1710.00433V1 [Math.DG] 1 Oct 2017 Fti Okwscridotwe Utown a Iiigthe Visiting Was Wang Taiwan

Home , Tubular neighborhood

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n (i) M is the total space of the cotangent bundle of a sphere, T ∗S (for n > 1), with the Stenzel metric [17], and Σ is the zero section; n (ii) M is the total space of the cotangent bundle of a complex projective space, T ∗CP , with the Calabi metric [3] and Σ is the zero section; (iii) M is the total space of one of the vector bundles S(S3), Λ2 (S4), Λ2 (CP2), and S (S4) − − − with the Ricci ﬂat metric constructed by Bryant–Salamon [2], where S is the spinor bundle and S is the spinor bundle of negative chirality, and Σ is the zero section of − the respective vector bundle.

These are essentially all metrics of special holonomy that are known to be written in a closed form. Note that in all these examples, the metrics of the total space are Ricci ﬂat, and the zero sections are totally geodesic. Hence, the strongly stability in these examples is equivalent to the positivity of the operator . In [18], we proved uniqueness and dynamical stability theorems R for the corresponding calibrated submanifolds and the proofs rely on the explicit knowledge of the ambient metric, whose coeﬃcients are governed by solutions of ODE systems. A natural question was how general such rigidity phenomenon is. In this article, we discover that the strong stability condition is precisely the condition that makes everything work. Moreover, we identify more examples that satisfy the strong stability condition:

Proposition A. Each of the following pairs (Σ, M) of minimal submanifolds Σ and their ambient Riemannian manifolds M satisfy the strong stability condition (3.2) :

(i) M is any Riemannian manifold of negative sectional curvature and Σ a totally geodesic submanifold; (ii) M is any Kähler manifold and Σ is a complex submanifold whose normal bundle has positive holomorphic curvature. (iii) M is any Calabi–Yau manifold and Σ is a special Lagrangian with positive Ricci curvature; 2 (iv) M is any G2 manifold and Σ is a coassociative submanifold with positive definite s 2 2W + 3 on Λ . − − − For example (i), the strong stability can be checked directly. The examples (ii), (iii), and (iv) will be explained in 3.2 and Appendix A. § We now state the main results of this paper. The first one says that a strongly stable minimal submanifold is rather unique.

Theorem A. Let Σn (M, g) be a compact, oriented minimal submanifold which is strongly ⊂ stable in the sense of (3.2). Then there exists a tubular neighborhood U of Σ such that Σ is the only compact minimal submanifold in U with dimension no less than n.

The second one is on the dynamical stability of a strongly stable minimal submanifold.

Theorem B. Let Σ (M, g) be a compact, oriented minimal submanifold which is strongly ⊂ stable in the sense of (3.2). If Γ is a submanifold that is close to Σ in 1, the mean curvature C flow Γt with Γ =Γ exists for all time, and Γt converges to Σ smoothly as t . 0 →∞ The precise statements can be found in Theorem 4.2 (Theorem A) and Theorem 6.2 (Theorem B), respectively. The 1 dynamical stability of the mean curvature flow for those calibrated C submanifolds considered in [18] was proved in the same paper. In this regard, this theorem is a generalization of our previous result. Here are some remarks on the strong stability condition. In the viewpoint of the second variational formula, the condition is natural, and is stronger than the positivity of the Jacobi operator. The main results of this paper are basically saying that the strong stability has nice geometric consequences. In particular, the minimal submanifold Σ needs not be a totally geodesic, while most known results about the convergence of higher codimensional mean curvature flow are under the totally geodesic assumption, e.g. [22].

Acknowledgement. The authors would like to thank Yohsuke Imagi for helpful discussions.

2. Local geometry near a submanifold

2.1. Notations and basic properties. Let (M, g) be a Riemannian manifold of dimension n + m, and Σ M be a compact, embedded, and oriented submanifold of dimension n. We use ⊂ , to denote the evaluation of two tangent vectors by the metric tensor g. The notation , h· ·i h· ·i is also abused to denote the evaluation with respect to the induced metric on Σ. Denote by the Levi-Civita connection of (M, g), and by Σ the Levi-Civita connection of the induced ∇ ∇ metric on Σ. 3 Denote by NΣ the normal bundle of Σ in M. The metric g and its Levi-Civita connection induce a bundle metric (also denoted by , ) and a metric connection for NΣ. The bundle h· ·i connection on NΣ will be denoted by . ∇⊥ In the following discussion, we are going to choose a local orthonormal frame e , , en, { 1 · · · en , , en m for T M near a point p Σ such that the restriction of e , , en on Σ is an +1 · · · + } ∈ { 1 · · · } oriented frame for T Σ and the restrictions of en , , en m is a frame for NΣ. The indexes { +1 · · · + } i, j, k range from 1 to n, the indexes α, β, γ range from n + 1 to n + m, the indexes A,B,C range from 1 to n + m, and repeated indexes are summed. The convention of the Riemann curvature tensor is

R(eC , eD)eB = e e eB e e eB eB , ∇ C ∇ D −∇ D ∇ C −∇[eC ,eD] RABCD = R(eA, eB , eC , eD)= R(eC , eD)eB, eA . h i What follows are some basic properties of the geometry of a submanifold. The details can be found in, for example [6, ch. 6].

(i) Σ is the projection of onto T Σ T M , and is the projection of onto ∇ ∇ ⊂ |Σ ∇⊥ ∇ NΣ T M . Their curvatures are denoted by ⊂ |Σ Σ Σ Σ Σ Σ Σ R = el el el, ek , klij h∇ei ∇ej −∇ej ∇ei −∇[ei,ej ] i R⊥ = ⊥ ⊥ eβ ⊥ ⊥eβ ⊥ eβ, eα , αβij h∇ei ∇ej −∇ej ∇ei −∇[ei,ej ] i (ii) Given any two tangent vectors X,Y of Σ, the second fundamental form of Σ in M is

defined by II(X,Y ) = ( X Y ) , where ( ) : T M NΣ is the projection onto the ∇ ⊥ · ⊥ → normal bundle. The mean curvature of Σ is the normal vector field defined by H =

tr II. With a normal vector V , II(X,Y,V ) is deﬁned to be II(X,Y ),V = X Y,V . Σ h i h∇ i In terms of the frame,

hαij = II(ei, ej, eα) and H = hαii eα .

(iii) For any tangent vectors X,Y,Z of Σ and a normal vector V , the Codazzi equation says that

R(X,Y )Z,V = ( X II)(Y,Z,V ) ( Y II)(X,Z,V ) (2.1) h i ∇ − ∇ where

Σ Σ ( X II)(Y,Z,V )= X (II(Y,Z,V )) II( Y,Z,V ) II(Y, Z,V ) II(X,Y, ⊥ V ) . (2.2) ∇ − ∇X − ∇X − ∇X

In terms of the frame, denote ( e II)(ej, ek, eα) by hαjk i, and (2.1) is equivalent to that ∇ i ; Rαkij = hαjk;i hαik;j. − 4 2.2. Geodesic coordinate and geodesic frame. For any p Σ, we can construct a “partial” ∈ geodesic coordinate and a geodesic frame on a neighborhood of p in M as follows:

(i) Choose an oriented, orthonormal basis e , , en for TpΣ. The map { 1 · · · } 1 n Σ j F : x = (x , ,x ) exp (x ej) 0 · · · 7→ p parametrizes an open neighborhood of p in Σ, where expΣ is the exponential map of

the induced metric on Σ. For any x of unit length, the curve γ(t)= F0(tx) is called a Σ radial geodesic on Σ (at p). By using to parallel transport e , , en along these ∇ { 1 · · · } radial geodesics, we get a local orthonormal frame for T Σ on a neighborhood of p in

Σ. The frame is still denoted by e , , en . { 1 · · · } (ii) Choose an orthonormal basis en , , en m for NpΣ. By using to parallel trans- { +1 · · · + } ∇⊥ port en , , en m along radial geodesics on Σ, we obtain a local orthonormal frame { +1 · · · + } for NΣ on a neighborhood of p in Σ. This frame is still denoted by en , , en m . { +1 · · · + } It is clear that e , , en, en , , en m is a local orthonormal frame for T M . { 1 · · · +1 · · · + } |Σ (iii) The map

1 n n+1 n+m α F : (x, y)= (x , ,x ), (y , ,y ) exp (y eα) · · · · · · 7→ F0(x) parametrizes an open neighborhood of p in M. The map exp is the exponential map

of (M, g). For any y of unit length, the curve σ(t)= F (x,ty) = expF0(x)(ty) is called a normal geodesic for Σ M. ⊂ (iv) For any x, step (ii) gives an orthonormal basis e , , en m for T M. By using { 1 · · · + } F (x,0) to parallel transport it along normal geodesics, we have an orthonormal frame for ∇ T M on a neighborhood of p in M. This frame is again denoted by e , , en m . { 1 · · · + } The freedom in the above construction is the choice of e , , en and en , , en m at p, { 1 · · · } { +1 · · · + } which is SO(n) O(m). A particular choice will be made later on. × Σ i β Remark 2.1. We will consider the curves s exp (x ei +sej) and s exp (y eβ +seα) in 7→ p 7→ F0(x) the following discussion. They will be abbreviated as F0(x+sej) and F (x, y+seα), respectively.

Remark 2.2. The frames e , , en, en , , en m are constructed by parallel transport { 1 · · · +1 · · · + } along radial geodesics on Σ and then normal geodesic for Σ. They are indeed smooth. We briefly ∂ explain the smoothness of e1, , en on a neighborhood of p in Σ. Write ei = Sij(x) ∂xj . The { · · · } l smoothness of the frame is equivalent to the smoothness of Sij(x). Let Γjk(x) be the Christoffel Σ Σ ∂ l ∂ l symbols of , i.e. ∂ ∂xk =Γjk(x) ∂xl . The Christoffel symbols Γjk(x) are smooth functions. ∇ ∇ ∂xj Since ei is parallel along radial geodesics, ∂S (x) ∂ Σ l ij l x j x xl ∂ ei =0= x l + x Sik( )Γik( ) j . ∇ ∂xl ∂x ∂x 5 To avoid confusion, fix ξ = (ξ1, ,ξn) Rn. Let γ(t) = tξ for t [0, 1]. Since d f(γ(t)) = · · · ∈ ∈ dt 1 (xl ∂ f(x)) , t ∂xl |γ(t) dS (γ(t)) ij = ξl S (γ(t)) Γj (tξ) . dt ik ik dS In other words, [Sij(ξ)] is the solution to the ODE system dt = F (S,t,ξ) at t = 1 with identity as the initial condition. Therefore, Sij(ξ) is smooth in ξ.

2.2.1. The tubular neighborhood Uε and the distance function.

Deﬁnition 2.3. For any δ > 0, let Uδ be the image of V NΣ V < δ under the exponential { ∈ | | | } map along Σ. By the implicit function theorem, there exists ε> 0, which is determined by the

geometry of Σ and M, such that the following statements hold for Uε:

(1) The map exp : V NΣ V < 2ε U ε is a diﬀeomorphism. { ∈ | | | }→ 2 1 n n+1 n+m (2) There exist the local coordinate system (x , x ,y , y ) and the frame e , , en m · · · · · · { 1 · · · + } constructed in the last subsection. α 2 (3) The function α(y ) is a well-deﬁned smooth function on Uε. α 2 (4) On Uε, the squareP root of α(y ) is the distance function to Σ.

(5) For any q Uε, there exists a unique p Σ such that there is a unique normal geodesic ∈ P ∈ in Uε connecting p and q.

α 2 We now analyze the gradient of the function α(y ) . To avoid confusion, let

ξ = (ξ1, ,ξn) Rn and Pη = (ηn+1, , ηn+m) Rm · · · ∈ · · · ∈ be constant vectors. Consider the normal geodesic σ(t) = F (ξ,tη); its tangent vector field is α ∂ α α σ′(t)= η ∂yα . On the other hand, σ′(0) is also equal to η eα, and η eα is defined and parallel α ∂ α along σ(t). Thus, η ∂yα = η eα on σ(t). Since the y-coordinate of σ(t) is tη, we find that

α ∂ α ∂ α y = tη = tη e = tσ′(t) ; (2.3) ∂yα σ(t) ∂yα σ(t) α

at t = 1 it gives ∂ yα = yαe . (2.4) ∂yα α

α ∂ By modifying the standard geodesic argument [4, p.4–9], the vector field y ∂yα σ(t) is half of α 2 α |∂ α ∂ the gradient vector field of α(y ) . In addition, note that (2.4) implies that y ∂yα ,y ∂yα = α 2 α ∂ β ∂ α α h i (y ) . The Gauss lemma implies that y α ,s =0 if y s = 0. By considering α P h ∂y ∂yβ i α the first variational formula of the one-parameter family of geodesics σ(t,s) = exp (tη), P P F0(ξ+sej ) 6 α ∂ ∂ one finds that y α , = 0. It follows from these relations that h ∂y ∂xj i ∂ (yα)2 = 2yα . (2.5) ∂yα ∇ α ! X For a locally defined smooth function near p, the following lemma establishes its expansion in terms of the coordinate system constructed above.

Lemma 2.4. Let Uε be a neighborhood of p Σ in M as in Deﬁnition 2.3 with the coordinate 1 n n+1 n+m ∈ system (x, y) = (x , x ,y , y ) and the frame e , , en, en , en m . Then, · · · · · · { 1 · · · +1 · · · + } any smooth function f(x, y) on Uε has the following expansion:

i α 2 2 f(x, y)= f(0, 0)+ x ei(f) p + y eα(f) p + ( x + y ) . | | O | | | | i α 2 2 More precisely, it means that f(x, y) f(0, 0) x ei(f) p y eα(f) p c( x + y ) for 2 − − | − | ≤ | | | | some constant c determined by the -norm of f and the geometry of M and Σ. C

Proof. Let q Uε be any point. To avoid confusion, denote the coordinate of q by (ξ, η), ∈ where ξ Rn and η Rm are regarded as constant vectors. Let q Σ be the point with ∈ ∈ 0 ∈ normal coordinate (ξ, 0), and consider the radial geodesic on Σ joining q0 and p, σ0(t)= F0(tξ). Applying Taylor’s theorem on f(σ0(t)) gives d 1 d2 f(σ (t)) f(ξ, 0)= f(0, 0)+ f(σ (t)) + (1 t2) 0 . dt t=0 0 − dt2 Z0 i Since σ0′ (t)= ξ ei, we ﬁnd that 1 i i j f(ξ, 0)= f(0, 0)+ ξ ei(f) p + ξ ξ (1 t) (ej (ei(f)))(σ (t)) dt . (2.6) | − 0 Z0

Next, consider the normal geodesic joining q and q0, σ(t)= F (ξ,tη). Remember that σ′(t)= α η eα. By considering f(σ(t)), 1 α α β f(ξ, η)= f(ξ, 0)+ η (eα(f)) q0 + η η (1 t) (eβ(eα(f)))(σ(t)) dt . (2.7) | − Z0 j 1 Similar to (2.6), (eα(f)) q0 = (eα(f)) p + ξ 0 ej(eα(f))(σ0(t)) dt. Putting these together ﬁn- | | ishes the proof of this lemma. R

2.2.2. The expansions of coordinate vector ﬁelds.

Lemma 2.5. Let Uε be a neighborhood of p Σ in M as in Deﬁnition 2.3 with the coordinate 1 n n+1 n+m ∈ system (x, y) = (x , x ,y , y ) and the frame e , , en, en , en m . Write · · · · · · { 1 · · · +1 · · · + } ∂ ∂ ∂ ∂ = , eA eA and = , eA eA , ∂xi h∂xi i ∂yµ h∂yµ i 7 ∂ ∂ then i , eA and µ , eA , considered as locally deﬁned multi-indexed functions, has the fol- h ∂x i h ∂y i lowing expansions:

∂ α 2 2 , ej = δij y hαij + ( x + y ) , h∂xi i (x,y) − p O | | | | (2.8) ∂ 2 2 , eβ = δµβ + ( x + y ) , h∂yµ i (x,y) O | | | | ∂ ∂ 2 2 and both i , eβ and µ , ej are of the order x + y . By inverting the matrices, h ∂x i (x,y) h ∂y i (x,y) | | | |

∂ α ∂ 2 2 ∂ 2 2 ei = + y hαij + ( x + y ) and eα = + ( x + y ) . (2.9) ∂xi ∂xj O | | | | ∂yα O | | | |

Proof. We apply Lemma 2.4 to these locally defined functions. ∂ i ∂ i By construction, i , ej p = δij. With a similar argument as that for (2.4), x i = x ei on h ∂x i| ∂x Σ Uε. It follows that ∩ j ℓ ∂ x = x , ej . h∂xℓ i Differentiating the above equation first with respect to xi and then with respect to xk, and then evaluating at p which has xℓ = 0 for all ℓ, we obtain ∂ ∂ ∂ ∂ , ej + , ej = 0 . ∂xk h∂xi i ∂xi h∂xk i p p

Σ On the other hand, it follows from the construction that ( ej) p = 0, and ∇ | ∂ ∂ Σ ∂ Σ ∂ ∂ ∂ k i , ej = ∂ i , ej = ∂ k , ej = i k , ej . ∂x h∂x i h∇ ∂xk ∂x i h∇ ∂xi ∂x i ∂x h∂x i p p p p

∂ ∂ Hence, k i , ej is zero at p. ∂x h ∂x i Since ej is parallel with respect to along normal geodesics, ( e ej) p = 0. It follows that ∇ ∇ α | ∂ ∂ ∂ ∂ ∂ α i , ej = ∂ i , ej = ∂ α , ej = α , ∂ ej = hαij p ∂y h∂x i h∇ ∂yα ∂x i h∇ ∂xi ∂y i − h∂y ∇ ∂xi i − p p p p

∂ where the third equality follows from the fact that α , ej 0 on Σ Uε. h ∂y i≡ ∩ ∂ Note that i , eβ vanishes on Σ Uε. Since eβ is parallel with respect to along normal h ∂x i ∩ ∇ geodesics, ( e eβ) p = 0, and then ∇ α | ∂ ∂ ∂ ∂ α i , eβ = ∂ i , eβ = ∂ α , eβ . ∂y h∂x i h∇ ∂yα ∂x i h∇ ∂xi ∂y i p p p

∂ ∂ ∂ By construction, α = eα on Σ U ε and ( ⊥eα) p = 0. Therefore, α i , eβ is zero at p. ∂y ∩ ∇ | ∂y h ∂x i ∂ µ ∂ The term µ , ej also vanishes on Σ Uε. It follows from (2.4) that y µ , ej = 0. h ∂y i ∩ h ∂y i Diﬀerentiating the above equation ﬁrst with respect to yα and then with respect to yβ, we 8 obtain ∂ ∂ ∂ ∂ , ej + , ej = 0 . ∂yα h∂yβ i ∂yβ h∂yα i p p

Since e ej = 0, the above two terms are always equal to each other, and thus both vanish. ∇ ν ∂ ∂ For ∂yµ , eβ , it follows from the construction that ∂yµ , eβ = δµβ on Σ Uε. According h µ i ν ∂ h i ∂ ∂ ∩∂ ∂ to (2.4), y = y ν , eµ . By a similar argument as that for k i , ej , ν µ , eβ also h ∂y i ∂x h ∂x i ∂y h ∂y i vanishes at p.

2.2.3. The expansions of connection coeﬃcients.

Proposition 2.6. Let Uε be a neighborhood of p Σ in M as in Definition 2.3 with the co- 1 n n+1 n+m ∈ ordinate system (x, y) = (x , x ,y , y ) and the frame e , , en, en , en m . · · · · · · { 1 · · · +1 · · · + } Let B C B C θ = e eA, eB ω = θ (eC )ω A h∇ C i A A n+m be the connection 1-forms of the frame fields on Uε, where ω A=1 is the dual coframe of n+m B { } eA . Then, at a point q Uε with coordinates (x, y), θ (eC ), considered as locally defined { }A=1 ∈ A multi-indexed functions, has the following expansions:

Σ where Rjilk p, Rjiαk p, Rjiαβ p hαij p, hαij;k p, Rαiβj p, Rαiγβ p,Rαβji⊥ p, Rαβγi p, Rαβδγ p all represent the evaluation of the corresponding tensors at p and with respect to the frame ﬁelds n n+m ei and eα . { }i=1 { }α=n+1

n Σ Proof. Since the restriction of the frame ei i=1 on Σ is parallel with respect to along the k j { } ∇ radial geodesics, x θ (ek) = 0 for any i, j 1,...,n . It follows that i (x,0) ∈ { } j j l ∂θi (el) j θ (ek) = x and thus θ (ek) = 0 . (2.13) i (x,0) − ∂xk (x,0) i p 9

By taking the partial derivative in xl and evaluating at p = (0, 0), we ﬁnd that j j ∂θi (ek) ∂θi (el) j j = , or equivalently, el(θ (ek)) p = ek(θ (el)) p (2.14) ∂xl p − ∂xk p i | − i | n+m Similarly, since the restriction of eµ µ=n+1 on Σ is parallel with respect to ⊥ along radial k ν { } ∇ geodesics, x θ (ek) = 0 for any µ,ν n + 1,...,n + m . It follows that µ (x,0) ∈ { } ν θµ(ek) p = 0 and (2.15) ν ν el(θ (ek)) = ek(θ (el)) . (2.16) µ p − µ p n+m µ B Since the frame eA is parallel with respect to along normal geodesics, y θ (eµ) = 0 { }i=1 ∇ A and it follows that B B ν ∂θA (eν ) B θ (eµ)= y θ (eµ) = 0. (2.17) A − ∂yµ ⇒ A (x,0)

By taking partial derivatives, ∂θB(e ) ∂2 θB(e ) ∂θB(e ) ∂θB(e ) ∂2 θB(e ) A µ = yν A ν and A µ = A ν yδ A δ . (2.18) ∂xk − ∂xk∂yµ ∂yν − ∂yµ − ∂yν∂yµ ∂ n n Note that on Σ, i and ei are both bases for T Σ. Therefore, { ∂x }i=1 { }i=1 B ek(θA (eµ)) (x,0) = 0 . (2.19) ∂ By construction, ∂yµ = eµ on Σ, and thus B B eν(θ (eµ)) = eµ(θ (eν )) . (2.20) A (x,0) − A (x,0)

In terms of the connection 1-forms, the components of the Riemann curvature tensor are

RABCD

= e e eB e e eB eB, eA h∇ C ∇ D −∇ D ∇ C −∇[eC ,eD] i A A E A A E A E = eC (θ (eD)) eD(θ (eC )) (θ θ )(eC , eD) θ (eE)θ (eC )+ θ (eE )θ (eD) . (2.21) B − B − B ∧ E − B D B C With these preparations, we proceed to prove all the expansion formulae: j (The expansion of θi (ek)) It follows from (2.13) that the zeroth order term is zero. By (2.14), the coeﬃcient of xl in the expansion is

j 1 j j 1 Σ el(θi (ek)) p = el(θi (ek)) ek(θi (el)) = Rjilk . | 2 − p 2 p h i Σ Note that for Rαβji, all the indices of summation in (2.21) go from 1 to n. Due to (2.19), the coeﬃcient of yα in the expansion is

j j j eα(θi (ek)) = eα(θi (ek)) ek(θi (eα)) = Rjiαk . p − p |p h i 10

j l (The expansion of θi (eβ)) By (2.17), the zeroth order term is zero, and the coeﬃcient of x in the expansion is zero. According to (2.20), the coeﬃcient of yα in the expansion is

j 1 j j eα(θi (eβ)) = eα(θi (eβ)) eβ(θi (eα)) = Rjiαβ . p 2 − p |p h i α α (The expansion of θ (ej)) On Σ Uε, θ (ej)= e ei, eα = hαij . Its derivative along ek is i ∩ i h∇ j i Σ Σ ek(II(ei, ej, eα)) = ( e II)(ei, ej, eα) + II( ei, ej , eα) + II(ei, ej, eα) + II(ei, ej, ⊥ eα) . ∇ k ∇ek ∇ek ∇ek Due to (2.13) and (2.15), the last three terms vanish at p. It follows that ek(II(ei, ej, eα)) p is | equal to hαij k p. ; | β α The coeﬃcient of y is eβ(θ (ej) p. By (2.19) and (2.17), i | α α k α Rαiβj = eβ(θi (ej)) + θi (ek)θβ(ej) = [eβ(θi (ej)) hαikhβkj] . |p p − |p h i α (The expansion of θi (eβ)) According to (2.17), the zeroth order term is zero, and the coeﬃ- cient of xl in the expansion is zero. By (2.20) and (2.17),

α 1 α α 1 eγ (θ (eβ)) = [eγ(θ (eβ)) eβ(θ (eγ ))] = Rαiγβ . i |p 2 i − i |p 2 |p α (The expansion of θβ (ei)) By (2.15), the zeroth order term vanishes. With (2.16), (2.15) and (2.13),

β 1 β β 1 ej(θα(ei)) = ej(θα(ei)) ei(θα(ej)) = Rαβji⊥ . p 2 − p 2 p h i

Note that for Rαβji⊥ , the index of summation in the third term of (2.21) goes from n + 1 to n + m, and the indices of summation in the last two terms of (2.21) go from 1 to n. By (2.19), (2.17) and (2.15),

β β β eγ (θα(ei)) = eγ (θα(ei)) ei(θα(eγ )) = Rαβγi . p − p |p h i α α (The expansion of θ (eγ )) Due to (2.17), θ (eγ ) vanishes on Σ Uε. According to (2.20) and β β ∩ (2.17),

α 1 α α 1 eδ(θ (eγ )) = eδ(θ (eγ )) eγ (θ (eδ)) = Rαβδγ . β p 2 β − β p 2 |p This ﬁnishes the proof of this proposition.

2.2.4. Horizontal and vertical subspaces. For any q Uε M, there exists a unique p Σ ∈ ⊂ ∈ such that there is a unique normal geodesic inside Uε connecting q and p. Any tensor defined on Σ can be extended to Uε by parallel transport of along normal geodesics. Here are some ∇ notions that will be used in this paper. The parallel transport of T Σ along normal geodesics defines an n-dimensional distribution of T M Uε , which is called the horizontal distribution, and is denoted by . Its orthogonal | 11 H complement in T M is called the vertical distribution, and is denoted by . It is clear that = V H span e1, , en and = span en+1, , en+m . The parallel transport of the volume form of { · · · } V { · · · } 1 n Σ along normal geodesics defines an n-form on Uε, which is denoted by Ω. Let ω , ,ω , n+1 n+m { · · · ω , ,ω be the dual coframe of e , , en, en , , en m . In terms of the coframe, · · · } { 1 · · · +1 · · · + } Ω= ω1 ωn . (2.22) ∧···∧

For any q Uε and any oriented n-plane L TqM, consider the orthogonal projection ∈ ⊂ onto q, π , and the evaluation of Ω on L. Suppose that Ω(L) > 0. By the singular value V V decomposition, there exist oriented orthonormal basis e , , en for q, orthonormal basis { 1 · · · } H en , , en m for q and angles φ , , φn [0, π/2) such that { +1 · · · + } V 1 · · · ∈ n e˜j = cos φj ej + sin φj en j (2.23) { + }j=1

constitutes an oriented, orthonormal basis for L. If n>m, φj is set to be zero for j>m. It follows that

Ω(L) = cos φ cos φn , (2.24) 1 · · · and the operator norm of π is V

s(L) := π L = max sin φ1, , sin φn . (2.25) V | op { · · · }

Remark 2.7. The construction (2.23) works for Ω(L) = 0 as well, and some of the angles would be π/2. The formulae (2.24) and (2.25) remain valid. We brieﬂy explain this linear-algebraic construction. Consider the orthogonal projection onto q, π . Let L = ker(π : L q); it H H V H →H is a linear subspace of q. Let L′ be the orthogonal complement of L in L. Then, L = L′ L , V V ⊕ V and π : L′ q is injective. Note that π (L′) is orthogonal to L . The linear subspace L′ is H →H V V the graph of a linear map from π (L′) q to q. The basis (2.23) is constructed by applying H ⊂H V the singular value decomposition to this linear map together with an orthonormal basis for L . V

It is easy to see that the orthogonal complement of L has the following orthonormal basis:

m e˜α = sin φα eα n + cos φα eα α=n+1 (2.26) { − − }

where φα = φα n. If m>n, φα is set to be zero for α > 2n. The following estimates will be − needed later, and are straightforward to come by:

n n j k α j (ω ω )(˜ei, e˜i) n , (ω ω )(˜ei, e˜i) ns (2.27) ⊗ ≤ ⊗ ≤ i=1 i=1 X X 12 and

1 n (ω ω )(˜eα, e˜ , , e˜i , , eñ) s , ∧···∧ 1 · · · · · · ≤ 1 n 2 (ω ω )(˜eα, e˜β, e˜ , , e˜i , , e˜j , eñ) s , ∧···∧ 1 · ·c · · · · · · · ≤ α 1 i n (ω ω ω ω )(˜e , , eñ) ns , (2.28) ∧ ∧···∧ ∧···∧ c1 · · · c ≤ α 1 i n (ω ω ωb ω )(˜eβ , e˜1, , e˜j , , eñ) 1 , ∧ ∧···∧ ∧···∧ · · · · · · ≤ α β 1 i j n 2 (ω ω ω b ω ω ω )(˜e1, , eñ) n(n 1)s ∧ ∧ ∧···∧ ∧···∧ ∧···∧ c · · · ≤ − for any i, j, k 1,...,n and α,b β n +c 1,...,n + m . ∈ { } ∈ { } The above estimates are the zeroth order estimate. For the first and third inequalities of (2.28), a more refined version will also be needed. It follows from (2.23) and (2.26) that

n 1 n i sin φi (ω ω )(˜eα, e˜1, , e˜i , , eñ) = ( 1) δα(n+i) cos φk . (2.29) ∧···∧ · · · · · · − cos φi k Y=1 1 n n+1 n+m c Letω ˜ , , ω˜ , ω˜ , , ω˜ be the dual basis ofe ˜ , , eñ, eñ , , eñ m. According to · · · · · · 1 · · · +1 · · · + (2.23) and (2.26),

j j n+j α α n α ω = cos φj ω˜ sin φj ω˜ and ω = sin φα ω˜ − + cos φα ω˜ . − Hence,

α 1 i n (ω ω ω ω )(˜e , , e˜n) ∧ ∧···∧ ∧···∧ 1 · · · α n 1 \ i n = (sin φαω˜ − ) (cos φ1ω˜ ) (cos φiω˜ ) (cos φnω˜ ) (˜e1, , e˜n) ∧ b ∧···∧ ∧···∧ · · · (2.30) n i+1 sin φi = ( 1) δα(n+i) cos φk . − cos φi k Y=1

3. Minimal submanifolds and stability conditions

3.1. The stability of a minimal submanifold. A submanifold Σ (M, g) is said to be ⊂ minimal if its mean curvature vanishes, H = 0. It means that Σ is a critical point of the volume functional. A minimal submanifold Σ is said to be stable if the second variation of the volume functional is positive at Σ. We now recall the second variational formula of the volume functional. The detail can be found in [15, 3.2]. § Suppose that V is a normal vector field on Σ. There are two linear operators on NΣ in the second variation formula. The first one is the partial Ricci operator defined by

(V )=tr R( ,V ) ⊥ R Σ · · 13 where R is the Riemann curvature tensor of (M, g). The second one is basically the norm-square of the second fundamental form along V . The shape operator along V is a symmetric map from T Σ to itself, and is deﬁned by

T Σ V (X)= ( X V ) = X V + ⊥ V , or equivalently V (X),Y = II(X,Y ),V S − ∇ −∇ ∇X hS i h i for any tangent vectors X and Y of Σ. By regarding as a map from NΣ to Sym2(T Σ), deﬁne S (V )= t (V ) A S ◦ S where t : Sym2(T Σ) NΣ is the transpose map of . S → S With this understanding, the second variation of the volume functional in the direction of V is

2 ⊥V + (V ),V (V ),V (3.1) |∇ | hR i−hA i ZΣ Therefore, Σ is stable if and only if ( ) + is a positive operator. Note that ( ) ∇⊥ ∗∇⊥ R−A ∇⊥ ∗∇⊥ is always non-negative deﬁnite, and is a linear map on NΣ. Hence, the positivity of R−A is a condition easier to check, and implies the stability of Σ. R−A Deﬁnition 3.1. A minimal submanifold Σ (M, g) is said to be strongly stable if is a ⊂ R−A (pointwise) positive operator on NΣ.

In terms of the notations introduced in 2.1, Σ is strongly stable if there exists a constant § c0 > 0 such that

α β α β α 2 Riαiβv v hαijhβijv v c0 (v ) (3.2) − − ≥ α α,β,iX α,β,i,jX X for any (vn+1, , vn+m) Rm. · · · ∈ In particular, for a hypersurface Σ, the condition is

Ric(ν,ν) A 2 c , − − | | ≥ 0 where ν is a unit normal and A 2 = h2 . | | i,j ij P 3.2. Proof of Proposition A. It is easy to see that (3.2) holds for a totally geodesic submanifold in a manifold with negative sectional curvature. When the geometry has special properties, the condition (3.2) is equivalent to some natural curvature condition on the minimal submanifold. 14 3.2.1. Complex submanifolds in Kähler manifolds. Let (M 2n,g,J,ω) be a Kähler manifold, and Σ2p M be a complex submanifold. The submanifold Σ is automatically minimal. In fact, ⊂ the second variation (3.1) is always non-negative. In this case, the operator was studied R−A by Simons in the famous paper [15, 3.5]. We briefly summarize his results. The condition is § equivalent to that p 2 J R⊥(ei,fi)(V ) ,V c V − ≥ 0 | | * i ! + X=1 where e , , ep,f , ,fp is an orthonormal frame for T Σ with fi = Jei. In other words, { 1 · · · 1 · · · } the normal bundle curvature contracting with ωΣ is positive definite. It implies that the normal bundle of Σ admits no non-trivial holomorphic cross section.

3.2.2. Minimal Lagrangians in Kähler–Einstein manifolds. Let (M 2n,g,ω) be a Kähler–Einstein manifold, where ω is the Kähler form. Denote the Einstein constant by c, i.e. Ric = c g. A n half-dimensional submanifold L M is said to be Lagrangian if ω L vanishes. Suppose that ⊂ | L is both minimal and Lagrangian. Then, (3.2) is equivalent to the condition that

RicL c is a positive definite operator on T L , (3.3) − L where Ric is the Ricci curvature of g L. For completeness, the derivation is included in | Appendix A.1. We remark that when c< 0, a minimal Lagrangian is always stable. That is to say, the second variation (3.1) is strictly positive for any non-identically zero V ; see [5,13]. A case of particular interest is special Lagrangians in a Calabi–Yau manifold; see [8, III]. The § constant c = 0 for a Calabi–Yau manifold, and the strong stability condition (3.2) is equivalent to the positivity of RicL. By the Bochner formula, it implies that the first Betti number of L is zero. According to the result of McLean [12, Corollary 3.8], L is infinitesimally rigid as a special Lagrangian submanifold.

3.2.3. Coassociatives in G2 manifolds. A G2 manifold (M, g) is a 7-dimensional Riemannian manifold with G2 holonomy. A coassociative submanifold is a special class of minimal, 4- dimensional submanifold in M. A complete story can be found in [8, IV] and [10, ch.11–12], § and a brief summary is included in Appendix A.2. Suppose that Σ4 M is coassociative. The strong stability condition (3.2) is equivalent to ⊂ that s 2W + is a positive definite operator on Λ2 , (3.4) − − 3 − where W anti-self-dual part of the Weyl curvature of g Σ, and s is the scalar curvature of − | g . The computation bears its own interest in G geometry, and is included in Appendix |Σ 2 A.2. According to the Weitzenböck formula for anti-self-dual 2-forms [7, Appendix C], (3.4) 15 implies that Σ has no non-trivial anti-self-dual harmonic 2-forms. Due to [12, Corollary 4.6], Σ is infinitesimally rigid as a coassociative submanifold.

3.3. The Codazzi equation on a minimal submanifold. Suppose that Σ is a minimal submanifold. Choose a local orthonormal frame e , , en m such that the restriction of { 1 · · · + } e , , en on Σ are tangent to Σ and the restriction of en , , en m to Σ are normal to { 1 · · · } { +1 · · · + } Σ. Consider the following equation on Σ:

Σ hαii k = ek(hαii) 2 ei, ek hαki ⊥ eα, eβ hβii . ; − h∇ej i − h∇ej i Since the mean curvature vanishes, the ﬁrst and third terms are zero. For the second term, Σ ei, ek is skew-symmetric in i and k, and hαki is symmetric in i and k. Hence, the second h∇ej i term is also zero. By combining it with the Codazzi equation (2.1),

Rαiij = hαji;i . (3.5)

4. The convexity of ψ and a local uniqueness theorem of minimal submanifolds

α 2 Suppose that Σ is a minimal submanifold in (M, g) and consider the function ψ = α(y ) on the tubular neighborhood Uε of Σ as in 2.2.1. Similar to [18], the strong stability of Σ is § P closely related to the positivity of the trace of Hess(ψ) over an n-dimensional subspace.

Proposition 4.1. Let Σn (M, g) be a compact, oriented minimal submanifold that is strongly ⊂ stable in the sense of (3.2). There exist positive constants ε1 and c which depend on the geometry

of M and Σ and which have the following property. For any q Uε1 and any oriented n-plane ∈ L TqM, ⊂ 2 trL Hess(ψ) c s(L) + ψ(q) (4.1) ≥ where s(L) is deﬁned by (2.25).

Proof. Let p Σ be the point such that there is a normal geodesic in Uε connecting p and ∈ q. To calculate Hess(ψ), take the frame e , , en, en , , en m constructed in 2.2. Let { 1 · · · +1 · · · + } § ω1, ,ωn,ωn+1, ,ωn+m be the dual coframe. According to (2.4) and (2.5), dψ = 2yαωα, { · · · · · · } and thus ej(ψ) 0. By (2.11), ≡

Hess(ψ)(ei, ej) α α = ei(ej(ψ)) ( e ej)(ψ)= 2y θ (ei) − ∇ i − j 3 α α k α β α β 2 2 = 2y hαij 2y x hαij k 2y y Rαjβi 2y y (hαjkhβik) + (( x + y ) 2 ) . − p − ; p − p − p O | | | | 16

By (2.9), (2.11) and (2.12),

β β Hess(ψ)(eα, ei)= eα(ei(ψ)) ( e ei)(ψ)= 2y θ (eα) − ∇ α − i = ( x 2 + y 2) , O | | | | β γ γ Hess(ψ)(eα, eβ)= eα(eβ(ψ)) ( eα eβ)(ψ) = 2eα(y ) 2y θβ(eα) − ∇ − (4.2) 2 2 = 2δαβ + ( x + y ) . O | | | | We choose the frame so that L has an oriented, orthonormal basis of the form (2.23) and j evaluate trL Hess(ψ); note that all the x -coordinate of q are zero:

trL Hess(ψ)= Hess(ψ)(cos φj ej + sin φj en+j, cos φj ej + sin φj en+j) j X 2 α α β α β 2 = 2 cos φj y hαjj y y Rαjβj y y (hαjkhβjk) + 2sin φj − p − p − p j X h i + ( y 3)+ s(L) ( y 2) O | | · O | | α β α β 2 2 y y Rαjβj y y (hαjkhβjk) + 2 sin φj ≥ − p − p j,α,β j X h i X 2 α β α β 2 3 + 2 sin φj y y Rαjβj + y y (hαjkhβjk) s (L) c′ y p p − − | | j,α,β X h i for some c′ > 0. By using the strong stability condition (3.2), this ﬁnishes the proof of the proposition.

By the same argument as in [18], the convexity of ψ implies the following local uniqueness theorem of minimal submanifolds near Σ.

Theorem 4.2. (Theorem A) Let Σn (M, g) be a compact minimal submanifold which is ⊂ strongly stable in the sense of (3.2). Then, there exists a tubular neighborhood U of Σ such that any compact minimal submanifold Γ in U with dimΓ n must be contained in Σ. In other ≥ words, Σ is the only compact minimal submanifold in U with dimension no less than n.

Proof. It basically follows from [18, Lemma 5.1] and Proposition 4.1. The only point to check is that the estimate of Proposition 4.1 holds for dimension greater than n. Namely, it remains ¯ to show that for any q Uε1 and anyn ¯-plane L TqM withn>n ¯ , ∈ ⊂

tr¯ Hess(ψ) c L ≥ 0 for some positive constant c0.

The argument is similar to Remark 2.7. Pick an (¯n n)-subspace of ker(π : L¯ q). − H → H Denote it by L . Note that L belong to q. Let L be the orthogonal complement of L in V V V V 17 L¯. The dimension of L is n. By Proposition 4.1 and (4.2), the trace of the Hessian of ψ over L¯ has the following lower bound:

trL¯ Hess(ψ)=trL Hess(ψ)+trLV Hess(ψ)

c ψ(q)+ 2(¯n n) c′ψ(q) . ≥ − − Thus, the quantity is positive when ψ(q) is suﬃciently small.

Remark 4.3. In this rigidity theorem, it is not hard to see that the minimal submanifold Σ needs not to be orientable. However, in order to have ψ to be well-deﬁned on a tubular neighborhood, Σ has to be embedded.

5. Further estimates needed for the stability theorem

From now on, Σ is taken to be a strongly stable minimal submanifold and we see in the last

section that the distance function ψ to Σ defined on Uε satisfies a convexity condition. To study the dynamical stability of mean curvature flows near Σ, we need to measure how close a nearby submanifold is to Σ. The distance function ψ gives such a measurement in C0. In order to obtain measurements in higher derivatives, we extend the volume form and

the second fundamental form of Σ to the tubular neighborhood Uε. In particular, in 2.2.4, § the volume form of Σ is extended to an n-form Ω on Uε. The restriction of Ω to another n-dimensional submanifold Γ, which is denoted by Ω, measures how close Γ is to Σ in C1. ∗ The evolution equation of Ω along the mean curvature ﬂow plays an essential role for the ∗ estimates. The equation naturally involves the restriction of the covariant derivatives/second covariant derivatives of Ω on Γ. In this section, we derive estimates of these quantities in preparation for the proof of the stability theorem.

5.1. Extension of auxiliary tensors to Uε. We adopt the frame and coordinate constructed in 2. § The second fundamental form of Σ can also be extended to Uε by parallel transport along normal geodesics, as explained in 2.2.4. Denote the extension by IIΣ, which, in terms of the § frames, is given by

Σ i j II = hαij ω ω eα . (5.1) ⊗ ⊗ In other words, for any q Uε, hαij(q) = hαij(p) where p Σ is the unique point such that ∈ ∈ there is a normal geodesic in Uε connecting p and q, see Definition 2.3. To avoid introducing Σ Σ more notations, we use the metric g to lower the indices of II , and then II = hαij ei ej eα. ⊗ ⊗ Suppose that Γ is an oriented, n-dimensional submanifold in Uε M with Ω(Γ) > 0. With ⊂ the above extension, we can compare the second fundamental form of Γ with that of Σ. For any q Γ, choose a local orthonormal frame e˜1, , eñ,e ñ+1, , eñ+m on a neighborhood ∈ {18 · · · · · · } of q in M such that the restriction of e˜ , , eñ on Γ form an oriented frame for T Γ, and the { 1 · · · } restriction of eñ , , eñ m on Γ form a frame for NΓ. With this, the second fundamental { +1 · · · + } form of Γ is

Γ II = h˜αij e˜i e˜j e˜α where h˜αij = e e˜j, e˜α . (5.2) ⊗ ⊗ h∇˜i i As explained in 2.2.4, we may assume that these frames are of the form (2.23) and (2.26) at § q. The inverse transform reads

ej = cos φj e˜j sin φj e˜n+j and eα = sin φα e˜α n + cos φα e˜α . (5.3) − − It follows that

Σ II = hαij(p) (cos φi e˜i sin φi e˜n+i) (cos φj e˜j sin φj e˜n+j) (sin φα e˜α n + cos φα e˜α). q − ⊗ − ⊗ − (5.4)

Hence,

Γ Σ II , II = cos φi cos φj cos φα h˜αij hαij(p) . (5.5) h i q α,i,j X

In the above expression, hαij(p) depends only on p Σ, while φi, φα, and h˜αij all depend on ∈ q Γ. ∈ We extend another tensor which is related to the strong stability condition (3.2). Consider the parallel transport of the following tensor on Σ along normal geodesics:

i j α β (Rαiβj + hαikhβjk) (ω ω ) (e e ) , ⊗ ⊗ ⊗ which is considered to be deﬁned on Uε. Pairing the last component with ψ/2 produces a ∇ tensor of the same type as IIΣ, which is denoted by SΣ:

Σ β i j S q = y (Rαiβj(p) + (hαikhβjk)(p)) ω ω eα (5.6) | ⊗ ⊗ where p Σ is the point such that there is a unique normal geodesic in Uε connecting p and q. ∈ Similarly,

Γ Σ β II ,S = cos φi cos φj cos φα h˜αij y Rαiβj(p)+ (hαikhβjk)(p) . (5.7) h i q α,β,i,j k ! X X

Again in the above expression, Rαiβj(p)+ k(hαikhβjk)(p) depends only on p Σ, while β ∈ φi, φα,y , and h˜αij all depend on q Γ. ∈ P 1 Γ Σ Γ Σ In the rest of this subsection, we assume Ω(TqΓ) > and estimate II , II and II ,S . 2 h i q h i q We assume that TqΓ has an oriented frame of the form (2.23), and NqΓ has a frame of the form 1 (2.26). Since Ω(TqΓ) > 2 , it follows from (2.24) that 1 cos φj cos φ cos φn > for j 1,...,n . ≥ 1 · · · 2 ∈ { } 19 A direct computation shows that

2 2 0 < 1 cos φj = 1 1 sin φj 2 (s(TqΓ)) , − − − ≤ q2 (5.8) 1 sin φj 2 cos φj = < 2 (s(TqΓ)) . cos φ − cos φ j j

s Suppose that in (2.25) is achieved at φ1, and then

2 1 2 Ω(TqΓ) cos φ = 1 sin φ 1 sin φ ≤ 1 − 1 ≤ − 2 1 q 1 2 1 Ω(TqΓ) (s(TqΓ)) . (5.9) ⇒ − ≥ 2 On the other hand,

n 2 2 2 1 Ω(TqΓ) 1 (Ω(TqΓ)) = 1 (1 sin φj) c(n) (s(TqΓ)) (5.10) − ≤ − − − ≤ j Y=1 for some dimensional constant c(n). Applying the estimate (5.8) to (5.5) and (5.7), we obtain

Γ Σ 2 Γ II , II h˜αij hαij (p) c (s(q)) II , h i q − ≤ | | α,i,j X (5.11)

Γ Σ β 2 Γ II ,S h˜αij y Rαiβj(p)+ (hαikhβjk)(p) c (s(q)) ψ(q) II h i q − ≤ | | α,β,i,j k ! X X p

for some constant c depending on the geometry of Σ and M.

5.2. Estimates involving the derivatives of Ω. In this subsection, we derive estimates that involve derivatives of Ω, which are needed in the proof of Theorem B. In the following three lemmas, we estimate quantities that appear naturally in the evolution equation of Ω (6.1). ∗ Let Γ be an n-dimensional submanifold in the tubular neighborhood of Σ. The function

Ω is the Hodge star of Ω Γ with respect to the induced metric on Γ, and is the same as ∗ | 1 Ω(TqΓ). We assume throughout this subsection that Ω(q) > for any q Γ. For each q Γ, ∗ 2 ∈ ∈ let p Σ be the point such that there is a unique normal geodesic in Uε connecting p and ∈ q; see Deﬁnition 2.3. We use the coordinate and frame constructed in 2.1 to carry out the § computation. Moreover, we assume that TqΓ has an oriented frame of the form (2.23), and NqΓ has a frame of the form (2.26). For q Γ, ψ(q) is a 0 order quantity. Ω(q) and s(q) are both 1 ∈ C ∗ order quantities that depend on the tangent space TqΓ at q, where s(q)= s(TqΓ) is deﬁned C in (2.25). 20 5.2.1. The restriction of the derivative of Ω to Γ. To compute Ω, it is convenient to introduce ∇ the following shorthand notations:

j j+1 1 j n Ω = ι(ej)Ω = ( 1) ω ω ω , (5.12) − ∧···∧ ∧···∧ j+k 1 k j n jk ( 1) ω c ω ω ω if kj.  − ∧···∧c ∧···∧c ∧···∧ The covariant derivative of Ω is c c Ω = ( ωj) Ωj = θα (ωα Ωj) . (5.14) ∇ ∇ ∧ j ⊗ ∧ Lemma 5.1. Let Σn (M, g) be a compact, oriented minimal submanifold. Then, there exist ⊂ a positive constant c which depends on the geometry of M and Σ and which has the following 1 property. Suppose that Γ Uε is an oriented n-dimensional submanifold with Ω(q) > for ⊂ ∗ 2 any q Γ. Then, ∈

j Γ Σ Σ ( 1) h˜αjk ( e Ω)(˜eα, e˜ , , e˜j , , e˜n) + ( Ω) II , II + S − ∇˜k 1 · · · · · · ∗ h i α,j,k X h i c c s(q) 2 + ψ(q) IIΓ ≤ at any q Γ. The summation is indeed a contraction between IIΓ and Ω, and is independent ∈ ∇ of the choice of the orthonormal frame.

Proof. By (5.14),

j ˜ ˜ α cos φα ( 1) hαjk ( e˜k Ω)(˜eα, e˜1, , e˜j , , e˜n) = hαjkθj (˜ek) ( Ω) . − ∇ · · · · · · q − cos φj ∗ α,j,k h i α,j,k X X According to (2.11), (2.23) and the factc that the xj-coordinates of q are all zero,

α β 2 θ (˜ek) cos φk hαjk(p) cos φk y (Rαjβk + hαjlhβkl)(p) c s(q) + ψ(q) (5.15) j q − − ≤ 1 at q . Combining this with (5.8) and (5.11) ﬁnishes the proof of this lemma.

5.2.2. The restriction of the second derivative of Ω to Γ. Since ωα = θα ωi θα ωβ and ∇ − i ⊗ − β ⊗ Ωj = ( ωk) Ωjk = θk Ωk + θα (ωα Ωjk) , ∇ ∇ ∧ j ⊗ k ⊗ ∧ the covariant derivative of (5.14) is 2 α α α β β α jk Ω= (θi θi ) Ω + (θk θj ) (ω ω Ω ) ∇ − ⊗ ⊗ ⊗ ⊗ ∧ ∧ (5.16) + θα + θα θβ + θi θα (ωα Ωi) ∇ i β ⊗ i k ⊗ k ⊗ ∧ α where θi is the covariant derivative of a local section of T ∗M. ∇ 21 Lemma 5.2. Let Σn (M, g) be a compact, oriented minimal submanifold. Then there exists ⊂ a positive constant c which depends on the geometry of M and Σ and which has the following 1 property. Suppose that Γ Uε is an oriented n-dimensional submanifold with Ω(q) > for ⊂ ∗ 2 any q Γ. Then, ∈

2 i Σ Σ 2 ( Ω)(˜e , , e˜n) ( 1) Ω(˜eα, e˜ , , e˜i , , e˜n)R + ( Ω) II + S ∇e˜k,e˜k 1 · · · − − 1 · · · · · · α˜k˜k˜˜i ∗ k α,i,k X X h i 2 c c s (q)+ ψ(q) ≤ at any q Γ, where R = R(˜ek, e˜j )˜ek, e˜α are components of the restriction of the curvature ∈ α˜k˜k˜˜j h i tensor of M along Γ. Note that the two summations are independent of the choice of the orthonormal frame.

Proof. We examine the components on the right hand side of (5.16). Due to (2.11) and (2.12), j s2 θi (˜ek) c2 (q)+ ψ(q) , | |≤ (5.17) β 2 p θ (˜ek) c s (q)+ ψ(q) | α |≤ 2 for any i, j, k 1,...,n and α, β n + 1,...,n +pm . With (5.15) and the third and ﬁfth ∈ { } ∈ { } line of (2.28),

α β β α ij 2 θ (˜ek) θ (˜ek) (ω ω Ω )(˜e , , eñ) c s(q) , j i ∧ ∧ 1 · · · ≤ 3 β β α j 2 θ (˜ek) θ (˜ek) (ω Ω )(˜e , , eñ) c s(q) + ψ(q) , (5.18) α j ∧ 1 · · · ≤ 3 j α α j 2 θ (˜ek) θ (˜ek) (ω Ω )(˜e , , eñ) c s(q) + ψ(q) . i i ∧ 1 · · · ≤ 3 According to (5.15) and (5.8),

α 2 Σ Σ 2 2 θ (˜ek) II + S c (s(q)) + ψ(q) . (5.19) j − | | ≤ 4 α,j,k X

By (5.16), (5.19) and (5.18),

2 α α i Σ Σ 2 ( Ω)(˜e , , e˜n) (( θ )(˜ek, e˜k)) (ω Ω )(˜e , , e˜n) + ( Ω) II + S ∇e˜k,e˜k 1 · · · − ∇ i ∧ 1 · · · ∗

c (s(q))2 + ψ(q) . ≤ 5 (5.20)

The next step is to compute θα: ∇ i α α j α β θi = θi (ej ) ω + θi (eβ) ω α α j α β α j α β θ = d(θ (ej)) ω + d(θ (eβ)) ω + θ (ej) ω + θ (eβ) ω . (5.21) ⇒ ∇ i i ⊗ i ⊗ i ∇ i ∇ 22 By (5.17) and (5.15), we have the following estimate at q:

j j i α α ( ω )(˜ek, e˜k) = θ (˜ek) ω (˜ek) θ (˜ek) ω (˜ek) c s(q)+ ψ(q) , ∇ i − j ≤ 6 β β j β γ p ( ω )(˜ek, e˜k) = θ (˜ek) ω (˜ek)+ θ (˜ek) ω (˜ek) c . ∇ j γ ≤ 6

Together with (2.11),

α j α β θ (ej) ω + θ (eβ) ω (˜ek, e˜k) c s(q)+ ψ(q) . (5.22) i ∇ i ∇ ≤ 7 It follows from (2.11) and (2.8) that p

α k β d(θ (ej)) (hαij k) (p)ω + (Rαiβj + hαikhβjk) (p)ω c ψ(q) , i − ; ≤ 8 (5.23) α 1 γ p d(θ (eβ)) Rαiγβ(p)ω c i − 2 ≤ 8

where the norm on the left hand side is induced by the Riemannian metric g. By combining

(5.21), (5.22) an (5.23),

α 2 ( θ )(˜ek, e˜k) cos φk hαik k(p) c s(q)+ ψ(q) . ∇ i − ; ≤ 9 p It together with (2.30) and (5.8) gives that α α i ( θi )(˜ek, e˜k) (ω Ω )(˜e1, , e˜n) + ( Ω)sin φi h(n+i)ik;k(p) − ∇ ∧ · · · ∗ (5.24) c (s(q))2 + ψ(q) . ≤ 10 It remains to calculate the second term in the asserted inequality of the lemma. By (2.29),

i sin φi ( 1) Ω(˜eα, e˜1, , e˜i , , e˜n)Rα˜k˜k˜˜i = ( Ω) R(˜en+i, e˜k, e˜k, e˜i) . − · · · · · · ∗ cos φi α,i,k i,k X X With (2.23), (2.26) and (5.8), c

i 2 ( 1) Ω(˜eα, e˜ , , e˜i , , e˜n)R + ( Ω) sin φiR c (s(q)) . − 1 · · · · · · α˜k˜k˜˜i ∗ (n+i)kki ≤ 11 α,i,k i,k X X c Since R q R p c ψ(q), we have (n+i)kki| − (n+i)kki| ≤ 12 p i 2 ( 1) Ω(˜eα, e˜ , , e˜i , , e˜n)R + ( Ω) sin φiR (p) c (s(q)) + ψ(q) . − 1 · · · · · · α˜k˜k˜˜i ∗ (n+i)kki ≤ 13 α,i,k i,k X X c (5.25)

By (5.20), (5.24), (5.25) and the Codazzi equation (3.5), it ﬁnishes the proof of the lemma.

Remark 5.3. The tensor SΣ is needed for Lemma 5.2; otherwise the error term would be bigger. However, SΣ will only be used in some intermediate steps in the proof of Theorem B. 23 5.2.3. The derivative of Ω along Γ. The following lemma relates the derivative of Ω along Γ ∗ ∗ and the second fundamental form of Γ.

Lemma 5.4. Let Σn (M, g) be a compact, oriented minimal submanifold. Then, there exist ⊂ a positive constant c which depends on the geometry of M and Σ and which has the following 1 property. Suppose that Γ Uε is an oriented n-dimensional submanifold with Ω(q) > for ⊂ ∗ 2 any q Γ. Then, ∈ 2 Γ( Ω) 2 c (s(q)( Ω))2 IIΓ IIΣ 2 + c (s(q))2 + ψ(q) |∇ ∗ | ≤ ∗ | − | for any q Γ. ∈

Proof. We compute

Γ j ( Ω) = [˜ej(Ω(˜e , , eñ))]ω ˜ ∇ ∗ 1 · · · n j = ( e˜j Ω)(˜e1, , eñ)+ Ω(˜e1, , e˜i 1, e˜j e˜i, e˜i+1, , eñ) ω˜ ∇ · · · · · · − ∇ · · · " i # X=1 n ˜ j = ( e˜j Ω)(˜e1, , eñ)+ hαijΩ(˜e1, , e˜i 1, e˜α, e˜i+1, , eñ) ω˜ . ∇ · · · · · · − · · · " i # X=1 Note that the expression is tensorial, and we use the frame (2.23) and (2.26) to proceed. Due to (5.14) and (2.30),

α α i ( e Ω)(˜e , , e˜n)= θ (˜ej ) (ω Ω )(˜e , , e˜n) ∇˜j 1 · · · i ∧ 1 · · · n+i n+i sin φi = cos φj θi (ej)+sin φj θi (en+j) ( Ω) . cos φi ∗ By (2.11) and (5.8), at q,

n sin φi ( e˜j Ω)(˜e1, , e˜n) ( Ω) cos φn+i cos φi cos φjh(n+i)ij (p) ∇ · · · − ∗ cos φi i=1 X c (s(q))2 + ψ(q) . ≤ 1 According to (2.29),

n n ˜ sin φi ˜ hαijΩ(˜e1, , e˜i 1, e˜α, e˜i+1, , eñ)= ( Ω) h(n+i)ij . · · · − · · · − ∗ cos φ i i i X=1 X=1 24 To sum up, n Γ 2 2 ( Ω) = [˜ej(Ω(˜e , , eñ))] ∇ ∗ 1 · · · j=1 X n n 2 sin φi 2 ( e˜j Ω)(˜e1, , eñ) ( Ω) cos φn+i cos φi cos φjh(n+i)ij (p) ≤ ∇ · · · − ∗ cos φi j=1 i=1 X X n n 2 2 sin φi + 2( Ω) cos φn i cos φi cos φjh (p) h˜ ∗ cos φ + (n+i)ij − (n+i)ij j i i X=1 X=1 2 2 2 4nc (s(q)) + ψ(q) ≤ 1 2 2 2 + 8n (s(q)) ( Ω) h˜ cos φn i cos φi cos φjh (p) ∗ (n+i)ij − + (n+i)ij i,j X By (5.2) and (5.4),

2 Γ Σ 2 II II h˜αij cos φi cos φj cos φαhαij(p) | − | ≥ − α,i,j X This completes the proof of this lemma.

6. Stability of the mean curvature flow

After the preparation in the last sections, we consider the mean curvature ﬂow. We ﬁrst recall the following proposition from [23, Proposition 3.1].

Proposition 6.1. Along the mean curvature flow Γt in M, Ω = Ω(˜e , , eñ) satisfies ∗ 1 · · · d Ω=∆Γt Ω+ Ω( h˜2 ) dt ∗ ∗ ∗ αik α,i,k X ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ 2 [Ωαβ3 nhα1khβ2k + Ωα2β nhα1khβ3k + + Ω1 (n 2)αβhα(n 1)khβnk] − ··· ··· · · · ··· − − α,β,k (6.1) X 2( e Ω)(˜eα, , eñ)h˜α k 2( e Ω)(˜e , , e˜α)h˜αnk − ∇˜k · · · 1 −···− ∇˜k 1 · · · ˜ ˜ 2 [Ωα2 nRα˜k˜k˜1˜ + + Ω1 (n 1)αRα˜k˜kñ˜] ( e˜k,e˜k Ω)(˜e1, , eñ) − ··· · · · ··· − − ∇ · · · α,k X

Γt where ∆ denotes the time-dependent Laplacian on Γt, Ω˜ αβ3 n = Ω(˜eα, e˜β, e˜3, , eñ) etc., ··· · · · and R = R(˜ek, e˜ )˜ek, e˜α , etc. are the coefficients of the curvature operators of M. α˜k˜k˜1˜ h 1 i When Ω is a parallel form in M, Ω 0, this recovers an important formula in proving the ∇ ≡ long time existence result of the graphical mean curvature flow in [22]. 25 6.1. Proof of Theorem B. A finite time singularity of the mean curvature flow happens exactly when the second fundamental becomes unbounded; see Huisken [9], also [23]. The following theorem shows that if we start with a submanifold which is 1 close to a strongly C stable minimal submanifold Σ, then the mean curvature flow exists for all time, and converges smoothly to Σ.

Theorem 6.2. (Theorem B) Let Σn (M, g) be a compact, oriented, strongly stable minimal ⊂ submanifold. Then, there exist positive constants κ << 1 and c which depend on the geometry

of M and Σ and which have the following signiﬁcance. Suppose that Γ Uε is an oriented ⊂ n-dimensional submanifold satisfying

sup (1 ( Ω) + ψ) < κ . (6.2) q Γ − ∗ ∈ t Then, the mean curvature ﬂow Γt with Γ0 = Γ exists for all t> 0. Moreover, supq Γt II c t ∈ | | ≤ for any t> 0, where II the second fundamental form of Γt, and Γt converges smoothly to Σ as t . →∞

2 1 Proof. The constant κ will be chosen to be smaller than ε and 2 ; its precise value will be determined later. Suppose that the condition (6.2) holds for all Γt 0 t

d Γt Γt 2 ψ = Ht(ψ)=∆ ψ tr Hess ψ ∆ ψ c (ψ + s ) . (6.3) dt − Γt ≤ − 1 By applying Lemma 5.1, Lemma 5.2 and the second line of (2.28) to (6.1),

d ( Ω) ∆Γt ( Ω)+( Ω) IIt 2 2c s2 IIt 2 dt ∗ ≥ ∗ ∗ | | − 2 | | 2( Ω) IIt, IIΣ + SΣ c (s2 + ψ) IIt − ∗ h i− 2 | | + ( Ω) IIΣ + SΣ 2 c (s2 + ψ) ∗ | | − 2 ∆Γt ( Ω)+( Ω) IIt IIΣ SΣ 2 c (s2 + ψ) IIt 2 c (s2 + ψ) ≥ ∗ ∗ | − − | − 3 | | − 3 1 ∆Γt ( Ω) + ( Ω) IIt IIΣ 2 ( Ω) SΣ 2 ≥ ∗ 2 ∗ | − | − ∗ | | 2c (s2 + ψ) IIt IIΣ 2 c (s2 + ψ)(1 + 2 IIΣ 2) . − 3 | − | − 3 | | If κ 1/(48c ), it follows from (5.9) that 2c (s2 + ψ) ( Ω)/6. Since SΣ 2 c ψ and ≤ 3 3 ≤ ∗ | | ≤ 4 IIΣ 2 c , | | ≤ 4 d 1 ( Ω) ∆Γt ( Ω) + ( Ω) IIt IIΣ 2 c (s2 + ψ) . (6.4) dt ∗ ≥ ∗ 3 ∗ | − | − 5 26 By combining it with (6.3), (5.9) and (5.10), we have d 1 (1 ( Ω) + c ψ) ∆Γt (1 ( Ω) + c ψ) ( Ω) IIt IIΣ 2 c (1 ( Ω) + ψ) dt 6 6 3 7 − ∗ ≤ − ∗ − ∗ | − | − − ∗ (6.5) Γt 1 t Σ 2 c7 ∆ (1 ( Ω) + c6ψ) ( Ω) II II (1 ( Ω) + c6ψ) ≤ − ∗ − 3 ∗ | − | − c6 − ∗ where c =1+ c /c . By the maximum principle, max (1 ( Ω) + c ψ) is non-increasing. 6 5 1 Γt − ∗ 6 The evolution equation for the norm of the second fundamental form for a mean curvature flow is derived in [20, Proposition 7.1]. In particular, IIt 2 = h˜2 satisfies the following | | α,i,k αik equation along the flow: P

d t 2 Γt t 2 Γt t 2 II =∆ II 2 II + 2 ( e R) + ( e R) h˜αij dt| | | | − |∇ | ∇˜k α˜˜i˜jk˜ ∇˜j α˜k˜˜ik˜ h i 4R h˜αlkh˜αij + 8R h˜βikh˜αij 4R h˜αljh˜αij + 2R h˜βijh˜αij − ˜l˜i˜jk˜ α˜β˜˜jk˜ − ˜lk˜˜ik˜ α˜k˜β˜k˜ (6.6) 2 2

+ 2 h˜αikh˜γjk h˜αjkh˜γik + 2 h˜αijh˜αkl . − α,γ,i,j ! α ! X Xk i,j,k,lX X It follows that d IIt 2 ∆Γt IIt 2 2 Γt IIt 2 + c IIt 4 + IIt 2 + 1 . (6.7) dt| | ≤ | | − |∇ | 8 | | | | The quartic term IIt 4 could potentially lead to the ﬁnite time blow-up of IIt . We apply the | | | | same method in [22]: use the evolution equation of ( Ω)p to help. Let p be a constant no less ∗ than 1, whose precise value will be determined later. According to (6.4),

d p p 1 d ( Ω) = p( Ω) − ( Ω) dt ∗ ∗ dt ∗ p 1 Γt p p t Σ 2 2 p( Ω) − ∆ ( Ω) + ( Ω) II II c p(s + ψ) ≥ ∗ ∗ 3 ∗ | − | − 5 Γt p p 2 Γt t 2 p p t Σ 2 2 =∆ ( Ω) p(p 1)( Ω) − II + ( Ω) II II c p(s + ψ) . ∗ − − ∗ |∇ | 3 ∗ | − | − 5 After an appeal to Lemma 5.4, d p ( Ω)p ∆Γt ( Ω)p + 1 c p s2 ( Ω)p IIt IIΣ 2 c p2(s2 + ψ) . dt ∗ ≥ ∗ 3 − 9 ∗ | − | − 9 If κ 1/(24c p), it follows from (5.9) that c p s2 1/12. It together with (6.3) gives that ≤ 9 9 ≤ d p ( Ω)p Kp2ψ ∆Γt ( Ω)p Kp2ψ + ( Ω)p Kp2ψ IIt IIΣ 2 (6.8) dt ∗ − ≥ ∗ − 4 ∗ − | − | p 2 where K = c9/c1. The maximum principle implies that if ( Ω) Kp ψ > 0 on Γ, then p 2 ∗ − minΓt ( Ω) Kp ψ is non-decreasing. Moreover, for any p 1, we may choose κ such that ∗ − p 2 ≥ (6.2) implies that ( Ω) Kp ψ > 1/2. ∗ − 27 Denote ( Ω)p Kp2ψ by η. Due to (6.7) and (6.8), ∗ −

d 1 t 2 1 Γt t 2 1 Γt t 2 1 t 4 t 2 (η− II ) η− ∆ II 2η− II + c η− II + II + 1 dt | | ≤ | | − |∇ | 8 | | | | 2 t 2 Γt p t Σ 2 η− II ∆ η + η II II . − | | 4 | − | Since ∆Γt (η 1 IIt 2)= η 1∆Γt IIt 2 η 2 IIt 2∆Γt η 2η 1 Γt η, Γt (η 1 IIt 2) and IIt IIΣ 2 − | | − | | − − | | − − h∇ ∇ − | | i | − | ≥ 1 IIt 2 c , we have 2 | | − 10

d 1 t 2 Γt 1 t 2 1 Γt Γt 1 t 2 (η− II ) ∆ (η− II ) + 2η− η, (η− II ) dt | | ≤ | | h∇ ∇ | | i p 1 t 4 1 t 4 t 2 η− II + c η− II + (p + 1) II + 1 − 8 | | 11 | | | | Γt 1 t 2 1 Γt Γt 1 t 2 ∆ (η− II ) + 2η− η, (η− II ) ≤ | | h∇ ∇ | | i (6.9) p 1 t 2 2 1 t 2 2c (η− II ) + c (η− II ) + 2c . − 8 − 10 | | 11 | | 10 Choose p 16c +1. It follows from the maximum principle that η 1 IIt 2 is uniformly bounded, ≥ 11 − | | and hence there is no ﬁnite time singularity.

The 0 convergence is easy to come by. The diﬀerential inequality (6.3) implies that ψ C converges to zero exponentially. Similarly, it follows from (6.10) that 1 ( Ω)+ c ψ converges − ∗ 6 to zero exponentially. Therefore, Ω converges to 1 exponentially, and we conclude the 1 ∗ C convergence.

For the 2 and smooth convergence, consider η = ( Ω)p Kp2ψ. It follows from the above C ∗ − discussion that η has a positive lower bound. It is clear that η 1. Moreover, η converges to ≤ 1 as t . Integrating (6.8) gives →∞ t Σ 2 dη II II dµt c dµt . | − | ≤ 12 dt ZΓt ZΓt 2 Recall that the Lie derivative of dµt in Ht is H dµt; see [20, 2]. It follows that −| t | § 1 t Σ 2 d 2 II II dµt η dµt + η Ht dµt (6.10) c | − | ≤ dt | | 12 ZΓt ZΓt ZΓt We claim that the improper integral of the right hand side for 0 t < converges. To d 2 ≤ ∞ start, note that dµt = Ht dµt 0. Thus, vol(Γt) = dµt is positive and non- dt Γt − Γt | | ≤ Γt increasing, and must converge as t . For the ﬁrst term on right hand side of (6.10), R R →∞ R t d η dµs ds = η dµt η dµ ds − 0 Z0 ZΓs ZΓt ZΓ0 28 Since η converges to 1 (uniformly) and vol(Γt) converges as t , η dµt converges as → ∞ Γt t . For the second term on the right hand side of (6.10), →∞ R t t t 2 2 d η Hs dµs ds Hs dµs ds = dµs ds | | ≤ | | −ds Z0 ZΓs Z0 ZΓs Z0 ZΓs = vol(Γ ) vol(Γt) vol(Γ ) . 0 − ≤ 0 It is bounded from above, and is clearly non-decreasing in t. Therefore, it converges as t . →∞ It follows from the claim and (6.10) that

∞ t Σ 2 II II dµt dt< . (6.11) | − | ∞ Z0 ZΓt On the other hand, IIt IIΣ 2 obeys a differential inequality of the same form as (6.7): | − | d IIt IIΣ 2 ∆Γt IIt IIΣ 2 + c IIt IIΣ 4 + IIt IIΣ 2 + 1 . (6.12) dt| − | ≤ | − | 13 | − | | − | The derivation for this inequality is in Appendix B. By (6.12) and the uniform boundedness of t d t Σ 2 II , II II dµt is bounded from above uniformly. Due to Lemma 6.3, which is proved | | dt Γt | − | at the endR of this subsection, we find that t Σ 2 lim II II dµt = 0 . (6.13) t 0 | − | → ZΓt Since IIt is uniformly bounded, in view of (6.12), we find that d IIt IIΣ 2 ∆Γt IIt IIΣ 2 | | dt | − | − | − | is bounded from above, independent of t. This together with (6.13) implies that IIt IIΣ 2 0 | − | → as t in the sup norm. →∞ We can then write Γt as a graph (in the geodesic coordinate defined in 2.2) over Σ defined § by yα = f α, α = n + 1, n + m for f α functions on Σ. The above estimates imply that f α t · · · t t converges to 0 in 2 as t . As a mean curvature flow, f α satisfies a second order quasilinear C →∞ t α parabolic system, and standard arguments lead to the smooth convergence of ft .

Lemma 6.3. Let a > 0 and f(t) be a smooth function for t (a, ). Suppose that f(t) 0, ∈ ∞ ≥ ∞ f(t)dt converges, and f (t) C for some constant C > 0. Then, f(t) 0 as t . a ′ ≤ → →∞ RProof. It follows from f (t) C that f(t) f(t ) C(t t) for any t > t > a. Since f(t) 0 ′ ≤ ≥ 1 − 1 − 1 ≥ and ∞ f(t)dt < , given any ǫ (0, 1), there exists an Aǫ > a such that ∞ f(t)dt<ǫ. a ∞ ∈ Aǫ Thus,R for any t1 > Aǫ + 1 > Aǫ + √ǫ, R t1 t1 ǫ> f(t)dt (f(t1) C(t1 t)) dt t1 √ǫ ≥ t1 √ǫ − − Z − Z − 1 = √ǫ (f(t ) Ct )+ C √ǫt ǫ . 1 − 1 1 − 2 1 It follows that f(t) < (1 + 2 C)√ǫ for any t > Aǫ + 1. 29 Appendix A. Computations related to strong stability

For minimal Lagrangians in a K¨ahler–Einstein manifold and coassociatives in a G2 manifold, the 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:

Σ Rijkℓ R = hαiℓhαjk hαikhαjℓ . (A.1) − ijkℓ −

A.1. Minimal Lagrangians in Kähler–Einstein manifolds. Let (M 2n,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,

RACBC = RicAB = c gAB . C X n 2n A submanifold L M is Lagrangian if ω L vanishes. It implies that J induces an iso- ⊂ | morphism between its tangent bundle TL and normal bundle NL. In terms of the notations introduced in 2.1, the correspondence is § i i v ei v Jei . (A.2) ←→

In particular, if e , , en is an orthonormal frame for TL, Je , , Jen is an orthonor- { 1 · · · } { 1 · · · } mal frame for NL. Denote Jek by eJ(k), and let

Ckij = h = e ej, Jek . J(k)ij h∇ i i

Since J is parallel, it is easy to verify that Ckij is totally symmetric. Now, suppose that L is also minimal. By using the correspondence (A.2), the strong stability condition (3.2) can be rewritten as follows.

k ℓ k ℓ k ℓ k ℓ k ℓ R v v Ckij Cℓij v v = c gkℓ v v + R v v Ckij Cℓij v v − iJ(k)iJ(ℓ) − − J(i)J(k)J(i)J(ℓ) − 2 k ℓ k ℓ = c v + Rikiℓ v v Ckij Cℓij v v − | | − 2 L k ℓ k ℓ k ℓ = c v + R v v + Cjki Cjℓi v v Ckij Cℓij v v − | | ikiℓ − = 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 Ckij 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. 30 A.2.1. Four dimensional Riemannian geometry. The Riemann curvature tensor has a nice decomposition in 4 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

1 Σ R(ei ej)= R ek eℓ . ∧ 2 kℓij ∧ 2 2 2 In 4 dimensions, Λ decomposes into self-dual, Λ+, and anti-self-dual part, Λ . In terms of the 2 2 2 R − decomposition Λ = Λ+ Λ , the curvature map has the form ⊕ − W + s I B R = + 12 . BT W + s I " − 12 # Σ Here, s = Rijij 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 e e e e , e e + e e , e e e e , the lower-right { 1 ∧ 2 − 3 ∧ 4 1 ∧ 3 2 ∧ 4 1 ∧ 4 − 2 ∧ 3} block W + s I is − 12 Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ R1212 + R3434 2R1234 R1213 + R1224 R3413 R3424 R1214 R1223 R3414 + R3423 − − − − − 1 Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ R1312 R1334 + R2412 R2434 R1313 + R2424 +2R1324 R1314 R1323 + R2414 R2423 . 2 − − − −  Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ  R1412 R1434 R2312 + R2334 R1413 + R1424 R2313 R2324 R1414 + R2323 2R1423   − − − − −   (A.3) The operator will be needed is W s I = (W + s I) s I. One-fourth of the scalar curvature − − 6 − 12 − 4 is s 1 = RΣ + RΣ + RΣ + RΣ + RΣ + RΣ . (A.4) 4 2 1212 3434 1313 2424 1313 2424 A.2.2. G2 geometry. A 7-dimensional Riemannian manifold M with G2 holonomy can be char- acterized by the existence of a parallel, positive 3-form ϕ. A complete story can be found in [10, ch.11]. In terms of a local orthonormal coframe, the 3-form and its Hodge star are ϕ = ω567 + ω125 ω345 + ω136 + ω246 + ω147 ω237 , − − (A.5) ϕ = ω1234 ω1267 + ω3467 + ω1357 + ω3457 ω1456 + ω2356 ∗ − − where ω123 is short for ω1 ω2 ω3. It is known that the holonomy is G if and only if ϕ = 0, ∧ ∧ 2 ∇ which is also equivalent to dϕ =0=d ϕ. ∗ Remark A.1. There are two commonly used conventions for the 3-form; see [11] for instance. The convention here is the same as that in [12]; the deformation of coassociatives will then be determined by anti-self-dual harmonic forms. If one use the convention in [10], the deformation of coassociatives will be determined by self-dual harmonic forms. 31 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, e e = e . Since ϕ and the metric tensor are both parallel, is parallel as well. 1 × 2 5 × As a consequence,

R(eA, eB)(e e )= R(eA, eB )e e + e R(eA, eB)e , 1 × 2 1 × 2 1 × 2 and its e -component gives R AB R AB R AB = 0 for any A, B 1,..., 7 . In total, 3 53 − 62 − 71 ∈ { } the parallelity of leads to following seven identities: ×

R52AB + R63AB + R74AB = 0 , R67AB + R12AB R34AB = 0 , R51AB R64AB + R73AB = 0 , − − R57AB + R13AB + R24AB = 0 , (A.6) R54AB + R61AB R72AB = 0 , − − R56AB R14AB R23AB = 0 . R AB + R AB + R AB = 0 , − − − 53 62 71

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

A.2.3. Coassociative geometry. According to [8, IV], an oriented, 4-dimensional submanifold § Σ of a G manifold is said to be coassociative if ϕ coincides with the volume form of the 2 ∗ |Σ induced metric. Harvey and Lawson also proved that being coassociative is equivalent to that ϕ vanishes. Similar to the Lagrangian case, the normal bundle of a coassociative submanifold |Σ is canonically isomorphic to an intrinsic bundle. The following discussion is basically borrowed from [12, 4]. § Orthonormal frame. Suppose that Σ M is coassociative. One can ﬁnd a local orthonormal ⊂ frame e , , e such that e , e , e , e are tangent to Σ, e , e , e are normal to Σ, and { 1 · · · 7} { 1 2 3 4} { 5 6 7} ϕ takes the form (A.5) in this frame. Here is a sketch of the construction. Start with a unit normal vector, e , and a unit tangent vector, e , of Σ. Let e = e e . Then, set e to be a 5 1 2 5 × 1 3 unit vector tangent to Σ and orthogonal to e , e . Finally, let e = e e , e = e e and { 1 2} 4 3 × 5 6 1 × 3 e = e e . 7 3 × 2 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:

V (V y ϕ) . (A.7) 7→ |Σ In terms of the above frame, e corresponds to ω12 ω34, e corresponds to ω13 + ω24, and e 5 − 6 7 corresponds to ω14 ω23. − 32 As shown in [8], a coassociative submanifold must be minimal. In fact, its second fundamental form has certain symmetry. For instance,

h i = e e , e = e , e (e e ) 51 h∇ i 1 5i −h 1 ∇ i 6 × 7 i = e , ( e e ) e e , e ( e e ) −h 1 ∇ i 6 × 7i − h 1 6 × ∇ i 7 i = e , e e + e , e e = h i h i . −h 4 ∇ i 6i h 3 ∇ i 7i 64 − 73 What follows are all the relations:

h52i + h63i + h74i = 0 , h54i + h61i h72i = 0 , − (A.8) h i h i + h i = 0 , h i + h i + h i = 0 51 − 64 73 − 53 62 71 for any i 1, 2, 3, 4 . These relations imply that the mean curvature vanishes. They can be ∈ { } encapsulated as ej II(ei, ej ) = 0. j × P A.2.4. Strong stability for coassociatives. For any sections of NΣ, v, denote the symmetric bilinear form on the left hand side of (3.2) by Q(v, v). Under the identiﬁcation (A.7), Q˜(v, v)= 2 vT W v + s v 2 is also a symmetric bilinear form. − − 3 | | We now check that Q(v, v)= Q˜(v, v) for any unit vector v NpΣ 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.3) and (A.4) that

˜ Σ Σ Σ Σ Σ Q(v, v)= R1313 + R2424 + R1313 + R2424 + 2R1234 .

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

2 Q(v, v)= Ri i (h ij) − 5 5 − 5 i i,j X X 2 = R + R (h ij ) 6565 7575 − 5 i,j X 2 = R + R + R + R 2R + 2R (h ij ) 1313 2424 1313 2424 − 1423 1324 − 5 i,j X 2 = R + R + R + R + 2R (h ij ) . 1313 2424 1313 2424 1234 − 5 i,j X The second equality follows from Ricci ﬂatness. The third equality uses (A.6). The last equality is the ﬁrst Bianchi identity. With the Gauss equation and some simple manipulation,

Q(v, v) Q˜(v, v) − 2 2 2 = (hα + hα ) + (hα hα ) (hα + hα )(hα + hα ) (h ij) . (A.9) 14 23 13 − 24 − 11 22 33 44 − 5 α i,j X X 33 By appealing to (A.8), h + h = h h = h + h , h h = h h , 614 623 − 522 − 544 511 533 613 − 624 − 512 − 534 h + h = h + h , h h = h + h = h h , 714 723 − 512 534 713 − 724 − 522 533 − 511 − 544 and h + h = h h = h + h , 611 622 − 633 − 644 − 514 523 h + h = h h = h + h . 711 722 − 733 − 744 513 524 By using these relations, it is not hard to verify that (A.9) vanishes. Therefore, the strong stability condition (3.2) is equivalent to the positivity of 2W + s . − − 3 As a ﬁnal remark, this equivalence can also be seen by combining [12, Theorem 4.9] and the Weitzenb¨ock formula [7, Appendix C]. Nevertheless it is nice to derive the equivalence directly by highlighting the geometry of G2.

Appendix B. Evolution equation for tensors

Suppose that Ψ be 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) t . The connection naturally induces a connection ˜ on this bundle. The only ∗ |Γ ∇ ∇ 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) t ∇ ∗ ⊗ ∗ ⊗ ∗ |Γ 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, d IIt Ψ 2 ∆Γt IIt Ψ 2 ˜ (IIt Ψ) 2 + c( IIt Ψ 4 + IIt Ψ 2 + 1) (B.1) dt| − | ≤ | − | − |∇ − | | − | | − | 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,ǫ) M. For any 1 × n → p Γ0 and t0 [0,ǫ), choose a geodesic coordinate for Γ0 at p: x˜ , , x˜ . We also choose ∈ ∈ 34 { · · · } a local orthonormal frame e˜α for NΓt. The following computations on derivatives are always { } evaluated at the point (p,t0).

Let H = h˜αe˜α be the mean curvature vector of Γt. The components of the second fundamental form and its covariant derivative are denoted by ˜ hαij = ∂˜∂ ˜, e˜α = II(∂˜, ∂ ˜, e˜α) , h∇ i j i i j ˜ hαij,k = ( ∂ ˜ II)(∂˜, ∂ ˜, e˜α) , ∇ k i j ˜ hα,k = ∂⊥˜ H, e˜α . h∇ k i

At (p,t0), h˜α = h˜αkk and h˜α,i = h˜αkk,i.

Note that on Γ [0,ǫ), H is ∂t, and thus commutes with ∂ . It follows that the evolution 0 × ˜i of the metric is: d g˜ij = H ∂ , ∂ = H ∂ , ∂ + ∂ , H ∂ dt h ˜i ˜ji h∇ ˜i ˜ji h ˜i ∇ ˜ji ˜ ˜ = H, ∂˜∂ ˜ ∂ ˜ ∂˜,H = 2hαhαij , (B.2) −h ∇ i ji − h∇ j i i − d g˜ij = 2h˜ h˜ . (B.3) dt α αij

The covariant derivative of ∂˜i ande ˜α along H can be expressed as follows:

H ∂ = H ∂ , ∂ ∂ + H ∂ , e˜α e˜α ∇ ˜i h∇ ˜i ˜ji ˜j h∇ ˜i i = h˜αh˜αij ∂ + h˜α,i e˜α , (B.4) − ˜j H e˜α = H e˜α, ∂ ∂ + H e˜α, e˜β e˜β ∇ h∇ ˜ii ˜i h∇ i = h˜α,i ∂ + H e˜α, e˜β e˜β . (B.5) − ˜i h∇ i 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.

H Ψ= ⊥ Ψ= Σ Ψ+ Ψ ( ∂ ˜ ∂ ˜j ) ∂ ∂ ˜j ∂ ˜ ∂ ˜j ∇ ∇ ∇ j −∇∇ ˜j ∇∇ j

= ˜ ˜ Ψ ˜ Σ Ψ + Ψ+ Ψ ∂ j˜ ∂ j˜ ∂ ˜ ∂ ˜j ∂ ˜j ∂ ∂ ˜ ∇ ∇ − ∇ ∂ ˜ j −∇ ∇ ∇ ˜j j ∇ j ∇ 2 = ˜ ∗ ˜ Ψ+tr ( Ψ) . (B.6) −∇ ∇ Γt ∇ Σ ∗ Indeed, ∂ ˜ ∂ ˜j is zero at (p,t0). The tensor Ψ is deﬁned in the ambient space, and has ∇ j ∇ ∇ 2 nothing to do with the submanifold Γt. It follows from (B.6) that the evolution of of Ψ is | | d 2 Ψ = H Ψ, Ψ = 2 H Ψ, Ψ dt| | h i h∇ i 2 = 2 ˜ ∗ ˜ Ψ, Ψ + 2 tr ( Ψ), Ψ − h∇ ∇ i h Γt ∇ i 2 =∆Γt Ψ 2 2 ˜ Ψ 2 + 2 tr ( Ψ), Ψ (B.7) | | − |∇ | h Γt ∇ i 35 t ik jl The next task is to calculate the evolution equation for II , Ψ =g ˜ g˜ h˜αklΨ˜ αij where Ψ˜ αij = h i Ψ(∂˜i, ∂ ˜j, e˜α). According to (B.4) and (B.5),

d Ψ˜ = H Ψ(∂ , ∂ , e˜ ) dt αij ˜i ˜j α = ( H Ψ)(∂ , ∂ , e˜α)+Ψ(∂ , ∂ , H e˜α)+Ψ( H ∂ , ∂ , e˜α)+Ψ(∂ , H ∂ , e˜α) ∇ ˜i ˜j ˜i ˜j ∇ ∇ ˜i ˜j ˜i ∇ ˜j ˜ ˜ ˜ = ( H Ψ)(∂˜i, ∂ ˜j, e˜α) hα,kΨkij + H e˜α, e˜β Ψβij ∇ − h∇ i (B.8) h˜γ h˜γikΨ˜ αkj + h˜γ,iΨ˜ αγj h˜γ h˜γjkΨ˜ αik + h˜γ,jΨ˜ αiγ − −

The diﬀerence between ˜ ˜ IIt and IIt is: ∇∗∇ ∇∗∇ ˜ ˜ t t ˜ ˜ t t ∗ II ∗ II = ∂ ˜ ∂ ˜ II ∂ ˜ ∂ ˜ II (B.9) ∇ ∇ −∇ ∇ ∇ k ∇ k −∇ k ∇ k Since

˜ t t ( ∂ ˜ II )( , , ) = ( ∂ ˜ II )( , , ) ∇ k · · · ∇ k · · · t t T t T t = ∂ ˜ II ( , , ) II ( ∂ ˜ ) , , II , ( ∂ ˜ ) , II , , ( ∂ ˜ )⊥ , k · · · − ∇ k · · · − · ∇ k · · − · · ∇ k · ˜ t the tensor ∂ ˜ II has only the following components: ∇ k ˜ t t ˜ ( ∂ ˜ II )(∂˜, ∂ ˜, e˜α) = ( ∂ ˜ II )(∂˜, ∂ ˜, e˜α)= hαij,k , ∇ k i j ∇ k i j ˜ t ˜ ˜ ( ∂ ˜ II )(∂˜i, ∂ ˜j, ∂ ˜l)= hαijhαkl , ∇ k − (B.10) ˜ t ˜ ˜ ( ∂ ˜ II )(˜eβ, ∂ ˜, e˜α)= hβkihαij , ∇ k j ˜ t ˜ ˜ ( ∂ ˜ II )(∂˜, e˜β, e˜α)= hβkjhαij . ∇ k i

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

˜ ˜ t ˜ t ˜ t ( ∂ ˜ ∂ ˜ II )(∂˜, ∂ ˜, e˜α)= ∂ ˜ ( ∂ ˜ II )(∂˜, ∂ ˜, e˜α) ( ∂ ˜ II ) ∂˜, ∂ ˜, ∂ ˜ e˜α ∇ k ∇ k i j k ∇ k i j − ∇ k i j ∇ k ˜ t ˜ t ( ∂ ˜ II ) ∂ ˜ ∂˜, ∂ ˜, e˜α ( ∂ ˜ II ) ∂˜, ∂ ˜ ∂ ˜, e˜α − ∇ k ∇ k i j − ∇ k i ∇ k j t ˜ t T = ( ∂ ˜ ∂ ˜ II)(∂˜, ∂ ˜, e˜α) ( ∂ ˜ II ) ∂˜, ∂ ˜, ( ∂ ˜ e˜α) ∇ k ∇ k i j − ∇ k i j ∇ k ˜ t ˜ t ( ∂ ˜ II ) ( ∂ ˜ ∂˜)⊥, ∂ ˜, e˜α ( ∂ ˜ II ) ∂˜, ( ∂ ˜ ∂ ˜)⊥, e˜α − ∇ k ∇ k i j − ∇ k i ∇ k j t ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ = ( ∂ ˜ ∂ ˜ II)(∂˜, ∂ ˜, e˜α) hβij hβklhαkl hβklhαljhβki hβkj hβklhαil . ∇ k ∇ k i j − − − Use (B.9) to rewrite the above computation as

t t ( ˜ ∗ ˜ II ∗ II )(∂ , ∂ , e˜α)= h˜βijh˜βklh˜αkl + h˜βklh˜αljh˜βki + h˜βkjh˜βklh˜αil (B.11) ∇ ∇ −∇ ∇ ˜i ˜j 36 The tensor IIt does not have other components. However, ˜ ˜ IIt does. ∇∗∇ ∇∗∇

˜ ˜ t ˜ ˜ t ( ∗ II )(∂˜, ∂ ˜, ∂ ˜)= ( ∂ ˜ ∂ ˜ II )(∂˜, ∂ ˜, ∂ ˜) ∇ ∇ i j l − ∇ k ∇ k i j l ˜ t ˜ t = ∂ ˜ ( ∂ ˜ II )(∂˜, ∂ ˜, ∂ ˜) + ( ∂ ˜ II )( ∂ ˜ ∂˜, ∂ ˜, ∂ ˜) − k ∇ k i j l ∇ k ∇ k i j l ˜ t ˜ t + ( ∂ ˜ II )(∂˜, ∂ ˜ ∂ ˜, ∂ ˜) + ( ∂ ˜ II )(∂˜, ∂ ˜, ∂ ˜ ∂ ˜) ∇ k i ∇ k j l ∇ k i j ∇ k l ˜ ˜ ˜ ˜ = ∂ k˜(hαijhαkl)+ hαij,khαkl

= 2h˜αij,kh˜αkl + h˜αijh˜αkl,k ˜ ˜ ˜ ˜ ˜ = 2hαij,khαkl + hαijhα,l + hαijRα˜k˜k˜l˜ . (B.12)

˜ ˜ ˜ ˜ The second last equality uses the fact that 0 = ∂⊥˜ e˜α, e˜β hβij hαkl ∂⊥˜ e˜α, e˜β hαij hβkl. The h∇ k i − h∇ k i last equality uses the Codazzi equation (2.2). Similarly,

t ( ˜ ∗ ˜ II )(˜eβ, ∂ , e˜α)= 2h˜αlj,kh˜βkl h˜βkl,kh˜αjl ∇ ∇ ˜j − − = 2h˜αlj,kh˜βkl h˜β,lh˜αjl R h˜αjl − − − β˜k˜k˜l˜ t ( ˜ ∗ ˜ II )(∂ , e˜β, e˜α)= 2h˜αil,kh˜βkl h˜βkl,kh˜αil ∇ ∇ ˜i − − ˜ ˜ ˜ ˜ ˜ = 2hαil,khβkl hβ,lhαil Rβ˜k˜k˜˜lhαil − − − (B.13) t ( ˜ ∗ ˜ II )(∂ , e˜α, ∂ )= 2h˜αklh˜βilh˜βkj ∇ ∇ ˜i ˜j − t ( ˜ ∗ ˜ II )(˜eβ, e˜γ , e˜α)= 2h˜βklh˜γkjh˜αlj ∇ ∇ − t ( ˜ ∗ ˜ II )(˜eβ , ∂ , ∂ ) = 2h˜βklh˜αljh˜αki ∇ ∇ ˜j ˜i t ( ˜ ∗ ˜ II )(˜eβ, e˜α, ∂ ) = 0 ∇ ∇ ˜j 37 The evolution equation for h˜αij was derived in [20, Proposition 7.1]. With (B.8) and (B.6), we have

d t d ik jl II , Ψ = g˜ g˜ h˜αklΨ˜ αij dth i dt d d = 2h˜ h˜ h˜ Ψ˜ + 2h˜ h˜ h˜ Ψ˜ + h˜ Ψ˜ + h˜ Ψ˜ βik β αkj αij βjl β αil αij dt αij αij αij dt αij t = 2h˜βikh˜βh˜αkjΨ˜ αij + 2h˜βjlh˜βh˜αilΨ˜ αij ∗ II , Ψ − h∇ ∇ i + ( e R) Ψ˜ αij + ( e R) Ψ˜ αij 2R h˜αlkΨ˜ αij + 2R h˜βikΨ˜ αij ∇˜k α˜˜i˜jk˜ ∇˜j α˜k˜˜ik˜ − ˜l˜i˜jk˜ α˜β˜˜jk˜ + 2R h˜βjkΨ˜ αij R h˜αljΨ˜ αij R h˜αliΨ˜ αij + R h˜βijΨ˜ αij α˜β˜˜ik˜ − ˜lk˜˜ik˜ − ˜lk˜˜jk˜ α˜k˜β˜k˜ h˜αil h˜βljh˜β h˜βlkh˜βjk Ψ˜ αij h˜αlk h˜βljh˜βik h˜βlkh˜βij Ψ˜ αij − − − − h˜βik h˜βljh˜αlk h˜βlkh˜αlj Ψ˜ αij h˜αjkh˜βikh˜βΨ˜ αij + h˜βij e˜β, H e˜α Ψ˜ αij − − − h ∇ i t t II , ˜ ∗ ˜ Ψ + II , tr ( ∗ Ψ) h˜αijh˜α,kΨ˜ kij + h˜αij H e˜α, e˜β Ψ˜ βij − h ∇ ∇ i h Γt ∇ ∇ i− h∇ i h˜αij h˜γh˜γikΨ˜ αkj + h˜αijh˜γ,iΨ˜ αγj h˜αijh˜γ h˜γjkΨ˜ αik + h˜αijh˜γ,jΨ˜ αiγ − − t t t 2 = ˜ ∗ ˜ II , Ψ II , ˜ ∗ ˜ Ψ + II , tr ( Ψ) −h∇ ∇ i − h ∇ ∇ i h Γt ∇ i + ( e R) Ψ˜ αij + ( e R) Ψ˜ αij 2R h˜αlkΨ˜ αij + 2R h˜βikΨ˜ αij ∇˜k α˜˜i˜jk˜ ∇˜j α˜k˜˜ik˜ − ˜l˜i˜jk˜ α˜β˜˜jk˜ + 2R h˜βjkΨ˜ αij R h˜αljΨ˜ αij R h˜αliΨ˜ αij + R h˜βijΨ˜ αij α˜β˜˜ik˜ − ˜lk˜˜ik˜ − ˜lk˜˜jk˜ α˜k˜β˜k˜ + R h˜αijΨ˜ lij R h˜αjkΨ˜ αβj R h˜αilΨ˜ αiβ α˜k˜k˜˜l − β˜k˜k˜˜l − β˜k˜k˜˜l 2h˜αil h˜βljh˜β h˜βlkh˜βjk Ψ˜ αij 2h˜βik h˜βljh˜αlk h˜βlkh˜αlj Ψ˜ αij − − − − 2h˜αjkh˜βikh˜βΨ˜ αij 2h˜αklh˜βilh˜βkjΨ˜ jiα 2h˜βklh˜γkjh˜αljΨ˜ αβγ + 2h˜βklh˜αljh˜αkiΨ˜ iβj − − − + 2h˜αij,kh˜αklΨ˜ lij 2h˜αlj,kh˜βklΨ˜ αβj h˜αil,kh˜βklΨ˜ αiβ . − − The last equality uses (B.11), (B.12) and (B.13) to replace IIt by ˜ ˜ IIt. ∇∗∇ ∇∗∇ By the Cauchy–Schwarz inequality, d 1 IIt, Ψ ∆Γt IIt, Ψ + 2 ˜ IIt, ˜ Ψ IIt 2 + c( IIt 4 + IIt 2 + 1) . dth i− h i h∇ ∇ i ≤ 2|∇ | | | | |

This together with (6.7) and (B.7) imply that

d t 2 Γt t 2 t 2 t 2 t 4 t 2 II Ψ ∆ II Ψ 2 ˜ (II Ψ) + II + c′( II + II + 1) . dt| − | ≤ | − | − |∇ − | |∇ | | | | | According to (B.10),

t 2 2 2 t 2 t 4 2 ˜ (II Ψ) ˜ II ˜ Ψ II + c′′ II Ψ . |∇ − | ≥ |∇ | − |∇ | ≥ |∇ | | | − |∇ | Hence,

d t 2 Γt t 2 t 2 t 4 t 2 II Ψ ∆ II Ψ ˜ (II Ψ) + c′′′( II + II + 1) dt| − | ≤ | − | − |∇ − | | | | | 38 By the triangle inequality IIt 2 IIt Ψ 2 + Ψ 2, it ﬁnishes the proof of the proposition. | | ≤ | − | | |

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Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan E-mail address: [email protected]

Department of Mathematics, Columbia University, New York, NY 10027, USA E-mail address: [email protected]