Rigidity of complete spacelike translating solitons in pseudo-Euclidean space

J. Math. Anal. Appl. 477 (2019) 692–707

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Rigidity of complete spacelike translating solitons in pseudo-Euclidean space ✩ Ruiwei Xu ∗ , Tao Liu School of Mathematics and Information Science, Henan Normal University, Xinxiang, Henan 453007, PR China

a r t i c l e

i n f o

Article history: Received 29 October 2018 Available online 26 April 2019 Submitted by S.G. Krantz Keywords: Translating soliton Rigidity theorem Pseudo-Euclidean space

a b s t r a c t In this paper, we investigate the parametric version and non-parametric version of rigidity theorem of spacelike translating solitons in pseudo-Euclidean space Rm+n . n Firstly, we classify m-dimensional complete spacelike translating solitons in Rm+n , n and prove the only complete spacelike translating solitons in Rm+n are the spacelike n m-planes. Secondly, we generalize the rigidity theorem of entire spacelike Lagrangian translating solitons to spacelike translating solitons with general codimensions. © 2019 Elsevier Inc. All rights reserved.

1. Introduction The mean curvature flow (MCF) in pseudo-Euclidean space is a one-parameter family of spacelike immersions Xt = X(·, t) : M m → Rm+n with the corresponding image Mt = Xt (M ) such that n 

d dt X(x, t)

= H(x, t), X(x, 0) = X(x),

x ∈ M,

(1.1)

is satisfied, here H(x, t) is the mean curvature vector of Mt at X(x, t) in Rm+n . Mean curvature flow in the n ambient Minkowski space and Lorentzian manifold has been studied extensively (cf. [1,6,10–12,14–16,20,24] for example). Translating soliton and self-shrinker play an important role in analysis of singularities in MCF. A spacelike submanifold M m is said to be a translating soliton in Rm+n if the mean curvature vector H n satisfies H = T ⊥, ✩

Research partially supported by National Natural Science Foundation of China (No. 11871197 and No. 11671121).

* Corresponding author. E-mail addresses: [email protected] (R. Xu), [email protected] (T. Liu). https://doi.org/10.1016/j.jmaa.2019.04.057 0022-247X/© 2019 Elsevier Inc. All rights reserved.

(1.2)

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where T ∈ Rm+n is a non-zero constant vector, which is called a translating vector, and T ⊥ denotes the n orthogonal projection of T onto the normal bundle of M . M m is said to be a self-shrinker in Rm+n if it n satisfies a quasi-linear elliptic system H = −X ⊥ ,

(1.3)

where X ⊥ is the normal part of X. They are two important classes of solutions to (1.1). There is a plenty of works on the classification and uniqueness problem for translating soliton and self-shrinker in Euclidean space (cf. [2,4,7,9,13,19,23,25–27,32] for example). On the other hand, there are many works on the rigidity problem for complete spacelike submanifolds in pseudo-Euclidean space. Calabi [3] proposed the Bernstein problem for spacelike extremal hypersurfaces in Minkowski space Rm+1 and proved such hypersurfaces have to be hyperplanes when m ≤ 4. In [8] Cheng-Yau 1 solved the problem for all dimensions. Later, Jost-Xin [22] generalized the results to higher codimensions. . If M m is closed with respect to the Theorem 1. [22] Let M m be a spacelike extremal submanifold in Rm+n n Euclidean topology, then M has to be a linear subspace. Translating soliton and self-shrinker can be regarded as generalizations of extremal submanifold. Thus it is natural to study the corresponding rigidity problem for spacelike translating soliton and self-shrinker. Here we only investigate the rigidity of spacelike translating soliton in Rm+n . Now, we give a roughly brief n review the rigidity theorems of spacelike translating solitons. For complete spacelike Lagrangian translating solitons in R2n n , Xu-Huang [33] proved the following rigidity theorem. Theorem 2. [33] Let f (x1 , ..., xn ) be a strictly convex C ∞ -function defined on a convex domain Ω ⊂ Rn . If the graph M∇f = {(x, ∇f (x))} in R2n n is a complete spacelike translating soliton, then f (x) is a quadratic polynomial and M∇f is an affine n-plane. From Theorem 2 above, it is easy to see that the corresponding translating vector must be spacelike. In [6], Chen-Qiu proved a nonexistence theorem for complete spacelike translating solitons in Rm+n by n establishing a very powerful generalized Omori-Yau maximum principle. Theorem 3. [6] There exists no complete m-dimensional spacelike translating soliton (with a timelike translating vector). In fact, such nonexistence conclusion still holds for the lightlike translating vector case. Motivating by papers [6,33], we classify m-dimensional complete spacelike translating solitons in Rm+n by affine technique n and classical gradient estimates, and obtain the following Bernsterin theorem. , then it is an affine m-plane. Theorem 4. Let M m be a complete spacelike translating soliton in Rm+n n A more precise statement of the assertion in Theorem 4 says that there exists an m-dimensional complete spacelike translating soliton in Rm+n only if the translating vector is spacelike. It provides another proof n of the nonexistence theorem of translating soliton in [6], and a simple proof of the Bernstein theorem for translating soliton in [33]. Here we mention that it is valid if one shall use the generalized Omori-Yau maximum principle in [6] to prove Theorem 4 above. For non-parametric Lagrangian solitons of mean curvature flow, there are several interesting rigidity n theorems. As to entire self-shrinker for mean curvature flow in R2n n with the indefinite metric i=1 dxi dyi , Huang-Wang [17] and Chau-Chen-Yuan [5] used different methods to prove the rigidity of entire self-shrinker under a decay condition on the induced metric (D2 f ). Later, using an integral method, Ding and Xin [12]

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removed the additional decay condition and proved any entire smooth convex self-shrinking solution for mean curvature flow in R2n n is a quadratic polynomial. In [18], Huang and Xu investigated the rigidity problem of entire spacelike translating soliton graph in R2n n with some symmetry conditions. Motivating by [5,12,17,18], Xu-Zhu [34] proved a rigidity theorem of entire convex translating solutions for mean curvature 2 flow in R2n n under a decay condition on the induced metric (D f ) and provided a class of nontrivial entire spacelike Lagrangian translating solitons. Theorem 5. [34] Let f (x) be an entire smooth strictly convex function on Rn (n ≥ 2) and its graph M∇f = {(x, ∇f (x))} be a translating soliton in R2n n . If there exists a number  > 0 such that the induced metric 2 (D f ) satisfies (D2 f ) >

 I, as |x| → ∞, |x|2

(1.4)

then f (x) must be a quadratic polynomial and M∇f is an affine n-plane. By studying the distribution of the Gauss map, they obtained Bernstein theorems of translating solitons in Euclidean space (see [2], [23] and [32]). Here we shall use this tool to investigate spacelike graphic translating solitons in pseudo-Euclidean space Rm+n . n Theorem 6. Let uα (1 ≤ α ≤ n) be smooth functions defined everywhere in Rm and their graph M = (x, u1 (x), u2 (x), · · · , un (x)) be a spacelike translator in Rm+n . If one of the following conditions holds: n 1. if the Gauss image of M is bounded; 2. if there exists a number  > 0 such that the induced metric (gij ) satisfies det(gij ) >

 , as |x|

|x| → ∞,

(1.5)

then u1 (x), · · · , un (x) are linear functions on Rm , and M is an affine m-plane in Rm+n . n Remark. It is necessary that there is a restriction on the induced metric (gij ) for the rigidity of translating solitons. If not, there exist nontrivial entire smooth spacelike translating solitons, which are not planes. For example, submanifold (x, y, ln(1 + exp{2x}) − x, μy) ,

|μ| < 1,

(x, y) ∈ R2

(1.6)

is an entire spacelike translator with the translating vector (0, 1, 1, μ) in R42 . 2. Preliminaries The pseudo-Euclidean space Rm+n is the linear space Rm+n endowed with the metric n ds2 =

m m+n   (dxi )2 − (dxα )2 . i=1

(2.1)

α=m+1

Let X : M m → Rm+n be a spacelike m-submanifold in Rm+n with the second fundamental form B defined n n by BY W := (∇Y W )⊥

(2.2)

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for Y, W ∈ Γ(T M ), where ∇ denotes the connection on Rm+n . Let (·) and (·)⊥ denote the orthogonal n projections into the tangent bundle T M and the normal bundle N M , respectively. Let ∇ and ∇⊥ be connections on the tangent bundle and the normal bundle of M , respectively. Choose a local Lorentzian frame field {ei , eα }(i = 1, · · · , m; α = m + 1, · · · , m + n) such that {ei } are tangent vectors to M . The mean curvature vector of M in Rm+n is defined by n H=



H α eα =



α

Bii = −

i



hα ii eα ,

(2.3)

i,α

m+n where hα . We have the Gauss equation ii = Bii , eα . Here, ·, · is the canonical inner product in Rn

Rijkl = −

 α α α (hα ik hjl − hil hjk ),

(2.4)

α

and the Ricci curvature Rij = −

 α α α (hα kk hij − hki hkj ).

(2.5)

α,k

For a spacelike translating soliton M m , by definition we can decompose the translating vector T into a tangential part V and a normal part H on M m , namely T = V + H. Define H2 = − H, H = −|H|2 , where H2 is absolute value of the norm square of the mean curvature. Similarly define B2 = − B, B = −|B|2 . From (2.3) and the inequality of Schwartz, we have B2 ≥

1 H2 . m

(2.6)

Note that when the spacelike manifold M m is a Lagrangian gradient graph (x, ∇f ) in R2n n with the indefinite n metric i=1 dxi dyi , the functions H2 and B2 are the norm of Tchebychev vector field Φ and the Pick invariant J in relative geometry respectively (see [33]), up to a constant. Therefore we can use some affine technique to estimate functions H2 and B2 . Let z = X, X be the pseudo-distance function on M . Then we have (see also [30]) z,i = ei (z) = 2 X, ei , X, hα ij eα ),

z,ij = Hess(z)(ei , ej ) = 2(δij −  Δz = z,ii = 2m + 2 X, H .

(2.7) (2.8) (2.9)

In the following we set up the basic notations and formula for an m-dimension spacelike graphic sub. Let manifold in Rm+n n M := {(x1 , · · · , xm , u1 , · · · , un ); xi ∈ R, uα (x) = uα (x1 , · · · , xm )}, where i = 1, · · · , m and α = 1, · · · , n. Denote x = (x1 , · · · , xm ) ∈ Rm ; u = (u1 , · · · , un ) ∈ Rn .

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Let EA (A = 1, · · · , m + n) be the canonical Lorentzian basis of Rm+n . Namely, every component of the n vector EA is 0, except that the A-th component is 1. Then ei = Ei +



i ∈ {1, · · · , m}

uα i Em+α ,

α ∂uα ∂xi .

give a tangent frame on M . Here, uα i = on spacelike submanifold M is

In pseudo-Euclidean space with index n, the induced metric

gij = ei , ej = δij −



α uα i uj .

α

Then there are n linear independent unit normal vectors,  i

eα =

uα i Ei + Em+α 1

(1 − |Duα |2 ) 2

α ∈ {1, · · · , n},

,

α where Duα = (uα 1 , · · · , um ). Thus the Levi-Civita connection with respect to the induced metric has the Chistoffel symbols

Γkij =

1 kl ∂gil ∂gjl ∂gij α g ( + − ) = −g kl uα ij ul . 2 ∂xj ∂xi ∂xl

(2.10)

Put a nonzero constant vector T :=





ai Ei +

bα Em+α .

By the definition of the translator (1.2), for each eα , there holds T, eα = −H α . By calculation, we have H, eα = g ij ∇ei ej , eα = g ij uβij Em+β , eα

=

−g ij uα ij 1

(1 − |Duα |2 ) 2

,

and T, eα =

α ai uα i −b 1 . (1 − |Duα |2 ) 2

Then (1.2) is equivalent to the following elliptic system i α α g ij uα ij = −a ui + b ,

α ∈ {1, · · · , n}.

3. Calculation of ΔH2 and ΔB2 Using the idea and technique in [6] and [30], we have the following propositions.

(2.11)

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Proposition 3.1. For the spacelike translating soliton M m in Rm+n , the following estimate holds n 2 H4 − T, ∇H2 . m

ΔH2 ≥

Proof. Let {ei } be a local orthonormal normal frame field at the considered point of M . From (1.2), we derive 

 ⊥ ∇⊥ ej H = (∇ej H) =

∇ej

T−



⊥ T, ek ek

=−

k

 T, ek Bjk ,

(3.1)

k

and ⊥ ∇⊥ ei ∇ej H = −

  T, ek ∇⊥ H, Bik Bjk . ei Bjk − k

(3.2)

k

Hence, using the Codazzi equation ⊥ ∇⊥ ei Bjk = ∇ek Bji ,

we have Δ|H|2 = 2|∇⊥ H|2 + 2 H, Δ⊥ H

= 2|∇⊥ H|2 − 2 H, ∇⊥ V H − 2

 H, Bik 2 .

(3.3)

i,k

It follows that ΔH2 = 2∇⊥ H2 + 2 H, ∇⊥ V H + 2

 H, Bik 2 i,k

(3.4)

2 H4 . ≥ 2 H, ∇⊥ V H + m Note that 2 2 2 H, ∇⊥ V H = ∇V H, H = −∇V H = − T, ∇H .

(3.5)

Then (3.4) and (3.5) together give Proposition 3.1. 2 In order to prove the completeness of entire graph spacelike translator with respect to the induced metric, we need the following type estimate for ΔB2 . Proposition 3.2. For the spacelike translating soliton M in Rm+n , the Laplacian of the second fundamental n form B satisfies ΔB2 ≥

2 B4 − T, ∇B2 . n

Proof. Let {ei } be a local tangent orthonormal frame field on M and {eα } a local normal orthonormal frame field on M such that ∇eα = 0 at the considered point. From [30], we have

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ΔB2 = Δ



2 (hα ij ) = 2[



2 α α α α β β (hα ij,k ) + hij hkk,ij − hij hli hlj hkk

α,i,j

(3.6)

β α β +hα ij hij hlk hlk

+

β (hα ij hik

−

α β hβlj hα kl )(hqj hqk

−

hβqj hα kq )].

Let Sαβ =



β hα ij hij ,

N (hα ) =



i,j

2 (hα ij ) ,

i,j

then B2 =



Sαα .

α

So (3.6) becomes ΔB2 = 2[



2 α α α α β β (hα ij,k ) − hij H,ij + hij hli hlj H  + (Sαβ )2 + N (hα hβ − hβ hα )].

(3.7)

α,β

Note that 

(Sαβ )2 ≥

α,β

1  1 ( Sαα )2 = B4 , n α n

N (hα hβ − hβ hα ) ≥ 0. From (3.2) and Codazzi equation, we get 

α hα ij H,ij =

α,i,j



⊥ Bij , ∇⊥ ei ∇ej H = −

  Bij , ∇⊥ H, Bik Bij , Bjk

V Bij −

i,j

i,j,k

 1 α β β hα = − V, ∇B2 + ij hjk hik H . 2 Then substituting these inequalities into (3.7) completes the proof of Proposition 3.2.

2

4. Proof of Theorem 4 To gain Bernstein theorem, we are about to prove H2 ≡ 0 on M . If H2 = 0, then there exists a point p0 ∈ M such that H2 (p0 ) > 0. Set H2 (p0 ) = λ. Denote by r(p0 , p) the geodesic distance function from p0 ∈ M with respect to the induced metric g. For any positive number a, let Ba (p0 ) := {p ∈ M | r(p0 , p) ≤ a}. We consider the function Ψ := (a2 − r2 )2 H2

(4.1)

defined on Ba (p0 ). Obviously, Ψ attains its supremum at some interior point p∗ . We may assume that r2 is a C 2 -function in a neighborhood of p∗ . Choose an orthonormal frame field on M around p∗ with respect to the metric g. Then, at p∗ , we have ∇Ψ = 0,

ΔΨ ≤ 0.

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Hence 2(r2 ),i (H2 ),i + = 0, a2 − r 2 H2

(4.2)

2Δr2 2|∇r2 |2 ΔH2 (∇H2 )2 − 2 + − ≤ 0, 2 2 2 2 2 a −r (a − r ) H H4

(4.3)

− −

where “,” denotes the covariant derivative with respect to the metric g. Inserting Proposition 3.1 into (4.3), we get −

2Δr2 2|∇r2 |2 T, ∇H2

2 (∇H2 )2 2 H − − + − ≤ 0. a2 − r 2 (a2 − r2 )2 H2 m H4

(4.4)

Combining (4.2) with (4.4), we have 0≥−

2Δr2 24r2 T, ∇H2

2 − − + H2 . 2 2 2 2 2 2 a −r (a − r ) H m

(4.5)

By (4.2) and the inequality of Schwarz, we obtain T,

∇H2 ∇H2

= V,

H2 H2 ≤ |V |2 +

r2 4  (a2 − r2 )2

= (C0 + H2 ) +

(4.6)

r2 4 , 2  (a − r2 )2

where C0 = T, T and  is a small positive constant to be determined later. Inserting (4.6) into (4.5), we have 0≥−

r2 2Δr2 4 2 ) · − (24 + + ( − )H2 − C0 . 2 2 2 2 2 a −r  (a − r ) m

(4.7)

Now we shall use the technique in [21] to calculate the term Δr2 . In the case p0 = p∗ , we denote a = r(p0 , p∗ ) > 0 and assume that ∗

max H2 = H2 (˜ p).

Ba∗ (p0 )

By Proposition 3.1 we have max H2 =

Ba∗ (p0 )

max H2 .

∂Ba∗ (p0 )

Thus (a2 − r2 (p∗ ))2 H2 (˜ p) = (a2 − r2 (˜ p))2 H2 (˜ p) ≤ (a2 − r2 (p∗ ))2 H2 (p∗ ).

(4.8)

For any p ∈ Ba∗ (p0 ), by the Gauss equation we know that the Ricci curvature Ric(M, g) is bounded from below by 1 Rii (p) ≥ − H2 (˜ p). 4

(4.9)

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By the Laplacian comparison theorem, we get Δr2 ≤ 2m + (m − 1)rH(˜ p).

(4.10)

2 a2 a − )H2 (p∗ ) ≤ c(m, ) 2 + C0 + 2(m − 1)H(˜ p) 2 , 2 2 m (a − r ) a − r2

(4.11)

Substituting (4.10) into (4.7), we have (

where c(m, ) is a positive constant depending only on m and . Multiplying both sides of (4.11) by (a2 − r2 )2 (p∗ ), we have (

2 − )Ψ ≤ c(m, )a2 + C1 a4 + 2(m − 1)a(a2 − r2 )(p∗ )H(˜ p), m

(4.12)

where  C0 , 0,

C1 =

if C0 ≥ 0 if C0 < 0.

By (4.8) and the inequality of Schwarz, we get Ψ ≤ c(m, )a2 +

mC1  4 a . 2 − 2m

(4.13)

In the case p0 = p∗ , we have r(p0 , p∗ ) = 0. It is easy to get (4.13) from (4.7). Therefore (4.13) holds on Ba (p0 ). Then, at any interior point q ∈ Ba (p0 ), we get H2 (q) ≤ c(m, )

a4 a2 mC1  + . (a2 − r2 )2 2 − 2m (a2 − r2 )2

(4.14)

Case 1. If C0 > 0, we choose  sufficiently small, such that λ mC1  ≤ . 2 − 2m 2 For a → ∞, at the point q, we get H2 ≤

2λ . 3

In particular, H2 (p0 ) ≤

2λ . 3

This contradicts to H2 = λ at p0 , so we can conclude that H2 ≡ 0 on M n . Therefore it is an affine m-plane by Theorem 1. Case 2. If C0 = 0, let a → ∞ in (4.14), we get H2 = 0.

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On the other hand, it is easy to see that H is a zero vector, since M is an m-dimensional spacelike submanifold. Thus T ∈ T M . This contradicts to the translating vector T is nonzero. Then there exists no complete m-dimensional spacelike translating soliton with a lightlike translating vector. Case 3. If C0 < 0, let a → ∞ in (4.14), we get H2 ≤ 0. This is impossible. Then there exists no complete m-dimensional spacelike translating soliton with a timelike translating vector. This completes the proof of Theorem 4. 2 Remark. Here we mention that we can replace H with B in the proof of Theorem 4 and use Proposition 3.2 to prove the second fundamental form B ≡ 0. It directly proves M m must be an m-plane. 5. Proof of Theorem 6 To gain Theorem 6, we will show that the graph spacelike translating soliton is complete with respect to the induced metric g. Thus from Theorem 4, we complete the proof of Theorem 6. Using the similar calculation of section 5 in [34], we prove the completeness of the induced metric g if B has an upper bound. To the simplicity, here we shall use theorem 2.1 of Prof. Xin in [29] to obtain the completeness. Proposition 5.1. Let uα be smooth functions defined everywhere in Rm . Suppose their graph M = (x, u(x)) . If the norm of the second fundamental form B has a bound, then M is is a spacelike translator in Rm+n n complete with respect to the induced metric. Proof. Without loss of generality, we assume that the origin 0 ∈ M . From Proposition 3.1 of [22], we know that the pseudo-distance function z = X, X on M is a non-negative proper function. By (2.6), we know that if B has an upper bound, H also has an upper bound. On the other hand, by (3.1) we get ∇⊥ H2 = − ∇⊥ H, ∇⊥ H = −

  T, ek Bjk , T, el Bjl

j

k

l

≤ |V | B ≤ (|T | + H )B . 2

2

2

2

2

Then ∇⊥ H also has a bound. By Theorem 2.1 in [29], we have for some k > 0, the set {z ≤ k} is compact, then there is a constant c depending only on the dimension m and the bounds of mean curvature and its covariant derivatives such that for all x ∈ M with z(x) ≤ k2 , |∇z| ≤ c(z + 1).

(5.1)

Let γ : [0, r] → M be a geodesic on M issuing from the origin 0. Integrating (5.1) gives z(γ(r)) + 1 ≤ exp(cr), which forces M to be complete with respect to the induced metric. 2 In the following we prove the bound of B. Proposition 5.2. Let uα (1 ≤ α ≤ n) be smooth functions defined everywhere in Rm and their graph M = . If there exists a number  > 0 such that the (x, u1 (x), u2 (x), · · · , un (x)) be a spacelike translator in Rm+n n induced metric (gij ) satisfies

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(gij ) >

 I, as |x|

|x| → ∞,

(5.2)

then the function B has an upper bound on Rm .  Proof. By assumption, there exists a large enough R0 such that (gij ) > |x| I in case |x| > R0 . Choose a m positive number a > R0 . Let Ba (0) := {x ∈ R | |x| ≤ a}. Consider the function below

F (x) := (a2 − |x|2 )2 B2 defined on Ba (0). Obviously, F attains its supremum at some interior point p∗ . We can assume that B(p∗ ) > 0. Then, at p∗ , (log F )i = 0,

(log F )ij ≤ 0.

In the following we calculate the first equation explicitly, and contract the second term above with the positive definite matrix (g ij ).



n 

g ij

i,j=1

4xi (B2 )i − 2 = 0, 2 B a − |x|2

(5.3)

(B2 )ij (B2 )i (B2 )j 8xi xj 4δij − − 2 − 2 2 4 2 2 B B (a − |x| ) a − |x|2

 ≤ 0.

(5.4)

From Proposition 3.2 and the inequality of Schwarz, we get  i,j

g ij

ΔB2  ij k (B2 )k (B2 )ij = + g Γij 2 B B2 B2 ≥

 2 ∇B2 (B2 )k B2 − V,

+ g ij Γkij 2 n B B2

(5.5)

i,j,k

≥

2 1 (∇B2 )2  ij k (B2 )k B2 − |V |2 − mn + g Γij . n mn B4 B2 i,j,k

In the following we shall estimate the term 

g ij Γkij

i,j,k

 i,j,k

2

)k g ij Γkij (B B2 . By (2.10) (2.11) and (5.3), we have

 (B2 )k 4xk α kl α = (ai uα . i − b )g ul 2 2 B a − |x|2 α

By the inequality of Cauchy, we get

12

12    i α α kl α i α α 2 kl α 2 (a ui − b )g ul xk ≤ (a ui − b ) (g ul xk ) . α

α

α

By the spacelike of M , we have  α

Then

2 (uα i ) < 1,

∀ 1 ≤ i ≤ m.

R. Xu, T. Liu / J. Math. Anal. Appl. 477 (2019) 692–707





12 (ai uα i

12



−b )



≤

α 2

α

α 2

(b )



 2 + ( ai u α i )

α

α





≤|b| + |a| ( 2

α

i

 2 =|b| + |a| ( uα i ) i

√ ≤|b| + m|a|,

12

12

2 (uα i ) )

i



703

12

α

where a = (a1 , ..., am ), b = (b1 , ..., bn ). On the other hand,



12 2 (g kl uα l xk )



12

 α ≤ (g kl uα l uk )

α

1

(g kl xl xk ) 2

α





≤ (

 2 g ii )( ( (uα k ) )) α

i





≤ (

g ii )(

 2 ( (uα k ) ))

i

k

 g ii ). ≤ ma( √

k

12 (

12 g kk )|x|2

k

12

α





(

1

g kk ) 2 |x|

k

i

Therefore 

g ij Γkij

i,j,k

 C1 a g ii (B2 )k ≤ , B2 a2 − |x|2

(5.6)

where C1 = 4m(|a| + |b|). Thus, inserting (2.6), (5.3) and (5.6) into (5.5), we get  i,j

g ij

 2 1 (∇B2 )2 C1 a g ii (B2 )ij 2 2 B |V | ≥ − − mn − B2 n mn B4 a2 − |x|2   1 1 a2 g ii C1 a g ii 2 C0 − 16mn 2 ≥ B − − 2 , n mn (a − |x|2 )2 a − |x|2

(5.7)

where C0 = T, T . Then, inserting (5.3) and (5.7) into (5.4), we have   1 a2 g ii a g ii 2 B ≤ C0 + (16mn + 28n) 2 + nC1 2 . m (a − |x|2 )2 a − |x|2 2

(5.8)

Multiplying both sides of (5.8) by (a2 − |x|2 )2 (p∗ ), we have (a2 − |x|2 )2 B2 (p∗ ) ≤ If p∗ ∈ BR0 (0), then for any x ∈ Ba (0),

 1 C0 a4 + [(16mn2 + 28n)a2 + nC1 a3 ] g ii . m

(5.9)

R. Xu, T. Liu / J. Math. Anal. Appl. 477 (2019) 692–707

704

F (x) ≤ a4 max B2 . BR0 (0)

If p∗ ∈ Ba (0)\BR0 (0), by assumption 

g ii ≤ m

|x| , 

we get [(a2 − |x|2 )2 B2 ](p∗ ) ≤

1 mnC1 4 C0 a4 + C2 a3 + a , m 

(5.10)

where C2 is a positive constant depending only on m, n and . Thus, in both cases, we always have F (x) ≤ [(a2 − |x|2 )2 B2 ](p∗ ) ≤ C3 a4 + C2 a3 ,

(5.11)

where C3 is a positive constant depending only on m, n, , C1 , C0 and maxBR0 (0) B2 . Then, at any interior point q ∈ Ba (0), we obtain [(a2 − |x|2 )2 B2 ](q) ≤ [(a2 − |x|2 )2 B2 ](p∗ ) ≤ C3 a4 + C2 a3 .

(5.12)

Dividing both sides of (5.12) by (a2 − |x|2 )2 , we obtain B2 (q) ≤ C3

a4 a3 + C . 2 (a2 − |x|2 )2 (a2 − |x|2 )2

(5.13)

Let a → ∞ in (5.13), we get B2 (q) ≤ C3 . This completes the proof of Proposition 5.2.

2

In the following we shall use a similar method in [31] to prove Theorem 6. Proof of Theorem 6. Let P be any spacelike m-subspace in Rm+n . All spacelike m-subspaces form the n pseudo-Grassmannian Gnm,n . It is a specific Cartan-Hadamard manifold. Define the generalized Gauss map γ : M → Gnm,n by γ(p) = Tp M ∈ Gnm,n via the parallel translation in Rm+n for ∀p ∈ M . n . Choose P0 as an As before, let {EA }(A = 1, · · · , m + n) be the canonical Lorentzian base of Rm+n n m-plane spanned by E1 ∧ · · · ∧ Em . At each point in M its image m-plane P under the Gauss map is spanned by ei := Ei +

 ∂uα α

∂xi

Em+α .

It follows that   ∂uα ∂uα |e1 ∧ · · · ∧ em | = det δij − ∂xi ∂xj 2

(5.14)

R. Xu, T. Liu / J. Math. Anal. Appl. 477 (2019) 692–707

705

and 

det(gij ) = |e1 ∧ · · · ∧ em |.

(5.15)

The m-plane P is also spanned by pi = [det(gij )]− 2m ei , 1

furthermore, |p1 ∧ · · · ∧ pm | = 1. Then we have P, P0 = det( pj , Ei ) = [det(gij )]− 2 . 1

Now, drawing a geodesic C(s) between P0 and P parametrized by arc length s in the pseudoGrassmannian Gnm,n . For simplicity, we can assume that m ≤ n. The case m > n can be treated similar. From [28], we know that the geodesic C(s) can be represented as P (s), which is spanned by hi = Ei +



z iα (s)Em+α ,

α

where

iα



z (s) =

Z1 (s) 0

for some constants λi such that



0 0

 ,

Z1 (s) = diag(tanh(λ1 s), · · · , tanh(λm s))

m×n

λ2i = 1. This means that P (s) is spanned by

h1 = E1 + tanh(λ1 s)Em+1 , · · · , hm = Em + tanh(λm s)E2m . Let ˜ m = cosh(λm s)hm . ˜ 1 = cosh(λ1 s)h1 , · · · , h h Since |hi |2 = Ei + z iα (s)Em+α , Ei + z iβ (s)Em+β

=1 − tanh2 (λi s) =

1 , cosh2 (λi s)

∀1 ≤ i ≤ m,

˜ 1, · · · , h ˜ m are orthonormal. Therefore, we can compute the inner product P0 , P again by the vectors h m

 1 ˜j = P0 , P = det Ei , h cosh(λi s) = [det(gij )]− 2 .

(5.16)

i=1

Case 1. If the distance s between P0 and P is bounded by a finite number R, then the above formula (5.16) yields

706

R. Xu, T. Liu / J. Math. Anal. Appl. 477 (2019) 692–707

det(gij ) ≥

m 

−2 cosh(λi R)

.

(5.17)

i=1

On the other hand, the induced Riemannian metric on M is ds2 = gij dxi dxj , where gij = δij −



α uα i uj .

It is obvious that the eigenvalues of the matrix (gij ) at each point are ≤ 1. Therefore, by equation (5.17), we know that there exists a number  > 0 such that (gij ) > I.

(5.18)

Case 2. By assumption, there exists a number  > 0 such that the induced metric (gij ) satisfies (gij ) >

 I, as |x|

|x| → ∞.

(5.19)

Thus, combining Propositions 5.2, 5.1 and Theorem 4 in both cases, we complete the proof of Theorem 6. 2 Acknowledgments We wish to express our sincere gratitude to Prof. Qun Chen and Prof. Xingxiao Li for their valuable and helpful discussions on the topic. References [1] M.R. Adames, Spacelike self-similar shrinking solutions of the mean curvature flow in pseudo-Euclidean spaces, Comm. Anal. Geom. 22 (5) (2014) 897–929. [2] C. Bao, Y.G. Shi, Gauss map of translating solitons of mean curvature flow, Proc. Amer. Math. Soc. 142 (2014) 4333–4339. [3] E. Calabi, Examples of Bernstein problems for some nonlinear equations, in: Proc. Symp. Global Analysis, UC Berkeley, 1968. [4] H.D. Cao, H.Z. Li, A gap theorem for self-shrinkers of the mean curvature flow in arbitrary codimension, Calc. Var. Partial Differential Equations 46 (3) (2013) 879–889. [5] A. Chau, J.Y. Chen, Y. Yuan, Rigidity of entire self-shrinking solutions to curvature flows, J. Reine Angew. Math. 664 (2012) 229–239. [6] Q. Chen, H.B. Qiu, Rigidity of self-shrinkers and translating solitons of mean curvature flows, Adv. Math. 294 (2016) 517–531. [7] Q.M. Cheng, Y.J. Peng, Complete self-shrinkers of the mean curvature flow, Calc. Var. Partial Differential Equations 52 (3) (2015) 497–506. [8] S.Y. Cheng, S.T. Yau, Maximal spacelike hypersurfaces in the Lorentz-Minkowski spaces, Ann. Math. 104 (1976) 407–419. [9] T.H. Colding, W.P. Minicozzi II, Generic mean curvature flow I: generic singularities, Ann. Math. 175 (2) (2012) 755–833. [10] Q. Ding, Entire spacelike translating solitons in Minkowski space, J. Funct. Anal. 265 (2013) 3133–3162. [11] Q. Ding, Z.Z. Wang, On the self-shrinking system in arbitrary codimensional spaces, arXiv:1012.0429v2, 2010. [12] Q. Ding, Y.L. Xin, The rigidity theorems for Lagrangian self-shrinkers, J. Reine Angew. Math. 692 (2014) 109–123. [13] Q. Ding, Y.L. Xin, L. Yang, The rigidity theorems of self shrinkers via Gauss maps, Adv. Math. 303 (2016) 151–174. [14] K. Ecker, On mean curvature flow of spacelike hypersurfaces in asymptotically flat spacetime, J. Aust. Math. Soc. A 55 (1) (1993) 41–59. [15] K. Ecker, Interior estimates and longtime solutions for mean curvature flow of noncompact spacelike hypersurfaces in Minkowski space, J. Differential Geom. 46 (3) (1997) 481–498. [16] K. Ecker, Mean curvature flow of spacelike hypersurfaces near null initial data, Comm. Anal. Geom. 11 (2) (2003) 181–205. [17] R.L. Huang, Z.Z. Wang, On the entire self-shrinking solutions to Lagrangian mean curvature flow, Calc. Var. Partial Differential Equations 41 (2011) 321–339. [18] R.L. Huang, R.W. Xu, On the rigidity theorems for Lagrangian translating solitons in pseudo-Euclidean space II, Internat. J. Math. 26 (9) (2015) 1550072. [19] G. Huisken, Local and global behaviour of hypersurfaces moving by mean curvature, in: Differential Geometry: Partial Differential Equations on Manifolds, Los Angeles, CA, 1990, in: Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, pp. 175–191.

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