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The mean curvature at the first singular time of the mean curvature flow

Nam Q. Le

a

, Natasa Sesum

b,,1

aDepartment of Mathematics, Columbia University, New York, USA bDepartment of Mathematics, University of Pennsylvania, Philadelphia, PA, USA Received 15 February 2010; received in revised form 12 July 2010; accepted 28 August 2010

Available online 15 September 2010

Abstract

Consider a family of smooth immersionsF (·, t):Mn→Rn+1of closed hypersurfaces inRn+1moving by the mean curvature flow ∂F (p,t )∂t = −H (p, t)·ν(p, t), fort∈ [0, T ). We prove that the mean curvature blows up at the first singular timeT if all singularities are of type I. In the casen=2, regardless of the type of a possibly forming singularity, we show that at the first singular time the mean curvature necessarily blows up provided that either the Multiplicity One Conjecture holds or the Gaussian density is less than two. We also establish and give several applications of a local regularity theorem which is a parabolic analogue of Choi–Schoen estimate for minimal submanifolds.

©2010 Elsevier Masson SAS. All rights reserved.

1. Introduction

LetMnbe a compactn-dimensional hypersurface without boundary, and letF0:Mn→Rn+1be a smooth immer- sion ofMnintoRn+1. Consider a smooth one-parameter family of immersions

F (·, t ):Mn→Rn+1 satisfyingF (·,0)=F0(·)and

∂F (p, t )

∂t = −H (p, t )ν(p, t ),(p, t )M× [0, T ). (1.1)

HereH (p, t )andν(p, t )denote the mean curvature and a choice of unit normal for the hypersurfaceMt=F (Mn, t ) atF (p, t ), respectively. We will sometimes also writex(p, t )=F (p, t )and refer to (1.1) as to the mean curvature flow equation. Furthermore, for any compactn-dimensional hypersurfaceMnwhich is smoothly embedded inRn+1 byF :Mn→Rn+1, let us denote byg=(gij)the induced metric,A=(hij)the second fundamental form,= det(gij) dxthe volume form,∇ the induced Levi-Civita connection. Then the mean curvature ofMnis given by H=gijhij.

* Corresponding author.

E-mail addresses:namle@math.columbia.edu (N.Q. Le), natasas@math.upenn.edu (N. Sesum).

1 Partially supported by NSF grant 0604657.

0294-1449/$ – see front matter ©2010 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2010.09.002

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Without any special assumptions onM0, the mean curvature flow (1.1) will in general develop singularities in finite time, characterized by a blow up of the second fundamental formA(·, t ).

Theorem 1.1(Huisken [11]). SupposeT <is the first singularity time for a compact mean curvature flow. Then supMt|A|(·, t )→ ∞astT.

By the work of Huisken and Sinestrari [14] the blow up ofH near a singularity is known for mean convex hy- persurfaces. They show that whenH0 one has a pinching curvature estimate stating that|A|2C1H2+C2, for uniform constants C1,C2. In [22] a similar pinching estimate has been proven for star shaped hypersurfaces. The present article establishes the blow up of the mean curvature in the case of type-I singularities.

Definition 1.1.We say that the mean curvature flow (1.1) develops a singularity of type I atT <∞if the blow-up rate of the curvature satisfies an upper bound of the form

maxMt

|A|2(·, t ) C0

Tt, 0t < T . (1.2)

In this paper, we prove the following

Theorem 1.2.Assume(1.2)for the mean curvature flow(1.1). If maxMt

|H|2(·, t )C0 (1.3)

then the flow can be extended past timeT.

In fact, the above theorem is a consequence of the following result.

Theorem 1.3.Assume(1.2)for the mean curvature flow(1.1). If for someαn+2

HLα(M×[0,T ))C0 (1.4)

then the flow can be extended past timeT.

The proofs of Theorems 1.2 and 1.3 are based on blow-up arguments using Huisken’s monotonicity formula, the classification of self-shrinkers and White’s local regularity theorem for mean curvature flow.

Remark 1.1.To some extent, the conditionαn+2 appearing in Theorem 1.3 is optimal as illustrated by the mean curvature flow of the standard sphereSn.

Our Theorems 1.2 and 1.3 left open the question on the possible blow up of the mean curvature at the first singular timeT for mean curvature flows with singularities other than type I. This seems to be a difficult question. However, assuming the validity of Multiplicity One Conjecture (see p. 7 of [15] and the precise statement in Conjecture 3.1 of the present article), we prove the following

Theorem 1.4.Let M2 be a compact, smooth and embedded2-dimensional manifold inR3. If the Multiplicity One Conjecture holds and if

maxMt

|H|2(·, t )C0 (1.5)

then the flow can be extended past timeT.

The next result is independent of the Multiplicity One Conjecture. It is in some sense a refinement of White’s local regularity theorem [26]. White gives uniform curvature bounds in regions of spacetime where the Gaussian density is close to one. We prove the following

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Theorem 1.5.LetM2be a compact, smooth and embedded2-dimensional manifold inR3. Suppose that(1.5)holds.

Lety0∈R3be a point reached by the mean curvature flow(1.1)at timeT. If

tlimT

ρy0,Tt:=lim

t T

1

[4π(T−t )]n/2exp

−|yy0|2 4(T −t )

t<2 (1.6)

then(y0, T )is a regular point of the mean curvature flow(1.1).

Remark 1.2.Our theorem says that for mean curvature flow of surfaces with Gaussian density limt T

ρy0,Tt below 2, for everyy0reached by the flow at timeT, the mean curvature must blow up at the first singular time. In [23], Stone calculated the Gaussian density on spheres and cylinders. On spheres, the density is 4/e≈1.47 and on cylinders it is√

2π/e≈1.52.

Remark 1.3.In a recent paper, Cooper [7] gave a new characterization of the singular time of the mean curvature flow, without assuming type-I singularities. Roughly speaking, his result says that at the first singular time of the mean curvature flow, the product of the mean curvature and the norm of the second fundamental form blows up. His method also gave a new proof of our Theorem 1.2.

We also give the following characterization of a finite time singularity of (1.1) that works in all dimensionsn2.

Theorem 1.6.Assume that for the mean curvature flow(1.1), we have the following integral bound on the second fundamental form

ALp,q(M×[0,T )):=

T 0 Mt

|A|q p/q

dt 1/p

<∞ (1.7)

wherep, q(0,)satisfy n

q +2 p =1.

Then the flow can be extended past timeT.

The previously mentioned results were all global characterizations ensuring that the flow cannot develop any sin- gularities as long as some global quantities are bounded uniformly in time. We also give a result regarding the local regularity theory.

Theorem 1.7.SupposeM=(Mt)is a smooth, properly embedded solution of the mean curvature flow inB(x0, ρ)× (t0ρ2, t0)which reachesx0at timet0. There existsε0=ε0(M0) >0such that if0< σρand

t0

t0σ2

MtB(x0,σ )

|A|n+2dμ dt < ε0 (1.8)

then

0maxδσ/2 sup

t∈[t0δ)2,t0)

sup

xB(x0δ)Mt

δ2|A|2(x, t ) < ε

n−2+2

0

t0

t0σ2

MtB(x0,σ )

|A|n+2dμ dt n+22

. (1.9)

Our theorem is a parabolic version of Choi–Schoen estimate [3] for minimal surfaces. Related results can be found in Ecker [8] and Ilmanen [15]. The precise estimate of the form (1.9) for the case of minimal submanifolds can be found in Shen–Zhu [20], Proposition 2.2 (see also [4]). Moreover, in [18,27,28], the authors showed that if theLn+2 norm in space–time of the second fundamental form (or the mean curvature but under various convexity assumptions) is finite then it is possible to extend the mean curvature flow beyond the time interval under consideration. Our theorem

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can be viewed as a local version of these results without imposing any convexity assumptions. It turns out that the conclusion of Theorem 1.7 also holds in the case when the ambient space is a complete Riemannian manifold with bounded geometry. We show that in Corollary 5.1. We would like to mention that, as far as regularity is concerned, ourε-regularity result in Theorem 1.7 is a direct corollary of Theorem 14 in [15], which works in any codimension.

However, the essence of our theorem is the precise estimate (1.9). Note that the constant in Theorem 1.7 does not depend on the initial hypersurface, but only on a dimension. Finally, up to a constant depending on the dimensionn and the initial hypersurfaceM0, our estimate (1.9) is implied as well by the estimate (27) in Theorem 14 of [15]. This can be seen using Hölder’s inequality and volume bound during the mean curvature flow as in Lemma 1.4 in Ecker [8].

The organization of the paper is as follows. In Section 2 we give the proofs of Theorems 1.2 and 1.3. In Section 3 we prove Theorems 1.4 and 1.5. The proof of Theorem 1.6 will be given in Section 4. We conclude the paper with Section 5 in which we prove Theorem 1.7 and give some applications to it.

2. Characterization of type-I singularities

This section is concerned with the proofs of Theorems 1.2 and 1.3.

Proof of Theorem 1.2. Without loss of generality, assume that MnB1(0)⊂Rn+1. Let y0∈Rn+1 be a point reached by the mean curvature flow (1.1) at time T, that is, there exists a sequence (yj, tj) withtj T so that yjMtj andyjy0. We show that(y0, T )is a regular point of (1.1).

Note that the distance estimate [9, Corollary 3.6] gives dist(Mt, y0)

2n(T −t ), fort < T . (2.1)

Consider the parabolic dilationDλ:Rn+1× [0, T )→Rn+1× [−λ2T ,0)of scaleλ >0 at(y0, T )defined by Dλ(y, t )= λ(yy0), λ2(tT )

. (2.2)

Denote the new time parameter bys. Thent=T+λs2. Let MsλMs(y0,T ),λ=Dλ(Mt)=λ(MT+s

λ2y0).

Then(Msλ)is a solution of the mean curvature flow inBλ(0)fors∈ [−λ2T ,0). Denote byλs the induced volume form onMsλ. Letρy0,T :Rn+1×(−∞, T )→Rbe the backward heat kernel at(y0, T ), i.e.,

ρy0,T(y, t )= 1

[4π(T −t )]n/2exp

−|yy0|2 4(T−t )

. (2.3)

The monotonicity formula of Huisken [12] says that d

dt

Mt

ρy0,Tt= −

Mt

ρy0,T

H+ F 2(T−t )

2t, (2.4)

from which it follows that the limit limtT

Mtρy0,Ttexists. HereF(·, t )is the normal component of the position vectorF (·, t )∈Rn+1in the normal space ofMt inRn+1. Via the parabolic dilation, (2.4) becomes

d ds

Msλ

ρ0,0λs = −

Msλ

ρ0,0

Hsλ(Fsλ) 2s

2λs. (2.5)

Fixs0<0. Integrating both sides of (2.5) froms0τ tos0forτ >0, we get

s0

s0τ

Msλ

ρ0,0

Hsλ(Fsλ) 2s

2λsds=

Msλ0−τ

ρ0,0λs

0τ

Msλ

0

ρ0,0λs

0. (2.6)

Lett1=T +λs02. Then, by the invariance of

Mtρy0,Tt under the parabolic scaling,

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Mt1

ρy0,Tt1=

Msλ

0

ρ0,0λs

0.

Lettingλ→ ∞, one hast1T and

λlim→∞

Msλ

0

ρ0,0λs

0 =lim

tT

Mt

ρy0,Tt.

Similarly,

λlim→∞

Msλ0−τ

ρ0,0λs

0τ =lim

tT

Mt

ρy0,Tt.

Therefore, by (2.6),

λlim→∞

s0

s0τ

Msλ

ρ0,0

Hsλ(Fsλ) 2s

2λsds=0. (2.7)

On the other hand, the second fundamental form ofMsλsatisfies max|A|2(·, s) Msλ

= 1

λ2max|A|2(·, t )(Mt)= −1

s(Tt )max|A|2(·, t )(Mt) and thus, by (1.2),

max|A|2(·, s) MsλC0

s ,s

λ2T ,0 .

In particular, for fixedδ(0,1/2), the inequality A(y)2C0

δ2 (2.8)

holds foryMsλBλands∈ [−λ2T ,δ2]and therefore foryMsλB1/δands∈ [−1/δ2,δ2]forλsufficiently large depending onδandT, sayλλδ,T. By the interior estimate [10], one has for allm0

mA(y)2C(C0, m, n)

δ2(m+1) (2.9)

foryMsλB1/2δands∈ [−1/4δ2,δ2]. Moreover, by (2.1), dist 0, Msλ

=λdist(y0, MT+s

λ2

2n −s

λ2

=√

−2ns

for the above timessandλλδ,T. By Arzela–Ascoli theorem combined with a diagonal sequence argument when lettingδ0 for local graph representations of(Msλ), we can find a subsequenceλi→ ∞such that(Msλi)converges smoothly on compact subsets ofRn+1×(−∞,0)to a smooth solution(Ms)s<0of mean curvature flow. From (2.7), one sees thatH=2s1FonMsfors(s0τ, s0).

Takes0→0 andτ→ ∞to see thatH=2s1FonMsfor−∞< s <0. In other words,(M)sis a self-shrinking mean curvature flow. Moreover, one deduces from (1.3) and|Hsλ| = |Hλt|thatH=0 onMs. ThusMsis a minimal cone for eachs <0; see Corollary 2.8 in [6]. BecauseMsis smooth, it is a hyperplane. Now, fixs0<0. One has, as i→ ∞,Msλ0iMs

0 ∼=Rnandλs0idxn. Thus

ilim→∞

Msλi0

ρ0,0λsi

0=

Ms

0

ρ0,0dxn=1.

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This implies that, forti=T +λs02 i

tlimiT

Mti

ρy0,Tti=1, (2.10)

and therefore

tlimT

Mt

ρy0,Tt=1.

By White’s regularity theorem [26], the second fundamental form|A|(·, t )ofMtis bounded astT and(y0, T )is a regular point. Thus, the flow can be extended past timeT. 2

Proof of Theorem 1.3. We will split the proof of Theorem 1.3 in the following two lemmas. 2 Lemma 2.1.Theorem1.3holds forα > n+2.

Proof. We use the same notations as in the proof of Theorem 1.2. Note that under the parabolic dilationsDλi, the inequality (1.4) becomes

C0α T

0

Mt

|H|αtdt=λαi(n+2) 0

λ2iT

Msλi

Hsλiαλsids.

Thus 0

λ2iT

Msλi

Hsλiαλsids C0α λαi(n+2)

. (2.11)

Now, lettingi→ ∞ as in the proof of Theorem 1.2, we get a self-shrinking mean curvature flow(M)s with the property that

0

−∞

Ms

|H|αdus ds=0 (2.12)

becauseα > n+2. ThereforeH=0 onMs. Now we can argue similarly as in the proof of Theorem 1.2. 2 Lemma 2.2.Theorem1.3holds forα=n+2.

Proof. We use the same notation as in the proof of Theorem 1.2. Under the parabolic dilationsDλi, the inequality (1.4) becomes

C0 T

0

Mt

|H|n+2tdt= 0

λ2iT

Mλis

Hsλin+2λsids.

Lettingi→ ∞as before we get a complete and smooth self-shrinkerMsin the limit with the property that 0

−∞

Ms

|H|n+2s dsC0<. (2.13)

Our self-shrinker satisfies

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H=x, ν (−2s),

which is equivalent to saying thatMs=√

sM1, whereMssatisfies the mean curvature flow. Notice that

Ms

|H|n+2(·, s) dμs=

M−1

|H|(·,−1)

√−s n+2

·(s)n21= 1 (s)·

M−1

|H|n+2(·,−1) dμ1= a (s), wherea:=

M1|H|n+2(·,−1) dμ1. Ifa >0 then 0

−∞

Ms

|H|n+2s ds=a· 0

−∞

ds (s) = ∞,

which contradicts (2.13). Therefore a =0, which implies H (·,−1)=0 on M1. Similar argument shows that H (·, s)=0 onMs for everys <0. To prove that (y0, T )is a regular point of the flow we argue as in the proof of Theorem 1.2. 2

3. Extension results for surfaces

In this section, we prove Theorems 1.4 and 1.5. First, we recall the Multiplicity One Conjecture. In [15] Ilmanen proposed the following conjecture.

Conjecture 3.1 (Multiplicity One Conjecture). If M02 is embedded in R3, then for any family of rescalings λj(Mλ2

j s+Ty0)withλj→ ∞, there is a subsequence smoothly converging and with multiplicity one to the blowup Nt, that is, there are no concentration points or multiple layers in the limit.

Theorem 1.4 assumes that the conjecture above holds and its proof is given below.

Proof of Theorem 1.4. In this proof,n=2. Without loss of generality, assume thatMnB1(0)⊂Rn+1. Lety0∈ Rn+1 be a point reached by the mean curvature flow (1.1) at timeT, that is, there exists a sequence(yj, tj)with tj T so thatyjMtj andyjy0. We show that(y0, T )is a regular point of (1.1).

As in the proof of Theorem 1.2, lett=T +λs2 and MsλMs(y0,T ),λ=λ(MT+s

λ2y0).

Then(Msλ)is a solution of the mean curvature flow inBλ(0)fors∈ [−λ2T ,0). For any setA⊂Rn+1, let us define the parabolically rescaled measures at(y0, T ):

μλs(A)=λnHnMsλ·A).

Letρy0,T :Rn+1×(−∞, T )→Rbe the backward heat kernel at(y0, T )as defined in (2.3). Then, a result on weak existence of blow ups of Ilmanen and White (see Lemma 8, p. 14 of [15] and also [24]) says that: there exists a subsequenceλj and a limiting Brakke flow [1]{νs}s<0(also known as atangent flow) such thatμλsj νsin the sense of Radon measures for alls <0 and the following statements hold:

(a) (self-similarity)νs(A)=νsλ(A)λnνλ2s·A), for alls <0 and for allλ >0;

(b) (tangent flow is a self-shrinker)ν1satisfies

H (x)+S(x)·x

2 =0, ν1a.e.x; (3.1)

(c) furthermore, Huisken’s integral converges

ρ0,0(x,−1) dνs(x)=lim

t T

ρy0,Tt, s <0. (3.2)

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Equivalently, a subsequence of rescaled solutionsMsλconverges weakly to a limiting flowXs that is called atangent flowat(y0, T ). We knowXsis a self shrinker. Ilmanen showed in [16] that it has to be smooth. Our proof will rely on this fact and the validity of the Multiplicity One Conjecture. Let us briefly explain the notations used in (b).

For a locallyn-rectifiable Radon measureμ, we define itsn-dimensional approximate tangent planeTxμ(which existsμ-a.e.x) by

Txμ(A)=lim

λ0λnμ(x+λ·A).

The tangent planeTxμis a positive multiple ofHnP for somen-dimensional planeP. LetS:Rn+1−→G(n+1, n) denotes the μ-measurable function that maps x to the geometric tangent plane, denoted byP above. An impor- tant quantity is the first variation ofμ, defined byδVμ(X):=

divS(x)X(x) dμ(x)forXCc (Rn+1,Rn+1). Here divSX=n

i=1DeiX.ei wheree1, . . . , enis any orthonormal basis ofS. We also denote bySthe orthogonal projection ontoSand thus divSX can be written asS:DX. Now, if the total first variationδVμis a Radon measure and is absolutely continuous with respect toμ, then we can define the generalized mean curvature vectorH=HμL1loc(μ) ofμas follows

divSX dμ=

H·X dμ (3.3)

for allXCc(Rn+1,Rn+1). For further information on geometric measure theory, we refer the reader to Simon’s lecture notes [21]. Note that whenμis the surface measure of a smoothn-dimensional manifoldM, the generalized mean curvature vectorHμofμexists and is also the classical mean curvature vector ofM; see Corollary 4. 3 in [19].

Therefore, we can apply (3.3) toμλs, which is the rescaled surface measure of the smooth manifoldMsλ. From (3.3) and the definition ofμλs, one sees that the mean curvature vectorHλs ofμλs is Hλt whereHt is the mean curvature vector ofMt wheret=T +λs2. The lower semicontinuity of

|H|asserts that

|Hs|slim inf

λ→∞ Hλsλs lim sup

λ→∞

C0

λ λs =0.

ThusHs =0 for alls <0. Now, becauseXs is smooth for alls <0, the weak mean curvature vectorHs coincides with the mean curvature vector in classical sense. Thus we have a smooth solution Xs that is a self-shrinker with H =0 and therefore by the result in [6] it has to be a hyperplane. Furthermore νs represents the surface measure of the planeXs with multiplicity one by the validity of the Multiplicity One Conjecture. Using the convergence of Huisken’s integral (3.2), we see that

tlimT

ρy0,Tt=1.

By White’s regularity theorem [26], the second fundamental form|A|(·, t )ofMtis bounded astT and(y0, T )is a regular point. Thus, the flow can be extended past timeT. 2

We conclude this section by the proof of Theorem 1.5, which can be viewed as a local regularity result without a smallness condition. Note that the Multiplicity One Conjecture is not assumed in this theorem.

Proof of Theorem 1.5. We will use the same notation as in the proof of Theorem 1.4. Note that, when n=2, by the fact that

Hs2 is bounded (follows from the Gauss–Bonnet theorem for surfaces) and Allard’s Compactness Theorem [21], each Radon measureνs is integer 2-rectifiable, that is

s=θs(x) dH2Xs

whereXsis anH2-measurable, 2-rectifiable set andθsis anH2Xs-integrable, integer valued “multiplicity function”.

Furthermore the mean curvature vectorHsofνssatisfiesHsLs).Here is the only place we wish to use(1.5).

The same argument as in the proof of Theorem1.4implies thatHs=0 andXs is a plane.

The key point of our proof is the following Constancy Theorem.

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Theorem 3.1(Constancy Theorem). Letμ=θHnMbe an integraln-varifold in the open setΩRn+m,MΩa connectedC1-n-manifold,θ:MN0beHn-measurable with weak mean curvatureHμL1loc(μ), that is

divμη dμ=

M

divMηθ dHn= −

Hμ, ηdμ,ηC01 Ω,Rn+m

. (3.4)

Thenθis a constant:θθ0N0. HereN0is the set of all nonnegative integers and·is the standard Euclidean inner product onRn+m.

Nowθs is a constant andXs is a plane. Thus by the convergence of Huisken’s integral (3.2), we see that

tlimT

ρy0,Tt=

ρ0,0(x,−1) dνs(x)=

ρ0,0(x,−1)θsdH2Xs=θs.

By (1.6) and Proposition 2.10 in [26], 1θs <2. It follows from the integrality ofθs thatθs ≡1. Now, our result follows from White’s local regularity theorem [26]. 2

Proof of Theorem 3.1. For the case of stationary varifoldμ, i.e.,Hμ=0 , the proof of Theorem 3.1 can be found in [21], Theorem 41.1. For the general case stated above, due to Schätzle, the proof can be found in [17], Theorem 4.3.

However, in our application, whenMis a plane andHμ=0 , Theorem 3.1 has a very simple proof which we include it here. ChooseηC01(Ω,Rn+m)such thatηT M-the tangent bundle ofM. Then (3.4) gives

M

(divMη)θ dHn=0.

Calculating in local coordinates, this yields∇Mθ=0 weakly. Henceθθ0is constant, asMis connected. 2 We conclude this section with the following result, which has been suggested to us by one of the referees. It serves to somewhat remove the Multiplicity One Conjecture in Theorem 1.4.

Corollary 3.1.LetM2be a compact, smooth and embedded2-dimensional manifold inR3. If maxMt

|H|2(·, t )C0

and if no quasi-static higher multiplicity planes occur as tangent flows at any point(y0, T )wherey0∈R3is a point reached by the mean curvature flow(1.1)at timeT, then the flow can be extended past timeT.

Remark 3.1.For a precise definition of quasi-static limit flow, we refer the reader to White [25]. In the course of the proof of Theorem 1.4, we showed (without the Multiplicity One Conjecture) that, under the boundedness of the mean curvature, any tangent flow at(y0, T )is a quasi-static plane, maybe with higher multiplicity. The Multiplicity One Conjecture corresponds to the case that all tangent flows (even non-flat ones) are of multiplicity one. In Corollary 3.1, the only requirement is that, if a tangent flow happens to be a quasi-static plane, it must have a point of multiplicity one. Thus, we do not require any information on the multiplicity of non-flat tangent flows.

Proof of Corollary 3.1. As in the proof of Theorem 1.5, the bound on the mean curvature implies that at any(y0, T ), the tangent flows are planes. By Theorem 3.1, each of these planes has constant multiplicity. But at one point, the plane has multiplicity one. Thus the whole plane must have multiplicity one. Now, our result follows from White’s local regularity theorem [26]. 2

4. Some global results on the extension of the mean curvature flow

In this section we give global conditions for extending a smooth solution to (1.1), which has been a subject of study in [18].

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Proof of Theorem 1.6. We argue by contradiction. Suppose thatT is the extinction time of the flow. Then, by Theorem 1.1,|A|is unbounded. Therefore, there exists a sequence of points(xi, ti)withxiMti such that

Qi:= |A|(xi, ti)= max

0tti

max

xMt

|A|(x, t )→ +∞. (4.1)

Consider the sequenceM˜tiof rescaled solutions fort∈ [0,1]defined by F˜i(·, t )=Qi

F

·, ti+t−1 Q2i

xi

.

The sequence of rescaled solutionsM˜ticonverges (see [2]) to a complete smooth solution to the mean curvature flow, call itM˜tfort∈ [0,1]with the property that

| ˜A|(0,1)=1. (4.2)

IfgandA:= {hj k}are the induced metric, and the second fundamental form ofMt, respectively, then the correspond- ing rescaled quantities are given by

˜

gi=Q2ig; | ˜Ai|2=|A|2 Q2i . We calculate

ilim→∞

1 0

(M˜i)tB(0,1)

| ˜Ai|q pq

dt 1

p

= lim

i→∞

ti

ti 1

Q2 i

MtB(0,1

Qi)

|A|q pq

dt 1

p

lim

i→∞

ti

ti 1

Q2 i

Mt

|A|q pq

dt 1

p

=0. (4.3)

The last step follows from the facts that T

0 Mt

|A|q p/q

dt 1/p

<∞; lim

i→∞

1 Q2i =0.

By Fatou’s lemma and (4.3) it follows that 1

0

M˜tB(0,1)

| ˜A|q p

q =0.

By the smoothness of M˜t, this implies | ˜A|(x, t )≡0 for all x ∈ ˜MtB(0,1) and all t ∈ [0,1]. This contra- dicts (4.2). 2

5. Some local regularity results and applications 5.1. ε-regularity theorem for the mean curvature flow

In this section we prove Theorem 1.7, which is parabolic version of the epsilon regularity theorem for minimal surfaces proven by Choi and Schoen [3].

Proof of Theorem 1.7. We may assume without loss of generality that the flowM=(Mt)t <t0 is smooth up to and including timet0, because we can first prove the theorem witht0replaced byt0αfor fixedα >0 and then letα0 afterwards.

(11)

Let

F (δ)= sup

t∈[t0δ)2,t0]

sup

xB(x0δ)Mt

δ2|A|2(x, t ).

Since our flow is smooth up to timet0,F (0)=0. Thus, there existsδ(0, σ/2]such thatF (δ)=max0δσ/2F (δ).

It suffices to show that F (δ) < ε

2 n+2

0

t0

t0σ2

MtB(x0,σ )

|A|n+2dμ dt n+22

ε01η 2

n+2. (5.1)

Suppose not, then F (δ) ε01ηn+22

. (5.2)

Because the flow is defined up to and including timet0, we can findt∈ [t0δ)2, t0]andxB(x0, σδ)Mt0 such that

δ2|A|2(x, t)=F (δ). (5.3)

It follows fromδ(0, σ/2]that B

x

2

B

x0, σδ 2

;

tδ2 4, t

t0

σδ 2

2

, t0

. (5.4)

By the choice ofδ,tandx, δ

2 2

sup

t∈[t0δ2)2,t0]

sup

xB(x0δ2)Mt

|A|2(x, t )F (δ)=δ2|A|2(x).

Hence

sup

t∈[t0δ2)2,t0]

sup

xB(x0δ2)Mt

|A|2(x, t )4|A|2(x, t) and thus, it follows from (5.4) that

sup

t∈[tδ42,t]

sup

xB(x,δ2)Mt

|A|2(x, t )4|A|2(x, t). (5.5)

We now rescale our mean curvature flow by setting F (˜ ·, t )=QF

·, t+t−1 Q2

, M˜t= ˜F Mn, t where

Q=2 ε0η1 1

n+2|A|(x, t).

Then we have a mean curvature flowM˜t onB(x, Qδ/2)fort∈ [0,1]. Letg˜=Q2gbe the induced metric onM˜t

and letB(x˜ , r)be the geodesic ball w.r.t. the metricg˜and centered atxwith radiusr. By (5.2) and (5.3), we have

2 1. (5.6)

This combined with (5.5) gives sup

t∈[0,1] sup

x∈ ˜B(x,1)∩ ˜Mt

| ˜A|2(x, t )4| ˜A|2(x,1)1. (5.7)

Note that the last inequality follows from the facts that

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