ¯ R n +1+ ) and (¯ B n +1 , δ ¯ B ).Here δ ¯ R n +1+ and δ ¯ B denote the restriction of the Euclidean metric to ¯ R n +1+ and ¯ B n +1 respectively. Precisely, f ∗ δ ¯ B = e w δ ¯ R n +1+ = 4( | z ′ | + (1 + z n +1 ) ) δ ¯ R n +1+ . In other words, (¯ B n +1 , δ ¯ B ) and ( ¯ R n +1+ , e w δ ¯ R n +1+ ) are isometric.In ¯ R n +1+ , we use the polar coordinates ( ρ, ϕ, θ ) ∈ [0 , ∞ ) × [0 , π ] × S n − , where ρ = | z ′ | + z n +1 , z n +1 = ρ cos ϕ and θ ∈ S n − is the spherical coordinate.By using ( ρ, ϕ, θ ) in ¯ R n +1+ , the mapping f can be rewritten as f ( ρ, ϕ, θ ) = ρ sin ϕ~θ ρ + 2 ρ cos ϕ , ρ −
11 + ρ + 2 ρ cos ϕ ! . (10)Here ~θ denotes the position vector of the point z ′ | z ′ | ∈ S n − . We also have f ∗ δ ¯ B = e w δ ¯ R n +1+ = 4(1 + ρ + 2 ρ cos ϕ ) ( dρ + ρ dϕ + ρ sin ϕg S n − ) , where w = w ( ρ, ϕ, θ ) = log 2 − log(1 + ρ + 2 ρ cos ϕ ) . One may also check that the conformal Killing vector field X n +1 on ¯ B + is transformedto ˜ X = ( f − ) ∗ ( X n +1 ) = − ρ∂ ρ on ¯ R n +1+ . (11) UAN-LI TYPE MEAN CURVATURE FLOW 5
The integral curves of ˜ X are clearly the rays in R n +1+ initiating from the origin.Let Σ ⊂ ¯ B n +1 be a properly embedded compact hypersurface with boundary,given by an embedding x : ¯ S n + → ¯ B n +1 . We associate Σ with a correspondinghypersurface ˜Σ ⊂ ¯ R n +1+ given by the embedding˜ x = f − ◦ x : ¯ S n + → ¯ R n +1+ . In view of (11), Σ is star-shaped with respect to E n +1 if and only if ˜Σ is star-shaped(with respect to the origin) in ¯ R n +1+ , that is, ˜Σ intersects each of the rays in R n +1+ initiating from the origin exactly once, or in other words, ˜Σ is a graph over ¯ S n + .Since (¯ B n +1 , δ ¯ B ) and ( ¯ R n +1+ , e w δ ¯ R n +1+ ) are isometric, a proper embedding x : ¯ S n + → ¯ B n +1 can be identified as an embedding ˜ x : ¯ S n + → ( ¯ R n +1+ , e w δ ¯ R n +1+ ) . In the following,we use˜to indicate the corresponding quantity for ˜ x : ¯ S n + → ( ¯ R n +1+ , e w δ ¯ R n +1+ ).Given a star-shaped hyersurface ˜Σ in ( ¯ R n +1+ , e w δ ¯ R n +1+ ), by using the polar coor-dinate ( ρ, ϕ, θ ) ∈ ¯ R n +1+ , we may write˜ x = ρ ( y ) y = ρ ( ϕ, θ ) y, y = ( ϕ, θ ) ∈ ¯ S n + . We use σ = dϕ + sin ϕdθ and ∇ σ to denote the round metric and the covariantderivative on ¯ S n + . Set γ = log ρ, and v = p |∇ σ γ | . We have the following correspondence for several geometric quantities.
Proposition 2.1. (i) x n +1 = h f (˜ x ) , E n +1 i = 12 ( ρ − e w . (ii) | X n +1 | = e w | − ρ∂ ρ | = ρe w . (iii) h X n +1 , ν i = e w h− ρ∂ ρ , ˜ ν i = ρe w v . (iv) The Weingarten transformation h ji = g jk h ik satisfies h ji = ˜ h ji = 1 ρve w ( σ kj − γ k γ j v ) γ ik + (cid:20) sin ϕγ ϕ v + ( ρ − ρv (cid:21) δ ji . (v) H = ˜ H = 1 ρve w ( σ ij − γ i γ j v ) γ ij + n sin ϕγ ϕ v + n ( ρ − ρv . Remark 2.1.
We see from (iii) that in case we have C estimate, a positive lowerbound for h X n +1 , ν i is equivalent to the gradient estimate for γ .Proof. (i) follows from (10) and (ii) follows from (11).It is clear that the unit outward normal is given by˜ ν = e − w ν δ = e − w ρ − ∇ σ γ − ∂ ρ v , (12) GUOFANG WANG AND CHAO XIA where ν δ is the unit outward normal of ˜Σ ⊂ ( ¯ R n +1+ , δ ¯ R n +1+ ). Then (iii) follows from(11) and (12).By a well-known transformation law for the Weingarten transformation under aconformal change, we know that ˜ h ji of Σ ⊂ ( ¯ R n +1+ , e w δ ¯ R n +1+ ) with respect to − ˜ ν isgiven by ˜ h ji = e − w (( h δ ) ji + ∇ δν δ wδ ji ) , (13)where ( h δ ) ji is the Weingarten transformation with respect to − ν δ of ˜Σ ⊂ ( ¯ R n +1+ , δ ¯ R n +1+ )and ∇ δ is the Euclidean derivative.It is known that ( h δ ) ji = − ρv δ ji + 1 ρv ( σ kj − γ k γ j v ) γ ik , (14)On the other hand, using e − w = (1 + ρ + 2 ρ cos ϕ ), we have ∇ δν δ ( e − w ) = (cid:28) ( ρ + cos ϕ ) ∂ ρ − ρ − sin ϕ∂ ϕ , ρ − ∇ σ γ − ∂ ρ v (cid:29) (15) = − v ( ρ + cos ϕ + sin ϕγ ϕ ) . (iv) follows from (13), (14) and (15). (v) follows from (iv) by taking trace. (cid:3) We return to the flow problem (1) in (¯ B n +1 , δ ¯ B ). By the identification using f ,the corresponding family of embeddings ˜ x : S n + → ( ¯ R n +1+ , e w δ ¯ R n +1+ ) satisfies(16) (cid:26) ∂ t ˜ x = ( n h f (˜ x ) , E n +1 i − ˜ He w h− ρ∂ ρ , ˜ ν i )˜ ν in S n + × [0 , T ) , h ˜ ν, ˜ µ ◦ ˜ x i = 0 , on ∂ S n + × [0 , T ) , with an initial surface ˜ x ( · ,
0) = ˜ x . Here ˜ µ is the downward unit normal of ( ¯ R n +1+ , e w δ ¯ R n +1+ ).As long as ˜ x ( · , t ) is star-shaped in ¯ R n +1+ , we may reduce (16) to a scalar flow.Using a standard argument (see [4], Eq. (2.4.21)) and Proposition 2.1, we seethat ∂ t γ = − vρe w (cid:18) n ρ − e w − ˜ H ρe w v (cid:19) (17) = 1 ρve w (cid:18) σ ij − γ i γ j v (cid:19) γ ij + n sin ϕγ ϕ v − n ( ρ − |∇ σ γ | ρv = div σ (cid:18) ∇ σ γρve w (cid:19) − n + 1 v σ (cid:18) ∇ σ γ, ∇ σ (cid:18) ρe w (cid:19)(cid:19) . The last line above follows from the fact σ (cid:18) ∇ σ γ, ∇ σ (cid:18) ρe w (cid:19)(cid:19) = ρ − ρ |∇ σ γ | − sin ϕγ ϕ . Next we examine the boundary condition. Note that µ ⊥ ∂ B n +1 . Since theconformal change f preserves angles, we have ˜ µ ⊥ ∂ R n +1+ and in turn˜ µ = − e − w ∂ ϕ . In view of (12), the boundary condition in (16) reduces to ∇ σ∂ ϕ γ = 0 on ∂ S n + . (18) UAN-LI TYPE MEAN CURVATURE FLOW 7
In summary, the flow problem (16) reduces to solve the scalar PDE(19) ∂ t γ = 1 ρve w (cid:18) σ ij − γ i γ j v (cid:19) γ ij + n sin ϕγ ϕ v − n ( ρ − |∇ σ γ | ρv , in S n + × [0 , T ) , with the initial and the boundary conditions γ ( · ,
0) = γ , in S n + , ∇ σ∂ ϕ γ = 0 , on ∂ S n + × [0 , T ) . where γ is the corresponding function for x .3. A priori estimates
The short time existence of the scalar flow (19) follows by the standard parabolicPDE theory. Next we show the C and C estimates for (19). The a priori C estimate follows directly from the maximum principle. Proposition 3.1.
Let γ : S n + × [0 , T ) → R solve (19) . Then min S n + γ ≤ γ ≤ max S n + γ . The key point is the following gradient estimate for γ . Proposition 3.2.
Let γ : S n + × [0 , T ) → R solve (19) . Then there exists a constant C , depending on k γ k C and min S n + γ such that |∇ σ γ | ≤ C. Moreover, if n ≥ , we have |∇ σ γ | ≤ C e − C t . Proof.
For notation simplicity, we use ∇ = ∇ σ in the proof. Denote F ( ∇ γ, ∇ γ, ρ, ϕ ) = 1 ρve w (cid:18) σ ij − γ i γ j v (cid:19) γ ij + n sin ϕγ ϕ v − n ( ρ − |∇ γ | ρv , and F ij = ∂F∂γ ij , F p = ∂F∂γ p , F ρ = ∂F∂ρ , F ϕ = ∂F∂ϕ . Then ∂ t |∇ γ | = 2 γ k ( γ t ) k = 2 F ij γ k γ ijk + F p ∇ p |∇ γ | + 2 F ρ ρ |∇ γ | + 2 F ϕ γ ϕ . (20)By a direct computation, we have F ij = 1 ρve w (cid:18) σ ij − γ i γ j v (cid:19) , (21) F ρ = ρ − ρ v (cid:18) σ ij − γ i γ j v (cid:19) γ ij − n ( ρ + 1)2 ρ v |∇ γ | , (22) F ϕ = − sin ϕ v (cid:18) σ ij − γ i γ j v (cid:19) γ ij + n cos ϕv γ ϕ . (23)Using the Ricci identity γ ijk = γ kij + γ j σ ki − γ k σ ij GUOFANG WANG AND CHAO XIA and (21), we have2 F ij γ k γ ijk = F ij ∇ ij |∇ γ | − ρve w (cid:18) σ ij − γ i γ j v (cid:19) γ ik γ jk − n − ρve w |∇ γ | = F ij ∇ ij |∇ γ | − ρve w |∇ γ | + 12 ρv e w (cid:12)(cid:12) ∇|∇ γ | (cid:12)(cid:12) − n − ρve w |∇ γ | . (24)Replacing (22), (23) and (24) into (20), we get ∂ t |∇ γ | = F ij ∇ ij |∇ γ | + F p ∇ p |∇ γ | − ρve w |∇ γ | + 12 ρv e w (cid:12)(cid:12) ∇|∇ γ | (cid:12)(cid:12) − n − ρve w |∇ γ | +2 (cid:20) ρ − ρ v (cid:18) σ ij − γ i γ j v (cid:19) γ ij − n ( ρ + 1)2 ρ v |∇ γ | (cid:21) ρ |∇ γ | +2 (cid:20) − sin ϕ v (cid:18) σ ij − γ i γ j v (cid:19) γ ij + n cos ϕv γ ϕ (cid:21) γ ϕ = F ij ∇ ij |∇ γ | + F p ∇ p |∇ γ | + (cid:18) sin ϕ − ρ − ρ |∇ γ | (cid:19) h∇ γ, ∇|∇ γ | i v − ρve w |∇ γ | + 12 ρv e w (cid:12)(cid:12) ∇|∇ γ | (cid:12)(cid:12) − n − ρve w |∇ γ | + ρ − ρv ∆ γ |∇ γ | − n ( ρ + 1) ρv |∇ γ | + 2 n cos ϕv γ ϕ − ϕv ∆ γγ ϕ . (25)Now we examine the boundary normal derivative of |∇ γ | and have ∇ ∂ ϕ |∇ γ | = 2( γ θ α γ θ α ϕ + γ ϕ γ ϕϕ ) = γ θ α [ ∇ ∂ θα ( γ ϕ ) − ( ∇ ∂ θα ∂ ϕ ) γ ] = 0 . (26)Here we used γ ϕ = 0 along ∂ S n + and the fact that ∇ ∂ θα ∂ ϕ = 0.Assume for t ∈ [0 , T ), max ¯ S n + |∇ γ | ( · , t ) = |∇ γ | ( x t , t ). If x t ∈ S n + , it follows fromthe maximum point condition that ∇|∇ γ | = 0 , ∇ |∇ γ | ≤ . (27)If x t ∈ ∂ S n + , we see from (26) that ∇ ∂ ϕ |∇ γ | = 0, and in turn we also have (27).Thus, for each t ∈ [0 , T ), at x t , we have (27). We choose at x t local coordinates x , · · · x n such that γ = |∇ γ | . One has γ i = 0 for all i by (27). By further rotatingthe { x , · · · , x n } coordinate, we can assume ∇ γ is diagonal. Then |∇ γ | ≥ n − γ ) . UAN-LI TYPE MEAN CURVATURE FLOW 9
It follows from (25) that at x t ,0 ≤ ∂ t |∇ γ | ( x t , t ) ≤ − ρve w |∇ γ | − n − ρve w |∇ γ | + ρ − ρv ∆ γ |∇ γ | − n ( ρ + 1) ρv |∇ γ | + 2 n cos ϕv γ ϕ − ϕv ∆ γγ ϕ ≤ − − ǫ )( n − ρve w (cid:18) ∆ γ − ( n − ρ − e w − ǫ ) |∇ γ | (cid:19) − ǫ ( n − ρve w (cid:18) ∆ γ + ( n − ρe w sin ϕ ǫ γ ϕ (cid:19) + 1 ρv (cid:18) ( n − ρ − e w − ǫ ) − n ( ρ + 1) (cid:19) |∇ γ | + 1 v (cid:18) − n − ρe w |∇ γ | + 2 n cos ϕγ ϕ + ( n − ρe w sin ϕ ǫ γ ϕ (cid:19) . (28)Choosing ǫ = , we have( n − ρ − e w − ǫ ) − n ( ρ + 1) < ne w ρ − − ( ρ + 1)(1 + ρ + 2 ρ cos ϕ )] ≤ − nρ e w and − n − ρe w |∇ γ | + 2 n cos ϕγ ϕ + ( n − ρe w sin ϕ ǫ γ ϕ ≤ (cid:18) − ( n − ρ + 2 ρ cos ϕ ) ρ + 2 n cos ϕ + 4( n − ρ ρ + 2 ρ cos ϕ (cid:19) |∇ γ | ≤ ( − n −
1) + 2 cos ϕ + 2( n − |∇ γ | ≤ ( − n + 103 ) |∇ γ | . Thus 0 ≤ ∂ t |∇ γ | ≤ − nρe w v |∇ γ | + ( − n + 103 ) 1 ρv |∇ γ | . (29)It follows from (29) that |∇ γ | ≤ C . Moreover, when n ≥
3, one sees from (29) that |∇ γ | ≤ C e − C t . (cid:3) Global convergence
We first prove the nice properties of (1), mentioned in the Introduction.
Proposition 4.1.
Flow (1) satisfies ddt
Vol(Ω t ) = 0(30) and ddt Area(Σ t ) = − n − Z Σ X i From (3), we get ddt Vol(Ω t ) = Z Σ ( nx n +1 − H h X n +1 , ν i ) dA t = 0 . The first variational formula gives ddt Area(Σ t ) = Z Σ H ( nx n +1 − H h X n +1 , ν i ) dA t . Using the Minkowski formula (4) Z Σ Hx n +1 − n − σ ( κ ) h X n +1 , ν i dA t = 0 , we get ddt Area(Σ t ) = − Z Σ (cid:18) H − nn − σ ( κ ) (cid:19) h X n +1 , ν i dA t = − n − Z Σ X i Proof of Theorem 1.1. In view of Proposition 2.1 (iii), the C and C estimates inPropositions 3.1 and 3.2 imply that h X n +1 , ν i ≥ c > 0, that is, the star-shapednessof Σ t is preserved under the flow (1).Now we are ready to prove the long time existence in Theorem 1.1. Since equation(19) is a quasilinear parabolic PDE of divergent form, the higher order a prioriestimates follows from the standard parabolic PDE theory, once we have the C and C estimates in Propositions 3.1 and 3.2. Hence we prove that (19) has a smoothsolution for all time. The exponential convergence for n ≥ n = 2. By integrating(4.1) over t ∈ [0 , ∞ ) and using the uniform estimate, we get Z ∞ Z S n + | κ ( y, t ) − κ ( y, t ) | h X n +1 , ν i dA t dt ≤ C. where κ i ( y, t ), i = 1 , y, t ). Itfollows from the uniform bound for h X n +1 , ν i and dA t that(32) max y ∈ ¯ S n + | κ − κ | ( y, t ) = o t (1) , where o t (1) denotes a quantity which goes to zero as t → ∞ . See the proof ofProposition 5.5 in [5]. With the help of the property (32), we can show the smoothconvergence of flow (1) when n = 2. This idea was used first by Guan-Li in [5].Let us go back to the estimate at x t , where max ¯ S n + |∇ γ | ( · , t ) = |∇ γ | ( x t , t ). Againwe choose the local coordinate around x t such that at x t , γ = |∇ γ | , γ = 0 . UAN-LI TYPE MEAN CURVATURE FLOW 11 In view of Proposition 2.1 (iv), the Weingarten transformation h ji is diagonal inthis coordinate which means the coordinate directions are the principal directionsof x ( · , t ) at x t . Thus the principal curvature κ i at x t is given by κ i = γ ii ρve w + sin ϕγ ϕ v + ( ρ − ρv , i = 1 , . It follows that at x t , | ∆ γ | = | γ + γ | = | γ − γ | = ρve w | κ − κ | = o t (1) . (33)Using (33) and the C estimate, we get at ( x t , t ), ∂ t |∇ γ | ≤ − ρve w |∇ γ | − n − ρve w |∇ γ | + ρ − ρv ∆ γ |∇ γ | − n ( ρ + 1) ρv |∇ γ | + 2 n cos ϕv γ ϕ − ϕv ∆ γγ ϕ ≤ − n ( ρ + 1) ρv |∇ γ | + 1 v (cid:18) − ρe w |∇ γ | + 4 cos ϕγ ϕ (cid:19) + o t (1) ≤ − C |∇ γ | + o t (1) . (34)Here we have used − ρe w |∇ γ | + 4 cos ϕγ ϕ ≤ (cid:18) − ρ + 2 ρ cos ϕρ + 4 cos ϕ (cid:19) |∇ γ | ≤ . Now we claim that |∇ γ | = o t (1) . The smooth convergence follows from this claim and the interpolation theorem. Weshow the claim in two steps.First, we show that there exists a sequence { t i } with t i → ∞ such thatmax ¯ S n + |∇ γ ( · , t i ) | → i → ∞ . Assume this is not true. Then there exists ǫ > T > ¯ S n + |∇ γ ( · , t ) | ≥ ǫ , for t > T . From (34) we have that for a large T > t > T , we have ddt max ¯ S n + |∇ γ | ≤ − C max ¯ S n + |∇ γ | + 12 Cǫ = − Cǫ , which is impossible.Second, we show that for any sequence { s i } with s i → ∞ , we havemax ¯ S n + |∇ γ ( · , s i ) | → i → ∞ . If not, there exists a sequence { s i } with s i → ∞ such thatmax ¯ S n + |∇ γ ( · , s i ) | ≥ ǫ 12 GUOFANG WANG AND CHAO XIA for any s i and for some positive constant ǫ . Without loss of generality, we mayassume that t i < s i . We consider the interval I i := [ t i , s i ] for sufficiently large i ,such that we have from (34) at a maximum point x t ∈ ¯ S n + (35) ddt max ¯ S n + |∇ γ | ≤ − C max ¯ S n + |∇ γ | + 12 Cǫ for any t ≥ t i . Let y i ∈ ¯ S n + and ¯ t i ∈ [ t i , s i ] such that |∇ γ ( y i , ¯ t i ) | = max t ∈ [ t i ,s i ] max ¯ S n + |∇ γ ( · , t ) | ≥ ǫ . By the first step, we may assume that ¯ t i = t i for i large. It follows that ddt max ¯ S n + |∇ γ | (¯ t i ) ≥ . Together with (35), implies that |∇ γ ( y i , ¯ t i ) | < ǫ , a contradiction. This proves the claim.From the claim, it follows easily that γ ( t ) converges smoothly to a constant γ and ρ → ρ smoothly for some constant ρ > 0, depending on the initial enclosedvolume of x .Next we show the exponential convergence in the case n = 2 and the enclosedvolume of x is not that of a half ball. In this case, ρ = 1. We return to (28). Bychoosing ǫ < ∂ t |∇ γ | ( x t , t ) ≤ ρv (cid:18) ( ρ − e w − ǫ ) − n ( ρ + 1) (cid:19) |∇ γ | + 1 v (cid:18) − ρe w + 4 cos ϕ + ρe w sin ϕ ǫ (cid:19) |∇ γ | ≤ ρv (cid:18) ( ρ − e w − ǫ ) − n ( ρ + 1) (cid:19) |∇ γ | + 1 v (cid:18) − (1 − ρ cos ϕ ) ρ + ρ sin ϕ (cid:18) ǫ (1 + ρ + 2 ρ cos ϕ ) − (cid:19)(cid:19) |∇ γ | ≤ C |∇ γ | − (cid:18) (1 − ρ cos ϕ ) ρ + Cρ sin ϕ (cid:19) |∇ γ | . As ρ converges to ρ = 1, − (cid:18) (1 − ρ cos ϕ ) ρ + Cρ sin ϕ (cid:19) ≤ − C for some C > t large. Then the exponential convergence follows. (cid:3) Proof of Corollary 1.1. 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Albert-Ludwigs-Universit¨at Freiburg, Mathematisches Institut, Ernst-Zermelo-Str.1, 79104 Freiburg, Germany E-mail address : [email protected] School of Mathematical Sciences, Xiamen University, 361005, Xiamen, P.R. China E-mail address ::