On the extension of axially symmetric volume flow and mean curvature

ON THE EXTENSION OF AXIALLY SYMMETRIC VOLUME FLOW AND MEAN CURVATURE FLOW

arXiv:1301.1125v1 [math.DG] 7 Jan 2013

SEVVANDI KANDANAARACHCHI A BSTRACT. We investigate conditions of singularity formation of mean curvature flow and volume preserving mean curvature flow in an axially symmetric setting. We prove that no singularities can develop during a finite time interval, if the mean curvature is bounded within that time interval on the entire surface. We prove this for volume preserving mean curvature flow as well as for mean curvature flow.

1. I NTRODUCTION Consider n-dimensional hypersurfaces Mt , defined by a one parameter family of smooth immersions xt : M n → Rn+1 . The hypersurfaces Mt are said to move by mean curvature, if xt = x(·, t) satisfies d (1.1) x(l, t) = −H(l, t)ν(l, t), l ∈ M n , t > 0 . dt By ν(l, t) we denote a smooth choice of unit normal on Mt at x(l, t) (outer normal in case of compact surfaces without boundary), and by H(l, t) the mean curvature with respect to this normal. If the evolving compact surfaces Mt are assumed to enclose a prescribed volume V the evolution equation changes as follows: d x(l, t) = − (H(l, t) − h(t)) ν(l, t), dt where h(t) is the average of the mean curvature, R Hdgt , h(t) = RMt Mt dgt

(1.2)

l ∈ M n , t > 0,

and gt denotes the metric on Mt . As the initial surface we choose a compact n-dimensional hypersurface M0 , with boundary ∂M0 6= ∅. We assume M0 to be smoothly embedded in the domain G = {x ∈ Rn+1 : a < x1 < b} , a, b > 0 , and ∂M0 ⊂ ∂G. Here we have a free boundary. We consider an axially symmetric surface contained in the region G between the two parallel planes x1 = a and x1 = b. Motivated by the fact that the stationary solution to the associated Euler Lagrange equation satisfies a Neumann boundary condition, we also assume the surface to meet the planes at right angles along its boundary. We consider the general question whether a singularity can develop if the mean curvature of a surface is bounded for a given time interval. Le and Sesum investigated related problems in [12]. In particular they proved that for mean curvature flow, if all singularities are of type I, the mean curvature blows

2010 Mathematics Subject Classification. 53C44, 35K93. 1

2

SEVVANDI KANDANAARACHCHI

up at the first singular time T . This paper is organised as follows. In sections 3 and 4, we study volume preserving mean curvature flow and investigate whether a singularity could develop if H is bounded. In sections 5 and 6 we study the same question for mean curvature flow. We prove the following two theorems. Theorem 1.1. For x(l, t) ∈ Mt satisfying (1.2), if |H(l, t)| ≤ c for all l ∈ M 2 and for all t ∈ [0, T ) then, there exists a constant c′ , such that |A|(l, t) ≤ c′ for t ∈ [0, T ) , i.e. the flow can be extended past time T . Theorem 1.2. For x(l, t) ∈ Mt satisfying (1.1), if |H(l, t)| ≤ c for all l ∈ M 2 and for all t ∈ [0, T ) then, there exists a constant c′ , such that |A|(l, t) ≤ c′ for t ∈ [0, T ) , i.e. the flow can be extended past time T . Acknowledgements : This is the work of the author’s PhD dissertation. The author would like to thank her supervisor Maria Athanassenas for her guidance and support and Gerhard Huisken for helpful discussions and for assisting a visit to the Max Planck Institute of Gravitational Physics. In particular Gerhard Huisken pointed out the technique of rescaling back to obtain a contradiction as done in both the theorems. 2. N OTATION We use the same notation as in [11], which was based upon Huisken’s [9] and Athanassenas’ [1] notation in describing the 2-dimensional axially symmetric hypersurface. Let ρ0 : [a, b] → R be a smooth, positive function on the bounded interval [a, b] with ρ′0 (a) = ρ′0 (b) = 0. Consider the 2-dimensional hypersurface M0 in R3 generated by rotating the graph of ρ0 about the x1 -axis. We evolve M0 along its mean curvature vector keeping its enclosed volume constant subject to Neumann boundary conditions at x1 = a and x1 = b. Equivalently, we could consider the evolution of a periodic surface defined along the whole x1 axis. By definition the evolution preserves axial symmetry. The position vector x of the hypersurface satisfies the evolution equation d x = −(H − h)ν = H + hν , dt where H is the mean curvature vector. Since ∆x = H, where ∆ denotes the Laplacian on the surface, we obtain, (2.1)

d x = ∆x + hν . dt

Let i1 , i2 , i3 be the standard basis in R3 , corresponding to x1 , x2 , x3 axes and τ1 (t), τ2 (t) be a local orthonormal frame on Mt such that hτ2 (t), i1 i = 0,

and

hτ1 (t), i1 i > 0 .

Let ω = |ˆxxˆ | ∈ R3 denote the unit outward normal to the cylinder intersecting Mt at the point ˆ = x − hx, i1 i i1 . Let x(l, t) , where x y = hx, ωi and v = hω, νi−1 .


3

Here y is the height function. We call v the gradient function. We note that v is a geometric p quantity, related to the inclination angle; in particular v corresponds to 1 + ρ′2 in the axially symmetric setting. The quantity v has facilitated results such as gradient estimates in graphical situations ( see for example [5, 6] ). We introduce the quantities ( see also [9] ) p = hτ1 , i1 i y −1 ,

(2.2)

q = hν, i1 i y −1 ,

so that p2 + q 2 = y −2 .

(2.3)

The second fundamental form has one eigenvalue equal to p = √ 1 ρ

1+ρ′2

and one eigenvalue equal

to

−ρ′′ . (1 + ρ′2 )3/2 We note that ρ(x1 , t) is the radius function such that ρ : [a, b] × [0, T ) → R, whereas y(l, t) is the height function and y : M 2 × [0, T ) → R.

k = ∇1 ν, τ1 =

3. E VOLUTION

EQUATIONS AND PRELIMINARY RESULTS

- VOLUME F LOW

Lemma 3.1. We have the following evolution equations: (i) (ii) (iii) (iv) (v)

d 1 dt y = ∆y − y + hpy ; v d 2 2 2 dt v = ∆v − |A| v + y 2 − v |∇v| ; d 2 2 2 dt k = ∆k + |A| k − 2q (k − p) − hk ; d 2 2 2 dt p = ∆p + |A| p + 2q (k − p) − hp ; d 2 dt H = ∆H + (H − h)|A| ;

Proof. Evolution equations (i), (ii), (iii) and (iv) are proved in [3] Lemma 5.1 and (v) is proved in [8]. Bounds on h. Getting bounds for h(t) is very important as h(t) is a global term. This enables us to use the maximum principles. Athanassenas ([2] Proposition 1.4) has obtained bounds for h(t) for an axially symmetric hypersurface between two parallel planes having orthonormal Neumann boundary data. The proof is of a geometric nature and works because of the axial symmetry. We state it here. Proposition 3.2. (Athanassenas) Assume {Mt } to be a family of smooth, rotationally symmetric surfaces, solving (1.2) for t ∈ [0, T ) . Then the mean value h of the mean curvature satisfies 0 < c2 ≤ h ≤ c3 , with c2 and c3 constants depending on the initial hypersurface M0 . Next we prove a height dependent gradient estimate. Lemma 3.3. There exists a constant c4 depending only on the initial hypersurface, such that vy < c4 , independent of time.

4


Proof. We calculate from Lemma 3.1 2 d (yv − c3 t) = ∆(yv) − h∇v , ∇(yv)i − yv|A|2 + h − c3 . dt v As h ≤ c3 we get by the parabolic maximum principle yv − c3 t ≤ max yv , M0

yv ≤ max yv + c3 T =: c4 . M0

Proposition 3.4. Assume Mt to be axially symmetric surfaces as described in Section 2 that evolve by (1.2). Then there is a constant c1 depending only on the initial hypersurface, such that the principal curvatures k and p satisfy kp < c1 , independent of time. Proof. Similar to equation (19) of [9] we calculate from Lemma 3.1 k k hk 2 q2 d k (p − k) . =∆ + ∇p , ∇ + 2 2 (p − k) (p + k) + dt p p p p p p k p

≥ 1 then (p − k) < 0. By the parabolic maximum principle we obtain k k ≤ max 1, max =: c1 . (3.1) M0 p p If

In the next Proposition we will get bounds for

|k| p

if the mean curvature is bounded for all t < T .

Proposition 3.5. If there exists a constant C such that |H|(l, t) ≤ C for all l ∈ M 2 and for all 2 t < T , then there exists another constant c0 such that |k| p (l, t) ≤ c0 for all l ∈ M and for all t < T. Proof. As and

1 p

−C ≤ H = k + p ≤ C , = vy ≤ c4 from Lemma 3.3 |k| ≤ 1 + Cc4 =: c0 . p

We note that the above proposition holds for a hypersurface evolving by mean curvature or by volume preserving mean curvature. 4. R ESCALED

SURFACE

- VOLUME F LOW

4.1. Rescaling procedure. We follow a rescaling technique similar to that of Huisken and Sinestrari used in [10]. Consider time intervals [0, T − 1i ] , for i ≥ 1 , i ∈ N , and determine the point li ∈ M 2 and the (latest in that interval) time ti , such that (4.1)

|A|(li , ti ) = max |A|(l, t) ; l∈M 2 t≤T − 1i


5

as it is an axially symmetric hypersurface we choose li ∈ M 2 , such that x(li , ti ) is on the (x1 , x3 ) plane. Let αi = |A|(li , ti ) and xi = x(li , ti ) . We consider the family of rescaled surfaces Mi,τ defined by the following immersions: ˜ i (· , τ ) = αi x(· , α−2 (4.2) x i τ + ti ) − hxi , i1 ii1 , 1 where τ ∈ [−α2i ti , α2i (T − ti − 1i )] . Here t = α−2 i τ + ti where t ∈ 0, T − i . the rescaled surface

b b

the original surface

xi b

b b

b

hxi , i1 ii1

0

The axis of rotation

F IGURE 1. The rescaling Note that we rescale from a point on the axis of rotation corresponding to the maximum curvature |A|2 . Thus the rescaled surfaces Mi,τ satisfy axial symmetry and we denote by ρ˜i,τ their generating curves - the cross section of Mi,τ with the (x1 , x3 ) plane . We look at a sequence of rescaled ˜i the second fundamental surfaces at different times ti where ti → T . We denote by |A˜i | and H ˜ i . By the definition of x ˜ i we have form and the mean curvature associated with the immersion x ˜i (· , τ ) = α−1 H(· , α−2 τ + ti ) , and |A˜i |(· , τ ) = α−1 |A|(· , α−2 τ + ti ) . H i i i i For t ≤ T − (4.3)

1 i

from the definition of αi we have −2 −1 |A|(· , α−2 i τ + ti ) ≤ |A|(li , ti ) , that is αi |A|(· , αi τ + ti ) ≤ 1 .

We observe that Mi,τ flow by mean curvature, preserving the enclosed volume of the rescaled surface: To see this, let R ˜ gτ Mi,τ Hi (· , τ )d˜ ˜ R hi = , gτ Mi,τ d˜

and note that the metric satisfies

q

det g˜ij (τ ) =

αni

q

det gij (t) ,

where (with a slight abuse of notation) we denoted by g˜τ the metric on Mi,τ , and by g˜ij (τ ) the components of g˜τ . Therefore ˜ i (τ ) = h and finally (4.4)

R

Mi,τ

−2 n α−1 i H(· , αi τ + ti )αi dgt −2 −1 R = α−1 i h(αi τ + ti ) = αi h(t) , n dg α t Mi,τ i

dx dt d ˜ ˜ ˜ i = αi x = −α−1 i (H − h)ν = −(Hi − hi )ν . dτ dt dτ

6


τ b

i i=1 b

τ = −α21 t1 b

τ = α21 (T − t1 − 1)

i=2 τ = −α22 t2 b

b

τ = α22 (T − t2 − 1/2)

i=j

b

τ = −α2j tj b

τ = α2j (T − tj − 1/j)

b

b

b

F IGURE 2. Schematic representation of the flow of the rescaled surfaces for different i Now we prove Theorem 1.1 Proof of Theorem 1.1. Assume a singularity develops at t = T , i.e. |A|2 → ∞ as t → T . We rescale the surface as discussed above in (4.2). As the rescaled flow satisfies (4.4), the curvature bound |A˜i |2 ≤ 1 for any i implies analogous bounds on all its covariant derivatives as in Theorem 4.1 in [8]. On the rescaled surfaces we have the maximum curvature achieved at (li , 0) where li ∈ M n , which means A˜i (li , 0) = 1 for all i . As |H|(l, t) is bounded for all t < T , we have |k| p ≤ c0 on Mt by Lemma 3.5. In regards to the curvature we have ; p |A| = k2 + p2 ≤ c5 p ≤ c5 y −1 , (4.5)

−1 −1 |A˜i | = α−1 = c5 (αi y)−1 = c5 y˜−1 . i |A| ≤ αi c5 y

We will show that for a fixed τ0 , a subsequence of the rescaled surfaces converges as i goes to infinity. We call the limiting hypersurface Mτ0 . We will explain this process in detail. Converging sequence of points. Without loss of generality let us assume that the singularity develops at the origin. We translate each generating curve to have hx(li , ti ) , i1 i = 0. We rename them again with the same ρ and work with them for the rest of the section. We choose ¯l ∈ M 2 such that x(¯l(t), t) is on the (x1 , x3 ) plane, and ¯l(t) = l ∈ M 2 : hx(l(t), t) , i1 i = 0 ,

We note that ¯l(ti ) = li . For a fixed τ0 as i goes to infinity, t goes to T making the maxMt |A| on the corresponding original hypersurfaces go to infinity. Therefore on the rescaled hypersurfaces for a fixed τ0 we can find N0 ∈ N such that for i > N0 1 |A˜i |(¯l(α−2 . i τ 0 + ti ) , τ 0 ) ≥ 2


7

As |A˜i |(· , τ0 )˜ y (· , τ0 ) ≤ c5 , from (4.5), we have y˜(¯l(α−2 i τ0 + ti ) , τ0 ) ≤ 2c5 for i > N0 . For τ = τ0 and i > N0 we look at the points on the rescaled surface generating curves ρ˜i,τ0 which are on the x3 axis. As 0 ≤ ρ˜i (· , τ0 ) x1 =0 ≤ 2c5 , xn+1 2c5 b b b b b b b b

b

x1

0

F IGURE 3. The convergence of points. we find a subsequence of points that converge. We call the subsequence of corresponding rescaled generating curves ρ˜1i,τ0 and the limiting point on the x3 axis c∗ := lim ρ˜1i,τ0 x1 =0 . i→∞

Converging sequence of tangent vectors. Now we look at the unit tangent vectors of ρ˜1i,τ0 x1 =0 . xn+1 b b b b b b b b

b

0

x1

F IGURE 4. The unit tangent vectors. We translate the unit tangent vectors to the origin (see figure 4). By identifying these unit vectors with points on the sphere, we observe that a subsequence of these points converges and so obtain a subsequence of tangent vectors that converges. By translating back each tangent vector to its original position, we find the corresponding subsequence of rescaled generating curves; we call it ρ˜2i,τ0 . Now we have found a subsequence of rescaled generating curves, where the points and the tangent vectors converge at x1 = 0. Convergence in a small ball. As |A˜i | ≤ 1 for every rescaled hypersurface, we can roll a ball of radius 1 over the entire hypersurface in such a way that when it touches any point, it does not

8


√

1

1−

√

1−

δ2

1 − δ2

δ

F IGURE 5. The unit ball sitting on the rescaled hypersurface intersect the curve anywhere else. Thus we have a gradient bound in a small ball of radius δ < 1 on the hypersurface. Now we look at the sequence of curves ρ˜2i,τ0 near x1 = 0.

b

2c5

2c5 b

c∗

B(0,c∗ ) (δ)

bc

After rotating

x1

x1 δ

δ

F IGURE 6. Convergence in a small ball We rotate each curve around the point on the curve where x1 = 0 , so that the tangent vector at x1 = 0 is parallel to the x1 axis(see figure 6). For |x1 | < δ , each of these rotated curves has its δ gradient bounded by the same constant ρ˜′ ≤ √1−δ . This gives equicontinuity. Also at x1 = 0 2 the height of each curve is bounded by 2c5 . Hence the height of all the rotated curves is bounded by √ 2c5 + (1 − 1 − δ2 ) for |x1 | < δ (see figure 5). Therefore our curves are uniformly bounded. By Arzela-Ascoli, there exists a subsequence that converges uniformly. Let us call this subsequence of 3 . rotated curves gi,τ 0 3 Thus we have C 0 convergence for the sequence gi,τ in the small ball B(0,c∗ ) (δ) , which is the ball 0 of radius δ centred at (0, c∗ ). We call the corresponding subsequence before rotation ρ˜3i,τ0 . We dif 3 , and call it g ′3 . We note that g ′3 is bounded when |x | < δ. As ferentiate the sequence gi,τ 1 i,τ0 i,τ0 0 |g′′3 | the curvature of the rescaled surfaces is bounded we have 3 ≤ 1 , giving uniform bounds (1+(g′3 )2 ) 2 on g′′3 for |x1 | < δ. Thus we have uniform boundedness and equicontinuity for the sequence ′3 . From Arzela-Ascoli there exists a subsequence that converges uniformly. Let us call it g ′4 . gi,τ i,τ0 0 4 . From the Theorem of uniform We call the corresponding subsequence before differentiation gi,τ 0


9

4 convergence and differentiability we have C 1 convergence for the sequence gi,τ in the small ball 0 4 B(0,c∗ ) (δ). We call the corresponding subsequence before rotation ρ˜i,τ0 .

We note that the subsequence before rotation ρ˜4i,τ0 converges as well, because the tangent vectors and the points at x1 = 0 converge. Therefore the unrotated curves converge in the ball B(0,c∗ ) (δ) as well. 4 Convergence in a compact set. On the subsequence of rotated curves gi,τ we find another set of 0 4 points as follows. We mark the points on the rotated curves gi,τ0 x1 =±ǫ where ǫ < δ such that only 4 . a finite number of points are outside the ball B(0,c∗ ) (δ) . Let us first look at the points gi,τ 0 x1 =−ǫ

b

2c5

b b

After unrotating

2c5

b

b

b

−δ ǫ

δ

x1

x1

F IGURE 7. Finding the points p4i These new points and their tangent vectors converge on the unrotated curves ρ˜4i,τ0 as shown previously. We call these new points p5i (see figure 7). Now we translate each curve ρ˜4i,τ0 along the x1 such that the point p5i is on the x3 axis. Then we rotate each curve as before, around p5i such that the tangent vector of each curve is parallel to the x1 axis. For |x1 | < δ , by Arzela-Ascoli, we find 4 5 . As previously by cona subsequence of gi,τ that converges in C 0 . We call this subsequence gi,τ 0 0 ′5 6 sidering the differentiated sequence gi,τ0 we find another subsequence gi,τ0 that convergence in C 1 . We call the corresponding subsequence before rotation ρ˜6i,τ0 . We note that the original subsequence ρ˜6i,τ0 converges. To summarise, we first found the limiting curve in a small ball B(0,c∗ ) (δ) . Then by considering a ball centred on the limiting curve in B(0,c∗ ) (δ) , we found another subsequence that converged in both these balls (see figure 8). In this manner, by considering a diagonal subsequence, we can extend our limiting curve along small balls centred on the limiting curve. Therefore given any compact set [α, β] × [0, γ] where γ > 2c5 , we repeat the above process, and find a subsequence uniformly converging in a union of small balls (see figure 9), which is a subset of the given compact set. This subsequence may not be unique. We get the associated hypersurface by rotating this curve around the x1 axis. ˜ i = α−1 H and as |H(l, t)| ≤ c for this theorem We note that αi goes to ∞ as i goes to ∞ . As H i 2 ˜ for all l ∈ M and for all t < T , Hi goes to 0 , as i goes to ∞. As 0 < c2 ≤ h ≤ c3 and d ˜ ˜ i − h˜i )˜ ˜ i = −(H h˜i = α−1 νi , we observe that i h , we have hi (τ0 ) going to 0 as well. As dτ x

10


xn+1

b b

b b

b b

b b

x1

F IGURE 8. Convergence in two small balls

b b b b

b

b b b b

b

b b

b

b

b

b b

b

b

b

F IGURE 9. Convergence in a compact set d ˜ i = 0 . Thus Mτ0 is a stationary solution and therefore independent of τ . We rename limi→∞ dτ x this stationary solution M .

The contradiction. The limiting solution M is a catenoid as it is the only axially symmetric minimal surface with zero mean curvature. However for large i if we rescale back Mi,τ0 , to get an understanding of the original surface, then we see that the estimate vy ≤ c does not hold on that surface. Therefore as we will show in the next paragraph we get a contradiction. We denote the quantities associated to the catenoid M by a hat ˆ. We obtain the catenoid M by rotating yˆ = c5 cosh(c−1 ˆ1 ) , around the x1 axis where x ˆ1 is the x1 coordinate of the limiting 5 x 1 surface M . As the convergence is C , for any ǫ1 > 0 and for any l0 ∈ M 2 and for any fixed τ0 we have |ˆ v (l0 )ˆ y (l0 ) − v˜i (l0 , τ0 )˜ yi (l0 , τ0 )| ≤ ǫ1

for large i ,

vˆ(l0 )ˆ y (l0 ) − ǫ1 ≤ v˜i (l0 , τ0 )˜ yi (l0 , τ0 ) for i > I0 , for some I0 ∈ N p −1 For the catenoid vˆ = 1 + yˆ′2 = cosh(c5 x ˆ1 ). Therefore


11

−2 c5 cosh2 (c−1 ˆ1 (l0 )) − ǫ1 ≤ αi v(l0 , α−2 5 x i τ + ti )y(l0 , αi τ + ti ) ,

ǫ1 c5 cosh(2c−1 ˆ1 ) + 1 − ≤ vy 5 x 2αi αi

(4.6) As

for i > I0 .

τ + t ) − hx , i ii x ˆ1 (l0 ) = hˆ x(l0 ), i1 i = lim hαj x(l0 , α−2 j j 1 1 , i1 i , j j→∞ , τ + t ) − x x ˆ1 (l0 ) = lim αj x1 (l0 , α−2 j 1j j j→∞

where x1j := hxj , i1 i , for any ǫ2 > 0 we have τ + t ) − x ˆ1 (l0 ) − αj x1 (l0 , α−2 x j 1j ≤ ǫ2 for large j , j

giving us (4.7) for j > J0 . ˆ1 (l0 ) ≤ ǫ2 +αj x1 (l0 , α−2 αj x1 (l0 , α−2 j τ + tj ) − x1j j τ + tj ) − x1j −ǫ2 < x For given ǫ1 , ǫ2 and for a fixed i > I0 , we pick large j > J0 , j >> i, such that 1 4αi ǫ1 ǫ2 + 2c5 log c4 + −1 x1j + ǫ j holds. Considering Mt as a periodic surface, we can find points on the hypersurface which lie an ǫ˜ distance away from x1j . Therefore 1 4αi ǫ1 > x + τ + t (4.9) x1 l0 , α−2 ǫ + 2c log c + − 1 . 1j 0 j 2 5 4 j αj c5 αi We will use the above inequality to arrive at the contradiction. By (4.7) and (4.9) 4αi ǫ1 −2 x ˆ1 (l0 ) > αj x1 (l0 , αj τ + tj ) − x1j − ǫ2 > 2c5 log c4 + − 1 > 0. c5 αi Therefore cosh

1 αj x1 (l0 , α−2 i τ0 + tj ) − x1j − ǫ2 2c5

< cosh

1 x ˆ1 (l0 ) for j > J0 , 2c5

By (4.6) c5 1 1 ǫ1 c5 ǫ1 (αj (x1 − x1j ) − ǫ2 ) + 1 − < x ˆ1 (l0 ) + 1 − < vy , cosh cosh 2αi 2c5 αi 2αi 2c5 αi 1 ǫ1 c5 (αj (x1 − x1j ) − ǫ2 ) + 1 − < vy , cosh 2αi 2c5 αi ǫ c5 2c1 (αj (x1 −x1j )−ǫ2 ) − 1 (α (x −x )−ǫ ) 1 − e 2c5 j 1 1j 2 + 2 − < vy , e 5 4αi αi

12


for ǫ1 , ǫ2 , i, j and l0 as previously chosen. As αj x1 (l0 , α−2 j τ + tj ) − x1j −ǫ2 > 0 by the choice of l0 ǫ ǫ c5 2c1 (αj (x1 −x1j )−ǫ2 ) c5 2c1 (αj (x1 −x1j )−ǫ2 ) − 1 (α (x −x )−ǫ ) 1 1 −1+2 − ≤ − e 2c5 j 1 1j 2 + 2 − , e 5 e 5 4αi αi 4αi αi giving us (4.10)

ǫ c5 2c1 (αj (x1 −x1j )−ǫ2 ) 1 ≤ vy . e 5 +1 − 4αi αi

But from (4.9), we have 1 4αi ǫ1 > x + τ + t x1 l0 , α−2 ǫ + 2c log c + − 1 , 1j 0 j 2 5 4 j αj c5 αi 4αi ǫ1 −2 αj x1 l0 , αj τ0 + tj − x1j − ǫ2 > 2c5 log c4 + −1 , c5 αi 1 ǫ1 4αi α x l ,α−2 j τ0 +tj )−x1j )−ǫ2 ) 2c5 ( j ( 1 ( 0 c4 + − 1, e > c5 αi ǫ c5 2c1 (αj (x1 (l0 ,α−2 1 j τ0 +tj )−x1j )−ǫ2 ) +1 − (4.11) > c4 . e 5 4αi αi From Lemma 3.3 we know that vy ≤ c4 . Therefore (4.10) and (4.11) contradict Lemma 3.3: that means if we examine the rescaled surfaces, we can see that the estimate vy ≤ c4 does not hold on the corresponding original, non-rescaled hypersurfaces, near the singular time T . Therefore our original assumption is wrong. Hence, there exists a constant c′ such that |A|(l, t) ≤ c′ for t ∈ [0, T ). As in Theorem 4.1 in [8] we get bounds for the covariant derivatives of |A| as well. Thus the flow can be extended past time T . 5. E VOLUTION

EQUATIONS AND PRELIMINARY RESULTS

- M EAN C URVATURE F LOW

We consider an axially symmetric, n dimensional hypersurface in Rn+1 with Neumann boundary data and use the same notation as in the volume preserving case. In essence we consider the same initial hypersurface, but evolving by mean curvature defined by d x(l, t) = −H(l, t)ν(l, t) . dt We consider the possibility of a singularity developing in the hypersurface Mt in which |H(l, t)| ≤ c , for all t < T , without imposing any additional conditions on H. In this section we prove that a singularity cannot develop in this case. 2 2 ′ 2 If H(l, t) > 0 on all of Mt , then from the evolution equation for |A| H 2 one can see that |A| ≤ c H (see [9]); thus no singularities develop if H(l, t) is bounded. However we cannot use this evolution equation when H = 0. As we do not impose the condition H > 0 on the initial hypersurface, we repeat the blow up technique used in volume preserving mean curvature flow in the last section. First we will prove the required preliminary results for mean curvature flow. Lemma 5.1. We have the following evolution equations:


(i) (ii) (iii) (iv) (v)

13

1 d dt y = ∆y − y ; 2 v d 2 2 dt v = ∆v − |A| v + y 2 − v |∇v| ; d 2 2 dt k = ∆k + |A| k − 2q (k − p) ; d 2 2 dt p = ∆p + |A| p + 2q (k − p) ; d 2 dt H = ∆H + (H − h)|A| ;

Proof. Evolution equations (i), (iii) and (iv) are proved in [9], (ii) is proved in [4] and (v) is proved in [7]. The next Lemma corresponds to Lemma 3.3 in the volume preserving case. Lemma 5.2. There exists a constant c′4 depending only on the initial hypersurface, such that vy < c′4 , independent of time. Proof. Similarly to Lemma 3.3, we obtain d 2 (yv) = ∆(yv) − h∇v , ∇(yv)i − yv|A|2 . dt v From the parabolic maximum principle yv ≤ max yv =: c′4 . M0

The following corresponds to Proposition 3.4 in the volume preserving setting. Proposition 5.3. There is a constant c1 depending only on the initial hypersurface, such that c1 in Mt for all t < T .

|k| p

1, i.e. k > 0 , we get the last term to be negative on Mt . Thus k k ≤ max 1, max =: c1 . (5.1) M0 p p

For

6. R ESCALED

SURFACE

- M EAN C URVATURE F LOW

Now we have our prerequisites. To show the equivalent of Theorem 1.1, we employ the same rescaling that proceeds (4.2). The only difference being (6.1)

d dx dt ˜ ˜ i = αi = −α−1 x i Hν = −Hi ν . dτ dt dτ

Proof of Theorem 1.2. This is the same proof as in Theorem 1.1, the only exceptions being that we use Lemma 5.2 instead of Lemma 3.3 and we refer to Proposition 2.3 in [9] for the analogous bounds on covariant derivatives of A˜ .

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