A strong stability condition on minimal submanifolds and its implications

Chung-Jun Tsai; Mu-Tao Wang

doi:10.1515/crelle-2018-0038

Article

A strong stability condition on minimal submanifolds and its implications

Published/Copyright: January 22, 2019

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2020 Issue 764

Abstract

We identify a strong stability condition on minimal submanifolds that implies uniqueness and dynamical stability properties. In particular, we prove a uniqueness theorem and a 𝒞1 dynamical stability theorem of the mean curvature flow for minimal submanifolds that satisfy this condition. The latter theorem states that the mean curvature flow of any other submanifold in a 𝒞1 neighborhood of such a minimal submanifold exists for all time, and converges exponentially to the minimal one. This extends our previous uniqueness and stability theorem [24] which applies only to calibrated submanifolds of special holonomy ambient manifolds.

Funding statement: Supported in part by Taiwan MOST grants 105-2115-M-002-012, 106-2115-M-002-005-MY2 and NCTS Young Theoretical Scientist Award (C.-J. Tsai). This material is based upon work supported by the National Science Foundation under Grants No. DMS-1405152 and No. DMS-1810856 (Mu-Tao Wang). Part of this work was carried out when Mu-Tao Wang was visiting the National Center of Theoretical Sciences at National Taiwan University in Taipei, Taiwan.

A Computations related to strong stability

For minimal Lagrangians in a Kähler–Einstein manifold and coassociatives in a G2 manifold, condition (3.2) can be rewritten as a curvature condition on the submanifold. One ingredient is the geometric properties of U⁢(n) and G2 holonomy. Another ingredient is the Gauss equation

Ri⁢j⁢k⁢ℓ-Ri⁢j⁢k⁢ℓΣ=hα⁢i⁢ℓ⁢hα⁢j⁢k-hα⁢i⁢k⁢hα⁢j⁢ℓ.

A.1 Minimal Lagrangians in Kähler–Einstein manifolds

Let (M2⁢n,g,J,ω) be a Kähler–Einstein manifold, where J is the complex structure and ω is the Kähler form. Denote the Einstein constant by c; namely,

∑CRA⁢C⁢B⁢C=RicA⁢B=c⁢gA⁢B.

A submanifold Ln⊂M2⁢n is Lagrangian if ω|L vanishes. It implies that J induces an isomorphism between its tangent bundle TL and normal bundle NL. In terms of the notations introduced in Section 2.1, the correspondence is

(A.1)vi⁢ei↔vi⁢J⁢ei.

In particular, if {e1,…,en} is an orthonormal frame for TL, {J⁢e1,…,J⁢en} is an orthonormal frame for NL. Denote J⁢ek by eJ⁢(k), and let

Ck⁢i⁢j=hJ⁢(k)⁢i⁢j=〈∇ei⁡ej,J⁢ek〉.

Since J is parallel, it is easy to verify that Ck⁢i⁢j is totally symmetric.

Now, suppose that L is also minimal. By using correspondence (A.1), the strong stability condition (3.2) can be rewritten as follows:

-Ri⁢J⁢(k)⁢i⁢J⁢(ℓ)⁢vk⁢vℓ-Ck⁢i⁢j⁢Cℓ⁢i⁢j⁢vk⁢vℓ

=-c⁢gk⁢ℓ⁢vk⁢vℓ+RJ⁢(i)⁢J⁢(k)⁢J⁢(i)⁢J⁢(ℓ)⁢vk⁢vℓ-Ck⁢i⁢j⁢Cℓ⁢i⁢j⁢vk⁢vℓ

=-c⁢|v|2+Ri⁢k⁢i⁢ℓ⁢vk⁢vℓ-Ck⁢i⁢j⁢Cℓ⁢i⁢j⁢vk⁢vℓ

=-c⁢|v|2+Ri⁢k⁢i⁢ℓL⁢vk⁢vℓ+Cj⁢k⁢i⁢Cj⁢ℓ⁢i⁢vk⁢vℓ-Ck⁢i⁢j⁢Cℓ⁢i⁢j⁢vk⁢vℓ

=-c⁢|v|2+RicL⁢(v,v).

The first equality uses the Kähler–Einstein condition. The second equality follows from the parallelity of J. The third equality uses the Gauss equation and the minimal condition. The last equality relies on the fact that Ck⁢i⁢j is totally symmetric. This computation says that (3.2) is equivalent to the condition that RicL-c is a positive definite operator on TL.

A.2 Coassociative submanifolds in G2 manifolds

In this case, the ambient space is 7-dimensional, and the submanifold is 4-dimensional.

A.2.1 4-dimensional Riemannian geometry

The Riemann curvature tensor has a nice decomposition in four dimensions. What follows is a brief summary of the decomposition; readers are directed to [1] for more.

Let Σ be an oriented, 4-dimensional Riemannian manifold. The Riemann curvature tensor in general defines a self-adjoint transform on Λ2 by

ℜ⁢(ei∧ej)=12⁢Rk⁢ℓ⁢i⁢jΣ⁢ek∧eℓ.

In four dimensions, Λ2 decomposes into self-dual, Λ+2, and anti-self-dual part, Λ-2. In terms of the decomposition Λ2=Λ+2⊕Λ-2, the curvature map ℜ has the form

ℜ=[W++s12⁢𝐈BBTW-+s12⁢𝐈].

Here, s=Ri⁢j⁢i⁢jΣ is the scalar curvature, W± is the self-dual and anti-self-dual part of the Weyl tensor, B is the traceless Ricci tensor, and 𝐈 is the identity homomorphism.

With respect to the basis {e1∧e2-e3∧e4,e1∧e3+e2∧e4,e1∧e4-e2∧e3}, the lower-right block W-+s12⁢𝐈 is

(A.2)12⁢[R1212Σ+R3434Σ-2⁢R1234ΣR1213Σ+R1224Σ-R3413Σ-R3424ΣR1214Σ-R1223Σ-R3414Σ+R3423ΣR1312Σ-R1334Σ+R2412Σ-R2434ΣR1313Σ+R2424Σ+2⁢R1324ΣR1314Σ-R1323Σ+R2414Σ-R2423ΣR1412Σ-R1434Σ-R2312Σ+R2334ΣR1413Σ+R1424Σ-R2313Σ-R2324ΣR1414Σ+R2323Σ-2⁢R1423Σ].

The operator will be needed is W--s6⁢𝐈=(W-+s12⁢𝐈)-s4⁢𝐈. One-fourth of the scalar curvature is

(A.3)s4=12⁢(R1212Σ+R3434Σ+R1313Σ+R2424Σ+R1313Σ+R2424Σ).

A.2.2 G2 geometry

A 7-dimensional Riemannian manifold M whose holonomy is contained in G2 can be characterized by the existence of a parallel, positive 3-form φ. A complete story can be found in [13, ch.11]. In terms of a local orthonormal coframe, the 3-form and its Hodge star are

(A.4)φ=ω567+ω125-ω345+ω136+ω246+ω147-ω237,

φ=ω1234-ω1267+ω3467+ω1357+ω3457-ω1456+ω2356,

where ω123 is short for ω1∧ω2∧ω3. It is known that the holonomy is G2 if and only if ∇⁡φ=0, which is also equivalent to d⁢φ=0=d*φ.

Remark A.1.

There are two commonly used conventions for the 3-form; see [14] for instance. The convention here is the same as that in [15]; the deformation of coassociatives will then be determined by anti-self-dual harmonic forms. If one use the convention in [13], the deformation of coassociatives will be determined by self-dual harmonic forms.

The 3-form φ determines a product map × for tangent vectors of M. For any two tangent vectors X and Y,

X×Y=(φ⁢(X,Y,⋅))♯.

For instance, e1×e2=e5. Since φ and the metric tensor are both parallel, × is parallel as well.

As a consequence,

R⁢(eA,eB)⁢(e1×e2)=(R⁢(eA,eB)⁢e1)×e2+e1×(R⁢(eA,eB)⁢e2),

and its e3-component gives R53⁢A⁢B-R62⁢A⁢B-R71⁢A⁢B=0 for any A,B∈{1,…,7}. In total, the parallelity of × leads to following seven identities:

(A.5)R52⁢A⁢B+R63⁢A⁢B+R74⁢A⁢B=0,R67⁢A⁢B+R12⁢A⁢B-R34⁢A⁢B=0,

R51⁢A⁢B-R64⁢A⁢B+R73⁢A⁢B=0,-R57⁢A⁢B+R13⁢A⁢B+R24⁢A⁢B=0,

R54⁢A⁢B+R61⁢A⁢B-R72⁢A⁢B=0,R56⁢A⁢B-R14⁢A⁢B-R23⁢A⁢B=0,

-R53⁢A⁢B+R62⁢A⁢B+R71⁢A⁢B=0.

These identities imply that a G2 manifold is always Ricci flat.

A.2.3 Coassociative geometry

According to [10, Section IV], an oriented, 4-dimensional submanifold Σ of a G2 manifold is said to be coassociative if *φ|Σ coincides with the volume form of the induced metric. Harvey and Lawson also proved that if φ|Σ vanishes, there is an orientation on Σ so that it is coassociative. Similar to the Lagrangian case, the normal bundle of a coassociative submanifold is canonically isomorphic to an intrinsic bundle. The following discussion is basically borrowed from [15, Section 4].

Orthonormal frame

Suppose that Σ⊂M is coassociative. One can find a local orthonormal frame {e1,…,e7} such that {e1,e2,e3,e4} are tangent to Σ, {e5,e6,e7} are normal to Σ, and φ takes the form (A.4) in this frame. Here is a sketch of the construction. Start with a unit normal vector, e5, and a unit tangent vector, e1, of Σ. Let e2=e5×e1. Then set e3 to be a unit vector tangent to Σ and orthogonal to {e1,e2}. Finally, let e4=e3×e5, e6=e1×e3 and e7=e3×e2.

Normal bundle and second fundamental form

The normal bundle of Σ is isomorphic to the bundle of anti-self-dual 2-forms of Σ via the following map:

(A.6)V↦(V⁢⌟⁢φ)|Σ.

In terms of the above frame, e5 corresponds to ω12-ω34, e6 corresponds to ω13+ω24, and e7 corresponds to ω14-ω23.

As shown in [10], a coassociative submanifold must be minimal. In fact, its second fundamental form has certain symmetry. For instance,

h51⁢i=〈∇ei⁡e1,e5〉=-〈e1,∇ei⁡(e6×e7)〉

=-〈e1,(∇ei⁡e6)×e7〉-〈e1,e6×(∇ei⁡e7)〉

=-〈e4,∇ei⁡e6〉+〈e3,∇ei⁡e7〉=h64⁢i-h73⁢i.

What follows are all the relations:

(A.7)h52⁢i+h63⁢i+h74⁢i=0,h54⁢i+h61⁢i-h72⁢i=0,

h51⁢i-h64⁢i+h73⁢i=0,-h53⁢i+h62⁢i+h71⁢i=0

for any i∈{1,2,3,4}. These relations imply that the mean curvature vanishes. They can be encapsulated as ∑jej×𝕀⁢(ei,ej)=0.

A.2.4 Strong stability for coassociatives

For any sections of N⁢Σ, 𝐯, denote the symmetric bilinear form on the left-hand side of (3.2) by Q⁢(𝐯,𝐯). Under the identification (A.6),

Q~⁢(𝐯,𝐯)=-2⁢𝐯T⁢W-⁢𝐯+s3⁢|𝐯|2

is also a symmetric bilinear form.

We now check that

Q⁢(𝐯,𝐯)=Q~⁢(𝐯,𝐯)

for any unit vector 𝐯∈Np⁢Σ at any p∈Σ. As explained above, we may take e5=𝐯 and construct the other orthonormal vectors. With respect such a frame, it follows from (A.2) and (A.3) that

Q~⁢(𝐯,𝐯)=R1313Σ+R2424Σ+R1313Σ+R2424Σ+2⁢R1234Σ.

The quantity Q⁢(𝐯,𝐯) can be rewritten as follows:

Q⁢(𝐯,𝐯)=-∑iRi⁢5⁢i⁢5-∑i,j(h5⁢i⁢j)2

=R6565+R7575-∑i,j(h5⁢i⁢j)2

=R1313+R2424+R1313+R2424-2⁢R1423+2⁢R1324-∑i,j(h5⁢i⁢j)2

=R1313+R2424+R1313+R2424+2⁢R1234-∑i,j(h5⁢i⁢j)2.

The second equality follows from Ricci flatness. The third equality uses (A.5). The last equality is the first Bianchi identity. With the Gauss equation and some simple manipulation,

(A.8)Q(𝐯,𝐯)-Q~(𝐯,𝐯)=∑α((hα⁢14+hα⁢23)2+(hα⁢13-hα⁢24)2

-(hα⁢11+hα⁢22)(hα⁢33+hα⁢44))-∑i,j(h5⁢i⁢j)2.

By appealing to (A.7),

h614+h623=-h522-h544=h511+h533,h714+h723=-h512+h534,h613-h624=-h512-h534,h713-h724=-h522+h533=-h511-h544,

and

h611+h622=-h633-h644=-h514+h523,

h711+h722=-h733-h744=h513+h524.

By using these relations, it is not hard to verify that (A.8) vanishes. Therefore, the strong stability condition (3.2) is equivalent to the positivity of -2⁢W-+s3.

As a final remark, this equivalence can also be seen by combining [15, Theorem 4.9] and the Weitzenböck formula [9, Appendix C]. Nevertheless, it is nice to derive the equivalence directly by highlighting the geometry of G2.

B Evolution equation for tensors

Suppose that Ψ is a tensor defined on M of type (0,3). The main purpose of this section is to calculate its evolution equation along the mean curvature flow. Since there will be some different connections, we denote the Levi-Civita connection of (M,g) by ∇¯ to avoid confusions.

Let Γt be the mean curvature flow at time t. The tensor Ψ is a section of

(T*⁢M⊗T*⁢M⊗T*⁢M)|Γt.

The connection ∇¯ naturally induces a connection ∇~ on this bundle. The only difference between ∇¯ and ∇~ is that the direction vector in ∇~ must be tangent to Γt.

connection	bundle and base
∇¯	Levi-Civita connection of (M,g)
∇Γt	Levi-Civita connection of Γt with the induced metric
∇⊥	connection of the normal bundle of Γt
∇~	connection of (T⁢M⊗T⁢M⊗T*⁢M)\|Γt
∇	connection of T⁢Γt⊗T⁢Γt⊗N*⁢Γt defined by (2.2)

From the construction, ∇ is the composition of ∇~ with the orthogonal projection.

Proposition B.1.

Let Ψ be a tensor of type (0,3) defined on the ambient manifold M. Along the mean curvature flow Γt in M,

dd⁢t⁢|𝕀t-Ψ|2≤ΔΓt⁢|𝕀t-Ψ|2-|∇~⁢(𝕀t-Ψ)|2+c⁢(|𝕀t-Ψ|4+|𝕀t-Ψ|2+1),

where c>0 is determined by the Riemann curvature tensor of M and the sup-norm of Ψ, ∇¯⁢Ψ, ∇¯2⁢Ψ.

Proof.

The mean curvature flow can be regarded as a map from Γ0×[0,ε)→M. For any p∈Γ0 and t0∈[0,ε), choose a geodesic coordinate for Γ0 at p: {x~1,…,x~n}. We also choose a local orthonormal frame {e~α} for N⁢Γt. The following computations on derivatives are always evaluated at the point (p,t0).

Let H=h~α⁢e~α be the mean curvature vector of Γt. The components of the second fundamental form and its covariant derivative are denoted by

h~α⁢i⁢j=〈∇¯∂i~⁢∂j~,e~α〉=𝕀⁢(∂i~,∂j~,e~α),

h~α⁢i⁢j,k=(∇∂k~⁡𝕀)⁢(∂i~,∂j~,e~α),

h~α,k=〈∇∂k~⊥⁡H,e~α〉.

At (p,t0), h~α=h~α⁢k⁢k and h~α,i=h~α⁢k⁢k,i.

Note that on Γ0×[0,ε), H is ∂t, and thus commutes with ∂i~. It follows that the evolution of the metric is

dd⁢t⁢g~i⁢j=H⁢〈∂i~,∂j~〉=〈∇¯H⁢∂i~,∂j~〉+〈∂i~,∇¯H⁢∂j~〉

=-〈H,∇¯∂i~⁢∂j~〉-〈∇¯∂j~⁢∂i~,H〉=-2⁢h~α⁢h~α⁢i⁢j,

dd⁢t⁢g~i⁢j=2⁢h~α⁢h~α⁢i⁢j.

The covariant derivative of ∂i~ and e~α along H can be expressed as follows:

(B.1)∇¯H⁢∂i~=〈∇¯H⁢∂i~,∂j~〉⁢∂j~+〈∇¯H⁢∂i~,e~α〉⁢e~α

=-h~α⁢h~α⁢i⁢j⁢∂j~+h~α,i⁢e~α,

(B.2)∇¯H⁢e~α=〈∇¯H⁢e~α,∂i~〉⁢∂i~+〈∇¯H⁢e~α,e~β〉⁢e~β

=-h~α,i⁢∂i~+〈∇¯H⁢e~α,e~β〉⁢e~β.

The last part of the preparation is to relate the covariant derivative of Ψ in H to its Bochner–Laplacian in the ambient manifold M. We have

(B.3)∇¯H⁢Ψ=∇¯(∇¯∂j~⁢∂j~)⊥⁢Ψ=-∇¯∇∂j~Σ⁡∂j~⁢Ψ+∇¯∇¯∂j~⁢∂j~⁢Ψ

=(∇~∂j~⁢∇~∂j~⁢Ψ-∇~∇∂j~Σ⁡∂j~⁢Ψ)+(-∇¯∂j~⁢∇¯∂j~⁢Ψ+∇¯∇¯∂j~⁢∂j~⁢Ψ)

=-∇~*⁢∇~⁢Ψ+trΓt⁡(∇¯2⁢Ψ).

Indeed, ∇∂j~Σ⁡∂j~ is zero at (p,t0). The tensor ∇¯*⁢∇¯⁢Ψ is defined in the ambient space, and has nothing to do with the submanifold Γt. It follows from (B.3) that the evolution of |Ψ|2 is

(B.4)dd⁢t⁢|Ψ|2=H⁢(〈Ψ,Ψ〉)=2⁢〈∇¯H⁢Ψ,Ψ〉

=-2⁢〈∇~*⁢∇~⁢Ψ,Ψ〉+2⁢〈trΓt⁡(∇¯2⁢Ψ),Ψ〉

=ΔΓt⁢|Ψ|2-2⁢|∇~⁢Ψ|2+2⁢〈trΓt⁡(∇¯2⁢Ψ),Ψ〉.

The next task is to calculate the evolution equation for

〈𝕀t,Ψ〉=g~i⁢k⁢g~j⁢l⁢h~α⁢k⁢l⁢Ψ~α⁢i⁢j,

where Ψ~α⁢i⁢j=Ψ⁢(∂i~,∂j~,e~α). According to (B.1) and (B.2),

(B.5)dd⁢t⁢Ψ~α⁢i⁢j=H⁢(Ψ⁢(∂i~,∂j~,e~α))

=(∇¯H⁢Ψ)⁢(∂i~,∂j~,e~α)+Ψ⁢(∂i~,∂j~,∇¯H⁢e~α)

+Ψ⁢(∇¯H⁢∂i~,∂j~,e~α)+Ψ⁢(∂i~,∇¯H⁢∂j~,e~α)

=(∇¯H⁢Ψ)⁢(∂i~,∂j~,e~α)-h~α,k⁢Ψ~k⁢i⁢j+〈∇¯H⁢e~α,e~β〉⁢Ψ~β⁢i⁢j

-h~γ⁢h~γ⁢i⁢k⁢Ψ~α⁢k⁢j+h~γ,i⁢Ψ~α⁢γ⁢j-h~γ⁢h~γ⁢j⁢k⁢Ψ~α⁢i⁢k+h~γ,j⁢Ψ~α⁢i⁢γ.

The difference between ∇~*⁢∇~⁢𝕀t and ∇*⁡∇⁡𝕀t is

(B.6)∇~*⁢∇~⁢𝕀t-∇*⁡∇⁡𝕀t=∇~∂k~⁢∇~∂k~⁢𝕀t-∇∂k~⁡∇∂k~⁡𝕀t.

Since

(∇~∂k~⁢𝕀t)⁢(⋅,⋅,⋅)=(∇¯∂k~⁢𝕀t)⁢(⋅,⋅,⋅)

=∂k~(𝕀t(⋅,⋅,⋅))-𝕀t((∇¯∂k~⋅)T,⋅,⋅)

-𝕀t(⋅,(∇¯∂k~⋅)T,⋅)-𝕀t(⋅,⋅,(∇¯∂k~⋅)⊥),

the tensor ∇~∂k~⁢𝕀t has only the following components:

(B.7)(∇~∂k~⁢𝕀t)⁢(∂i~,∂j~,e~α)=(∇∂k~⁡𝕀t)⁢(∂i~,∂j~,e~α)=h~α⁢i⁢j,k,

(∇~∂k~⁢𝕀t)⁢(∂i~,∂j~,∂l~)=-h~α⁢i⁢j⁢h~α⁢k⁢l,

(∇~∂k~⁢𝕀t)⁢(e~β,∂j~,e~α)=h~β⁢k⁢i⁢h~α⁢i⁢j,

(∇~∂k~⁢𝕀t)⁢(∂i~,e~β,e~α)=h~β⁢k⁢j⁢h~α⁢i⁢j.

The above four equations hold everywhere, but not only at (p,t0). It follows that

(∇~∂k~⁢∇~∂k~⁢𝕀t)⁢(∂i~,∂j~,e~α)=∂k~⁡((∇~∂k~⁢𝕀t)⁢(∂i~,∂j~,e~α))-(∇~∂k~⁢𝕀t)⁢(∂i~,∂j~,∇¯∂k~⁢e~α)

-(∇~∂k~⁢𝕀t)⁢(∇¯∂k~⁢∂i~,∂j~,e~α)-(∇~∂k~⁢𝕀t)⁢(∂i~,∇¯∂k~⁢∂j~,e~α)

=(∇∂k~⁡∇∂k~⁡𝕀t)⁢(∂i~,∂j~,e~α)-(∇~∂k~⁢𝕀t)⁢(∂i~,∂j~,(∇¯∂k~⁢e~α)T)

-(∇~∂k~⁢𝕀t)⁢((∇¯∂k~⁢∂i~)⊥,∂j~,e~α)

-(∇~∂k~⁢𝕀t)⁢(∂i~,(∇¯∂k~⁢∂j~)⊥,e~α)

=(∇∂k~⁡∇∂k~⁡𝕀t)⁢(∂i~,∂j~,e~α)-h~β⁢i⁢j⁢h~β⁢k⁢l⁢h~α⁢k⁢l

-h~β⁢k⁢l⁢h~α⁢l⁢j⁢h~β⁢k⁢i-h~β⁢k⁢j⁢h~β⁢k⁢l⁢h~α⁢i⁢l.

Use (B.6) to rewrite the above computation as

(B.8)(∇~*⁢∇~⁢𝕀t-∇*⁡∇⁡𝕀t)⁢(∂i~,∂j~,e~α)=h~β⁢i⁢j⁢h~β⁢k⁢l⁢h~α⁢k⁢l+h~β⁢k⁢l⁢h~α⁢l⁢j⁢h~β⁢k⁢i

+h~β⁢k⁢j⁢h~β⁢k⁢l⁢h~α⁢i⁢l.

The tensor ∇*⁡∇⁡𝕀t does not have other components. However, ∇~*⁢∇~⁢𝕀t does.

(B.9)(∇~*⁢∇~⁢𝕀t)⁢(∂i~,∂j~,∂l~)=-(∇~∂k~⁢∇~∂k~⁢𝕀t)⁢(∂i~,∂j~,∂l~)

=-∂k~⁡((∇~∂k~⁢𝕀t)⁢(∂i~,∂j~,∂l~))

+(∇~∂k~⁢𝕀t)⁢(∇¯∂k~⁢∂i~,∂j~,∂l~)

+(∇~∂k~⁢𝕀t)⁢(∂i~,∇¯∂k~⁢∂j~,∂l~)

+(∇~∂k~⁢𝕀t)⁢(∂i~,∂j~,∇¯∂k~⁢∂l~)

=∂k~⁡(h~α⁢i⁢j⁢h~α⁢k⁢l)+h~α⁢i⁢j,k⁢h~α⁢k⁢l

=2⁢h~α⁢i⁢j,k⁢h~α⁢k⁢l+h~α⁢i⁢j⁢h~α⁢k⁢l,k

=2⁢h~α⁢i⁢j,k⁢h~α⁢k⁢l+h~α⁢i⁢j⁢h~α,l+h~α⁢i⁢j⁢Rα~⁢k~⁢k~⁢l~.

The second last equality uses the fact that

0=〈∇∂k~⊥⁡e~α,e~β〉⁢h~β⁢i⁢j⁢h~α⁢k⁢l-〈∇∂k~⊥⁡e~α,e~β〉⁢h~α⁢i⁢j⁢h~β⁢k⁢l.

The last equality uses the Codazzi equation (2.1). Similarly,

(B.10)(∇~*⁢∇~⁢𝕀t)⁢(e~β,∂j~,e~α)=-2⁢h~α⁢l⁢j,k⁢h~β⁢k⁢l-h~β⁢k⁢l,k⁢h~α⁢j⁢l

=-2⁢h~α⁢l⁢j,k⁢h~β⁢k⁢l-h~β,l⁢h~α⁢j⁢l-Rβ~⁢k~⁢k~⁢l~⁢h~α⁢j⁢l,

(∇~*⁢∇~⁢𝕀t)⁢(∂i~,e~β,e~α)=-2⁢h~α⁢i⁢l,k⁢h~β⁢k⁢l-h~β⁢k⁢l,k⁢h~α⁢i⁢l

=-2⁢h~α⁢i⁢l,k⁢h~β⁢k⁢l-h~β,l⁢h~α⁢i⁢l-Rβ~⁢k~⁢k~⁢l~⁢h~α⁢i⁢l,

(∇~*⁢∇~⁢𝕀t)⁢(∂i~,e~α,∂j~)=-2⁢h~α⁢k⁢l⁢h~β⁢i⁢l⁢h~β⁢k⁢j,

(∇~*⁢∇~⁢𝕀t)⁢(e~β,e~γ,e~α)=-2⁢h~β⁢k⁢l⁢h~γ⁢k⁢j⁢h~α⁢l⁢j,

(∇~*⁢∇~⁢𝕀t)⁢(e~β,∂j~,∂i~)=2⁢h~β⁢k⁢l⁢h~α⁢l⁢j⁢h~α⁢k⁢i,

(∇~*⁢∇~⁢𝕀t)⁢(e~β,e~α,∂j~)=0.

The evolution equation for h~α⁢i⁢j was derived in [25, Proposition 7.1]. With equations (B.5) and (B.3), we have

dd⁢t⁢〈𝕀t,Ψ〉=dd⁢t⁢(g~i⁢k⁢g~j⁢l⁢h~α⁢k⁢l⁢Ψ~α⁢i⁢j)

=2⁢h~β⁢i⁢k⁢h~β⁢h~α⁢k⁢j⁢Ψ~α⁢i⁢j+2⁢h~β⁢j⁢l⁢h~β⁢h~α⁢i⁢l⁢Ψ~α⁢i⁢j+(dd⁢t⁢h~α⁢i⁢j)⁢Ψ~α⁢i⁢j+h~α⁢i⁢j⁢(dd⁢t⁢Ψ~α⁢i⁢j)

=2⁢h~β⁢i⁢k⁢h~β⁢h~α⁢k⁢j⁢Ψ~α⁢i⁢j+2⁢h~β⁢j⁢l⁢h~β⁢h~α⁢i⁢l⁢Ψ~α⁢i⁢j-〈∇*⁡∇⁡𝕀t,Ψ〉+(∇¯e~k⁢R)α~⁢i~⁢j~⁢k~⁢Ψ~α⁢i⁢j

+(∇¯e~j⁢R)α~⁢k~⁢i~⁢k~⁢Ψ~α⁢i⁢j-2⁢Rl~⁢i~⁢j~⁢k~⁢h~α⁢l⁢k⁢Ψ~α⁢i⁢j+2⁢Rα~⁢β~⁢j~⁢k~⁢h~β⁢i⁢k⁢Ψ~α⁢i⁢j

+2⁢Rα~⁢β~⁢i~⁢k~⁢h~β⁢j⁢k⁢Ψ~α⁢i⁢j-Rl~⁢k~⁢i~⁢k~⁢h~α⁢l⁢j⁢Ψ~α⁢i⁢j-Rl~⁢k~⁢j~⁢k~⁢h~α⁢l⁢i⁢Ψ~α⁢i⁢j

+Rα~⁢k~⁢β~⁢k~⁢h~β⁢i⁢j⁢Ψ~α⁢i⁢j-h~α⁢i⁢l⁢(h~β⁢l⁢j⁢h~β-h~β⁢l⁢k⁢h~β⁢j⁢k)⁢Ψ~α⁢i⁢j

-h~α⁢l⁢k⁢(h~β⁢l⁢j⁢h~β⁢i⁢k-h~β⁢l⁢k⁢h~β⁢i⁢j)⁢Ψ~α⁢i⁢j-h~β⁢i⁢k⁢(h~β⁢l⁢j⁢h~α⁢l⁢k-h~β⁢l⁢k⁢h~α⁢l⁢j)⁢Ψ~α⁢i⁢j

-h~α⁢j⁢k⁢h~β⁢i⁢k⁢h~β⁢Ψ~α⁢i⁢j+h~β⁢i⁢j⁢〈e~β,∇¯H⁢e~α〉⁢Ψ~α⁢i⁢j-〈𝕀t,∇~*⁢∇~⁢Ψ〉

+〈𝕀t,trΓt⁡(∇¯*⁢∇¯⁢Ψ)〉-h~α⁢i⁢j⁢h~α,k⁢Ψ~k⁢i⁢j+h~α⁢i⁢j⁢〈∇¯H⁢e~α,e~β〉⁢Ψ~β⁢i⁢j

-h~α⁢i⁢j⁢h~γ⁢h~γ⁢i⁢k⁢Ψ~α⁢k⁢j+h~α⁢i⁢j⁢h~γ,i⁢Ψ~α⁢γ⁢j-h~α⁢i⁢j⁢h~γ⁢h~γ⁢j⁢k⁢Ψ~α⁢i⁢k+h~α⁢i⁢j⁢h~γ,j⁢Ψ~α⁢i⁢γ

=-〈∇~*⁢∇~⁢𝕀t,Ψ〉-〈𝕀t,∇~*⁢∇~⁢Ψ〉+〈𝕀t,trΓt⁡(∇¯2⁢Ψ)〉

+(∇¯e~k⁢R)α~⁢i~⁢j~⁢k~⁢Ψ~α⁢i⁢j+(∇¯e~j⁢R)α~⁢k~⁢i~⁢k~⁢Ψ~α⁢i⁢j-2⁢Rl~⁢i~⁢j~⁢k~⁢h~α⁢l⁢k⁢Ψ~α⁢i⁢j

+2⁢Rα~⁢β~⁢j~⁢k~⁢h~β⁢i⁢k⁢Ψ~α⁢i⁢j+2⁢Rα~⁢β~⁢i~⁢k~⁢h~β⁢j⁢k⁢Ψ~α⁢i⁢j-Rl~⁢k~⁢i~⁢k~⁢h~α⁢l⁢j⁢Ψ~α⁢i⁢j

-Rl~⁢k~⁢j~⁢k~⁢h~α⁢l⁢i⁢Ψ~α⁢i⁢j+Rα~⁢k~⁢β~⁢k~⁢h~β⁢i⁢j⁢Ψ~α⁢i⁢j+Rα~⁢k~⁢k~⁢l~⁢h~α⁢i⁢j⁢Ψ~l⁢i⁢j-Rβ~⁢k~⁢k~⁢l~⁢h~α⁢j⁢k⁢Ψ~α⁢β⁢j

-Rβ~⁢k~⁢k~⁢l~⁢h~α⁢i⁢l⁢Ψ~α⁢i⁢β-2⁢h~α⁢i⁢l⁢(h~β⁢l⁢j⁢h~β-h~β⁢l⁢k⁢h~β⁢j⁢k)⁢Ψ~α⁢i⁢j

-2⁢h~β⁢i⁢k⁢(h~β⁢l⁢j⁢h~α⁢l⁢k-h~β⁢l⁢k⁢h~α⁢l⁢j)⁢Ψ~α⁢i⁢j-2⁢h~α⁢j⁢k⁢h~β⁢i⁢k⁢h~β⁢Ψ~α⁢i⁢j

-2⁢h~α⁢k⁢l⁢h~β⁢i⁢l⁢h~β⁢k⁢j⁢Ψ~j⁢i⁢α-2⁢h~β⁢k⁢l⁢h~γ⁢k⁢j⁢h~α⁢l⁢j⁢Ψ~α⁢β⁢γ+2⁢h~β⁢k⁢l⁢h~α⁢l⁢j⁢h~α⁢k⁢i⁢Ψ~i⁢β⁢j

+2⁢h~α⁢i⁢j,k⁢h~α⁢k⁢l⁢Ψ~l⁢i⁢j-2⁢h~α⁢l⁢j,k⁢h~β⁢k⁢l⁢Ψ~α⁢β⁢j-h~α⁢i⁢l,k⁢h~β⁢k⁢l⁢Ψ~α⁢i⁢β.

The last equality uses (B.8), (B.9) and (B.10) to replace ∇*⁡∇⁡𝕀t by ∇~*⁢∇~⁢𝕀t.

By the Cauchy–Schwarz inequality,

|dd⁢t⁢〈𝕀t,Ψ〉-ΔΓt⁢〈𝕀t,Ψ〉+2⁢〈∇~⁢𝕀t,∇~⁢Ψ〉|≤12⁢|∇⁡𝕀t|2+c⁢(|𝕀t|4+|𝕀t|2+1).

This together with (6.6) and (B.4) imply that

dd⁢t⁢|𝕀t-Ψ|2≤ΔΓt⁢|𝕀t-Ψ|2-2⁢|∇~⁢(𝕀t-Ψ)|2+|∇⁡𝕀t|2+c′⁢(|𝕀t|4+|𝕀t|2+1).

According to (B.7),

|∇~⁢(𝕀t-Ψ)|2≥|∇~⁢𝕀2|-|∇~⁢Ψ|2≥|∇⁡𝕀t|2+c′′⁢|𝕀t|4-|∇¯⁢Ψ|2.

Hence,

dd⁢t⁢|𝕀t-Ψ|2≤ΔΓt⁢|𝕀t-Ψ|2-|∇~⁢(𝕀t-Ψ)|2+c′′′⁢(|𝕀t|4+|𝕀t|2+1)

By the triangle inequality |𝕀t|2≤|𝕀t-Ψ|2+|Ψ|2, it finishes the proof of the proposition. ∎

C Moser iteration for 𝒞2 convergence

The main purpose of this appendix is to prove the 𝒞2 convergence part of Theorem 6.2, in particular |𝕀t-𝕀Σ|2→0. We already show that:

The mean curvature flow {Γt} exists for all time, and the second fundamental form 𝕀t is uniformly bounded.
Γt converges to Σ in 𝒞1.
The L2 convergence, (6.10), of 𝕀t: limt→0⁡∫Γt|𝕀t-𝕀Σ|2⁢dμt=0.

When 𝕀t is uniformly bounded, it is known that all the higher order derivatives of 𝕀t remains uniformly bounded. This can be proved by using the evolution equation of ∇(k)⁡𝕀t. See [2, Proposition 4.8].

With the 𝒞1 convergence, for t large enough Γt can be written as the graph of a section of the normal bundle, as in Section 6.2.1. Taking second order derivatives (in space) gives the evolution equation of 𝕀t-𝕀Σ in the non-parametric form. The strategy is to apply the Moser iteration argument [16, 23] to estimate its sup-norm in terms of the L2 norm. It together with (6.10) would lead to the 𝒞2 convergence.

The argument will be done over open balls of Σ, and will be demonstrated on the ball of radius 1.

C.1 Second fundamental form and second order derivative

Denote by Γ*** the Christoffel symbols of the ambient metric (6.12). It follows from (6.12) that there are constants ε and C which have the following significances.

For any section 𝐲⁢(𝐱) with |𝐲|𝒞0≤ε,
|Γi⁢jk-Γ¯i⁢jk|+|Γi⁢μk+g¯k⁢j⁢hμ⁢i⁢j|+|Γμ⁢νk|+|Γi⁢jγ-hγ⁢i⁢j|
+|Γi⁢μγ-Aμ⁢iγ|+|Γμ⁢νγ|≤C⁢|𝐲|𝒞0.
Here, we denote the induced metric on Σ by g¯i⁢j and its Christoffel symbols by Γ¯i⁢jk to avoid confusion. Underlined geometric quantities depend on 𝐱 only.
For any section 𝐲⁢(𝐱) with |𝐲|𝒞1≤ε, the orthogonal projection of the ambient coordinate vector field ∂∂⁡yμ to the normal of the section is surjective. Moreover,
|(∂∂⁡xi)⊥|+|(∂∂⁡yμ)⊥-∂∂⁡yμ|≤C⁢|𝐲|𝒞1.

Assume that 𝐲=𝐲⁢(𝐱) has small 𝒞1 norm. Denote its graph, {(𝐱,𝐲⁢(𝐱))}, by Γ. The tangent space of Γ is spanned by

F*⁢(∂∂⁡xi)=∂∂⁡xi+∂i⁡yμ⁢∂∂⁡yμ.

We compute

∇F*⁢(∂∂⁡xi)⁡F*⁢(∂∂⁡xj)=(∂i⁡∂j⁡yμ)⁢∂∂⁡yμ+Γi⁢jk⁢∂∂⁡xk+Γi⁢jμ⁢∂∂⁡yμ

+terms with coefficients ⁢(∂k⁡y),

and thus

|(∇F*⁢(∂∂⁡xi)⁡F*⁢(∂∂⁡xj))⊥-(∂i⁡∂j⁡yμ+hμ⁢i⁢j)⁢∂∂⁡yμ|≤C⁢|𝐲|𝒞1.

It follows that

|𝕀Γ-𝕀Σ-∂2⁡𝐲|≤C⁢|𝐲|𝒞1.

Now, suppose that Γt={𝐲=𝐲(𝐱,t)} is a mean curvature flow which converges to 0 uniformly in 𝒞1. To prove that 𝕀Γt-𝕀Σ converges to zero uniformly, it suffices to show that ∂2⁡𝐲 converges to zero uniformly.

C.2 The mean curvature flow equation

As in Section 6.2.1, we introduce the dummy variable plμ=∂k⁡yμ. In the following discussion, a function ℱ⁢(𝐱,𝐲,𝐩) is said to be 𝒪⁢(k) for some k∈ℕ∪{0} if there exist constants Cℓ and Cℓ,ℓ′ such that

(C.1)|δ𝐱ℓ⁢ℱ|≤Cℓ⁢(|𝐲|2+|𝐩|2)k2,

|δ𝐲ℓ⁢δ𝐩ℓ′⁢ℱ|≤Cℓ,ℓ′⁢(|𝐲|2+|𝐩|2)k-ℓ-ℓ′2 for any ⁢ℓ+ℓ′≤k

(for any 𝐱 in the region of consideration). The derivative in the variables (𝐱,𝐲,𝐩) is denoted by δ to avoid confusion.

Recall (6.15):

(C.2)g=g¯(1+12(g¯i⁢j(piμ+Aν⁢iμyν)(pjμ+Aγ⁢jμyγ)

-g¯i⁢jyμyν(Rj⁢μ⁢i⁢ν+g¯k⁢ℓhμ⁢i⁢khν⁢j⁢ℓ)))+ℰ(𝐱,𝐲,𝐩),

where ℰ⁢(𝐱,𝐲,𝐩) is 𝒪⁢(3) in the sense of (C.1).

With (C.2), the mean curvature flow equation (6.16) takes the following form:

(C.3)∂⁡yμ∂⁡t=(δμ⁢ν+𝒪⁢(2))⁢(dd⁢xi⁢(g¯i⁢j⁢(∂⁡yμ∂⁡xj+Aν⁢jμ⁢yν)+𝒪⁢(2))-𝒪⁢(1))

=g¯i⁢j⁢(𝐱)⋅∂2⁡yμ∂⁡xi⁢∂⁡xj+ℰνμ⁢i⁢j⁢(𝐱,𝐲,𝐩)⋅∂2⁡yν∂⁡xi⁢∂⁡xj+ℰμ⁢(𝐱,𝐲,𝐩),

where both ℰνμ⁢i⁢j and ℰμ are of 𝒪⁢(1).

C.3 The evolution equation for the second order derivatives

Denote ∂k⁡∂ℓ⁡yμ by qk⁢ℓμ. We are going to derive the evolution equation for qk⁢ℓμ. Remember that the 𝒞1 norm of 𝐲 converges to 0 as t→∞, and the 𝒞2 norm of 𝐲 is uniformly bounded.

For the first term on the right-hand side of (C.3),

∂2∂⁡xk⁢∂⁡xℓ⁢(g¯i⁢j⁢∂2⁡yμ∂⁡xi⁢∂⁡xj)=∂∂⁡xi⁢(g¯i⁢j⁢∂⁡qk⁢ℓμ∂⁡xj)+[∂⁡g¯i⁢j∂⁡xk⁢∂⁡qi⁢jμ∂⁡xℓ+∂⁡g¯i⁢j∂⁡xℓ⁢∂⁡qi⁢jμ∂⁡xk-∂⁡g¯i⁢j∂⁡xi⁢∂⁡qk⁢ℓμ∂⁡xj]

+∂2⁡g¯i⁢j∂⁡xk⁢∂⁡xℓ⁢qi⁢jμ

=∂∂⁡xi⁢(g¯i⁢j⁢∂⁡qk⁢ℓμ∂⁡xj)+𝒪⁢(0)⋅∂⁡𝐪+𝒪⁢(0)⋅𝐪.

For the second term on the right-hand side of (C.3), one performs the commuting derivatives to find that

∂2∂⁡xk⁢∂⁡xℓ⁢(ℰνμ⁢i⁢j⁢(𝐱,𝐲,𝐩)⋅∂2⁡yν∂⁡xi⁢∂⁡xj)=∂∂⁡xi⁢(ℰνμ⁢i⁢j⁢(𝐱,𝐲,𝐩)⁢∂⁡qk⁢ℓν∂⁡xj)

+ℰ~1⁢(𝐱,𝐲,𝐩,𝐪)⋅∂⁡𝐪+ℰ~0⁢(𝐱,𝐲,𝐩,𝐪)⋅𝐪

for some smooth function ℰ~1 and ℰ~0 in 𝐱,𝐲,𝐩=∂⁡𝐲,𝐪=∂2⁡𝐲. Since the 𝒞2 norm of 𝐲 is uniformly bounded, the coefficient functions ℰ~1 and ℰ~0 are uniformly bounded. For the last term on the right-hand side of (C.3),

∂2∂⁡xk⁢∂⁡xℓ⁢ℰμ=[δ2⁢ℰμδ⁢xk⁢δ⁢xℓ+δ2⁢ℰμδ⁢xℓ⁢δ⁢yν⁢∂⁡yν∂⁡xk+δ2⁢ℰμδ⁢xk⁢δ⁢yν⁢∂⁡yν∂⁡xℓ+δ⁢ℰμδ⁢yν⁢δ⁢yγ⁢∂⁡yγ∂⁡xk⁢∂⁡yν∂⁡xℓ]

+ℰ^1⁢(𝐱,𝐲,𝐩)⋅∂⁡𝐪+ℰ^0⁢(𝐱,𝐲,𝐩,𝐪)⋅𝐪.

Since the 𝒞1 norm of 𝐲 converges to zero uniformly and ℰμ=𝒪⁢(1), the first line on the right-hand side converges to zero uniformly as t→∞. Similar to above, ℰ^1 and ℰ^0 are uniformly bounded.

To sum up, write qk⁢ℓμ as uA. Its evolution equation takes the following form:

(C.4)∂⁡uA∂⁡t=∂∂⁡xi⁢(g¯i⁢j⁢∂⁡uA∂⁡xj+𝒫B⁢iA⁢j⁢∂⁡uB∂⁡xj)+𝒮BA⁢j⁢∂⁡uB∂⁡xj+𝒯BA⁢uB+ℛA,

where 𝒮,𝒯 are uniformly bounded, and 𝒫,ℛ converges to 0 uniformly as t→∞. The solution and the coefficient functions are all smooth. As mentioned in beginning of this appendix, ∇Γt⁡𝕀t is uniformly bounded, and thus ∂j⁡uA is uniformly bounded.

C.4 Moser iteration

We apply the Moser iteration to estimate the sup-norm of uA in terms of its L2 norm. The formulation here is modified from the argument of Trudinger [23].

Let B the open ball of radius 1, and let DT=B×[T-1,T]. Denote by Bt be the slice B×{t} for any t∈[T-1,T]. Introduce the V2⁢(DT) norm [23, (1.5)]:

∥f∥V2⁢(DT)=supt∈[T-1,T]⁡∥f∥L2⁢(Bt)+∥f∥L2⁢(DT).

The following Sobolev lemma is a fundamental tool for the iteration.

Lemma C.1 ([23, Lemma 1.1]).

There exist constants κ>1 and C>0 (which are independent of T) such that for any f with finite V2⁢(DT) norm and with compact support in Bt (a.e. t), f must have finite L2⁢κ⁢(DT) norm. Moreover,

(C.5)∥f∥L2⁢κ⁢(DT)≤C⁢∥f∥V2⁢(DT).

We proceed with the Moser iteration argument for solutions uA of (C.4). There exists c0>0 such that

∑i,jg¯i⁢j⁢vi⁢vj≥c0⁢|v|2

at any x∈B and for any vector {vi}∈ℝn. For any δ>0, consider

f=(∑A(uA)2+supDT⁡∑A|ℛA|2+supDT⁡∑i,j,A,B|𝒫B⁢iA⁢j|+δ)12.

For any β≥1, the partial derivative (in space) of its 12⁢(β+1) power is

∂i⁡fβ+12=β+12⁢fβ-32⁢∑A(uA⁢∂i⁡uA).

By the Cauchy–Schwarz inequality,

|∂i⁡fβ+12|≤β+12⁢fβ-32⁢(∑A(uA)2)12⁢(∑A(∂i⁡uA)2)12.

It follows that

|∇⁡fβ+12|2=(β+1)24⁢fβ-3⁢|12⁢∇⁡f2|2≤(β+1)24⁢fβ-1⁢∑A,i(∂i⁡uA)2.

Let B⁢(ρ) the open ball of radius ρ, and denote B⁢(ρ)×[T-ρ2,T] by Rρ. Note that R1=DT, and Rρ′⊂Rρ for any ρ′<ρ≤1. Fix ρ and ρ′ with 12≤ρ′<ρ≤1. Let η be a cut-off function which is 1 on B⁢(ρ′)×[T-(ρ′)2,∞), and vanishes outside B⁢(ρ)×[T-ρ2,∞). We compute

∬η2⁢∂∂⁡t⁢fβ+1=(β+1)⁢∬η2⁢fβ-1⁢(uA⁢∂t⁡uA)

=(β+1)∬η2fβ-1uA[∂i(g¯i⁢j∂juA+𝒫B⁢iA⁢j∂juB)

+𝒮BA⁢j∂juB+𝒯BAuB+ℛA].

We estimate each term on the right-hand side of the last expression. For the first term, we consider

-∬η2⁢fβ-1⁢uA⁢∂i⁡(g¯i⁢j⁢∂j⁡uA)

=∬∂i⁡(η2⁢fβ-1⁢uA)⁡g¯i⁢j⁢∂j⁡uA

=∬η2⁢[fβ-1⁢g¯i⁢j⁢∂i⁡uA⁢∂j⁡uA+(β-1)⁢fβ-3⁢g¯i⁢j⁢∂i⁡(f22)⁢∂j⁡(f22)]

+2⁢η⁢∂i⁡η⁢g¯i⁢j⁢fβ-1⁢uA⁢∂j⁡uA

≥1c1⁢∬η2⁢[1β+1⁢|∇⁡fβ+12|2+fβ-1⁢∑A|∇⁡uA|2]-c1⁢∬|∇⁡η|2⁢fβ+1.

For the second term on the right-hand side,

∬η2⁢fβ-1⁢uA⁢∂i⁡(𝒫B⁢iA⁢j⁢∂j⁡uB)

=-∬∂i⁡(η2⁢fβ-1⁢uA)⁡𝒫B⁢iA⁢j⁢∂j⁡uB

=-∬η2⁢[(β-1)⁢fβ-3⁢uC⁢∂i⁡uC⁢uA⁢𝒫B⁢iA⁢j⁢∂j⁡uB+fβ-1⁢∂i⁡uA⁢𝒫B⁢iA⁢j⁢∂j⁡uB]

≤c2⁢(β+1)⁢∬η2⁢fβ+1,

where we have use the uniform boundedness of ∂j⁡uA. For the rest terms on the right-hand side,

∬η2⁢fβ-1⁢uA⁢[𝒮BA⁢j⁢∂j⁡uB+𝒯BA⁢uB+ℛA]

≤1c1⁢∬η2⁢fβ-1⁢∑A|∇⁡uA|2+c3⁢∬η2⁢fβ+1.

Putting these computations together gives

(C.6)∬∂∂⁡t⁢(η2⁢fβ+1)+∬|∇⁡(η⁢fβ+12)|2

≤c4⁢(β+1)2⁢∬(η2+|∇⁡η|2+η⁢∂t⁡η)⁢fβ+1.

By definition, there exists t′∈(T-1,T) such that

∫Bt′η2⁢fβ+1>12⁢supt∈[T-1,T]⁡∫Btη2⁢fβ+1.

By considering (C.6) on B×[T-1,t′],

supt∈[T-1,T]⁡∫Btη2⁢fβ+1≤2⁢c4⁢(β+1)2⁢∬Rρ(η2+|∇⁡η|2+η⁢∂t⁡η)⁢fβ+1.

Combining it with (C.6) for Rρ gives

supt∈[T-1,T]⁡∫Btη2⁢fβ+1+∬DT|∇⁡(η⁢fβ+12)|2

≤3⁢c4⁢(β+1)2⁢∬DT(η2+|∇⁡η|2+η⁢∂t⁡η)⁢fβ+1.

Due to (C.5) and the choice of η, we find that

∥fβ+12∥L2⁢κ⁢(Rρ′)≤c5⁢β+1ρ-ρ′⁢∥fβ+12∥L2⁢(Rρ)

for some κ>1. Equivalently,

∥f∥Lκ⁢(β+1)⁢(Rρ′)≤(c5⁢β+1ρ-ρ′)2β+1⁢∥f∥Lβ+1⁢(Rρ).

Let ρn=12+2-(n+1) for n∈{0,1,2,…}. Denote by Φ⁢(n) the L2⁢κn norm of f on Rρn). It follows that

Φ⁢(n+1)≤(c5⁢ 2n+2⁢κn)1κn⁢Φ⁢(n)≤⋯≤(4⁢c5)∑j=0n1κj⁢(2⁢κ)∑j=0njκj⁢Φ⁢(0).

It follows that the sup-norm of f on B⁢(12)×[T-14,T] is bounded by some multiple of its L2 norm on B⁢(1)×[T-1,T]. Since δ>0 is arbitrary, the L2 norm of uA, ℛA and 𝒫B⁢iA⁢j are uniformly small, this implies that uA converges to zero uniformly.

Acknowledgements

The authors are grateful to Professor Gerhard Huisken for his comments on the stability of the mean curvature flow and for pointing out the reference [7]. The authors would like to thank Yohsuke Imagi for helpful discussions, and to thank the anonymous referee for helpful comments on the earlier version of this paper.

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Received: 2017-10-06

Revised: 2018-09-25

Published Online: 2019-01-22

Published in Print: 2020-07-01

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https://doi.org/10.1515/crelle-2018-0038