{'fraction_non_alphanumeric': 0.07345676162334365, 'fraction_numerical': 0.0266863659448728, 'mean_word_length': 3.6055709121837127, 'pattern_counts': {'":': 0, '<': 5, '<?xml version=': 0, '>': 10, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 10, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Classification and rigidity of self-shrinkers in the Mean curvature flow

In this paper, we first use the method of Colding and Minicozzi [5] to show that K. Smoczyk's classification theorem [16] for complete self-shrinkers in higher codimension also holds under a weaker condition. Then as an application, we give some rigidity results for self-shrinkers in arbitrary codimension.


Introduction
An immersion x : M n → R n+p of a smooth n-dimensioal manifold M into the Euclidean space is called a self-shrinker if it satisfies the quasilinear elliptic system: 1) where H denotes the mean curvature vector of the immersion and ⊥ is the projection onto the normal bundle of M .
Self-shrinkers play an important role in the study of the mean curvature flow. Not only they correspond to self-shrinking solutions to the mean curvature flow, but also they describe all possible blow ups at a given singularity of the mean curvature flow. We refer the readers to [7,[11][12][13]26] and references therein for more information on self-shrinkers and singularities of mean curvature flow.
There are many results about the classification of self-shrinkers. In the curve case, U. Abresch and J. Langer [1] gave a complete classification of all solutions to (1.1). These curves are now called Abresch-Langer curves, and the only simple closed one is the circle.
In higher dimension and codimension one, Huisken [11,12] (see also [28]) proved a classification theorem for smooth mean convex self-shrinkers M n in R n+1 with polynomial volume growth. Suppose further in the noncompact case |(∇) k A| are uniformly bounded for k = 0, 1, 2, as well as |A| 2 ≤ CH 2 everywhere on M . Then M are isometric to Γ × R n−1 or S k ( √ k) × R n−k (0 < k ≤ n). Here, Γ is a Abresch-Langer curve and S k ( √ k) is a kdimensional sphere. Recently, Colding and Minicozzi II [7] generalized this result and showed that Huisken's classification theorem still holds without the assumption on the bounds for derivatives of the second fundamental form. Moreover, they showed that the only smooth embedded entropy stable self-shrinkers with polynomial volume growth in R n+1 are hyperplanes, n-spheres, and cylinders. By imposing symmetries, Kleene and Møller [16] classified the complete n-dimensional embedded self-shrinkers of revolution in R n+1 .
In arbitrary codimension, the situation becomes more complicated. K. Smoczyk [25] proved the following results: Theorem A(Theorem 1.1 in [25]). Let x : M n → R n+p be a closed selfshrinker, then M is a minimal submanifold of the sphere S n+p−1 ( √ n) if and only if H = 0 and ∇ ⊥ ν = 0, where ν = H/|H| is the principal normal.
Theorem B(Theorem 1.2 in [25]). Let x : M n → R n+p be a complete noncompact connected self-shrinker with H = 0 and ∇ ⊥ ν = 0. Suppose further that M has uniformly bounded geometry, that is, there exists constants c k such that |(∇) k A| ≤ c k for any k ≥ 0. Then M must belong to one of the followings: Here, Γ is one of the Abresch-Langer curves andM r is a complete minimal submanifold of the sphere S r+p−1 ( √ r) ⊂ R p+r , where 0 < r = rank(A ν ) ≤ n denotes the rank of the principal second fundamental form A ν =< ν, A >.
Note that any blowup of a Type-I singularity of the mean curvature flow forming on a compact submanifold will automatically be complete with uniformly bounded geometry, therefore Theorem B may be applied to those blowup limits. In Smoczyk's proof of Theorem B, the uniformly bounded geometry is needed in the integrating by parts with respect to the Gauss kernel ρ(x) = e − 1 2 |x| 2 . In the first part of this paper, we will use the method of Colding and Minicozzi II [7] to show that Theorem B also holds under a weaker condition: Theorem 1.1. Let x : M n → R n+p be a complete non-compact connected self-shrinker with H = 0 and ∇ ⊥ ν = 0. Suppose further that M has polynomial volume growth and satisfies |A| 2 − |A ν | 2 ≤ c for some constant c, where A ν =< ν, A > is the principal second fundamental form. Then M must belong to one of the followings: When p = 1 and the self-shrinker is embedded, Theorem A and Theorem 1.1 reduce to Colding-Minicozzi II's result (Theorem 0.17 in [7]): ×R n−k are the only complete embedded self-shrinkers without boundary, with polynomial volume growth, and H ≥ 0 in R n+1 .
In the statement of Theorem 1.1, We say that a submanifold M n in R n+p has polynomial volume growth if there exists constants C and d such that for all r ≥ 1, there holds where B r denotes an Euclidean ball with radius r. By using Huisken's monotonicity formula [11], Colding-Minicozzi II [7] proved that any selfshrinker which arises as the blow up at a given singularity in the mean curvature flow must have polynomial volume growth. Their result is proved for self-shrinkers in the hypersurface case, but it also holds for arbitrary codimension.
In the second part of this paper, we shall apply Theorem 1.1 to give some rigidity properties of self-shrinkers with higher codimension. Recall that the first gap theorem for self-shrinkers was proved by N. Q. Le and N. Sesum [18] for hypersurface case. Later this was generalized by H. -D. Cao and H. Li [4] to self-shrinkers with arbitrary codimension, they showed that Theorem C. ) Let x : M n → R n+p be an n-dimensional complete self-shrinker without boundary and with polynomial volume growth, if then either (i) |A| 2 ≡ 0 and M is the hyperplane, or (ii) Remark 1.2. We remark that Theorem C is independent of the dimension and codimension of the self-shrinker. In [5], the authors got a related result without condition "polynomial volume growth".
Although classifying the self-shrinker with higher codimension is complicated, we can also apply Theorem A and Theorem 1.1 to give some results for self-shrinkers in some special situations. In the following we will consider the self-shrinkers with codimension 2, 2-dimension and with flat normal bundle respectively. Theorem 1.3. Let X : M n → R n+2 be a complete embedded self-shrinker without boundary and with polynomial volume growth. If H = 0, ∇ ⊥ ν = 0, where ν is the principal normal, and then there are two possibilities: (ii) |A| 2 ≡ 2 and M is one of the self-shrinkers in Example 1.1 below.
be the Clifford minimal hypersurfaces in the sphere S r+1 ( √ r). Then is a complete embedded self-shrinker without boundary and with polynomial volume growth, with parallel principal normal and |A| 2 ≡ 2.
For the 2-dimensional self-shrinkers, we have the following two rigidity results.
Theorem 1.5. Let X : M 2 → R 2+p be a 2-dimensional complete embedded self-shrinker without boundary and with polynomial volume growth. If H = 0, ∇ ⊥ ν = 0, where ν is the principal normal, and then there are two possibilities: and M is the self-shrinker in Example 1.3 below. Theorem 1.6. Let X : M 2 → R 2+p be a 2-dimensional complete embedded self-shrinker without boundary and with polynomial volume growth. If H = 0, ∇ ⊥ ν = 0, where ν is the principal normal, and then there are two possibilities: (i) |A| 2 ≡ 5 3 and M is the self-shrinker in Example 1.3 below. (ii) |A| 2 ≡ 11 6 and M is the self-shrinker in Example 1.4 below.
For self-shrinkers with the higher codimension, the normal bundle is complicated, which would influence the submanifold properties. Now we consider the simplest case, i.e., the normal bundle is flat. We will prove the following gap theorem: Theorem 1.7. Let x : M n → R n+p be a complete immersed self-shrinker without boundary and with polynomial volume growth, assume (i) flat normal bundle, that is, then |A| 2 ≡ 0 and M n is the hyperplane, or |A| 2 ≡ p and M n is Remark 1.3. As noted in [25], let Γ 1 , Γ 2 , · · · , Γ m be the Abresch-Langer curves, then Γ 1 × · · · Γ m in R 2m is a self-shrinker with |H| > 0 and flat normal bundle, so the condition (1.8) of Theorem 1.7 is necessary.
Note that the condition (1) and (2) imply the closed self-shrinker to be a minimal submanifold in sphere have been proved by Smoczyk [25] and Cao-Li [4]. In section 8, we will prove that the condition (3) can also imply the self-shrinker is a minimal submanifold in sphere. Then by applying the well-known theorems on the minimal submanifolds in sphere by Ejiri [10], H. Li [19], Itoh [14,15] and Yau [27], Proposition 1.8 will imply three simple characterizations for closed self-shrinkers, see Theorem 6.1 -6.3.

Preliminaries
Let x : M → R n+p be an n-dimensional submanifold of an (n + p)dimensional Euclidean space R n+p . Let {e 1 , · · · , e n } be a local orthonormal basis of M with respect to the induced metric, and {θ 1 , · · · , θ n } be their dual 1-forms. Let e n+1 , · · · , e n+p be the local unit orthonormal normal vector fields. In this paper we make the following conventions on the range of indices: Then we have the following structure equations (see [3,4,20]) where h α ij denote the the components of the second fundamental form of M .
ii )e α is the mean curvature vector field, and H = |H| is the mean curvature of M .
The Gauss equations are given by The Codazzi equations are given by where the covariant derivative of h α ij is defined by If we denote by R αβij the curvature tensor of the normal connection θ αβ in the normal bundle of x : M → R n+p , then the Ricci equations are By exterior differentiation of (2.4), we have the following Ricci identities: We define the first and second covariant derivatives, and Laplacian of the mean curvature vector field H = α H α e α in the normal bundle N (M ) as Let f be a smooth function on M , we define the covariant derivatives f i , f ij , and the Laplacian of f as follows Now we assume the submanifold M n satisfies the self-shrinker equation (1.1). The following equations have been derived in [4].
The self-shrinker equation (1.1) is equivalent to 10) The first and second covariant derivative of H have the following components: If H = 0, we can choose local orthogonal frame {e α } for the normal bundle N M such that e n+p is parallel to the mean curvature vector H, that is, Proof. By use of (2.3), (2.6), Lemma 2.2, (2.1) and (2.12), we have the following calculations: Concerning the term |Z| 2 , we have the following inequalities which holds for all submanifolds with parallel principal normal (may not be a selfshrinker).
where the inequality used the Cauchy-Schwarz inequality, this proves (2.14).
To show (2.15), we have by (2.14) where we used the algebraic fact ( which completes the proof.

Some integral estimates
Recall the following operator L which was introduced and studied firstly on self-shrinkers by Colding and Minicozzi (see (3.7) in [7]): where ∆, ∇ and div denote the Laplacian, gradient and divergent operator on the self-shrinker respectively, < ·, · > denotes the standard inner product in R n+p . The operator L is self-adjoint in a weighted L 2 space. The next two results were proved by Colding-Minicozzi [7] for hypersurface self-shrinkers but can be stated in the same way for self-shrinkers in arbitrary codimension. Using the operator L, Lemma 2.3 has the following Corollary, Remark 3.1. We note that our assumption "H > 0" implies "|Z| > 0" because of |Z| 2 ≥ H 2 /n. Proof. The two equations just follow from (2.13) and Lemma 2.1.
Proof. Suppose that η is a function with compact support, from the selfadjointness of L and Lemma 3.4 we have Combining this with the Cauchy-Schwarz inequality Now we choose a sequence of cut-off function η j which satisfies is the Euclidean ball of radius j centered at the origin. Applying the above inequality with η = η j ϕ, letting j → ∞, and using the dominated convergence theorem, we complete the proof of the Lemma.
The next proposition gives weighted estimates for the principle normal second fundamental form and its covariant derivatives.

Corollary 3.3 and Lemma 2.4 give the inequality
Integrating this with 1 2 η 2 , it follows from the self-adjointness of L that Using the inequality 2ab Assume |η| ≤ 1 and |∇η| ≤ 1, combining (3.5) and (3.6) gives Choose ǫ > 0 small, such that 1+ǫ where the second inequality is due to (2.11) and Lemma 2.1. Since H > 0, Lemma 3.5 and the polynomial volume growth give that M |Z| 2 (1 + Now we will prove the weighted integral estimates that will be needed in the next section, to guarantee the self-adjointness of L can apply on complete self-shrinkers. So the second part follows from Proposition 3.6.

Proof of Theorem 1.1
In this section, we will give the proof of Theorem 1.1. First we prove two geometric identities, which is the key for proving the classification.
Proof. By (3.8), we can apply Corollary 3.2 to |Z| 2 and log H to get by use of Lemma 3.4 Similarly, by (3.9), we apply Corollary 3.2 to two copies of |Z| to get by use of Corollary 3.3 and (2.14) So we conclude that ∇|Z| ≡ |Z|∇ log H, therefore, |Z| = βH for some constant β > 0. And the inequality in (4.4) must be equality, so we have By (4.2), the above two inequalities must be equalities, so we have: (i) For each k, there exists a constant α k such that h n+p ii,k = α k λ i for every i. (ii) If i = j, then h n+p ij,k = 0.
By the Codazzi equation, (ii) implies that ii,j = α j λ i so we have α j = 0. If the rank of (h n+p ij ) is at least two at q, then α j = 0 for all j ∈ {1, · · · , n}. Thus (i),(ii) imply H n+p ,k = 0 for all k ∈ {1, · · · , n} and h n+p ij,k = 0 for all i, j, k ∈ {1, · · · , n}. If the rank of (h n+p ij ) is one at q, then H is the only nonzero eigenvalue of (h n+p ij ), and then |Z| 2 = H 2 .
Next we will show that if the rank of (h n+p ij ) is at least two at some q, then the rank of (h n+p ij ) is at least two everywhere. For each x ∈ M , let λ 1 (x) and λ 2 (x) be the two eigenvalues of (h n+p ij (x)) that are largest in absolute value and define the set Then Ω is nonempty since q ∈ Ω. λ i (x) are continuous in x, so Ω is closed.
For any x ∈ Ω, the rank of (h n+p ij ) is at least two at x, this is an open condition, so there is an open set U containing x where the rank of (h n+p ij ) is at least two, then h n+p ij,k ≡ 0 on this set U , and the eigenvalues of (h n+p ij ) are constant on U . This implies U ⊂ Ω, and therefore Ω is open. Since M is connected, we conclude that Ω = M , therefore the rank of (h n+p ij ) is at least two everywhere. This implies the Case (i).
Since H > 0, the remaining case is where the rank of (h n+p ij ) is exactly one at every point, this implies Case (ii). This completes the proof of Lemma 4.2.
Proof of Theorem 1.1. We can treat the two cases in Lemma 4.2 separately by following the argument of K. Smoczyk in [25] to complete the proof. Note that in Case I, h n+p ij,k ≡ 0 implies |A ν | 2 = |Z| 2 ≡ constant, combining with the assumption |A| 2 − |A ν | 2 ≤ c, we can complete the proof of Case I. While for the Case II, the assumption |A| 2 − |A ν | 2 ≤ c is sufficient for us to complete the proof by following the Smoczyk's arguments in [25]. 5. Rigidity of Self-shrinkers in higher codimension 5.1. Self-shrinkers with codimension two. Assume x : M n → R n+2 is a complete imbedded self-shrinker without boundary and with polynomial volume growth in codimension two, we will give the proof of Theorem 1.
where Γ is one of the Abresch-Langer curves, or whereM r ֒→ S r+1 ( √ r) is a minimal hypersurface. Note that the only simple closed one of the Abresch-Langer curves is the circle, so the first case (5.1) is M n = S 1 (1) × R n−1 with |A| 2 ≡ 1. Then we consider the second case (5.2). Denote A the second fundamental form of M n → R n+2 , andÃ the second fundamental form ofM r ֒→ S r+1 ( √ r). Then where∆,∇ denote Laplacian and covariant derivatives with respect to the induced metric onM . The Simons' equality (5.4) can be found in Simons' paper [24]. For convenience of readers, we give a proof here: Denoteh ij ,R ijkl the components of the second fundamental form and the curvature tensor of M r in S r+1 ( √ r). Since the sphere S r+1 ( √ r) has constant curvature 1 r , the Ricci identities and Gauss-Codazzi equations give that where we used thatM r ֒→ S r+1 ( √ r) is a minimal hypersurface in the last two equalities. Then it follows that 1 2∆ Since x : M n → R n+2 is a embedded self-shrinker without boundary and with polynomial volume growth, by Cheng-Zhou (see Theorem 4.1 in [6]), x : M n → R n+2 is proper, thus we have thatM r is closed. Therefore (5.4) implies that if 0 ≤ |Ã| 2 ≤ 1, (5.5) we have either |Ã| ≡ 0 andM r is totally geodesic in S r+1 ( √ r), that is . From (5.3), the condition (5.5) is equivalent to 1 ≤ |A| 2 ≤ 2. (5.6) So we conclude that if we have (5.6), then there are two possibilities: (1) |A| 2 ≡ 1 and M n = S m ( √ m) × R n−m with 1 ≤ m ≤ n; (2) |A| 2 ≡ 2 and M n = S k ( √ k) × S r−k ( √ r − k) × R n−r with 2 ≤ r ≤ n and 1 ≤ k ≤ r − 1.
Proof of Theorem 1.4. As the proof of Theorem 1.3, Theorem A and Theorem 1.1 imply M n must be one of the two cases (5.1) and (5.2). Note that the case (5.1) has |A| 2 ≡ 1, which violates with the assumption (1.5), so M must be the case (5.2), that is 5.2. Self-shrinkers of two dimension. In this subsection, we assume x : M 2 → R 2+p is a 2-dimensional complete imbedded self-shrinker without boundary and with polynomial volume growth, and we will give the proof of Theorem 1.5 and 1.6.
Proof of Theorem 1.5. Theorem A and Theorem 1.1 imply that is an embedded self-shrinker without boundary and with polynomial volume growth,M 2 ֒→ S p+1 ( √ 2) are closed minimal submanifolds. DenoteÃ the second fundamental form ofM 2 ֒→ S p+1 ( √ 2), then where A is the second fundamental form of x : M 2 → R 2+p . Then the condition (1.6) is equivalent to A well-known theorem (cf. Theorem B in [17]) implies either is a Veronese surface. Therefore in terms of |A| 2 , we have two possibilities: (1) |A| 2 ≡ 1 and M 2 = S 2 ( √ 2) in R 3 (2) |A| 2 ≡ 5 3 and the self-shrinker has the form x : Note that M 2 = S 1 (1) × R ֒→ R 2+p also has |A| 2 ≡ 1, this completes the proof of Theorem 1.5.
Proof of Theorem 1.6. Theorem A and Theorem 1.1 imply that is a closed minimal submanifold in the sphere S p+1 ( √ 2). Denote its second fundamental form byÃ , then where A is the second fundamental form of x : M 2 → R 2+p . Then the condition (1.7) is equivalent to (5.10) A well-known theorem (cf. Theorem C in [17]) implies either is a canonical immersion. Therefore in terms of |A| 2 , we have two possibilities: (i) |A| 2 ≡ 5 3 and the self-shrinker x : (ii) |A| 2 ≡ 11 6 and the self-shrinker x : M 2 = S 2 ( √ 12) → S 6 ( √ 2) ֒→ R 5 is a canonical immersion.
That is (5.14) Therefore our assumption (1.8) implies either |A| 2 ≡ 0 and M n is a plane; or |A| 2 ≡ p and |∇A| ≡ 0, noting our assumption σ αβ = 1 p |A| 2 δ αβ , we can conclude that M is This completes the proof of Theorem 1.7.

Further remarks on closed self-shrinkers
Proof of Proposition 1.8. The conditions (1) and (2) imply the closed selfshrinker is a minimal submanifold in the sphere S n+p−1 ( √ n) have been proved by Smoczyk [25] and Cao-Li [4] respectively. Now we only need to prove that the condition (3) can also imply the closed self-shrinker must be the minimal submanifold in the sphere.
Therefore H = −x and M n is a minimal submanifold in the sphere S n+p−1 ( √ n).
By applying the well-known theorems on the minimal submanifold in sphere by Ejiri [10], H. Li [19], Itoh [14,15] and Yau [27], Proposition 1.8 implies the following three results. Theorem 6.1. Let M n (n ≥ 3) be a closed self-shrinker in R n+p which satisfies one of the three conditions in Proposition 1.8. If the Ricci curvature of M satisfies Ric ≥ n−2 n , then M n must be one of the followings: (1) M n = S n ( √ n); (2) M n = S k ( n 2 ) × S k ( n 2 ) ֒→ S n+1 ( √ n), n = 2k, p = 2; ( denotes the complex projective space with sectional curvature smaller than 1 3 .
Theorem 6.2. Let M n be a closed self-shrinker in R n+p which satisfies one of the three conditions in Proposition 1.8. If the sectional curvature of M satisfies K ≥ 1 2(n+1) , then M n must be one of the followings: (1) M n = S n ( √ n); (2) M n = S n ( 2(n + 1)) ֒→ S n+p−1 ( √ n).
Theorem 6.3. Let M n be a closed self-shrinker in R n+p which satisfies one of the three conditions in Proposition 1.8. If the sectional curvature of M satisfies K ≥ p−2 2p−3 , then M n must be one of the followings: (2) M 2 = S 2 ( √ 6) ֒→ S 4 ( √ 2), n = 2, p = 3.
Appendix A. Two formulas of Simons' type and the weighted integral estimates In this appendix, we give two formulas of Simons' type for self-shrinkers and the weighted integral estimates of the first and the second covariant derivatives of the second fundamental form of self-shrinkers, which we used in subsection 5.3.
Proposition A.1. Let x : M n → R n+p be an immersed self-shrinker, then we have the following Simons' type formula, is the curvature of the normal bundle and |R ⊥ | 2 = R 2 βαkl . In particular, for hypersurface selfshrinkers, that is p = 1, we have Remark A.1. In the hypersurface case, the formulas (A.3) and (A.4) were derived by Colding-Minicozzi II [7] and Ding-Xin [9] respectively. In arbitrary codimension, the formula (A.1) was also proved by Ding-Xin [9].
Proof. For an immersion x : M n → R n+p , we have the following formulas of Simons' type (for example, see [9,20] So for the self-shrinker, substituting (2.12) into the above equation, we obtain This is (A.1). Now we prove (A.2), recall that for the covariant derivatives of the second fundamental form, we have the following Ricci identities (see [3,20]) where we used in last equality For self-shrinkers, we have by use of (2.10), ( Then we obtain + 2h α rjl R rikl + 2h α ril R rjkl + 2h β lij R βαkl + h α rj (R rikl ) l + h α ir (R rjkl ) l + h β ij (R βαkl ) l +h α ri (R rljl ) k + h α rl (R rijl ) k + h β li (R βαjl ) k