Area-preserving mean curvature flow of rotationally symmetric hypersurfaces with free boundaries
aa r X i v : . [ m a t h . DG ] D ec AREA-PRESERVING MEAN CURVATURE FLOW OFROTATIONALLY SYMMETRIC HYPERSURFACES WITH FREEBOUNDARIES
KUNBO WANG
Abstract.
In this paper, we consider the area-preserving mean curvature flowwith free Neumann boundaries. We show that for a rotationally symmetric n -dimensional hypersurface in R n +1 between two parallel hyperplanes will con-verge to a cylinder with the same area under this flow. We use the geometricproperties and the maximal principle to obtain gradient and curvature esti-mates, leading to long-time existence of the flow and convergence to a constantmean curvature surface. Introduction and the main results
A Hypersurface M t in Eucilidean space is said to be evolving by mean curvatureflow if each point X ( · ) of the surface moves, in time and space, in the direction ofits unit normal with speed equal to the mean curvature H at that point. That is(1.1) ∂X∂t = − Hν ( x, t ) , where ν ( x, t ) is the outer unit normal. It was first studied by Brakle in [6] from theviewpoint of geometric measure theory. In [12], G. Huisken showed that any com-pact, convex hypersurface without boundary converges asymptotically to a roundsphere in a finite time interval. Mean curvature flow is also the steepest descentflow for the area functional, evolving to minimal surfaces. In [13], Huisken initiatedthe idea of considering the following volume-preserving mean curvature flow,(1.2) ∂∂t X ( x, t ) = ( h ( t ) − H ( x, t )) ν ( x, t ) , where h ( t ) = ´ Mt Hdµ t ´ Mt dµ t is the average of the mean curvature on M t . Huisken provedif the initial hypersurface M n ( n ≥
2) is uniformly convex, then the evolutionequation (1.2) has a smoothly solution M t for all times 0 ≤ t < ∞ and M t convergesto a round sphere enclose the same volume as M in the C ∞ -topology as t → ∞ .In [20], Pihan studied the following area-preserving mean curvature flow.(1.3) ∂∂t X ( x, t ) = (1 − h ( t ) H ( x, t )) ν ( x, t ) . Here h ( t ) = ´ Mt Hdµ t ´ Mt H dµ t . Pihan showed that if the initial hypersurface is compactwithout boundary, (1.3) has a unique solution for a short time under the assumption h (0) >
0. For n = 1, Pihan also showed that an initially closed, convex curve in the Date : December 16, 2017.2000
Mathematics Subject Classification.
Primary 35R35, 53C21, 53C44.
Key words and phrases. area-preserving mean curvature flow, free boundary. plane converges exponentially to a circle with the same length as the initial curve.For n ≥
2, Mccoy in [16] showed that if the initial n -dimensional hypersurface M is strictly convex then the evolution equation (1.3) has a smooth solution M t forall time 0 ≤ t < ∞ , and M t converge, as t → ∞ , in the C ∞ -topology, to a spherewith the same surface area as M . In [11], Huang and Lin use the idea of iterationof Li in [14] and Ye in [23], in cases of volume preserving mean curvature flow andRicci flow, respectively. And obtained the same result, by assuming that the initialhypersurface M satisfies h (0) > ´ M | A | − n H dµ ≤ ǫ. In this paper, we study the area preserving mean curvature flow with free bound-aries, where a restriction on the angles of boundaries with fixed hypersurfaces inEuclidean space are imposed. In this setting, there are some interesting works (cf.[7], [19] and [22]). In these papers, the authors study the mean curvature flow withNeumann and Dirichlet free boundaries. Let Σ be a fixed hypersurface smoothlyembedded in R n +1 . We say X ( x, t ) is evolved by mean curvature flow with freeNeumann bounary condition on Σ, if ∂X∂t = − Hν, ∀ ( x, t ) ∈ M n × [0 , T ) , h ν M t , ν Σ ◦ X i ( x, t ) = 0 , ∀ ( x, t ) ∈ ∂M n × [0 , T ) ,X ( · ,
0) = M ,X ( x, t ) ⊂ Σ , ∀ ( x, t ) ∈ ∂M n × [0 , T ) . The volume-preserving mean curvature flow with free Neumann boundaries wasfirst studied by Athanassenas in [2].Let M be a complete n -dimensional hypersurface with boundary ∂M = ∅ . Assume M is smoothly embedded in the domain G = { x ∈ R n +1 : 0 < x n +1 < d, d > } , We denote by Σ i ( i = 1 , , the two parallel hyperplanes bounding the domain G ,and assume ∂M ⊂ Σ i ( i = 1 , Theorem 1.1.
Let
V, d ∈ R be given two positive constants. M ⊂ G is a smooth,rotationally symmetric, initial hypersurface which intersects Σ i ( i = 1 , orthog-onally at the boundaries which encloses the volume V . Then the free Neumannboundaries problem for equation (1.2) has a unique solution on [0 , + ∞ ) , whichconverges to a cylinder C ⊂ G of volume V under assumption | M | ≤ Vd as t → ∞ . Other works on this problem were investigated in [3], [4], [17] and [18]. In thispaper, we consider the following problem for area-preserving mean curvature flowwith free Neumann boundaries.
Problem 1.1. ∂∂t X ( x, t ) = (1 − h ( t ) H ( x, t )) ν ( x, t ) , ∀ ( x, t ) ∈ M n × [0 , T ) , h ν M t , ν Σ i ◦ X i ( x, t ) = 0 , ∀ ( x, t ) ∈ ∂M n × [0 , T ) , i = 1 , ,X ( · ,
0) = M ,X ( x, t ) ⊂ Σ i , ∀ ( x, t ) ∈ ∂M n × [0 , T ) , i = (1 , . REA-PRESERVING MEAN CURVATURE FLOW OF ROTATIONALLY SYMMETRIC HYPERSURFACES WITH FREE BOUNDARIES3
We prove the following main theorem for Problem 1.1.
Main Theorem.
Let
V, d ∈ R + be given two constants and M ⊂ G to be asmooth, rotationally symmetric, mean convex initial hypersurface which intersectsΣ i ( i = 1 ,
2) orthogonally at the boundaries. Then the solution to Problem 1.1exists for all times t > M under the assumption | M | ≤ Vd . Remark 1.1.
We say M is mean convex if the mean curvature is positive every-where. The condition of mean convex will be used to prove the equation (1.3) isstrictly parabolic. In [20], Pihan shows that the equation (1.3) is strictly parabolicfor a short time if h (0) > . And as the case of volume-preserving mean curva-ture flow in [3], Problem 1.1 is a Neumann boundary problem for strictly parabolicequation, from which we obtain the short time existence. And also see [19] and [22]for details of general cases. This paper is organized as follows. In Section 2, we give some definitions andpreliminaries. In Section 3, we show some basic properties of this flow. We provethat the property of mean convexity can be preserved under equation (1.3) and thesurfaces do not pinch off under the condition of our main theorem. In Section 4,we use the property of mean convexity and maximal principle to show the curva-ture estimates. Gradient and curvature estimates lead to long time existence to aconstant curvature surface. And we prove our main theorem in Section 5.The methods we use here are those introduced by Athanassenas in [2], Ecker andHuisken in [9]. We use the free Neumann boundary condition to convert the bound-ary estimates to interior estimates (see Lemma 3.4, Lemma 4.1 and Theorem 4.1).We put the condition of mean convexity here is to give an upper and lower boundsfor h ( t ) and v ( x, t ), which is crucial for our curvature estimates.2. Preliminaries
We adopt the similar notations of Huisken in [12]. Let M be an n -dimensionalRiemannian manifold. Vectors on M are denoted by X = { X i } , covectors by Y = { Y i } and mixed tensors by T = { T ikj } . The induced metric and the secondfundamental form on M are denoted by g = g ij and A = { h ij } respectively. Thesurface area element of M is given by µ = q det ( g ij ) . For tensors T ijkl and U ijkl on M, we have the inner product( T ijkl , U ijkl ) = g iα g jβ g kγ g lδ T ijkl U αβγδ and the norm | T ijkl | = ( T ijkl , T ijkl ) . The trace of the second fundamental form, H = g ij h ij , is the mean curvature of M , and | A | = g ik g jl h ij h kl is the square of the norm of the second fundamentalform on M . We also denote˜ C = tr ( A ) = g ij g kl g mn h ik h jm h ln . If X : M n ֒ → R n +1 smoothly embeds M n into R n +1 , then the induced metric g is given by g ij = h ∂X∂x i ( x ) , ∂X∂x j ( x ) i and the second fundamental forms h ij = KUNBO WANG h ∂X∂x i ( x ) , ∂ν∂x i ( x ) i , where h· , ·i is the ordinary scalar product of vectors in R n +1 . Thematrix of the Weingarten map of M is h ij ( x ) = g ik ( x ) h kj ( x ) . The eigenvalues ofthis matrix are the principal curvatures of M . The induced connection on M isgiven via the Christoffel Symbols.Γ kij = 12 g kl ( ∂g jl ∂x i + ∂g il ∂x j − ∂g ij ∂x l ) . The covariant derivative, for a vector field v = v j ∂∂x j is given by ∇ i v j = ∂v j ∂x i + Γ jik v k , ∇ i v j = ∂v j ∂x i − Γ jik v k . The Laplacian of T is △ T = g ij ∇ i ∇ j T. The Riemannian curvature tensor on M can be given through the Gauss Equations R ijkl = h ik h jl − h il h jk . We denote | M t | to be the surface area of M t . We assume M is mean convex androtationally symmetric about an axis which intersects Σ i orthogonally. We denotethis axis by x n +1 and use the parametrization ρ ( x n +1 ) : [0 , d ] R for the generating curve of a surface of revolution, which is a radius function.3. Basic properties
From now on, we write [0 , T ) to indicate the maximal time interval for whichthe flow exists. First we verify that the surface area does indeed remain fixedunder the area-preserving flow (1.3), while the enclose volume does not decrease.The rotationally symmetric property is preserved under the equation (1.3). Thisis clear from the evolution equation, since the mean curvature and the normal aresymmetric.
Lemma 3.1.
The surface area of M t remains constant throughout the flow, that is ddt ˆ M t dµ t ≡ . Proof.
We apply the first variation of area formula to the vector field ∂X∂t , extendedappropriately, and the divergence theorem, ddt ˆ M t dµ t = ˆ M t div M t ( ∂X∂t ) dµ t = − ˆ M t (1 − hH ) Hdµ t ≡ . (cid:3) Lemma 3.2.
The volume enclosed by M t does not decrease throughout the flow.That is, if E t ⊂ R n +1 is the ( n + 1) -dimensional set enclosed by M t and the twoparallel planes Σ i , then ddt V ol ( E t ) ≥ REA-PRESERVING MEAN CURVATURE FLOW OF ROTATIONALLY SYMMETRIC HYPERSURFACES WITH FREE BOUNDARIES5
Proof.
We first extend ∂X∂t to a vector field on the whole of E t , then apply the firstvariation of area formula and divergence theorem, ddt V ol ( E t ) = ˆ E t div R n +1 ( ∂X∂t ) dV = ˆ ∂E t < ∂X∂t ν > dµ t = ˆ M t < ∂X∂t ν > dµ t + ˆ Σ i < ∂X∂t ν > dµ t = ˆ M t dµ t − ( ´ M t Hdµ t ) ´ M t H dµ t ≥ . Here we have use the free Neumann boundary condition to obtain the integral onΣ i is 0. (cid:3) As in [16] (Section 4), we have the following evolution equations.
Lemma 3.3.
We have(1) ∂∂t g ij = 2(1 − hH ) h ij ; (2) ∂∂t g ij = − − hH ) h ij ;(3) ∂∂t ν = h ∇ H ;(4) ∂∂t h ij = h △ h ij + (1 − hH ) h mi h mj + h | A | h ij ; (5) ∂∂t h ij = h ( △ h ij + | A | h ij ) − h im h mj ; (6) ∂∂t H = h △ H − (1 − hH ) | A | ; (7) ∂∂t | A | = h ( △| A | − |∇ A | + 2 | A | ) − C ; (8) ( ∂∂t − h △ ) H = − h |∇ H | − − hH ) H | A | . Lemma 3.4.
The mean curvature is positive on M t , t ∈ [0 , T ) . Furthermore onthe boundaries ∂M t , we have lim x n +1 → H ( x, t ) = a ( t ) > , lim x n +1 → d H ( x, t ) = b ( t ) > . Proof.
Since we consider the hypersurface has free boundaries, we can not directlyuse the maximal principle. Suppose the first time H ( x, t ) = 0 is attained at aninterior point of M t , then from Lemma 3.3, we have( ∂∂t − h △ ) H = − h |∇ H | − − hH ) H | A | . From the maximal principle, we know H ( x, t ) = 0 can not be attained at aninterior point of M t . If t is the first time such that lim x n +1 → H ( x, t ) = 0 orlim x n +1 → d H ( x, t ) = 0. By a reflection of Σ and Σ , we can define two pieces ofnew hypersurfaces outside the boundary, which satisfies the free Neumann bound-aries conditions. Denote ˜ M t to be the new hypersurface and ˜ ρ ( x n +1 ) its radiusfunction . Precisely,˜ ρ ( x n +1 ) = ρ (2 d − x n +1 ) , d ≤ x n +1 ≤ dρ ( x n +1 ) , ≤ x n +1 < dρ ( − x n +1 ) , − d ≤ x n +1 < H ( x , x , · · · , x n , , t ) = lim x n +1 → H ( x, t ), ˜ H ( x , x , · · · , x n , d, t ) = lim x n +1 → d H ( x, t ).Then at the boundary points, ∂∂t ˜ H ( x, t ) ≤ △ ˜ H ( x, t ) >
0. So the maximal prin-ciple can still be applied, which proves the lemma. (cid:3)
Now we will show that the radius of the hypersurface M t has uniform lower andupper bounds for any time t ∈ [0 , T ). The method follows from [2](Lemma 1). KUNBO WANG
Lemma 3.5.
Under the conditions of the Main Theorem, there exist constant r and R only depending on n , d , V and | M | such that r ≤ ρ ( x n +1 , t ) ≤ R for any t ∈ [0 , T ) .Proof. Given an initial surface M , we denote by C the cylinder with the sameenclosed volume V as M in G . Assume that there is some t > M t pinches off. We project M t onto the plane Σ , using the natural projection π : R n +1 → R n . Then | M t | ≥ | π ( M t ) | . Any M t has to intersect the cylinder C at least once by the volume constrain, thatthe volume of M t is not decreasing. Therefore | M | = | M t | > | π ( M t ) | > | π ( C ) | = ω n ρ nC = Vd .
Here ρ C is the radius of C , and ω n is the volume of unit ball in R n , then we obtaina contradiction. For the upper bound, we assume that ρ ( x n +1 , t ) max = R ( t ), thenwe have | M | = | M t | > ω n · ( R ( t ) − ρ C ) n , which implies R ( t ) < ρ C + ( | M | ω n ) n . (cid:3) Curvature estimates
Let ˆ x = ( x , · · · , x n , ω = ˆ x | ˆ x | to denote the unit outer normal to thecylinder intersecting M t at the point X ( P, t ). As in [2] and [21], we define theheight function of M t to be u = h X, ω i . And define v ( x, t ) = h ω, ν i − = p ( ˙ ρ ) + 1,where ˙ ρ is the derivative of ρ about x n +1 . From Lemma 3.5, we can obtain theheight estimate r ≤ u ( x, t ) ≤ R. Now we show that v has an upper bound under theassumption T < ∞ . The kernel ideal is that according to our parametrization, thepoints where v tends to infinity are not the singular points of the evolving surface,and the | A | tends to zero at these points. Lemma 4.1. If T < ∞ , then v ( x, t ) ≤ M T < + ∞ for any t ∈ [0 , T ] , in thelimitation sense when t = T , i.e. v ( x, T ) = lim t → T v ( x, t ) . Here M T is a constantdepending on n, d, T, r, R, V and | M | .Proof. Since M t is rotationally symmetric we have H = κ + ( n − κ , where κ and κ denote the principle curvatures. If we parameterize M by its radius function ρ ∈ C ∞ ([0 , d ]), then clearly H = − ¨ ρ (1 + ˙ ρ ) + n − ρ (1 + ˙ ρ ) . Suppose t is the first time such that lim x n +1 → s v ( x, t ) = lim x n +1 → s p ( ˙ ρ ) ( x n +1 , t ) + 1 =+ ∞ for some s ∈ (0 , d ). Since lim x n +1 → s ˙ v =
12 2 ˙ ρ ¨ ρ √ ( ˙ ρ ) +1 = 0, we have lim x n +1 → s ¨ ρ ( x n +1 , t ) =0, then we have H = 0 at this point, which is a contradiction with H > t → T,x n +1 → s v ( x, t ) = + ∞ , then lim t → T,x n +1 → s | A | ( x, t ) = (¨ ρ ) ( x n +1 ,t )[( ˙ ρ ) ( x n +1 ,t )+1] + n − ρ [( ˙ ρ ) ( x n +1 ,t )+1] = 0 which implies X ( x , · · · , x n , s, T ) is not a singular point. So REA-PRESERVING MEAN CURVATURE FLOW OF ROTATIONALLY SYMMETRIC HYPERSURFACES WITH FREE BOUNDARIES7 the maximal principal method in Lemma 3.4 can still be applied and we must have
H > v ( x, t ) max < + ∞ forall t ∈ [0 , T ], by the continuity of v , we have v ≤ M T for some constant dependingon T . (cid:3) Next, we will show an estimate of h ( t ). First, we prove the following lemma. Lemma 4.2.
Under the assumption of the main theorem, we have C ≤ ´ M t Hdµ t ≤ C , for all time t ∈ [0 , T ) . Here C and C are positive constants only dependingon n, d, r, R, V and | M | .Proof. First we show that ´ M t Hdµ t ≥ C for some constant C . This is a directconsequence of the first variation formula and mean curvature is positive. Since n | M | = n | M t | ≤ ˆ M t H < X, ν > dµ t ≤ ˆ M t H | X | dµ t ≤ p d + R ˆ M t Hdµ t , the lower bound for ´ M t Hdµ t is obtained. Next we show there is an upper boundfor ´ M t Hdµ t , that there is a constant C such that ´ M t Hdµ t ≤ C . We stillparameterize M t by its radius function ρ ( x n +1 , t ). We denote ω n to be the volumeof unit ball in R n , and its surface area is nω n , then we have H = − ¨ ρ (1 + ˙ ρ ) + n − ρ (1 + ˙ ρ ) . | M t | = nω n ˆ d ρ n − p ρ dx n +1 . ˆ M t Hdµ t = nω n ˆ d ( − ¨ ρ (1 + ˙ ρ ) ρ n − + ( n − ρ n − ) dx n +1 . On one hand, we have ´ d ( n − ρ n − dµ t ≤ ( n − dR n − . On the other hand, by our boundary conditons, ˆ d − ¨ ρ (1 + ˙ ρ ) ρ n − dx n +1 = ˆ d − ρ n − d (arctan ˙ ρ ) dx n +1 = − ρ n − · (arctan ˙ ρ | d ) + ( n − ˆ d ρ n − · ˙ ρ · arctan ˙ ρdx n +1 ≤ ( n − π · ˆ d ρ n − p ρ dx n +1 ≤ ( n − π r ˆ d ρ n − p ρ dx n +1 = ( n − π nω n | M t | r = ( n − π nω n | M | r . Thus the upper bound is obtained. (cid:3)
Corollary 4.1.
Under the assumption of the main theorem, we have < h ( t ) ≤ C for all time t ∈ [0 , T ) . Here C is a constant only depending on n, d, r, R, V and | M | . KUNBO WANG
Proof.
Now we use the Cauchy-Schwarz inequality( ´ M t Hdµ ) ( ´ M t H dµ )( ´ M t dµ ) ≤ . From which we have 0 < ´ Mt Hdµ t ´ Mt H dµ t ≤ ´ Mt dµ t ´ Mt Hdµ t = | M | ´ Mt Hdµ t ≤ C . (cid:3) Now we show that | A | is bounded for any finite time interval. Theorem 4.1.
If the maximal time interval [0 , T ) is finite, i.e. T < + ∞ , then wehave | A | ( x, t ) ≤ C T , where C T is a constant depending only on T , n, d, r, R, V and | M | .Proof. First we compute the evolution equation of v ( x, t ) = < ω, ν > − . Clearly wehave ∂∂t v = − v h w, ∂∂t ν i = − v · h h ∇ H, ω i . From [2] , we have △ v = | A | v − v < ω, ∇ H > + 2 |∇ v | v − n − u · v. Then we obtain ( ∂∂t − h △ ) v = − h | A | v − h |∇ v | v + ( n − hvu , and ( ∂∂t − h △ ) v = 2 v · ( − h | A | v − h |∇ v | v + ( n − hvu ) − h |∇ v | = − h |∇ v | − hv | A | + 2( n − hv u . We considering | A | v as in [12] and divide the points in M t into three sets. S t = { P ∈ M t | ¨ ρ ≥ } .I t = { P ∈ M t | ¨ ρ < , κ κ < α } .J t = { P ∈ M t | ¨ ρ < , κ κ ≥ α } . Here α is a positive constant large enough. We will show that for all points in S t and I t , | A | v has uniform upper bounds for any t ∈ [0 , T ). We split our proof intothree cases. Case (1) . If P ∈ S t , from H = − ¨ ρ (1+ ˙ ρ ) + n − ρ (1+ ˙ ρ ) >
0, we have¨ ρ < n − ρ [1 + ( ˙ ρ ) ] . Then we have | A | v = (¨ ρ ) [1 + ( ˙ ρ ) ] + n − ρ ≤ ( n − ρ + n − ρ ≤ C. REA-PRESERVING MEAN CURVATURE FLOW OF ROTATIONALLY SYMMETRIC HYPERSURFACES WITH FREE BOUNDARIES9
From now on, we denote by C to any constant depending on n, V, d, r, R and M . Case (2) . If P ∈ I t , then κ κ < α . We have − ρ · ¨ ρ ( ˙ ρ ) +1 < α , and − ¨ ρ ( ˙ ρ ) +1 < αr . Thus | A | v = (¨ ρ ) [1 + ( ˙ ρ ) ] + n − ρ ≤ C.Case (3) . For points in J t , we use the technique of maximal principle. First we have( ∂∂t − h △ ) | A | v = | A | · ( − h |∇ v | − hv | A | + 2( n − hv u )+ v · ( − h |∇ A | + 2 h | A | − C ) − h ∇| A | · ∇ v = | A | · ( − h |∇ v | − hv | A | + 2( n − hv u )+ v · ( − h |∇ A | + 2 h | A | − C ) + h ( −∇| A | ∇ v − v | A |∇| A |∇ v )= | A | · ( − h |∇ v | − hv | A | + 2( n − hv u )+ v · ( − h |∇ A | + 2 h | A | − C ) + h [ − v − ∇ v ∇ ( | A | v ) + v − |∇ v | | A | − v | A |∇| A |∇ v ] + 2( n − h | A | v u ≤ − h | A | |∇ v | − hv |∇| A || − Cv − hv − ∇ v ∇ ( | A | v )+ 4 h |∇ v | | A | − hv | A |∇| A |∇ v + 2( n − h | A | v u ≤ − hv ∇ v ∇ ( | A | v ) + 2( n − h | A | v u − Cv . We have used |∇| A || ≤ |∇ A | and Cauchy-Schwarz inequality. Since2( n − h | A | u ˜ C ≤ C · κ + ( n − κ κ + ( n − κ = C · κ + ( n − κ ( κ κ ) n − κ κ ) ≤ C · κ . Then, if κ > C , we have n − h | A | v u − Cv <
0. Thus | A | v can not attain amaximal value by the maximal principle. And if κ ≤ C , we have | A | = k + n − ρ [1 + ( ˙ ρ ) ] ≤ C, and | A | v ≤ CM T . Therefore, | A | ≤ CM T v min = C T . (cid:3) Corollary 4.2.
Under the assumption of the above theorem, we have a lower boundfor h ( t ) , namely, h ( t ) ≥ m T . Here, m T is a constant depending on T , n , V , d, r, α, R and | M | .Proof. It is a direct consequence of H ≤ n | A | , Lemma 4.2 and Theorem 4.1. (cid:3) Next we give the higher derivative estimates as Hamilton in [10].
Corollary 4.3.
Under the assumption of Theorem 4.1, we have the following higherderivative estimates |∇ m A | ≤ C m ( T ) Proof.
First, we have ∂∂t |∇ m A | = h △|∇ m A | − h |∇ m +1 A | + X i + j + k = m ∇ i A ∗ ∇ j A ∗ ∇ k A ∗ ∇ m A + X r + s = m ∇ r A ∗ ∇ s A ∗ ∇ m A. We assume when l ≤ m , we have |∇ l A | ≤ C l ( T ). Then for n = m + 1, we have ∂∂t |∇ m +1 A | ≤ h △|∇ m +1 A | + C ( T ) · ( |∇ m +1 A | + 1) . We choose f = |∇ m +1 A | + N |∇ m A | , where N is a constant large enough. Then ∂∂t f ≤ h △|∇ m +1 A | + C ( T ) · ( |∇ m +1 A | + 1)+ N h △|∇ m A | − hN |∇ m +1 A | + C ( T ) ≤ h △ f − C ( T ) |∇ m +1 A | + C ( T ) = h △ f − C ( T ) ( f − N |∇ m A | ) + C ( T ) ≤ h △ f − C ( T ) f + C ( T ) . Thus f ≤ C T . (cid:3) Corollary 4.4. T = + ∞ . Proof of the main theorem
Since the upper bound we derived above is a constant depending on T , | A | maybe unbounded when t tends to infinity. We will show that this will not happen andthe initial hypersurface converges to a constant mean curvature surface. Theorem 5.1.
The mean curvature H of the evolving surfaces converge to a con-stant as t → ∞ .Proof. Since ddt
V ol ( E t ) = ´ M t (1 − hH ) dµ t . Then we have ˆ ∞ ddt V ol ( E t ) dt = ˆ ∞ ˆ M t (1 − hH ) dµ t dt = V ol ( E ∞ ) − V ol ( E ) ≤ C. Therefore, lim t →∞ ˆ M t (1 − hH ) dµ t = 0 . Thus, lim t →∞ ˆ M t dµ t = lim t →∞ ( ´ M t Hdµ t ) ´ M t H dµ t . Then by Cauchy-Schwarz inequality, H = C for some constant. (cid:3) REA-PRESERVING MEAN CURVATURE FLOW OF ROTATIONALLY SYMMETRIC HYPERSURFACES WITH FREE BOUNDARIES11
Proof.
Rotationally symmetric hypersurfaces of constant mean curvature in R n +1 are plane, sphere, cylinder, catenoid, unduloid and nodoid, they are known as theDelaunay surfaces (see [8]). Our boundary conditions excludes the possibilities ofplane, sphere, catenoid and nodoid. In [2] (see Section 1), Athanassenas use thecondition | M | ≤ Vd to exclude the existence of unduloids in G . So our possibilitycan only be the cylinder. Thus the Main Theorem is proved. (cid:3) Acknowledgement.
The result of this paper is surveyed under the supervision ofProfessor Sheng WeiMin at Zhejiang University. The author would like to expresssincere gratitude to Professor Sheng WeiMin for his guide in these years. His carefulreading and comments have led to an improvement of this paper.
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