The Stability of Self-Shrinkers of Mean Curvature Flow in Higher Codimension

In this paper, we generalize Colding and Minicozzi's work \cite{CM} on the stability of self-shrinkers in the hypersurface case to higher co-dimensional cases. The first and second variation formulae of the $F$-functional are derived and an equivalent condition to the stability in general codimension is found. Moreover, we show that the closed Lagrangian self-shrinkers given by Anciaux in \cite{An} are unstable.


Introduction
Let X : Σ → R m be an isometric immersion of an n-dimensional manifold Σ in the Euclidean space R m . The mean curvature flow of X is a family of immersions X t : Σ → R m that satisfies where H(x, t) is the mean curvature vector of X t (Σ) at X t (x) and v ⊥ denotes the projection of v into the normal space of X t (Σ). Mean curvature flow of a submanifold in a Riemannian manifold can also be defined similarly. Because the mean curvature vector points in the direction in which the area decreases most rapidly, mean curvature flow is thus a canonical way to construct minimal submanifolds. It also improves the geometric properties of a object along the flow (e.g., see [7]) A submanifold Σ in R m is called a self-shrinker if its position vector X : Σ → R m satisfies The terminology comes from the fact that √ 1 − tX(Σ) is a solution of mean curvature flow, i.e., a self-shrinker evolves homothetically along mean curvature flow in a shrinking way. Moreover, self-shrinkers describe all possible central blow-up limits of a finite-time singularity of the mean curvature flow. This follows from Huisken's monotonicity formula [8], and its generalization to type II singularity by Ilmanen [10] and White. Singularities will occur in general along mean curvature flow and are obstacles to continue the flow. It is therefore an important issue to understand singularities and the candidates of their blow-up limits, self-shrinkers.
Standard sphere S n ( √ 2n) and cylinder S k ( √ 2k) × R n−k are simple examples of self-shrinkers in R m . Abresch and Langer [1] found all immersed closed self-shrinkers in the plane. For other complete hypersurfaces case, Huisken [9] classified all self-shrinkers with nonnegative mean curvature and bounded geometry. The bounded geometry condition is later weakened to polynomial volume growth by Colding and Minicozzi in [5]. On the other hand, many other different co-dimension one self-shrinkers are found (e.g., see [3]), and a classification of all self-shrinkers is not expected. Our understanding on self-shrinkers in higher co-dimension is even more limited. Smoczyk obtained a classification for self-shrinkers with parallel principal normal ν ≡ H/|H| and bounded geometry in [12]. Various different families of Lagrangian selfshrinkers, which are of middle dimension, are constructed in [2], [11] and [6].
Adapted from the back heat kernel introduced by Huisken in [8], Colding and Minicozzi [5] defined a functional F by for any submanifold X : Σ n → R n+1 , x ∈ R n+1 and t > 0. One of the main properties of this functional is that (Σ, x 0 , t 0 ) is a critical point of F iff Σ satisfies H = − (X−x 0 ) ⊥ 2t 0 . Especially, it is a self-shrinker when x 0 = 0 and t 0 = 1. They proved that if an n-dimensional complete smooth embedded self-shrinker Σ n without boundary and with polynomial volume growth in R n+1 is F -stable with respect to compactly supported variations, then it must be the round sphere or a hyperplane. Here F -stable means that for every compactly supported smooth variation Σ s with Σ 0 = Σ, there exist variations x s of 0 and t s of 1 such that ∂ 2 ∂s 2 F (Σ s , x s , t s ) ≥ 0 at s = 0. The importance of the study is that roughly speaking, the blow-up near type-I singularity of mean curvature flow for generic initial data gives stable selfshrinkers (see [5] for exact statement.) In this paper, we intend to generalize Colding and Minicozzi's work [5] to higher co-dimensional cases. The domain of the functional F is now (Σ, x, t) for Σ n ⊂ R m , x ∈ R m and t > 0. Colding and Minicozzi's classification on stable self-shrinkers in co-dimension one is first to conclude that the mean curvature function h is the first eigenvalue of an elliptic operator, it then implies h ≥ 0, and Huisken's classification of self-shrinkers with nonnegative h will lead to the conclusion. Although the counter part of Huisken's result in higher co-dimension is still not available, we can also pin down the stability of self-shrinkers in higher co-dimension to the mean curvature vector being the first vector-valued eigenfunction for an elliptic system. More precisely, the equivalent condition of stabilities is as in the following Theorems. Theorem 4 Suppose Σ ⊂ R m is an n-dimensional smooth closed selfshrinker H = − X ⊥ 2 . The following statements are equivalent: 4 dµ ≥ 0 for any smooth normal vector field V which satisfies is a second order elliptic operator and A ij is the second fundamental form as definition in (2), and ∇ ⊥ is the normal connection of Σ.
For the complete case, we define Note that V ∈ S Σ is not necessarily with compact support. The equivalent condition for the stability of F in the complete case becomes Theorem 5 Let Σ ⊂ R m be an n-dimensional smooth complete self-shrinker H = − X ⊥ 2 without boundary. Suppose that the second fundamental form A of Σ is of polynomial growth and Σ has polynomial volume growth. The following statements are equivalent: Using Theorem 4 and 5, we immediately can conclude that the product of any two non-trivial self-shrinkers, which is also a self-shrinker, is F -unstable.
where X i is the position vector of Σ i . Suppose that each Σ i has polynomial volume growth and the second fundamental form of Σ i is of polynomial growth. Then Σ 1 × Σ 2 ⊂ R m 1 +m 2 is a self-shrinker and is F-unstable.
Note that we in fact allow the self-shrinkers to be immersed in our discussion. The examples S n ( √ 2n) and R n are still stable self-shrinkers in R m , but the situation for all other higher co-dimensional examples is not clear. We employ the above equivalent condition to investigate the F -stability of the Lagrangian self-shrinkers constructed by Anciaux in [2] in Section 4. These n-dimensional self-shrinkers in C n , n ≥ 2, are expressed as γ(s)ψ(σ), where ψ : M n−1 → S 2n−1 ⊂ C n is a minimal Legendrian immersion and γ is a complex-valued function that satisfies the system of ordinary differential equations (27). We prove that Theorem 6 Anciaux's closed examples as described in Lemma 1 is Funstable.
Since Anciaux's examples are Lagrangian in C n , it is natural to ask whether these examples are still F -unstable under the restricted Lagrangian variations. We have the following Theorem 7 Anciaux's closed examples is F -unstable under Lagrangian variations for the following cases (i) n = 2 or n ≥ 7, where E and E max are described in (28).
Acknowledgements: The authors are grateful to Mu-Tao Wang for his constant support and interest in this work. The first author also like to thank Jacob Bernstein's discussions.
2 The 1st and 2nd variation formulae of F

Notations and preliminaries
Let X : Σ n → R m be a smooth isometric immersion of a submanifold of codimension m − n. If {e i } and {e α } are orthonormal frames for the tangent bundle T Σ and the normal bundle NΣ, respectively, then the coefficients of the second fundamental form and the mean curvature vector are defined to be and where by convention we are summing over repeated indices and ∇ is the standard connection of the ambient Euclidean space. For a submanifold B in an ambient manifold C, we use A B,C and H B,C to denote the associated second fundamental form and mean curvature vector, respectively. When the ambient space is C n , we denote them as A B (or A) and H B (or H) for simplicity. Given a normal vector field V in NΣ, A, V is a (2, 0)−tensor and | A, V | 2 is defined as A ij , V 2 . When Σ is a hypersurface, the mean curvature vector H and the second fundamental form reduce to the function h = − H, n and the 2-tensor h ij = − A ij , n , respectively. Here n is the unit outer normal vector of Σ.
Definition 1. Let Σ be a submanifold in R m and B r (0) be the geodesic ball in R m with radius r. Σ is said to have polynomial volume growth if there are constants C 1 , C 2 and k ∈ N so that for all r ≥ 0 Definition 2. A normal vector field V (or the second fundamental form A) of Σ is of polynomial growth if there are constants C 1 , C 2 and k ∈ N so that for all r ≥ 0 The space of all normal vector fields with polynomial growth is denoted by P Γ(NΣ). For any two normal vector fields V and W in P Γ(NΣ), its weighted inner product, denoted as V, W e , is defined to be Σ V, W e − |X| 2 4 dµ. The space (P Γ(NΣ), ·, · e ) is called the weighted inner product space.

The first variation formula of F
Colding and Minicozzi derived the first and second variation formulae of the F functionals of a hypersurface in [5]. These can be generalized to higher co-dimensional cases by similar calculation. We derive the first variation formula of F in the following Theorem. where V has compact support. Then where X s is the position vector of Σ s and H s is its mean curvature vector.
Proof. From the first variation formula for area, we know that The variation of the weight 1 √ 4πts n e −|Xs−xs| 2 /4ts have terms coming from the variation of X s , the variation of x s and the variation of t s , respectively. Using the following equations Combining this with (4) gives (3).
Definition 3. We will call (Σ, x 0 , t 0 ) a critical point of F if it is critical with respect to all normal variations which have compact support in Σ and all variations in x and t.
and it is easy to see the following property: is a critical point of F.
Therefore, we only consider the case x 0 = 0, t 0 = 1. In the case of hypersurfaces, Colding and Minicozzi proved that, (Σ, 0, 1) is a critical point of F if Σ satisfies that h = X,n 2 . Their result, when written in the vector form H = − X ⊥ 2 , also holds for higher co-dimensional cases. The proof needs following propositions.
Here X i is the i-th component of the position vector X, i.e., X i = X, ∂ i and the linear operator Lf = ∆f − 1 2 X, ∇f = e ∇f ).
These propositions were proved by Colding and Minicozzi in the case of hypersurfaces (see Lemma 3.20 and Lemma 3.25 in [5]). We omit the proofs here because the argument is similar. Combining (3), (5) and (7), we get

The general second variation formula of F
Theorem 2. Let Σ be an n-dimensional complete manifold without boundary which has polynomial volume growth. Suppose that Σ s is a normal variation of Σ, x s , t s are variations of x 0 and t 0 , and Proof. Apply one more derivative on equation (3), it gives , y Similar to the derivation of the second variation formula for the area, we have On the other hand, since [V, Using ∂Xs ∂s = V , ∂t −1 s ∂s = −τ t −2 s and ∂xs ∂s = y, we simplify where the second equality is from (11), (12), and the definition of L ⊥ x 0 ,t 0 . The second term in (10) is given by For the third term in (10), observe that Combining these gives the theorem.

The second variation formula at a critical point
For convenience, from now on we denote D 2 (V,y,τ ) F as ∂ 2 F ∂s 2 (Σ, 0, 1) in (9). When (Σ, 0, 1) is a critical point of F , we have H = − X ⊥ 2 , the second variation formula of F at the point can be simplified as the following equation (13).
Theorem 3. Let Σ be a complete manifold without boundary which has polynomial volume growth. Suppose that Σ s is a normal variation of Σ, x s , t s are variations of x 0 = 0 and t 0 = 1, and where V has compact support. If (Σ, 0, 1) is a critical point of F , then Here the operator L ⊥ = L ⊥ 0,1 , and Proof. Since (Σ, 0, 1) is a critical point of F , by (3) we have that It follows from (7) that Theorem 2 (with x 0 = 0 and t 0 = 1) gives where we use (15) and (16) to conclude the vanishing of a few terms in (9). Note that y is a constant vector and τ is a constant. Squaring out the last term of D 2 (V,y,τ ) F and using (15) and (16) again leads to Using the equality (6) and Stokes' theorem, we have that Combining (7) and (8), the second variation D 2 (V,y,τ ) F can be further simplified as In [5], Colding and Minicozzi defined the following concept.

An equivalent condition for F-stability
Starting from this section, we assume that Σ satisfies H = − 1 2 X ⊥ .

Vector-valued eigenfunctions and eigenvalues of L ⊥
From equation (13), the second order operator L ⊥ is the important term of second variation of F . When Σ is a hypersurface with h = X,n 2 , Colding and Minicozzi [5] showed that the mean curvature function h and the translations y, n are eigenfunctions of L with eigenvalues 1 and 1 2 , respectively. Here y is a constant vector in R n+1 , n is the outer unit normal vector of Σ, and This property can also be generalized to the higher co-dimensional case.
Proposition 4. Assume that Σ ⊂ R m is a smooth submanifold satisfying H = − X ⊥ 2 , then the mean curvature vector H and the normal part y ⊥ of a constant vector field y are vector-valued eigenfunctions of L ⊥ with where L ⊥ is as in (14). Moreover, if Σ is compact, then L ⊥ is self-adjoint in the weighted space defined in Definition 2 and Proof. Fix p ∈ Σ and choose an orthonormal frame {e i } such that ∇ e i e j (p) = 0, g ij = δ ij in a neighborhood of p.
In the second equality of (19), we used X ⊤ = X, e j e j . Taking another covariant derivative at p, it gives where we used (19), ∇ e k e j (p) = 0, and the Codazzi equation in the last equality. Taking the trace of (20) and using H = − 1 2 X ⊥ , we conclude that Therefore, For a constant vector y in R m , the covariant derivative of y ⊥ is Taking another covariant derivative at p, it gives by ∇ e k e j (p) = 0 and the Codazzi equation. Taking the trace of (22) and using (19), (21), we conclude that Therefore, The equation (18) follows from the divergence theorem that In the case that Σ is compact, since the operator L ⊥ is self-adjoint in the weighted inner product space with respect to Σ and L ⊥ H = H, L ⊥ y ⊥ = 1 2 y ⊥ , we have Hence H, y ⊥ e = 0, for any constant vector y. Since H, y ⊥ e = H, y e , it gives When Σ is complete, (23) is still true provided that Σ has polynomial volume growth and the second fundamental form of Σ is of polynomial growth.

An equivalent condition
In the following theorems, we give an equivalent condition for F (Σ, 0, 1) to be stable. It is inspired by the proof of Lemma 4.23 of Colding and Minicozzi in [5]. dµ < 0. For any real value τ and constant vector y in R m , using (13), we have where the second equality follows from the conditions (24). This contradicts the stability of F .
is a Hilbert space with the weighted inner product that is spanned by E ⊥ 1 , ..., E ⊥ m where {E i } is the standard basis in R m . Given a smooth normal vector field V , it can be decomposed as aH + z ⊥ + V 0 . Here aH and z ⊥ are the projections of V to H and N tr , respectively. Note that V 0 is a smooth normal vector field satisfying (24). For any real value τ and constant vector y ∈ R m , by plugging the decomposition of V into (13), we have where the condition (ii) is used in the last inequality. Choosing τ = −a and y = z, it gives D 2 (V,z,−a) F ≥ 0. That is, Σ is F -stable. For the complete case, we define S Σ = {V ∈ NΣ |V |(X) and |∇ ⊥ V |(X) are of polynomial growth }.
Note that V ∈ S Σ might not be of compact support. We can also find the following equivalent condition for the stability of F in the complete case.
Theorem 5. Let Σ ⊂ R m be an n-dimensional smooth complete self-shrinker, H = − X ⊥ 2 , without boundary. Suppose that the second fundamental form A of Σ is of polynomial growth and Σ has polynomial volume growth. The following statements are equivalent: Remark 2. When V ∈ S Σ , A is of polynomial growth, and Σ has polynomial volume growth, the integral Proof of Theorem 5. (i) ⇒ (ii) Assume the contrary that there is a smooth normal vector field V in S Σ satisfying Here V may not have a compact support. For j ∈ N, consider smooth functions φ j : and |φ ′ j | ≤ 1. Define cutoff functions ψ j (X) = φ j (ρ(X)), X ∈ Σ, where ρ(X) is the distance function from a fixed point p ∈ Σ to X with respect to the metric g ij . Let V j (X) = ψ j (X)V (X), then we have Here {e i } is an orthonormal basis for T X Σ. Using (25), (26), and the dominant convergence theorem, it follows that For any small positive ǫ, choose a sufficiently large j such that | V j , H e | < ǫ|H| e , and max For any real value τ and constant vector y in R m , we get Choosing ǫ 2 < 1 10 V, L ⊥ V e , we get D 2 F (V j ,y,τ ) < 0 for every τ and y. This contradicts the stability of F .
(ii) ⇒ (i) A compactly supported smooth normal vector field V can be decomposed as aH + z ⊥ + V 0 , where V 0 , H, and N tr are mutually orthogonal with respect to the weighted inner product. Because V , H, and z ⊥ belong to S Σ and S Σ is a linear vector space, V 0 belongs to S Σ , too. The remaining part of the proof is essentially the same as the proof of (ii) ⇒ (i) in Theorem 4.
We immediately have the following corollaries.
where X i are the position vectors of Σ i . Then Σ 1 × Σ 2 ⊂ R m 1 +m 2 is a self-shrinker and is F-unstable.
Proof. The mean curvature H of Σ 1 ×Σ 2 is expressed as (H 1 , H 2 ) ∈ R m 1 ×R m 2 and Σ 1 × Σ 2 is a self-shrinker because To prove this corollary, by Theorem 4, it suffices to construct a smooth normal vector field V such that (24) holds while Σ V, −L ⊥ V e − |X| 2 4 dµ < 0. Let V = (aH 1 , bH 2 ), where a and b would be chosen later. Note that V is not vanish since Σ 1 and Σ 2 are closed submanifolds in Euclidean spaces. The first integral in (24) is We can choose a and b to be nonzero constants such that (24) is equal to − → 0 because of the equation (23). The weighted inner product V, −L ⊥ V e can be computed as

<0.
Here the first equality follows from the fact that L ⊥ splits to L ⊥ 1 and L ⊥ 2 , and the equation (17).
where X i is the position vector of Σ i . Suppose that each Σ i has polynomial volume growth and the second fundamental form of each Σ i is of polynomial growth. Then Σ 1 ×Σ 2 ⊂ R m 1 +m 2 is a self-shrinker and is F-unstable.
Using Theorem 5, the proof of Corollary 2 is similar to the proof of Corollary 1. Riemannian metric is ·, · = Re ·, · = n i=1 (dx 2 i + dy 2 i ) and the symplectic form is ω(·, ·) = −Im ·, · = n i=1 dx i ∧ dy i . We have ω(·, ·) = J·, · , where J is the standard almost complex structure J( ∂ ∂x i ) = ∂ ∂y i and J( ∂ ∂y i ) = − ∂ ∂x i . Recall that an immersion ψ from a manifold M of dimension (n − 1) into S 2n−1 is said to be Legendrian if α| ψ(M ) = 0 for the contact 1-form α(·) = ω(X M , ·), where X M is the position vector and ω is the standard symplectic form on C n . Moreover, dα = 2ω and Jy, z = ω(y, z) = 1 2 dα(y, z) = 0, JX M , y = ω(X M , y) = α(y) = 0 for all y, z ∈ T ψ(M). It means that y, Jz, X M , and JX M are mutually orthogonal with respect to the standard metric g for any y, z ∈ T ψ(M). When ψ is a minimal immersion, the complex scalar product γψ of a smooth regular curve γ : I → C * and ψ is a Lagrangian submanifold in C n , i.e., ω| γψ ≡ 0. This was observed by Anciaux in [2]. Indeed, he proved by following Lemma.

Lemma 1. [2]
Let ψ : M → S 2n−1 be a minimal Legendrian immersion and γ : I → C * be a smooth regular curve parameterized by the arclength s. Then the following immersion is a Lagrangian. Moreover, γ * ψ satisfies the self-shrinker equation if and only if γ satisfies the following system of ordinary differential equations: where the curve γ is denoted as r(s)e iφ(s) and θ is the angle of the tangent and the x-axis. From (27), we have a conservation law where 0 < E ≤ E max = ( 2n e ) n/2 is a constant determined by the initial data (r(s 0 ), θ(s 0 ) − φ(s 0 )).

The unstability for general variations
Because the complete noncompact Lagrangian examples constructed by Anciaux in [2] do not have polynomial volume growth, the F -functional is not well-defined and hence we will only discuss the closed cases. That is, the corresponding curves γ are closed and the immersions ψ : M → S 2n−1 are closed. Theorem 6. Fix n ≥ 2. Let Σ be the image of the immersion γ(s) * ψ(σ) in Lemma 1. If Σ is closed, then Σ is F -unstable.
To prove the result, we first set up the notations and derive a few Lemmas. For a fixed point p ∈ Σ = γ * ψ(I × M), it can be represented by γ(s 0 )q for some s 0 ∈ I and q ∈ ψ(M). Choose a local normal coordinate system x 1 , ..., x n−1 at q. Denote u s = ∂X ∂s = γ ′ X M , e i = ∂X M ∂x i , and u i = ∂X ∂x i = γe i for i = 1, ..., n − 1, where X M is the position vector of ψ(M) and X = γX M . The matrix (g αβ ) of the induced metric of Σ with respect to the basis u 1 , ..., u n−1 , u s is g ss = 1, g js = g sj = 0, g jk = r 2 h jk , and h jk (q) = δ jk (29) for j, k = 1, ..., n − 1. The Levi-Civita connections on Σ and ψ(M) are denoted by ∇ and ∇ M , respectively. Define For V ∈ N 0 , the operator V, −L ⊥ V e can be simplified as below.
Lemma 2. Assume that Σ is a closed Lagrangian self-shrinker as in Lemma 1 and V ∈ N 0 is represented by J(γw). The second fundamental forms of Σ in C n and ψ(M) in S 2n−1 are denoted by A Σ and A M,S , respectively. Then we have Proof. (i) For V ∈ N 0 , it can be represented by J(γw) for some vector field w ∈ Γ(T ψ(M)). Using γγ = r 2 and γ ′ γ = re i(θ−φ) , we conclude that A Σ ss , V = Re γ ′′ X M , J(γw) = Re(γ ′′ γ X M , Jw ) = 0 for k, l = 1, .., n − 1. Here the second equalities of the second and third equations of (33) are followed by the fact that e k , Jw, X M , and JX M are mutually orthogonal. Combining (29) and (33), it gives (ii) Since Σ is a Lagrangian, {Ju α } α=1,...,n−1,s is an orthogonal basis at p for the normal bundle. We will calculate the normal projection of (∇ ⊥ uα J(γw)) α=1,...,n−1,s on Ju j and Ju s . Using the property that w, Je k , X M , and JX M are mutually orthogonal, γγ = r 2 and γ ′ γ = re i(θ−φ) , we conclude that From (34), it follows that Thus (iii) is proved.
To further simplify V, −L ⊥ V e , we now derive some integral properties of the curve γ. Recall that the linear operator Lf = ∆f − 1 2 X, ∇f = e |X| 2 4 div(e − |X| 2 4 ∇f ) in Proposition 1. It gives since ∂Σ = ∅. On the other hand, using equation (6) and ∇|X| 2 = 2X ⊤ gives Combining (37), (38), and using |X ⊤ | = Re re i(φ−θ) = r cos(θ − φ), one has is the projection of E β 0 into the tangent space of ψ(M). For fixed q ∈ ψ(M), choose a local normal coordinate system x 1 , ..., x n−1 at q. Denote ∂ j = ∂ ∂x j . We have where E ⊥ β is the normal part of E β . Since the map ψ is a Legendrian immersion into S 2n−1 , the span {∂ 1 , ..., ∂ n−1 , X M } is a Lagrangian plane in C n . It gives Using the equality (JA M,S ) ⊤ = JA M,S , the second term of f (E ⊤ β ) can be simplified as Combining (42) and (43), it gives for α = 1, ..., n. Summing (44) and (45) over α = 1, ..., n gives On the other hand, we have at q because ∂ 1 , ..., ∂ n−1 is an orthonormal basis for T q ψ(M). Plugging it into (46), we get Therefore, there exists a β 0 in {1, .., 2n} such that E ⊤ β 0 is a nonzero vector field and Which is the inequality in (39). Using (40), E β 0 , A M jk is symmetric for j, k, it follows that the vector field w 0 = E ⊤ β 0 satisfies both conditions in (39). Now we are ready to proved Theorem 6: Proof of Theorem 6. By Theorem 4, it suffices to construct a smooth normal vector field V such that (24) holds while Σ V, −L ⊥ V e − |X| 2 4 dµ < 0. Assume V = J(γw), where w ∈ Γ(T ψ(M)) would be chosen later. Because H is parallel to Ju s (see [2], p.40), we have Σ V, H e − |X| 2 4 dµ = 0 and the first condition in (24)  Recall that the construction of γ in [2] is made by m > 1 pieces Γ 1 , ..., Γ m which each corresponds one period of curvature function. (In particular, when γ is the circle S 1 ( √ 2n), we take m = 2.) Every piece Γ i is the same as Γ 1 up to a rotation. Suppose the rotation index of γ is l. Then we have We use (35) to conclude the equality above. For the case n = 2, the only minimal Legendrian curves in S 3 are great circles. They are totally geodesic in S 3 . Therefore, the weighted inner product V, −L ⊥ V e can be simplified as Here we use (35) again to get the last equality. Finally, by choosing w to be the tangent vector of the great circle, which is a parallel vector field, we can make the weighted inner product negative.

The unstability for Lagrangian variations
Since Anciaux's examples are Lagrangian, it is natural to investigate whether these examples are still unstable under the more restricted Lagrangian variations. That is, for variations from the deformation of Lagrangian submanifolds. A simple calculation shows that a vector field V induces a Lagrangian variation if and only if the associated one form α V = ω(V, ·) is closed, i.e.
where ∇ ⊥ is the normal connection on NΣ and X, Y ∈ T Σ. For the problem, we can prove Theorem 7. Let Σ be an n-dimensional closed Lagrangian self-shrinker as in Lemma 1. Then Σ is F -unstable under Lagrangian variations for the following cases (i) n = 2 or n ≥ 7, (ii) 2 < n < 7, and E ∈ [ 1 √ 2 E max , E max ], where E and E max are described in (28).
Because ∇ ⊥ us V, Ju j = ∇ ⊥ u j V, Ju s for V ∈ N 0 , it does not induce a Lagrangian variation. Thus to prove the theorem, we need to consider variations different from those in §4.2. We now define a new set N 1 as follows: