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016 The extension problem of the mean curvature flow (I)

Haozhao Li∗ and Bing Wang†

August 10, 2016

Abstract

We show that the mean curvature blows up at the first finite singular time for a closed smoothembedded mean curvature flow inR3.

Contents

1 Introduction 1

2 Preliminaries 5

3 Weak compactness of the mean curvature flow 63.1 The pseudolocality theorem . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 73.2 Energy concentration property . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 143.3 Weak compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 16

4 Multiplicity-one convergence of the rescaled mean curvature flow 174.1 Convergence away from singularities . . . . . . . . . . . . . . . .. . . . . . . . . . 184.2 Decomposition of spaces and a “monotone decreasing” quantity . . . . . . . . . . . 204.3 Construction of auxiliary functions . . . . . . . . . . . . . . . .. . . . . . . . . . . 234.4 Proof of the multiplicity-one convergence . . . . . . . . . . .. . . . . . . . . . . . 27

5 Proof of the extension Theorem 29

Appendix A The parabolic Harnack inequality 30

1 Introduction

Letx0 : Σ2 → R

3 be a closed smooth embedded surface inR3. A one-parameter family of immersions

x(p, t) : Σ2 → R3 is called a mean curvature flow, ifx satisfies the equation

∂x

∂t= −Hn, x(0) = x0, (1.1)

∗Supported by NSFC grant No. 11131007.†Supported by NSF grant DMS-1510401.

1

whereH denotes the mean curvature of the surfaceΣt := x(t)(Σ) andn denotes the unit normalvector field ofΣt. In [30], Huisken proved that if the flow (1.1) develops a singularity at timeT < ∞,then the fundamental form will blow up at timeT . A natural conjecture is that the mean curvaturewill blow up at the finite singular time of a mean curvature flow. There are many results toward thisconjecture(cf. Cooper [5][6], Le-Sesum [22][23][24], Lin-Sesum [25], Xu-Ye-Zhao [49] ...). Thisconjecture is also proposed in page 42 of Mantegazza’ book [36] as an open problem. In this paper,we confirm this conjecture in dimension three:

Theorem 1.1. If x(p, t) : Σ2 → R3(t ∈ [0, T )) is a closed smooth embedded mean curvature flow

with the first singular timeT < +∞, then

supΣ×[0,T )

|H|(p, t) = +∞.

It is interesting to compare the extension problem of mean curvature flow with Ricci flow. In[27] Hamilton proved that the Riemann curvature tensor willblow up at the finite singular time ofa Ricci flow. In [37] Sesum extended Hamilton’s result to the Ricci curvature by using Perelman’snoncollapsing theorem. In a series of papers [40][41][12] Wang and Chen-Wang provided severalconditions which can be used to extend Ricci flow, and they gave a different proof of Sesum’s result. Itis also conjectured that the Ricci flow can be extended if the scalar curvature stays bounded. Importantprogresses have been made by Zhang [50], Song-Tian [39], Bamler-Zhang [2] and Simon [38].

The mean curvature flow with convexity conditions has been well studied during the past severaldecades. If the initial hypersurface satisfies some convexity conditions, like mean convex or two con-vex, then the mean curvature flow has some convexity estimates (cf. Huisken [30], Huisken-Sinestrari[32][33], Haslhofer-Kleiner [28]). These convexity estimates are important for studying the surgeryof mean curvature flow (cf. Huisken-Sinestrari[34], Brendle-Huisken [3], Haslhofer-Kleiner [29]).In [44] [45] White also gave some important properties of thesingularities of a mean curvature flowwith mean convex initial hypersurfaces. However, all theseresults rely on the convexity condition ofinitial hypersurfaces, and it is very difficult to study general cases (cf. Colding-Minicozzi [16] [17],T. Ilmanen [35]). Theorem 1.1 can be viewed as an attempt to study the general singularities withoutassuming convexity conditions.

Now we sketch the proof of Theorem 1.1. Assume that the mean curvature is bounded along theflow (1.1) and the first singular timeT < +∞. Consider the corresponding rescaled mean curvatureflow

∂x

∂t= −

(

H − 1

2〈x,n〉

)

n, ∀ t ∈ [0,∞). (1.2)

Then the mean curvature decays exponentially to zero along the flow (1.2). We have to show that theflow (1.2) converges smoothly to a plane with multiplicity one. The proof consists of two steps:

Step 1. Convergence of the rescaled mean curvature flow with multiplicities.In this step, we first follow the ideas of Chen-Wang [9] [10] todevelop the weak compactness

theory of mean curvature flow with certain properties, whichare basically area doubling propertytogether with bounded mean curvatureH and bounded energy

∫

Σ |A|2. To prove such weak compact-ness, there are two main technical ingredients:

2

• The two-sided, long-time pseudolocality theorem.

• The energy concentration property.

The short-time forward pseudolocality theorem for mean curvature flow was studied under the localgraphic condition, which says that the surface can be written locally as a graph of a single-valuedfunction. See Ecker-Huisken [19] [20], M.T. Wang [43], Chen-Yin [4] and S.Brendle [3]. In our case,we need to remove this graphic condition since the flow may converge with multiplicities. Underthe assumption that the mean curvature is bounded, we will show a new short-time, two-sided pseu-dolocality type theorem(c.f. Theorem 3.7). Moreover, the pseudolocality theorem can be improvedas long-time version(c.f. Theorem 3.8) whenever the mean curvature is very small. Then the energyconcentration property follows from the pseudolocality theorem. Once we have the long-time, two-sided pseudolocality theorem and the energy concentrationproperty (cf. Lemma 3.10), we can showthe weak compactness of mean curvature flow and get the “flow” convergence of the rescaled meancurvature flow. Since the limit is both minimal and self-shrinking, we obtain the limit must be a planepassing through the origin, with possibly more than one multiplicity.

Step 2. Show that the multiplicity of the convergence is one.In this step, we use the ideas from the compactness of self-shrinkers and minimal surfaces (cf.

Colding-Minicozzi [14][15], L.Wang [42]) to show that the multiplicity is one. For otherwise we canconstruct a sequence of positive solutions to the corresponding parabolic equations on compact sets ofthe limit surface away from singularities. However, we shall show that the existence of such a solutioncontradicts the fact that the limit is a plane with multiplicities. To obtain the contradiction, there arealso two main technical ingredients:

• Uniform estimates of the sequence of positive solutions.

• TheL-stability of the limit surface across the singular set.

To show uniform estimates of these positive solutions, we introduce a new “almost” decreasing quan-tity along the rescaled mean curvature flow, and use this quantity with the parabolic Harnack inequalityto control the positive solutions. After normalization andtaking the limit, we get a positive solutionwith good estimates on the limit surface away from singularities. Using these estimates, we showthat the limit surface isL-stable away from singularities. Recall that a hypersurface Σ is calledL-stable(c.f. Colding-Minicozzi [15]) if for any compactly supported functionu, we have

∫

Σ−uLu e−

|x|2

4 ≥ 0,

whereL is the operatorLu = ∆u+ |A|2u− 12〈x,∇u〉+ u

2 defined by Colding-Minicozzi [16]. Fol-lowing the argument of Gulliver-Lawson [21] in minimal surfaces, we show that the limit surface isactuallyL-stable across the singular set. However, the limit surfaceis a plane which is notL-stable.Thus we obtain the desired contradiction. This contradiction forces that the convergence of (1.2) musthave multiplicity one and hence in the smooth topology.

3

Once the rescaled mean curvature flow (1.2) converges smoothly to a plane with multiplicity one,we use Huisken’s monotonicity formula and White’s regularity theorem to show that the mean curva-ture flow (1.1) actually has no singular points at timeT . This finishes the proof of Theorem 1.1.

The strategy of the proof of Theorem 1.1 is very similar to theone used in the study of theKahler Ricci flow on Fano manifolds by Chen-Wang [10] and Chen-Sun-Wang [8]. Actually, inChen-Wang [10], the flow weak-compactness was setup for the normalized Kahler Ricci flow, whichis comparable to the step 1 mentioned above. Then in Chen-Sun-Wang [8], the limit and convergencetopology can be improved to be “smooth” whenever the underlying manifold isK-stable. This is re-lated to our step 2 described above. However, here we used theL-stability of the limit surfaces, insteadof theK-stability of the the underlying manifolds to rule out the possible singularities. Not surpris-ingly, the strategy can also be applied to study the extension problem for the 4-dimensional Ricciflow with bounded scalar curvature. Actually, based on the calculation of M. Simon(c.f. [38]), wehave uniform bounded energy

∫

M|Rm|2 along the flow. Then the weak-compactness of the parabolic

normalized Ricci flows(compared with equation (1.2))

∂tg = −2(Ric− g)

follows from Chen-Wang [9](See also [2] for a different approach). The limit of the above flow isthen a Ricci-flat gradient shrinking soliton with finite singular points, which is nothing but a flat met-ric cone overS3/Γ for some finite subgroupΓ of SO(3). Similar to the step 2 mentioned above, the4-dimensional Ricci flow extension problem will be confirmedif one can develop methods to rule outthe nontrivialΓ’s.

We remark that the weak compactness of the mean curvature flow, i.e., step 1, is based on theobservation that locally the structure of the mean curvature flow with boundedH and energy

∫

Σ |A|2is modeled by the structure of minimal surfaces with uniformly bounded topology. Similar obser-vation is also the key for the weak compactness of the Ricci flow with bounded energy

∫

M|Rm|m2

and bounded scalar curvatureR. The two-sided, long-time pseudo-locality(c.f. Chen-Wang [9], [10]and [11]) and the energy concentration are the technical tools for writing down the observation inrigorous analysis estimates. It seems that such observation holds for many other geometric flows, e.g.,the harmonic map flow and the Calabi flow.

The organization of this paper is as follows. In Section 2 we recall some basic facts on meancurvature flow and minimal surfaces. In Section 3 we develop the weak compactness theory of meancurvature flow under some geometric conditions. In Section 4we show the rescaled mean curvatureflow with the exponential decay of mean curvature converges smoothly to a plane with multiplicityone. Finally, we finish the proof of Theorem 1.1 in section 5.

Acknowledgements: H. Z. Li would like to thank Professors T.H. Colding, W. P. Minicozzi IIand X. Zhou for insightful discussions. Part of this work wasdone while he was visiting MIT andhe wishes to thank MIT for their generous hospitality. B. Wang would like to thank Professors T.Ilmanen and L. Wang for helpful discussions.

4

2 Preliminaries

Let x(p, t) : Σn → Rn+1 be a family of smooth embeddings inRn+1. (Σn,x(t)), 0 ≤ t < T is

called a mean curvature flow ifx(t) satisfies

∂x

∂t= −Hn, ∀ t ∈ [0, T ), (2.1)

and called a rescaled mean curvature flow ifx(t) satisfies

∂x

∂t= −

(

H − 1

2〈x,n〉

)

n, ∀ t ∈ [0, T ), (2.2)

It is easy to check that the rescaled mean curvature flow is equivalent to the mean curvature flow afterrescalings in space and a reparameterization of time.

It is well known that the volume ratio is bounded from below along the flow (2.1). See, forexample, Lemma 2.9 of Colding-Minicozzi [16].

Lemma 2.1. Let (Σn,x(t)), 0 ≤ t < T be a mean curvature flow (2.1). Then there is a constantN = N(Vol(Σ0), T ) > 0 such that for allr > 0 andp0 ∈ R

n+1 we have

Vol(Br(p0) ∩ Σt) ≤ Nrn, ∀ t ∈ [0, T ).

A hypersurfacex : Σn → Rn+1 is called a self-shrinker, if it satisfies the equation

H =1

2〈x,n〉. (2.3)

By Corollary 2.8 of Colding-Minicozzi [16], we have

Lemma 2.2. (Corollary 2.8 of [16]) IfΣ is a self-shrinker and the mean curvature is zero, thenΣ isa minimal cone. In particular, ifΣ is also smooth and embedded, then it is a hyperplane through0.

Observe that the equation (2.1) is invariant under the rescaling

x(p, s) = λ(

x(p, T +s

λ2)− p0

)

, ∀ (p, s) ∈ Σ× [−Tλ2, 0). (2.4)

whereλ > 0. In fact, under the rescaling (2.4) we have the relations

Aij = λAij , gij = λ2gij , H =1

λH, |A| = 1

λ|A|,

whereA, g andH denote the second fundamental form, the induced metric and the mean curvature ofthe surfaceΣ = x(Σ) respectively. Moreover, by direct calculation we have

Lemma 2.3. Let Σ2 be a smooth surface inR3. The area ratio and theL2 norm of the secondfundamental form are invariant under (2.4). Namely, we have

Areag(Br(p) ∩ Σ)

πr2=

Areag(Br(p) ∩ Σ)

πr2

and∫

Br(p)∩Σ|A|2 dµ =

∫

Br(p)∩Σ|A|2 dµ,

wherer = λr, p = λ(p − p0) andΣ = λ(Σ− p0).

5

Now we recall some facts on compactness of immersed hypersurfaces inRn+1. For anyBR(p) ⊂Rn+1, we say an immersed (or embedded) hypersurfaceΣn ⊂ R

n+1 properly immersed (or embed-ded) inBR(p), if either Σ is closed or∂Σ has distance at leastR from the pointp. We say that asequenceΣi converges inCk,α topology to a hypersurfaceΣ, if for any p ∈ Σ eachΣi is locally(nearp) a graph over the tangent spaceTpΣ and the graph ofΣi converges to the graph ofΣ in theusualCk,α topology.

The following theorem is well known and it shows the convergence of properly immersed surfacesunder the second fundamental form bound. The readers are referred to [48] for the details.

Theorem 2.4(Compactness of minimal surfaces). LetΣni be a sequence of smooth, properly im-

mersed hypersurfaces inBR(p) ⊂ Rn+1. If for all i ≥ 1, the second fundamental formsupΣi∩BR(p) |Ai| ≤

Λ for a constantΛ > 0, then there is a subsequence ofΣni converges inC1, 1

2 topology, possibly withmultiplicities, to an immersed hypersurfaceΣ∞. Moreover, if the hypersurfacesΣn

i are minimal,then the convergence is in smooth topology.

To study the convergence of mean curvature flow, we define the convergence of a sequence ofone-parameter hypersurfaces as follows.

Definition 2.5. We say that a sequence of one-parameter smoothly immersed hypersurfacesΣni,t,−1 <

t < 1 in Rn+1 converges in smooth topology, possibly with multiplicities, to a limit flowΣ∞,t,−1 <

t < 1 away from a space-time singular setS ⊂ Rn+1 × (−1, 1), if for any t ∈ (−1, 1), anyp ∈

Σ∞,t\St and largei, there existsr > 0 andǫ > 0 such that the hypersurfaceΣi,s∩Br(p) with s ∈ [t−ǫ, t+ǫ] can be written as a collection of graphs of smooth functionsu1i (x, s), u2i (x, s), · · · , uNi (x, s)over the tangent plane ofΣ∞,t at the pointp. Moreover, for eachk ∈ 1, 2, · · · , N the functionsuki (x, s) converges smoothly inx ands asi → +∞.

In the above definition,St is defined bySt = x ∈ Rn+1 | (x, t) ∈ S. If St is independent oft,

then we can also replace the space-times singular setS simply bySt0 for somet0.

The following compactness result of mean curvature flow is well-known. See, for exmaple, page481-482 of [4] for a detailed proof.

Theorem 2.6 (Compactness of mean curvature flow). Let (Σni ,xi(t)),−1 < t < 1 be a se-

quence of mean curvature flow properly immersed inBR(0) ⊂ Rn+1(i.e. for eacht ∈ (−1, 1) the

hypersurfaceΣi,t is properly immersed inBR(0)). Suppose that

supΣi,t∩BR(0)

|A|(x, t) ≤ Λ, ∀ t ∈ (−1, 1)

for someΛ > 0. Then a subsequence ofΣi,t ∩ BR(0),−1 < t < 1 converges in smooth topologyto a smooth mean curvature flowΣ∞,t,−1 < t < 1 in BR(0).

3 Weak compactness of the mean curvature flow

In this section, we follow the arguments of Ricci flow by Chen-Wang in [9][10] and minimal surfacesby Choi-Schoen in [7] to study the weak compactness of mean curvature flow under some geometricconditions. This weak compactness result will be used to prove the convergence of rescaled meancurvature flow in the next section.

6

3.1 The pseudolocality theorem

The pseudolocality type results of the mean curvature flow were studied by Ecker-Huisken [19] [20],M. T. Wang [43], Chen-Yin [4] and Brendle-Huisken [3]. However, all these pseudolocality theoremsabove require the condition that the initial hypersurface can be locally written as a graph of a single-valued function. In our case, we need to remove this graphic condition since the flow may convergewith multiplicities. Here, we give a different type of pseudolocality theorem under the assumptionthat the mean curvature along the flow is uniformly bounded.

Definition 3.1. For anyr > 0, p ∈ Rn+1 andΣn ⊂ R

n+1, we denote byCx(Br(p)∩Σ) the connectedcomponent ofBr(p) ∩Σ containingx ∈ Σ.

Lemma 3.2. (cf. Lemma 7.1 of [4]) LetΣn ⊂ Rn+1 be properly embedded inBr0(x0) for some

x0 ∈ Σ with

|A|(x) ≤ 1

r0x ∈ Br0(x0) ∩Σ.

Let x1 · · · , xn+1 be coordinates of radiusr0 aroundx0 such thatTx0Σ = span∂x1 , · · · , ∂xn.Then there is a map

u :

x′ = (x1, · · · , xn)∣

∣

∣|x′| < r0

96

→ R

with u(0) = 0 and|∇u|(0) = 0 such that the connected component ofΣ∩|x′| < r096 can be written

as a graph(x′, u(x′)) | |x′| < r096 and

|∇u|(x′) ≤ 36

r0|x′|.

Using Lemma 3.2, we show that the local area ratio of the surface is very close to1.

Lemma 3.3. Suppose thatΣn ⊂ Br0(p) ⊂ Rn+1 is a hypersurface with∂Σ ⊂ ∂Br0(p) and

supΣ

|A| ≤ 1

r0.

For anyδ > 0, there is a constantρ0 = ρ0(r0, δ) such that for anyr ∈ (0, ρ0) and anyx ∈ B r02(p)∩Σ

we haveVolΣ(Cx(Br(x) ∩ Σ))

ωnrn≤ 1 + δ. (3.1)

Proof. By Lemma 3.2, for anyx ∈ B r02(p) ∩ Σ the componentCx(Bρ0(x) ∩ Σ) with ρ0 = r0

192

can be written as a graph of a functionu over the tangent plane atx, which we assume to beP =

(x1, · · · , xn, xn+1) ∈ Rn+1 | xn+1 = 0, with |∇u|(x′) ≤ 72

r0|x′| wherex′ = (x1, · · · , xn). For

anyr ∈ (0, ρ0), the volume ratio ofCx(Br(x) ∩ Σ) is given by

VolΣ(Cx(Br(x) ∩Σ))

ωnrn=

1

ωnrn

∫

Br(x)∩P

√

1 + |∇u|2 dµ ≤√

1 +5184

r20r2.

Thus, we can chooser sufficiently small such that (3.1) holds. The lemma is proved.

7

The next result shows that we can control the local volume ratio along the mean curvature flowwith bounded mean curvature.

Lemma 3.4. Let (Σn,x(t)),−1 ≤ t ≤ 1 be a smooth embedded mean curvature flow (2.1) withmaxΣt×[−1,1] |H(p, t)| ≤ Λ. Then for anyt1, t2 ∈ [−1, 1] we have

Volg(t2)(Cpt2(Br2(pt2) ∩ Σt2))

ωnrn2≤ f(t1, t2,Λ, r2)

Volg(t1)(Cpt1(Br1(pt1) ∩ Σt1))

ωnrn1(3.2)

wherept = xt(p, t) for somep ∈ Σ and

r1 = r2 + 2Λ|t2 − t1|,

f(t1, t2,Λ, c, r2) = eΛ2|t2−t1|

(

1 +2Λ

r2|t2 − t1|

)n

Proof. Since the mean curvature is bounded, for anyΩ ⊂ Σ we calculate∣

∣

∣

d

dsVolg(t)(Ω)

∣

∣

∣=

∣

∣

∣

∫

Ω|H|2 dµt

∣

∣

∣≤ Λ2Volg(t)(Ω),

which implies that

e−Λ2|t1−t2|Volg(t2)(Ω) ≤ Volg(t1)(Ω) ≤ eΛ2|t1−t2|Volg(t2)(Ω). (3.3)

Note that for anyp, q ∈ Σ, we have

∂

∂t|x(p, t)− x(q, t)|2 = 2〈x(p, t) − x(q, t),−H(p, t)n(p, t) +H(q, t)n(q, t)〉

≤ 4Λ|x(p, t) − x(q, t)|.

It follows that|x(p, t1)− x(q, t1)| ≤ |x(p, t2)− x(q, t2)|+ 2Λ|t1 − t2|.

Let pt = x(p, t). Then we have

x(t2)−1(Cpt2

(Br(pt2) ∩ Σt2)) ⊂ x(t1)−1(Cpt1

(Br+2Λ|t1−t2|(pt1) ∩ Σt1)).

Therefore, we have the area estimates

Volg(t2)(Cpt2(Br(pt2) ∩Σt2))

≤ Volg(t2)(x(t1)−1(Cpt1

(Br+2Λe|t1−t2|(pt1) ∩ Σt1)))

≤ eΛ2|t1−t2|Volg(t1)(Cpt1

(Br+2Λ|t1−t2|(pt1) ∩ Σt1)), (3.4)

where we used (3.3) in the last inequality. For anyx ∈ Σn ⊂ Rn+1 andr > 0, we have

Volg(t2)(Cpt2(Br2(pt2) ∩ Σt2))

ωnrn2

=Volg(t2)(Cpt2

(Br2(pt2) ∩ Σt2))

ωnrn2

≤ eΛ2|t1−t2| ·

(r1r2

)n

·Volg(t1)(Cpt1

(Br1(pt1) ∩Σt1))

ωnrn1

= eΛ2|t1−t2| ·

(r1r2

)n

·Volg(t1)(Cpt1

(Br1(pt1) ∩Σt1))

ωnrn1,

8

wherer1, r2 satisfy the following relations:

r1 = r2 + 2Λ|t1 − t2|.

The lemma is proved.

Using the idea of the monotonicity formula of minimal surfaces, we show that the volume ratio isalmost monotone if the mean curvature is bounded. See, for example, Proposition 1.12 in Colding-Minicozzi [13].

Lemma 3.5. Let Σn ⊂ Rn+1 be a properly embedded hypersurface inBr0(x0) with x0 ∈ Σ and

|H| ≤ Λ. Then for anys ∈ (0, r0) we have

VolΣ(Bs(x0) ∩ Σ)

ωnsn≤ eΛr0 · VolΣ(Br0(x0) ∩ Σ)

ωnr0n.

In particular, lettings → 0 we have

VolΣ(Br(x0) ∩Σ) ≥ e−Λrωnrn, ∀ r ∈ (0, r0].

Proof. Note that the functionf(x) = |x− x0| satisfies the identity

∆Σf2 = 2n− 2H〈x− x0,n〉.

By the Stokes’ theorem, we have

2nVol(f ≤ s) =

∫

f≤s∆Σ f2 + 2

∫

f≤sH〈x− x0,n〉

= 2

∫

f=s|(x− x0)

T |+ 2

∫

f≤sH〈x− x0,n〉. (3.5)

The coarea formula implies that

Vol(f ≤ s) =∫ s

0

∫

f=r|∇Σf |−1. (3.6)

Combining the identities (3.5)-(3.6), we have

d

ds

(

s−nVol(f ≤ s))

= −ns−n−1Vol(f ≤ s) + s−n

∫

f=s

|x− x0||(x− x0)T |

= s−n−1

∫

f=s

|(x− x0)N |2

|(x− x0)T |− s−n−1

∫

f≤sH〈x− x0,n〉

≥ −Λ · s−nVol(f ≤ s).

LetF (s) = s−nVol(f ≤ s). Then for anys ∈ (0, r0) we have

F (s) ≤ F (r0)eΛ(r0−s) ≤ F (r0)e

Λr0 .

Thus, the lemma is proved.

9

Lemma 3.6. For anyC > 0, there existsδ = δ(n,C) > 0 satisfying the following property. Anycomplete smooth minimal hypersurfaceΣn ⊂ R

n+1 with bounded second fundamental form|A| ≤ C

and volume ratioVolΣ(Br(p) ∩ Σ)

ωnrn< 1 + δ, ∀ r > 0 (3.7)

must be a hyperplane.

Proof. Suppose not, there exists a sequence of non-flat minimal hypersurfacesΣi with |Ai| ≤ C and

Vol(Br(pi) ∩ Σi)

ωnrn< 1 + δi, ∀ r > 0 (3.8)

wherepi ∈ Σi andδi → 0. SinceΣi are non-flat, we can assume that|Ai|(pi) = 1. By Theorem 2.4,a subsequence ofΣi = Σi− pi converges smoothly to a complete smooth minimal hypersurfacesΣ∞

with |A∞|(0) = 1 and volume ratio

Vol(Br(0) ∩Σ∞)

ωnrn= 1, ∀ r > 0. (3.9)

(3.9) implies thatΣ∞ is a hyperplane(c.f. Corollary 1.13 of Colding-Minicozzi [13]), which contra-dicts the equality|A∞|(0) = 1. Thus, the lemma is proved.

Combining the above results, we show the following pseudolocality theorem.

Theorem 3.7(Two-sided pseudolocality). For anyr0 ∈ (0, 1],Λ, T > 0, there existη = η(n,Λ), ǫ =

ǫ(n,Λ) > 0 satisfying

limΛ→0

η(n,Λ) = η0(n) > 0, limΛ→0

ǫ(n,Λ) = ǫ0(n) > 0 (3.10)

and the following properties. Let(Σn,x(t)),−T ≤ t ≤ T be a closed smooth embedded mean

curvature flow (2.1). Assume that

(1) the second fundamental form satisfies|A|(x, 0) ≤ 1r0

for anyx ∈ Cp0(Br0(p0) ∩ Σ0) wherep0 = x0(p) for somep ∈ Σ;

(2) the mean curvature of(Σn,xt),−T ≤ t ≤ T is bounded byΛ.

Then for any(x, t) satisfying

x ∈ Cpt(Σt ∩B 116

r0(p0)), t ∈

[

− ηr202(Λ + Λ2)

,ηr20

2(Λ + Λ2)

]

∩ [−T, T ] (3.11)

wherept = xt(p), we have the estimate

|A|(x, t) ≤ 1

ǫr0.

10

Proof. The proof consists of the following steps:Step 1.Without loss of generality, we assumer0 = 1. By Lemma 3.3 and the assumption (1), for

any fixedδ > 0 there exists a constantρ0 = ρ0(δ) ∈ (0, 12 ] such that for anyr ∈ (0, ρ0] we have

Volg(0)(Cy0(Br(y0) ∩ Σ0))

ωnrn≤ 1 + δ, ∀ y0 ∈ Cp0(B 1

2(p0) ∩ Σ0). (3.12)

For anyρ0 ∈ (0, ρ0], we define

η0(n, ρ0, δ) := sup

η ∈ (0,1

8]∣

∣

∣eη(

1 +2η

ρ0 − 2η

)n

(1 + δ) ≤ 1 + 2δ

. (3.13)

For any givenΛ > 0, we chooseρ0 = ρ0(n, δ,Λ) ∈ (0, ρ0] such that

ρ0(n, δ,Λ) := sup

ρ0 ∈ (0, ρ0]∣

∣

∣eΛ(ρ0−2η0(n,ρ0,δ))(1 + 2δ) ≤ 1 + 3δ

, (3.14)

whereη0 is defined by (3.13). Note thatρ0 andη0 have positive lower bounds depending only onn

andδ asΛ → 0.

Step 2.By Lemma 3.4, (3.12) and (3.13), for anyt ∈ [−T, T ] with Λ|t|+ Λ2|t| ≤ η0 we have

Volg(t)(Cyt(Bρ1(yt) ∩ Σt))

ωnρn1

≤ eΛ2|t|

(

1 +2Λ|t|ρ1

)nVolg(0)(Cy0(Bρ0(y0) ∩ Σ0))

ωnρn0

≤ 1 + 2δ, (3.15)

whereyt = xt(x−10 (y0)), y0 is any point inCp0(B 1

2(p0) ∩ Σ0) andρ1 := ρ0 − 2η0(n, ρ0, δ). Note

thatρ1 > 0 by (3.13). Letη1 =√

η02(Λ+Λ2)

. Then (3.15) holds for anyt ∈ [−2η21 , 2η21 ] ∩ [−T, T ].

By Lemma 3.5, (3.14), (3.15) and the definition ofρ1, for anys ∈ (0, ρ1] and anyt ∈ [−2η21 , 2η21 ] ∩

[−T, T ] we have

Volg(t)(Cyt(Bs(yt) ∩ Σt))

ωnsn≤ eΛρ1

Volg(t)(Cyt(Bρ1(yt) ∩ Σt))

ωnρn1≤ 1 + 3δ. (3.16)

Let ρ := 14 − η0. Then by the definition ofη0 we have1

8 ≤ ρ ≤ 14 . By the assumption (2), we have

pt ∈ Bρ(p0) ∩ Σt 6= ∅ for anyt ∈ [−2η21 , 2η21 ] ∩ [−T, T ] since

|pt − p0| ≤ Λ|t| ≤ 2Λη21 =η0

1 + Λ< η0 ≤

1

8≤ ρ.

Using the assumption (2) again, for anyqt ∈ Cpt(Bρ(p0) ∩ Σt) with t ∈ [−2η21 , 2η21 ] ∩ [−T, T ] we

haveq0 := x0(x−1t (qt)) ∈ Cp0(B 1

2(p0) ∩Σ0) since

|q0 − p0| ≤ |q0 − qt|+ |qt − p0| <1

8+ ρ ≤ 1

2.

Combining this with (3.16), for anyt ∈ [−2η21 , 2η21 ] ∩ [−T, T ] andq ∈ Cpt(Bρ(p0) ∩ Σt) we have

Volg(t)(Cq(Bs(q) ∩Σt))

ωnsn≤ 1 + 3δ, ∀ s ∈ (0, ρ1]. (3.17)

11

Step 3.Suppose there existΛ > 0, a sequence ofǫ → 0, ǫ ∈ (0, η1] and smooth solutions to themean curvature flowxt : Σ

n → Rn+1 for t ∈ [−T, T ] with T ≥ 2η21 such that|A|(x, 0) ≤ 1 for any

x ∈ Cp0(B1(p0) ∩ Σ0), and there exists(x1, t1) satisfyingt1 ∈ [−η21, η21 ] andx1 ∈ Cpt1

(B 116(p0) ∩

Σt1) such that

Q1 := |A|(x1, t1) >1

ǫ. (3.18)

Note thatpt1 ∈ B 116(p0) ∩Σt1 since

|pt1 − p0| ≤ Λη21 ≤ η02(1 + Λ)

≤ 1

16.

Fix K > 0 such thatK > 12Λǫ and2Kǫ < 1

16 . Check whether there exists a point

(x, t) ∈ Cx1,t(BKQ−11(x1) ∩ Σt)× [t1 −

1

2Q−2

1 , t1] (3.19)

satisfying|A|(x, t) > 2Q1. Here we definex1,t = xt(x−1t1

(x1)). Note thatx1,t ∈ BKQ−1

1(x1) ∩ Σt

since|x1 − x1,t| ≤ Λ|t| ≤ 1

2ΛQ−2

1 < KQ−11 , ∀ t ∈ [t1 −

1

2Q−2

1 , t1].

If there is no such point, then we stop. Otherwise, we can find apoint, which we denote by(x2, t2),satisfying (3.19) andQ2 := |A|(x2, t2) > 2Q1. Then we check whether there exists a point

(x, t) ∈ Cx2,t(BKQ−12(x2) ∩ Σt)× [t2 −

1

2Q−2

2 , t2] (3.20)

satisfying|A|(x, t) > 2Q2. We can also check thatx2,t ∈ BKQ−1

2(x2)∩Σt. If there is no such point,

then we stop. Otherwise, we can find a point which we denote by(x3, t3). Repeating the process, wecan find a sequence of points(xk, tk). Note that

tk ≥ t1 −1

2

(

Q−2k−1 +Q−2

k−2 + · · · +Q−21

)

≥ t1 − ǫ2 > −2η21

and the Euclidean distance

d(xk, p0) ≤ d(xk, xk−1) + d(xk−1, xk−2) + · · ·++d(x1, p0)

≤ K(

Q−1k−1 +Q−1

k−2 + · · ·+Q−11

)

+ d(x1, p0)

≤ 2Kǫ+1

16

≤ 1

8≤ ρ,

where we chooseK = ǫ−12 andǫ small. SinceQk := |A|(xk, tk) ≥ 2k−1Q1 → +∞ ask → +∞,

the process will stop at some finitek and we get a point(x, t) satisfying the following properties:

• (x, t) ∈ Cpt(Bρ(p0) ∩Σt)× (−2η21 , t1];

• |A|(x, t) ≤ 2|A|(x, t) for any point(x, t) ∈ Cx,t(BKQ−1(x) ∩ Σt) × [t − 12Q

−2, t], wherex,t := xt(x

−1t

(x)) andQ := |A|(x, t). Note that[t− 12Q

−2, t] ⊂ [−2η21 , t1].

12

Step 4. We rescale the flow by

x(p, s) = Q(

x(p, t+s

Q2)− x

)

, ∀ (p, s) ∈ Σ× [−(T + t)Q2, (T − t)Q2].

Then the rescaled flowΣs := xs(Σ) is a mean curvature flow satisfying the following properties:

• For any(x, s) ∈ Cxs(x

−10 (0))(Bǫ

− 12(0)∩Σs)×[−1

2 , 0], we have|AΣs|(x, s) ≤ 2 and|AΣ0

|(0, 0) =1;

• For anyr ∈ (0, ǫ−12 ) we have the volume ratio

Vol(C0(Br(0) ∩ Σ0))

ωnrn≤ 1 + 3δ.

Here we used (3.17), (3.18) and the facts thatKQ−1 ≤ KQ−11 ≤ ǫ

12 < ρ1 whenǫ is small.

• The mean curvature of the flowΣs satisfies|H| ≤ ΛQ−1.

Sinceǫ → 0 andQ → +∞, the flowCxs(x

−10 (0))(Bǫ

−12(0) ∩ Σs)× [−1

2 , 0] converges smoothly to acomplete smooth minimal surfaceΣ∞ with supΣ∞

|AΣ∞ | ≤ 2, |AΣ∞ |(0) = 1 and volume ratio

Vol(Br(0) ∩ Σ∞)

ωnrn≤ 1 + 3δ, ∀ r > 0.

If we chooseδ = 13δ0 whereδ0 = δ0(n) is the constant in Lemma 3.6, thenΣ∞ is a hyperplane,

which contradicts|AΣ∞ |(0) = 1. The theorem is proved.

A direct corollary of Theorem 3.7 is the following long time pseudolocality theorem. The long-time-pseudolocality type theorem originates from the study of the Kahler Ricci flow by Chen-Wang(c.f.Theorem 1.4 of Chen-Wang [10], or Proposition 4.15 and Remark 5.3 of Chen-Wang [11]). It will beinspiring to compare the following theorem with its KahlerRicci flow counterpart.

Theorem 3.8(Long-time, two-sided pseudolocality). For any r0 ∈ (0, 1], T > 0, there existδ =

δ(n, r0, T ), ǫ = ǫ(n) > 0 with the following properties. Let(Σn,x(t)),−T ≤ t ≤ T be a closedsmooth embedded mean curvature flow (2.1). Assume that

(1) the second fundamental form satisfies|A|(x, 0) ≤ 1r0

for anyx ∈ Cp0(Br0(p0) ∩ Σ0) wherep0 = x0(p) for somep ∈ Σ;

(2) the mean curvature of(Σn,xt),−T ≤ t ≤ T is bounded byδ.

Then for any(x, t) ∈ Cpt(Σt ∩B 116

r0(p0))× [−T, T ] wherept = xt(p), we have the estimate

|A|(x, t) ≤ 1

ǫr0.

13

Proof. We apply Theorem 3.7 forΛ = δ, then we get the constantη(n, δ) andǫ(n, δ). By (3.11), theconclusion holds for anyt ∈ [−T, T ] if

η(n, δ)r202(δ + δ2)

≥ 2T. (3.21)

Sinceη0(n) = limδ→0 η(n, δ) > 0, there exists a constantδ = δ(n, r0, T ) such that (3.21) holds.Note thatlimδ→0 ǫ(n, δ) = ǫ0(n) > 0 by (3.10). Thus, the theorem is proved.

3.2 Energy concentration property

In [7], Choi-Schoen showed the following energy concentration property for minimal surfaces. Thisproperty says that the energy near a point with large curvature cannot be small .

Lemma 3.9. (Choi-Schoen [7]) Fixρ ≤ 1. There is a numberǫ0 > 0 such that ifΣ2 ⊂ R3 is a

minimal surface with∂Σ ⊂ ∂Bρ(x) with |A|(x) ≥ ρ−1, then

∫

B ρ2(x)∩Σ

|A|2 dµ ≥ ǫ0.

Motivated by Choi-Schoen’s result, we show that the energy concentration property holds formean curvature flow with bounded mean curvature by using the pseudolocality theorem.

Lemma 3.10(Energy concentration). For anyΛ,K, T > 0, there exists a constantǫ(n,Λ,K, T ) >

0 with the following property. Let(Σn,x(t)),−T ≤ t ≤ T be a closed smooth embedded meancurvature flow (2.1). Assume thatmaxΣt×[−T,T ] |H|(p, t) ≤ Λ. Then we have

∫

Σ0∩BQ−1 (q)|A|n dµ0 ≥ ǫ(n,Λ,K, T ) (3.22)

wheneverq ∈ Σ0 with Q := |A|(q, 0) ≥ K.

Proof. Letx0 ∈ Σ0 such thatQ := |A|(x0, 0) ≥ K. We define the functionf(x) = |A|(x, 0)d(x, ∂Ω)on Ω := BQ−1(x0) ∩ Σ0. Hered denotes the Euclidean distance inRn+1. Note thatf = 0 on theboundary∂Ω, if ∂Ω 6= ∅, andf = 1 at the center pointx0.

Case 1.maxΩ f < 10. We rescale the flow byx(p, t) = Q(x(p,Q−2t) − x0) and letΩ :=

B1(0)∩ Σ0, whereΣ0 := x(0)(Σ). Note that the mean curvature ofΣt(−K2T ≤ t ≤ K2T ) satisfies

maxΣt×[−K2T,K2T ]

|H|(p, t) ≤ Λ

Q≤ Λ

K, (3.23)

andmaxΩ |A|d(x, ∂Ω) = maxΩ f < 10. Thus, we have|A|(x, 0) < 10d(x,∂Ω)

for anyx ∈ B1(0)∩ Σ0.

In particular, insideB 12(0) ∩ Σ0 we have|A| < 20. Theorem 3.7 implies that there existsδ0 =

δ0(n,Λ,K, T ) ∈ (0, 1) such that|A|(p, t) ≤ 1δ0

for anyt ∈ [−δ20 , δ20 ] andp ∈ Σt∩Bδ0(0). Therefore,

there existsδ1 = δ1(n,Λ,K, T ) ∈ (0, δ0) such that we have all higher order curvature estimates in

14

Bδ1(0) ∩ Σt for any t ∈ [−δ21 , δ21 ]. Note that|A|(0, 0) = 1, the higher order curvature estimates

implies that|A|(q, 0) ≥ 12 onBδ2(0) ∩ Σ0 for someδ2(n,Λ,K, T ) ∈ (0, δ1). Therefore, we have

∫

BQ−1 (x0)∩Σ0

|A|n dµ0 =

∫

B1(0)∩Σ0

|A|n dµ0 ≥∫

Bδ2(0)∩Σ0

|A|n dµ0

≥ 1

2nVolΣ0

(Bδ2(0) ∩ Σ0)

≥ 1

2n· ωne

−Λδ2K δn2 ,

where we used Lemma 3.5 in the last inequality.Case 2.maxΩ f ≥ 10. Let y0 be the point wheref achieve the maximum andQ′ := |A|(y0, 0).

Note thatQ′ ≥ 10

d(y0, ∂Ω)≥ 10Q ≥ 10K.

We rescale the flow byx(p, t) = Q′(x(p,Q′−2t)− x0) and we define

y0 := Q′(y0 − x0), Ω := BQ′Q−1(0) ∩ Σ0,

whereΣ0 := x(0)(Σ). Then the functionf(x, 0) := |A|(x, 0)d(x, ∂Ω) achieves the maximum atthe pointy0 and the mean curvature ofΣt is bounded by Λ

10K for anyt ∈ [−Q′2T,Q′2T ]. Moreover,|A|(y0, 0) = 1 andd(y0, ∂Ω) ≥ 10. For anyx ∈ B1(y0) ∩ Σ0, we have

|A|(x, 0) ≤ d(y0, ∂Ω)

d(x, ∂Ω)|A|(y0, 0) ≤

d(y0, ∂Ω)

d(y0, ∂Ω)− 1≤ 2.

Then using the backward pseudolocality as in case1, we obtain that∫

B1(y0)∩Σ0

|A|n dµ0 ≥ ǫ(n,Λ,K, T )

for someǫ(n,Λ,K, T ) > 0. Using the scaling invariance, we have∫

BQ′−1 (y0)∩Σ0

|A|n dµ0 > ǫ(n,Λ,K, T ).

By the definition ofy0, we haveQ′−1 ≤ 110d(y0, ∂Ω). Therefore,BQ′−1(y0) ⊂ BQ−1(x0) and we

have the inequality∫

BQ−1 (x0)∩Σ0

|A|n dµ0 > ǫ(n,Λ,K, T ).

The lemma is proved.

A direct corollary of Lemma 3.10 is the following result.

Corollary 3.11 (ǫ-regularity ). There existsǫ0(n) > 0 satisfying the following property. Let(Σn,x(t)),−1 ≤t ≤ 1 be a closed smooth embedded mean curvature flow (2.1). Suppose that the mean curvaturesatisfiesmaxΣt×[−1,1] |H|(p, t) ≤ 1. For anyq ∈ Σ0, if

∫

Σ0∩Br(q)|A|n dµ0 ≤ ǫ0(n)

for somer > 0, then we have

maxB r

2(q)∩Σ0

|A| ≤ max1, 2r. (3.24)

15

Proof. For anyp ∈ B r2(q) ∩ Σ0, if Q := |A|(p, 0) satisfiesQ−1 < r

2 andQ > 1, then by Lemma3.10 we have

∫

Br(q)∩Σ0

|A|n dµ0 ≥∫

BQ−1 (p)∩Σ0

|A|n dµ0 ≥ ǫ0(n),

whereǫ0 is the constant determined by choosingK = Λ = 1 in Lemma 3.10. Therefore, we haveQ ≤ 1 or Q−1 ≥ r

2 , which implies (3.24). The corollary is proved.

3.3 Weak compactness

In this subsection, we focus on the casen = 2. As in Ricci flow [9], we study the refined sequenceof mean curvature flow, which can be viewed as a sequence blownup from a rescaled mean curvatureflow with bounded mean curvature and bounded energy.

Definition 3.12 (Refined sequences). Let (Σ2i ,xi(t)),−1 ≤ t ≤ 1 be a one-parameter family of

closed smooth embedded surfaces satisfying the mean curvature flow equation (2.1). It is called arefined sequence if the following properties are satisfied for everyi :

(1) There exists a constantD > 0 such thatd(Σi,t, 0) ≤ D, whered(Σ, 0) denotes the Euclideandistance from the point0 ∈ R

3 to the surfaceΣ ⊂ R3.

(2) The mean curvature satisfies the inequality

limi→+∞

maxΣi,t×[−1,1]

|Hi|(p, t) = 0; (3.25)

(3) There is a uniform constantΛ such that∫

Σi,t

|Ai|2 dµi,t ≤ Λ, ∀ t ∈ [−1, 1];

(4) There is uniformN > 0 such that for allr > 0 andp ∈ R3 we have

Areagi(t)(Br(p) ∩ Σi,t) ≤ Nr2, ∀ t ∈ [−1, 1].

(5) There exist uniform constantsr, κ > 0 such that for anyr ∈ (0, r] and anyp ∈ Σi,t we have

Areagi(t)(Br(p) ∩ Σi,t) ≥ κr2, ∀ t ∈ [−1, 1].

Proposition 3.13 (Weak compactness of refined sequences). If (Σ2i ,xi(t)),−1 ≤ t ≤ 1 is a

refined sequence, then there exists a finite set of pointsS0 ⊂ R3 and a smooth embedded minimal

surfaceΣ∞ such that a subsequence of(Σ2i ,xi(t)),−1 < t < 1 converges in smooth topology,

possibly with multiplicity at mostN0, to Σ∞ away fromS0.

16

Proof. We follow the argument of compactness of minimal surfaces (cf. White [47][48], or Colding-Minicozzi [13]). Fix largeρ > 0 and letΩ = Bρ(0) ⊂ R

3. By Property (1) in Definition 3.12, wehaveΣi,0 ∩ Ω 6= ∅ for largeρ. For anyU ⊂ Ω, we define the measuresνi by

νi(U) =

∫

U∩Σi,0

|Ai|2 dµi,0 ≤ Λ.

The general compactness of Radon measures implies that there is a subsequence, which we still denoteby νi, converges weakly to a Radon measureν with ν(Ω) ≤ Λ. We define the set

S0 = x ∈ Ω | ν(x) ≥ ǫ0,

whereǫ0 is the constant in Corollary 3.11. It follows thatS0 contains at mostΛǫ0

points, which isindependent ofρ. Given anyy ∈ Ω\S0. There exists somes ∈ (0, 15) such thatB10s(y) ⊂ Ω andν(B10s(y)) < ǫ0. Sinceνi → ν, for i sufficiently large we have

∫

B10s(y)∩Σi,0

|Ai|2 dµi,0 < ǫ0.

Corollary 3.11 implies that fori sufficiently large we have the estimate

maxB5s(y)∩Σi,0

|A|(x, 0) ≤ max1, 1

5s ≤ 1

5s. (3.26)

Note that by Property (2) in Definition 3.12 the mean curvature ofΣi,t tends to zero. By Theorem 3.8there exists a universal constantǫ > 0 such that for largei and any smalls ∈ (0, 15) we have

maxB 1

16 r0(y)∩Σi,t

|A|(x, t) ≤ 1

ǫr0, ∀ t ∈ [−1, 1], (3.27)

wherer0 = 5s. Therefore, we have all higher order estimates of the second fundamental form atany point away from the singular set. By Theorem 2.6 and a diagonal sequence argument we canshow that a subsequence of(Σ2

i ,xi(t)),−1 < t < 1 converges in smooth topology, possiblywith multiplicities, to an embedded minimal surfaceΣ∞ away from the singular setS0. Property(4)-(5) imply that the multiplicity of the convergence is bounded by some constantN0. SinceΣ∞

is minimal and the convergence is smooth outsideS0, the same argument as in Proposition 7.14 ofColding-Minicozzi [13] shows thatΣ∞ ∪S0 is a smooth embedded minimal surface. The propositionis proved.

4 Multiplicity-one convergence of the rescaled mean curvature flow

In this section, we show that a rescaled mean curvature flow with mean curvature exponential decaywill converge smoothly to a plane with multiplicity one.

Theorem 4.1. Let(Σ2,x(t)), 0 ≤ t < +∞ be a rescaled mean curvature flow

∂x

∂t= −

(

H − 1

2〈x,n〉

)

n (4.1)

17

satisfyingd(Σt, 0) ≤ D, and max

Σt

|H(p, t)| ≤ Λ0e− t

2 (4.2)

for two constantsD,Λ0 > 0. Then there exists a sequence of timestj → +∞ such thatΣtj convergein smooth topology to a plane passing through the origin withmultiplicity one.

4.1 Convergence away from singularities

Lemma 4.2. Under the assumption of Theorem 4.1, for any sequenceti → +∞ there is a subse-quence, still denoted byti, such thatΣti+t,−T < t < T converges in smooth topology, possiblywith multiplicities at mostN0, to a plane passing through the origin away from the space-time singu-lar setS = (x, t) | t ∈ (−T, T ), x ∈ e

t2S0, whereS0 is the singular set in the convergence of the

surfacesΣti.

Proof. The proof divides into the following steps.Step 1. The energy ofΣt is uniformly bounded along the flow (4.1).In fact, we rescale the flow

Σt bys = T − e−t, Σs =

√T − sΣ− log(T−s) (4.3)

such thatΣs, T−1 ≤ s < T is a mean curvature flow satisfying (2.1) with mean curvatureboundedby Λ0. Note that the scalar curvatureS of Σs satisfiesS = H2 − |A|2, whereH andA denote themean curvature and the second fundamental form ofΣs respectively. By the Gauss-Bonnet theoremwe have

∫

Σs

|A|2 dµs =

∫

Σs

|H|2 dµs − 4πχ(Σ) ≤ Λ20Area(Σ0)− 4πχ(Σ),

where we used the fact thatArea(Σs) is non-increasing ins. Hereχ(Σ) denotes the Euler character-istic of Σ. Therefore, by Lemma 2.3 the energy ofΣt satisfies the inequality

∫

Σt

|A|2 dµt ≤ Λ20Area(Σ0)− 4πχ(Σ). (4.4)

Step 2. For each sequenceti → +∞, we can obtain a refined sequence converging to a limitΣ∞.For any sequenceti → +∞, we can rescale the flowΣt by

s = 1− e−(t−ti), Σi,s =√1− s Σti−log(1−s) (4.5)

such that for eachi the flowΣi,s, 1− eti ≤ s < 1 is a mean curvature flow satisfying (2.1) with thefollowing properties:

(a). For any smallλ > 0, the mean curvature ofΣi,s satisfies

limi→+∞

maxΣi,s×[1−eti ,1−λ]

|Hi|(p, s) = 0;

(b). The energy ofΣi,s satisfies (4.4);

(c). Uniform upper bound on the area ratio;

18

(d). Uniform lower bound on the area ratio;

(e). There exists a constantD′ > 0 such thatd(Σi,s, 0) ≤ D′ for anyi.

In fact, Property(a) and(e) follow from the assumption (4.2), and Property(b) follows from (4.4).Property(c) follows from Lemma 2.1 and Lemma 2.3, and Property(d) follows directly from Lemma3.5. Therefore, by Definition 3.12 for anyT > 0, smallλ > 0 and anys0 ∈ [−T + 1,−λ] thesequenceΣi,s0+τ ,−1 < τ < 1 is a refined sequence. By Proposition 3.13 a subsequence ofΣi,s0+τ ,−1 < τ < 1 converges in smooth topology, possibly with multiplicity at mostN0, to asmooth embedded minimal surfaceΣ∞ away from a finite set of pointsS = q1, · · · , ql.

Claim 4.3. Under the above assumptions, there exists a subsequence ofΣi,s,−T < s < 1−λ suchthat it converges in smooth topology, possibly with multiplicity at mostN0, to a limit minimal surfaceΣ∞ away from a singular setS = q1, · · · , ql.

Proof. We first lets0 = −T + 1 and we get a subsequence, denoted byΣi(1)k

,s0+τ,−1 < τ < 1,

such that it converges in smooth topology, possibly with multiplicity at mostN0, to a limit mini-mal surfaceΣ∞ away from a singular setS = q1, · · · , ql. Then we takes1 = −T + 2 andconsider the convergence of the sequenceΣ

i(1)k

,s1+τ,−1 < τ < 1. Note that the two sequences

Σi(1)k

,s1+τ,−1 < τ < 1 andΣ

i(1)k

,s0+τ,−1 < τ < 1 coincide on a nonempty time interval.

Therefore, we can take a further subsequence ofΣi(1)k

,s1+τ,−1 < τ < 1 such that it converges to

the same limit surfaceΣ∞ away from the same singular setS. Repeating this process at finite manytimes and we get a subsequence ofΣi,s,−T < s < 1 − λ such thatΣi,s converges toΣ∞ awayfrom S for all s ∈ (−T, 1− λ). The Claim is proved.

Step 3. Each limitΣ∞ must be a plane through the origin.In fact, by Huisken’s monotonicityformula, along the rescaled mean curvature flow (4.1) we have

d

dt

∫

Σt

e−|x|2

4 dµt = −∫

Σt

e−|x|2

4

∣

∣

∣H − 1

2〈x,n〉

∣

∣

∣

2dµt. (4.6)

This implies that∫ ∞

0

∫

Σt

e−|x|2

4

∣

∣

∣H − 1

2〈x,n〉

∣

∣

∣

2dµt < +∞.

For anyti → +∞, we rescale the flow (4.1) by (4.5) such that for eachi the flowΣi,s, 1− eti ≤ s <

1 is a mean curvature flow satisfying (2.1) and we denote the solution by xi,s. Therefore, for fixedT > 0, smallλ > 0 and largei we have

limi→+∞

∫ 1−λ

−T

ds

∫

Σi,s

e−

|xi,s|2

4(1−s)

∣

∣

∣Hi −

1

2(1 − s)〈xi,s,n〉

∣

∣

∣

2dµi,s

= limti→+∞

∫ ti−log λ

ti−log(1+T )

∫

Σt

e−|x|2

4

∣

∣

∣H − 1

2〈x,n〉

∣

∣

∣

2dµt = 0.

19

Thick partTK

Thin partTN

High curvature neighborhoodH

Figure 1: Decomposition of the space

Since(Σ, xi(p, s)),−T ≤ s ≤ 1−λ converges locally smoothly, possibly with multiplicity atmostN0, to (Σ∞, x∞) away fromS, we have

∫ 1−λ

−T

ds

∫

Σ∞

e−

|x∞|4(1−s)

∣

∣

∣H − 1

2(1− s)〈x∞,n〉

∣

∣

∣

2dµ∞,s = 0.

Therefore,(Σ∞, x∞(p, s)),−T < s < 1− λ satisfies the equation

H − 1

2(1− s)〈x∞,n〉 = 0

away from the singular setS. Since Σ∞ is a smooth embedded minimal surface, we haveH =

〈x∞,n〉 = 0. By Lemma 2.2Σ∞ must be a plane passing through the origin. Letxi(p, t) = x(p, ti+

t) andΣi,t = Σti+t. SinceΣi,s,−T < s < 1− λ converges locally smoothly toΣ∞ away fromS,the flowΣi,t,− log(1 + T ) < t < − log λ also converge locally smoothly to the planeΣ∞ awayfrom the singular setS = (x, t) | t ∈ (− log(1 + T ),− log λ), x ∈ e

t2 S. The lemma is proved.

4.2 Decomposition of spaces and a “monotone decreasing” quantity

To study the long-time behavior of the flow (4.1), we can decompose the space as follows.

Definition 4.4. (1). We define the setS = S(ǫ,Σt) = y ∈ Σt | |y| < ǫ−1, |A|(y, t) > ǫ−1.

(2). The ballBǫ−1(0) can be decomposed into three parts as follows:

• the high curvature partH, which is defined byH = H(ǫ,Σt) = x ∈ R3 | |x| <

ǫ−1, d(x,S) < ǫ2. Hered denotes the Euclidean distance inR3.

20

TK = Bǫ−1(0)\(

Σ∞ ∪B ǫ2(S0)

)

TN = Σ∞ ∩(

Bǫ−1(0)\B ǫ2(S0)

)

H = B ǫ2(S0)

S0

0 x1

x2

x3

Figure 2: Decomposition in the limit space

• the thick partTK, which is defined by

TK = TK(ǫ,Σt)

=

x ∈ R3∣

∣

∣|x| < ǫ−1, there is a continuous curveγ ⊂ Bǫ−1(0)\(H ∪ Σt)

connectingx and somey with B(y, ǫ) ⊂ Bǫ−1(0)\(H ∪ Σt)

.

• the thin partTN, which is defined byTN = TN(ǫ,Σt) = Bǫ−1(0)\(H ∪TK).

Intuitively, the high curvature partH is the neighborhood of points with large second fundamentalform(c.f. Figure 1). The thin partTN is the domain between the highest and lowest sheet. Thethick part is the union of path connected components of the domain “outside” the sheets. Note that inthe above definition, the existence of curveγ is to guarantee the path-connectedness. Because of theboundary issue, some points inTK may be very close to the points inTN or H.

Remark 4.5. The decomposition of space is motivated by the decomposition of Ricci flow time slicesin Chen-Wang [10].

Lemma 4.6. For each fixedǫ, we havelimt→∞

|TN(ǫ,Σt)| = 0.

Proof. For otherwise, we can find anǫ and a sequenceti → ∞ such that

limi→∞

|TN(ǫ,Σti)| ≥ κ0 > 0. (4.7)

However,Σti converges locally smoothly to a planeΣ∞ passing through the origin away from thesingular setS0 ⊂ Σ∞. Therefore, the setsS(ǫ,Σti) converge toS0 as i → +∞. Without loss ofgenerality, we can assume thatΣ∞ is determined byx3 = 0 and the singular setS0 = (1, 0, 0).Then the high curvature partsH(ǫ,Σti) converge toB ǫ

2((1, 0, 0))(c.f. Figure 2).

Note thatBǫ−1(0)\(Σ∞ ∪B ǫ2((1, 0, 0))) contains two partsB+ ∪ B−, where

B+ =

(x1, x2, x3)∣

∣

∣x21 + x22 + x23 < ǫ−2, (x1 − 1)2 + x22 + x23 >

ǫ2

4, x3 > 0

,

B− =

(x1, x2, x3)∣

∣

∣x21 + x22 + x23 < ǫ−2, (x1 − 1)2 + x22 + x23 >

ǫ2

4, x3 < 0

.

21

Fix an arbitrary pointx ∈ B+. It is not hard to see thatx can be connected to(0, 0, 1) by a continuouscurveγ ⊂ B+. Furthermore, it is clear thatBǫ((0, 0, 1)) ∩ Bǫ((1, 0, 0)) = ∅. Similar argumentapplies forB−. Therefore, we see thatB+∪ B− are contained in the limit ofTK(ǫ,Σti). Recall thatH(ǫ,Σti) converges toB ǫ

2((1, 0, 0)). Following from its definition, it is clear that the Hausdorff limit

of TN(ǫ,Σti) is a subset of(Bǫ−1(0) ∩ Σ∞)\B ǫ2((1, 0, 0)). In particular, we have

limi→∞

|TN(ǫ, ti)| = 0.

This contradicts the assumption (4.7).

It is not hard to observe that

Lemma 4.7. If Σ∞ has multiplicity more than one, then for sufficiently largeti we have

|TN(ǫ,Σti)| > 0.

Proof. SinceΣt is embedded andΣti converges locally smoothly to the limit planeΣ∞, all compo-nents of(Σti ∩ Bǫ−1(0))\H(ǫ,Σti ) lie in the ǫ

2 -neighborhood of the planeΣ∞ for largeti. By thedefinition ofTN, TN(ǫ,Σti) is nonempty and we have|TN(ǫ,Σti)| > 0.

However, in general we do not know whether|TN(ǫ,Σt)| is a continuous function oft, but wecan chooseti carefully such that|TN(ǫ,Σt)| is bounded on a time interval.

Lemma 4.8. There is a sequence of timesti → ∞ such that

supti≤t≤ti+i

|TN(ǫ,Σt)| ≤ 2|TN(ǫ,Σti)|. (4.8)

Proof. Fix s1 = 1, then search for timet ∈ [s1, s1 + 1] satisfying|TN(ǫ,Σt)| > 2|TN(ǫ,Σs1)|. Ifno such time exists, then we sett1 = s1. Otherwise, we choose such a time and denote it bys

(1)1 .

Then search the time interval[s(1)1 , s(1)1 + 1]. Inductively, we search[s(k)1 , s

(k)1 + 1]. If we have

supt∈[s

(k)1 ,s

(k)1 +1]

|TN(ǫ,Σt)| ≤ 2|TN(ǫ,Σs(k)1

)|,

then we denotet1 = s(k)1 and stop the searching process. Otherwise, choose a times

(k+1)1 ∈

[s(k)1 , s

(k)1 + 1] with more than doubled|TN| value and continue the process. Note that

|TN(ǫ,Σs(k)1)| ≥ 2k|TN(ǫ,Σs1)| → ∞, as k → ∞.

This process must stop in finite steps. After we findt1, sets(0)2 = t1 + 1 and continue the previousprocess to find time in[s(0)2 , s

(0)2 +2] with more than doubled|TN|-value, with slight change that the

time-interval has length2. Similarly, for some finitek, we have

supt∈[s

(k)2 ,s

(k)2 +2]

|TN(ǫ,Σt)| ≤ 2|TN(ǫ,Σs(k)2

)|.

Then we definet2 = s(k)2 . Inductively, after we findtl, we sets(0)l+1 = tl + l. Then start the process to

search time in[s(0)l+1, s(0)l+1 + l + 1] with more than doubled|TN| value. This process is well defined.

From its construction, it is clear thatti → ∞ and satisfies (4.8). The lemma is proved.

22

4.3 Construction of auxiliary functions

In this subsection, we will construct a function which will be used to show that the limit planeis L-stable. For any sequenceti → +∞, we assume that the multiplicity of the convergence ofΣi,t,−T < t < T is is at least two. Without loss of generality, we assume thatthe limit plane isΣ∞ = (x1, x2, x3) |x3 = 0 ⊂ R

3. By Lemma 4.2,Σi,t converges in smooth topology toΣ∞ awayfrom the singular setSt = e

t2S0, whereS0 consists of finite many points. Fix smallǫ > 0. We define

Ωǫ(t) = (Σ∞ ∩Bǫ−1(0))\ ∪p∈St Bǫ(p) (4.9)

and for any time intervalI ⊂ (−T, T ) we defineΩǫ(I) =⋂

t∈I Ωǫ(t). Moreover, we define

Ωǫ(t) = (x1, x2, x3) ∈ R3 | (x1, x2, 0) ∈ Ωǫ(t), x3 ∈ R.

For anyt ∈ (−T, T ) and largeti the surfaceΣi,t ∩ Ωǫ(t) is a union of graphs over the setΩǫ(t).By embeddedness and orientability, these graphs are ordered by height. Letu+i (x, t) andu−i (x, t) bethe functions representing the top and bottom sheets ( whichwe denote byΣ+

i,t andΣ−i,t respectively)

overΩǫ(t). By the convergence property of the flow(Σ,xi(t)),−T < t < T, for any time intervalI ⊂ (−T, T ) the functionsu+i (x, t) andu−i (x, t) can be extended to be smooth functions onΩǫ(I)×I.

By definition, the surfacesΣ+i,t andΣ−

i,t can be expressed by

xi,1(x, t) = x∞(x) + u+i (x, t)n∞(x), xi,2(x, t) = x∞(x) + u−i (x, t)n∞(x),

wherex ∈ Ωǫ(t) andn∞ denotes the unit normal vector field onΣ∞. Thenu+i (x, t) andu−i (x, t)satisfy the equation of rescaled mean curvature flow onΩǫ((−T, T ))× (−T, T ):

∂u

∂t=

√

1 + |∇u|2div( ∇u√

1 + |∇u|2)

− 1

2(x1ux1 + x2ux2) +

u

2

= ∆0u− ∇2u(∇u,∇u)

1 + |∇u|2 − 1

2〈x,∇u〉+ u

2, (4.10)

wherex = (x1, x2, 0) ∈ Ωǫ((−T, T )) and∆0 is the standard Laplacian operator on the planeΣ∞.We defineui = u+i − u−i . Let fi,s = u−i + sui for s ∈ [0, 1]. Thenfi,0 = u−i andfi,1 = u+i . Wedefine

G(s) =∇2fi,s(∇fi,s,∇fi,s)

1 + |∇fi,s|2.

Therefore, we have

G(1)−G(0) =

∫ 1

0

∂G(s)

∂sds = apqui,pq + bpui,p, (4.11)

where

apq =

∫ 1

0

(

(1 + |∇fi,s|)−1∇pfi,s∇qfi,s

)

ds, (4.12)

bp =

∫ 1

0

(

2(1 + |∇fi,s|)−1∇k∇pfi,s∇kfi,s

−2(1 + |∇fi,s|)−2∇pfi,s∇2fi,s(∇fi,s,∇fi,s))

ds. (4.13)

23

Combining (4.10) with (4.11), we get the equation ofui

∂ui∂t

= ∆0ui −1

2〈x,∇ui〉+

ui2

− apqui,pq − bpui,p (4.14)

for any (x, t) ∈ Ωǫ((−T, T )) × (−T, T ). Fix somex0 ∈ Σ∞\ ∪t∈(−T,T ) St. Then for sufficientlysmallǫ > 0 we havex0 ∈ Ωǫ((−T, T )). We define

wi(x, t) =ui(x, ti + t)

ui(x0, ti + ǫ′), ∀ (x, t) ∈ Ωǫ((−T, T ))× (−T, T ), (4.15)

whereǫ′ > 0 will be determined later. Thenwi(x, t) is a positive function withwi(x0, ǫ′) = 1 and

satisfies the equation onΩǫ((−T, T ))× (−T, T )

∂wi

∂t= ∆0wi −

1

2〈x,∇wi〉+

wi

2− apqwi,pq − bpwi,p, (4.16)

whereapq andbp are defined by (4.12) and (4.13). Note that for fixedǫ andT the coefficientsapq andbp are uniformly bounded asti is large and tend to zero asti → +∞.

Remark 4.9. Note thatui(x, t) can be extended to a smooth function, still denoted byui(x, t), onΩǫ,i(I)× I for any time intervalI ⊂ (−T, T ). Therefore, (4.14) and (4.16) also hold onΩǫ(I)× I.

Now we estimate|TN| of the surfaceΣti+t for anyt ∈ (−T, T ).

Lemma 4.10. For any sequenceti chosen in Lemma 4.2, there existstT > 0 such that for anyt ∈ (−T, T ) andti > tT we have

∫

Ωǫ(t)ui(x, t) dµ∞ ≤ |TN(ǫ,Σi,t)| ≤

∫

Ω ǫ5(t)

ui(x, t) dµ∞, (4.17)

wheredµ∞ denotes the standard volume form ofΣ∞.

Proof. Recall that we assumedΣ∞ = (x1, x2, x2) ∈ R3 | x3 = 0. For anyt ∈ (−T, T ) and large

ti, we define

Φ(ǫ,Σi,t) = (x1, x2, x3) ∈ R3 | x′ := (x1, x2, 0) ∈ Ωǫ(t), u

−i (x

′, t) ≤ x3 ≤ u+i (x′, t).

Then the Euclidean volume ofΦ(ǫ,Σi,t) can be calculated by

|Φ(ǫ,Σi,t)| =∫

Ωǫ(t)ui(x, t) dµ∞. (4.18)

We claim that for largeti and anyt ∈ (−T, T )

Φ(ǫ,Σi,t) ⊂ TN(ǫ,Σi,t) ⊂ Φ(ǫ

5,Σi,t). (4.19)

In fact, sinceΣi,t converges locally smoothly toΣ∞ away fromSt, for anyt ∈ (−T, T ) and largetiwe haveS(ǫ,Σi,t) ⊂ B ǫ

10(St) = ∪p∈StB ǫ

10(p). By the definition ofH, we have

H(ǫ,Σi,t) ⊂ B 34ǫ(St) ∩Bǫ−1(0). (4.20)

24

For anyx = (x1, x2, x3) ∈ Φ(ǫ,Σi,t), we have(x1, x2, 0) ∈ Ωǫ(t), which implies that(x1, x2, x3) 6∈Bǫ(St). Therefore,x 6∈ H(ǫ,Σi,t) and we haveΦ(ǫ,Σi,t) ⊂ TN(ǫ,Σi,t) whenti is large. On theother hand, for anyx ∈ TN(ǫ,Σi,t), we havex 6∈ TK(ǫ,Σi,t) andx 6∈ H(ǫ,Σi,t), which impliesthatd(x,S) ≥ ǫ

2 by the definition ofH. SinceSt consists of finite many points,d(x,St) is attainedby some pointq0 ∈ St. Thus, we have

d(x,St) = miny∈St

d(x, y) = d(x, q0) ≥ maxz∈S∩B ǫ

10(q0)

(d(x, z) − d(z, q0)) ≥ǫ

4, (4.21)

where we used the fact thatd(x, z) ≥ ǫ2 andd(z, q0) ≤ ǫ

10 . Moreover, sincex 6∈ TK(ǫ,Σi,t), xis sufficiently close toΣ∞. This together with (4.21) implies thatx ∈ Φ( ǫ5 ,Σi,t) when ti is large.Therefore, (4.19) is proved.

Thus, the inequality (4.17) follows from (4.18) and (4.19).The lemma is proved.

Using Lemma 4.8, we get some uniform estimates on the functionswi.

Lemma 4.11. Fix ǫ > 0 and T > 1. Let ti be the sequence chosen in Lemma 4.8. There existtT > 0 and ǫ′ = ǫ′(S0, ǫ) > 0 satisfying the following property. Letwi(x, t) be the function definedby (4.15) andǫ′(S0, ǫ). For anyti > tT , any time intervalI = [a, b] ⊂ [1, T − 1) and any setK ⊂⊂Ωǫ([a, b +

12 ]), there exist constantsC1 = C1(ǫ,K,S0, x0) > 0 andC2 = C2(ǫ,K,S0, a, b, x0) > 0

such that

C2(ǫ,K,S0, a, b, x0) < wi(x, t) ≤ C1(ǫ,K,S0, x0), ∀ (x, t) ∈ K × I. (4.22)

Moreover, at timet = a there existsC3 = C3(ǫ,K,S0, a, x0) > 0 independent ofb such that

wi(x, a) ≥ C3(ǫ,K,S0, a, x0) > 0, ∀ x ∈ K. (4.23)

Herex0 is the point chosen in (4.15).

Proof. Since the sequence(Σ,xi(t)),−T < t < T converges locally smoothly to the planeΣ∞

away from the singular setS, for largeti the coefficients of (4.16) satisfy

|apq|+ |bp| ≤ 1

2, ∀ (x, t) ∈ Ωǫ(I)× I, (4.24)

whereI is any time interval contained in(−T, T ). SinceSt = et2S0, there existsǫ′ = ǫ′(S0, ǫ) > 0

such thatΩ ǫ

5(0) ⊂ Ω ǫ

6(t), ∀ t ∈ [0, ǫ′]. (4.25)

For suchǫ′ we normalize the functionui(x, t) as in (4.15) and we havewi(x0, ǫ′) = 1. Now we apply

the Harnack inequality for the parabolic equation (4.16), and the readers are referred to Theorem A.2in the appendix or Corollary 7.42 of [26] for more details. Note that the equation (4.16) satisfies thestructure inequalities (A.2)-(A.6) withf = g = h = 0, we can apply Theorem A.2 to the functionwi(x, t) onΩ ǫ

5(0)× [0, ǫ′] and we have

∫

Ω ǫ5(0)

wi(x, 0) dµ∞ ≤ C(ǫ, ǫ′, x0)wi(x0, ǫ′) = C(ǫ,S0, x0), (4.26)

25

where we used the fact thatwi(x0, ǫ′) = 1. On the other hand, Lemma 4.8 and Lemma 4.10 imply

that for anyt ∈ [0, i] ∩ [0, T )

∫

Ωǫ(t)ui(x, t) dµ∞ ≤ |TN(ǫ,Σti+t)| ≤ 2|TN(ǫ,Σti)|

≤ 2

∫

Ω ǫ5(0)

ui(x, 0) dµ∞. (4.27)

Combining (4.27) with the definition (4.15) ofwi, we have the following inequality for anyt ∈[0, i] ∩ [0, T )

∫

Ωǫ(t)wi(x, t) dµ∞ ≤ 2

∫

Ω ǫ5(0)

wi(x, 0) dµ∞ ≤ C(ǫ,S0, x0), (4.28)

where we used (4.26) in the last inequality. FixK ⊂⊂ Ωǫ([a, b +12 ]), whereI = [a, b] ⊂ [1, T − 1)

by the assumption. We apply Theorem A.2 to the functionwi(x, t) onK × [a, b+ 12 ] and we get

wi(x, t) ≤ C(ǫ,K)

∫

Ωǫ(t+12)wi(x, t+

1

2) dµ∞ ≤ C(ǫ,K,S0, x0), ∀ (x, t) ∈ K × I,

where we used (4.28) in the last inequality. Therefore,wi(x, t) is bounded from above.We next show thatwi(x, t) has a positive lower bound onK × I. Note that there existsǫ′′ =

ǫ′′(ǫ,S0, a) > 0 such thatΩǫ([a, b]) ⊂ Ω ǫ

2([a− ǫ′′, b]).

Sincex0 ∈ Ωǫ(T ) ⊂ Ω ǫ2([a − ǫ′′, b]) and the setΩ ǫ

2([a − ǫ′′, b]) is path-connected, there exists a

continuous pathγ(s) ⊂ Ω ǫ2([a − ǫ′′, b]) connectingx0 and any pointy ∈ K with the property that

the distance fromγ(s) to the boundary ofΩ ǫ2([a − ǫ′′, b]) is at leastδ = δ(ǫ,S0,K, a, b, x0) > 0.

Applying Theorem A.2 forγ(s) andΩ′ = Ω ǫ2([a− ǫ′′, b]), we have

wi(x, t) ≥ C(ǫ,S0,K, a, b, x0)wi(x0, a− ǫ′′) ≥ C(ǫ,S0,K, a, b, x0)wi(x0, ǫ′)

= C(ǫ,S0,K, a, b, x0), ∀ (x, t) ∈ K × I,

where we used the fact thatwi(x0, ǫ′) = 1 and we assumed thatǫ′′ andǫ′ are small such thata− ǫ′′ >

ǫ′. HereC might be different from one another. Note that whent = a, the constantδ can be chosento be independent ofb and we have the estimates:

wi(x, a) ≥ C(ǫ,S0,K, a, x0)wi(x0, a− ǫ′′) ≥ C(ǫ,S0,K, a, x0)wi(x0, ǫ′)

= C(ǫ,S0,K, a, x0), ∀x ∈ K.

The lemma is proved.

Lemma 4.12. The same conditions as in Lemma 4.11. Asti → +∞, we can take a subsequence ofthe functionswi(x, t) such that it converges inC2 topology on any compact subsetK ⊂⊂ (Σ∞ ∩Bǫ−1(0))\ ∪t∈[a,b+ 1

2] St, where[a, b] ⊂ [1, T − 1), to a positive functionw(x, t) satisfying

∂w

∂t= Lw := ∆0w − 1

2〈x,∇w〉+ 1

2w, ∀ (x, t) ∈ K × [a, b]. (4.29)

26

Moreover, there exist constantsC1(ǫ,K,S0, x0), C2(ǫ,K,S0, a, b, x0) > 0 such that

C2(ǫ,K,S0, a, b, x0) ≤ w(x, t) ≤ C1(ǫ,K,S0, x0), ∀ (x, t) ∈ K × [a, b], (4.30)

and at timet = a there existsC3 = C3(ǫ,K,S0, a, x0) > 0 independent ofb such that

wi(x, a) ≥ C3(ǫ,K,S0, a, x0) > 0, ∀ x ∈ K. (4.31)

Proof. By Lemma 4.11, the functionwi(x, t) is uniformly bounded from above and below. By theinterior Holder estimate (cf. Theorem 4 in [1] or Theorem 12.3 in [26])wi(x, t) has local space-timeCα estimates. Thus, by Theorem 4.9 in [26]wi(x, t) has local space-timeC2,α estimates, whichimplies that a subsequence converges uniformly inC2 to a nonnegative functionw(x, t) on (x, t) ∈K × I satisfying (4.29)-(4.31). The lemma is proved.

4.4 Proof of the multiplicity-one convergence

In this subsection, we shall show Theorem 4.1, i.e., the limit plane is multiplicity one, under the specialconditionH decays exponentially fast. Our argument largely follows that of Colding-Minicozzi [15].However, some new technical difficulties need to be addressed. For example, the proof of the limitsurface to beL-stable(c.f. Lemma 4.13) is technically different from theclassical one. Due to theloss of general self-shinker equation, we need to delicately choose subsequence and apply parabolicversion of maximum principle. The final contradiction relies heavily on the discussion in previoussubsections.

Lemma 4.13. Fix ǫ > 0. Letti be the sequence of times in Lemma 4.8 andΣ∞ the limit plane ofthe sequence(Σ,x(ti + t)),−T < t < T. Then we have

−∫

Σ∞

(φLφ)e−|x|2

4 ≥ 0, (4.32)

for any functionφ ∈ W 1,20 (Σ∞\0), whereW 1,2

0 (Σ∞\0) denotes the set of all functionsφ ∈W 1,2(Σ∞) with compact support inΣ∞\0, andL is defined by (4.29).

Proof. Consider the singular setSt = et2S0 of the convergence of the sequence(Σ,x(ti+t)),−T <

t < T, whereti is the sequence of times in Lemma 4.8. SinceS0 consists of finite many points,all points inSt\0 will move further away from the point0 whent is increasing. Therefore, anycompact set will contain no points inSt\0 whent is large. LetminS0 := min|y| | y ∈ S0\0 >

0. Then for anyt > tǫ := 2 log 1ǫminS0

and anyy0 ∈ S0\0 we haveet2 |y0| > ǫ−1. This implies

that(St\0) ∩ Bǫ−1(0) = ∅ for anyt > tǫ. Note thattǫ is independent ofT and we chooseT largesuch thatT > tǫ.

Let A(ǫ, ǫ−1) = x ∈ R3 | ǫ < |x| < ǫ−1. For anyφ ∈ W 1,2

0 (Σ∞ ∩ A(ǫ, ǫ−1)), we haveK = Supp(φ) ⊂ Σ∞ ∩A(ǫ, ǫ−1). Let I = (tǫ, T ). By the definition oftǫ, we know thatK ⊂ Ωǫ(I).We apply Lemma 4.12 for the compact setK and the time intervalI, and we get the limit functionw(x, t) which is positive onK × I. Let v := logw. Thenv is bounded from above and below by(4.30). Note thatv satisfies the equation

∂v

∂t= ∆0v +

1

2− 1

2〈x,∇v〉 + |∇v|2, ∀ (x, t) ∈ K × I.

27

For anyt ∈ I, we have

0 =

∫

Σ∞

div(

φ2e−|x|2

4 ∇v)

=

∫

Σ∞

(

2φ〈∇φ,∇v〉 +(∂v

∂t− 1

2− |∇v|2

)

φ2)

e−|x|2

4

≤∫

Σ∞

(

|∇φ|2 − 1

2φ2 +

∂v

∂tφ2

)

e−|x|2

4 .

This implies that for anyt ∈ I,

−∫

Σ∞

(φLφ)e−|x|2

4 ≥ −∫

Σ∞

∂v

∂tφ2e−

|x|2

4 = − d

dt

∫

Σ∞

vφ2e−|x|2

4 . (4.33)

Integrating both sides of (4.33), we have

−∫ T

tǫ

dt

∫

Σ∞

(φLφ)e−|x|2

4 ≥∫

Σ∞

vφ2e−|x|2

4

∣

∣

∣

t=tǫ−

∫

Σ∞

vφ2e−|x|2

4

∣

∣

∣

t=T≥ −C,

whereC is independent ofT by (4.30) and (4.31). Thus, we have

−∫

Σ∞

(φLφ)e−|x|2

4 ≥ − C

T − tǫ. (4.34)

Note thattǫ is independent ofT . LettingT → +∞ in (4.34) we have the inequality (4.32) holds foranyφ ∈ W 1,2

0 (Σ∞ ∩A(ǫ, ǫ−1)). Sinceǫ is arbitrary and the limit planeΣ∞ is unique up to rotations,we know that (4.32) holds for anyφ ∈ W 1,2

0 (Σ∞\0). The lemma is proved.

Following the argument of Gulliver-Lawson [21], we show that Σ∞ isL-stable across the singularsetS. In other words, we have the following Lemma.

Lemma 4.14. The same assumption as in Lemma 4.13. For any smooth functionφ with compactsupport inΣ∞, we have

−∫

Σ∞

(φLφ)e−|x|2

4 ≥ 0. (4.35)

Proof. Choose0 < δ < R < 1 and define the function

η(x) =

logRlog |x| , 0 < |x| < R,

1, |x| ≥ R

and the functionβ(x) = β(|x|) such thatβ(x) = 0 for |x| < δ2 , β(x) = 1 for |x| ≥ δ, 0 ≤ β(x) ≤ 1

and|∇β| ≤ 3δ. Suppose thatφ is any function with compact support onΣ∞. Moreover, we can check

that∫

Σ∞

|∇η|2e−|x|2

4 ≤ C

| logR| , and∫

Σ∞

|∇β|2e−|x|2

4 ≤ C, (4.36)

whereC are universal constants. Note thatηβφ has compact support inΣ∞\S. Using the inequality

(a+ b+ c)2 ≤ 2(

1 +1

τ

)

(a2 + b2) + (1 + τ)c2, ∀ τ > 0,

28

we have|∇(ηβφ)|2 ≤ 2

(

1 +1

τ

)

φ2(

β2|∇η|2 + η2|∇β|2)

+ (1 + τ)η2β2|∇φ|2. (4.37)

We normalizeφ such that|φ| ≤ 1. Note thatηβφ ∈ W 1,20 (Σ∞\0). Combining (4.37) with Lemma

4.13, we have

0 ≤ −∫

Σ∞

(

ηβφ · L(ηβφ))

e−|x|2

4

=

∫

Σ∞

(

|∇(ηβφ)|2 − 1

2(ηβφ)2

)

e−|x|2

4

≤ (1 + τ)

∫

Σ∞

(

|∇φ|2 − 1

2φ2

)

e−|x|2

4 + C(

1 +1

τ

) 1

| logR| + C( logR

log δ

)2

+1

2

∫

Σ∞

(

1 + τ − (ηβ)2)

φ2 e−|x|2

4 ,

whereC are universal constants. Takingδ → 0 and thenR → 0, we have

0 ≤ (1 + τ)

∫

Σ∞

(

|∇φ|2 − 1

2φ2

)

e−|x|2

4 +τ

2

∫

Σ∞

φ2 e−|x|2

4 .

Letting τ → 0, we get the inequality∫

Σ∞

(

|∇φ|2 − 1

2φ2

)

e−|x|2

4 ≥ 0. (4.38)

The lemma is proved.

The rest of the argument is the same as that in Colding-Minicozzi [15]. If φ is identically one onBR and cuts off linearly to zero onBR+1\BR, then (4.38) implies that

0 ≤∫

Σ∞

(

|∇φ|2 − 1

2φ2

)

e−|x|2

4 ≤∫

Σ∞\BR

e−|x|2

4 − 1

2

∫

BR∩Σ∞

e−|x|2

4 . (4.39)

Note that the right hand side of (4.39) is negative whenR is sufficiently large. Therefore, we get acontradiction. Thus, for the sequence of timesti in Lemma 4.8, the multiplicity of the convergenceof (Σ,x(ti + t)),−T < t < T is one and the convergence is smooth everywhere. Theorem 4.1isproved.

5 Proof of the extension Theorem

For the convenience of readers, we copy down the statement ofour main extension theorem, i.e.,Theorem 1.1 as follows.

Theorem 5.1. If x(p, t) : Σ2 → R3(t ∈ [0, T )) is a closed smooth embedded mean curvature flow

with the first singular timeT < +∞, then supΣ×[0,T )

|H|(p, t) = +∞.

29

Proof. Suppose not, we can find(x0, T ) such that the mean curvature flowx(p, t) blows up atx0 ∈ R3

at timeT withΛ0 := sup

Σ×[0,T )|H|(p, t) < +∞. (5.1)

Then Corollary 3.6 of [18] implies that for allt < T we have

d(Σt, x0) ≤ 2√T − t, (5.2)

whered(Σt, x0) denotes the Euclidean distance from the pointx0 to the surfaceΣt. We can rescalethe flowΣt by

s = − log(T − t), Σs = es2

(

ΣT−e−s − x0

)

such that the flow(Σs, x(p, s)),− log T ≤ s < +∞ satisfies the following properties:

(1) x(p, s) satisfies the equation∂x

∂s= −

(

H − 1

2〈x,n〉

)

n; (5.3)

(2) the mean curvature ofΣs satisfies|H(p, s)| ≤ Λ0e− s

2 for someΛ0 > 0;

(3) d(Σs, 0) ≤ 2.

By Theorem 4.1, there exists a sequence of timessi → +∞ such thatΣsi converges smoothly to aplane passing through the origin with multiplicity one. Consider the heat-kernel-type function

Φ(x0,T )(x, t) =1

4π(T − t)e−

|x−x0|2

4(T−t) , ∀ (x, t) ∈ Σt × [0, T ).

Huisken’s monotonicity formula(cf. Theorem 3.1 in [31]) implies that

Θ(Σt, x0, T ) := limt→T

∫

Σt

Φ(x0,T )(x, t) dµt = limsi→+∞

1

4π

∫

Σsi

e−|x|2

4 dµsi = 1,

which implies that(x0, T ) is a regular point by Theorem 3.1 of B. White [46]. Thus, the unnormal-ized mean curvature flow(Σ,x(t)), 0 ≤ t < T cannot blow up at(x0, T ), which contradicts ourassumption. The theorem is proved.

Appendix A The parabolic Harnack inequality

In the appendix, we collect the results on the parabolic Harnack inequality of Aronson-Serrin in [1]for the reader’s convenience.

Let Ω be a bounded domain inRn. Consider the space-time cylinderQ = Ω × (0, T ) for somefixedT > 0. Letu(x, t) be a solution of the equation

∂u

∂t= divA(x, t, u, ux) + B(x, t, u, ux), (A.1)

30

whereA = (A1,A2, · · · ,An) is a vector function of(x, t, u, ux) andB is a scalar function of thesame variables. Assume that the functionsA andB satisfy the following inequalities

p · A(x, t, u, p) ≥ a|p|2 − b2u2 − f2, (A.2)

|B(x, t, u, p)| ≤ c|p|+ d|u|+ g, (A.3)

|A(x, t, u, p)| ≤ a|p|+ e|u|+ h, (A.4)

wherea, a are positive constants, and the coefficientsb, c, · · · , h are nonnegative smooth functionson Q. Note that in [1] the coefficients is assumed to be in some functional spaces, but here we onlyconsider the smooth case.

Theorem A.1. Let u be a nonnegative weak solution of (A.1) inQ. Suppose thatΩ′ is a convexsubdomain ofΩ which has a positive distanceδ from the boundary ofΩ. Then for anyx, y in Ω′ andall s, t satisfying0 < s < t < T we have

u(y, s) + k ≤ (u(x, t) + k)eC

(

|x−y|2

t−s+ t−s

R+1

)

, (A.5)

whereC depends only on the coefficients of (A.2)-(A.6),k = ‖f‖+‖g‖+‖h‖ andR = min(1, s, δ2).Here‖ · ‖ denotes theL∞ norm.

Note that in Theorem A.1 the subdomainΩ′ is assumed to be convex. However, we can slightlymodify Theorem A.1 such that it still holds for a general subdomainΩ′ in Ω. In fact, for any twopointsx, y ∈ Ω′ there is a continuous pathγ(s) ⊂ Ω′ connectingx andy, and we can apply TheoremA.1 on the small balls centered at each pointp ∈ γ(s). Therefore, we have the following result.

Theorem A.2. Letu be a nonnegative weak solution of (A.1) inQ. Suppose thatΩ′ is a subdomainof Ω which has a positive distanceδ from the boundary ofΩ. Then for anyx, y in Ω′ and all s, tsatisfying0 < s < t < T we have

u(y, s) + k ≤ (u(x, t) + k)eC

(

L(γ(x,y))2

t−s+ t−s

R+1

)

, (A.6)

whereC depends only on the coefficients of (A.2)-(A.6),k = ‖f‖+‖g‖+‖h‖, R = min(1, s, δ2) andL(γ(x, y)) denotes the length of a continuous pathγ(s) ⊂ Ω′ connectingx andy. Here‖ · ‖ denotestheL∞ norm.

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Haozhao Li, Key Laboratory of Wu Wen-Tsun Mathematics, Chinese Academy of Sciences,School of Mathematical Sciences, University of Science andTechnology of China, No. 96 JinzhaiRoad, Hefei, Anhui Province, 230026, China; hzli@ustc.edu.cn.

Bing Wang, Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706,USA; bwang@math.wisc.edu.

34