Linear Weingarten -Biharmonic Hypersurfaces in Euclidean Space

Annali di Matematica Pura ed Applicata (1923 -) (2020) 199:1533–1546 https://doi.org/10.1007/s10231-019-00930-0

Linear Weingarten -biharmonic hypersurfaces in Euclidean space

Dan Yang1 · Jingjing Zhang1 · Yu Fu2

Received: 13 September 2019 / Accepted: 12 November 2019 / Published online: 22 November 2019 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Abstract In order to derive the best possible estimates of the total mean curvature of a compact submanifold in a Euclidean space in terms of spectral geometry, in the late 1970s, Bang- Yen Chen introduced the theory of ﬁnite-type submanifolds, which could be viewed as λ-biharmonic submanifolds in the sense of λ-biharmonic maps. Interestingly, Chen proposed in 1991 the following problem (Chen in Soochow J Math 17:2:169–188, 1991, Problem 12): “Determine all submanifolds of Euclidean spaces which are of null 2-type. In particular, classify null 2-type hypersurfaces in Euclidean spaces.” In this paper, we give further support evidence to the above problem. We are able to prove that a linear Weingarten null 2-type or λ-biharmonic hypersurface Mn of a Euclidean space Rn+1 has constant mean curvature and constant scalar curvature provided n < 7.

Keywords λ-Biharmonic maps · Null 2-type submanifolds · Linear Weingarten submanifolds

Mathematics Subject Classiﬁcation 53D12 · 53C40 · 53C42

1 Introduction

Harmonic maps are defined as critical points of the energy E for all variations. A smooth map is harmonic if and only if its tension field vanishes identically. The bienergy E2 of a smooth map is defined as the total tension, i.e., the integral of the squared norm of the tension field.

B Yu Fu [email protected]; [email protected] Dan Yang [email protected] Jingjing Zhang [email protected]

1 School of Mathematics, Liaoning University, Shenyang 110044, People’s Republic of China 2 School of Mathematics, Dongbei University of Finance and Economics, Dalian 116025, People’s Republic of China 123 1534 D. Yang et al.

In this sense, E2 provides a measure for the extent to which the map fails to be harmonic. A smooth map is said to be biharmonic if it is a critical point of E2 for all variations. We now consider a constrained variational problem of E2. The critical points of E2 for all variations with fixed energy, equivalently, critical points of λ-bienergy E2,λ := E2 + λE for all variations are called λ-biharmonic maps, where λ is a Lagrange multiplier (c.f.[9,18,22]). A submanifold is called λ-biharmonic if the isometric immersion that defines the submanifold is a λ-biharmonic map. It is easy to see that any minimal submanifold is λ-biharmonic. In a special case, when λ = 0, λ-biharmonic submanifolds are automatically biharmonic. Let φ : Mn −→ Rm be an isometric immersion of an n-dimensional connected submanifold Mn into a Euclidean space Rm. In the sense of λ-biharmonic maps, λ-biharmonic submanifolds satisfy the following geometric condition: −→ −→ H = λ H . (1.1) The theory of finite-type submanifolds began in the late 1970s, when Bang-Yen Chen tried ([3,8,9]) to find the best possible estimates of total mean curvature of a closed submanifolds of Euclidean space. The family of finite-type submanifolds is huge and contains many important families of submanifolds, including all the minimal submanifolds of Euclidean space. In general, a submanifold of Rm is said to be finite type if the position vector x of Mn in Rm can be decomposed in the following form:

x = x0 + x1 +···+xk, where c0 is a constant vector and x1,...,xk are non-constant maps satisfying

xi = λi xi , i = 1,...,k.

In particular, if all of the eigenvalues λ1,...,λk are mutually different, then the submanifold n n M is said to be of k-type. In particular, if one of λ1,...,λk is zero, then M is said to be of null k-type. Obviously, null k-type immersions occur only when Mn is non-compact. It is well known that a 1-type submanifold of a Euclidean space Rm is either a minimal submanifold of Rm or a minimal submanifold of a hypersphere in Rm. By the definition, null 2-type submanifolds are the most simple ones of finite-type sub- m manifolds besides 1-type submanifolds. After choosing a coordinate system on R with c0 as its origin, we have the following simple spectral decomposition for a null 2-type submanifold Mn: x = x1 + x2,x1 = 0,x2 = λx2, (1.2) −→ where λ is nonzero constant. After applying Beltrami’s formula x =−n H ,(1.2) reduces to Eq. (1.1) as well. Hence, null 2-type submanifolds are λ-biharmonic submanifolds in a Euclidean space. Chen proposed in 1991 the following interesting problem [5, Problem 12]: “Determine all submanifolds of Euclidean spaces which are of null 2-type. In particular, classify null 2-type hypersurfaces in Euclidean spaces.” The first result on null 2-type submanifolds was obtained by the Chen in 1988 [3]. The author proved that every null 2-type surface in R3 is an open portion of a circular cylinder. Ferrândez and Lucas [17] proved that a null 2-type hypersurface in Rn+1 with at most two distinct principal curvatures is a spherical cylinder. In 1995, Hasanis and Vlachos [21] proved that null 2-type hypersurfaces in R4 have constant mean curvature and constant scalar curvature (cf. [14]). In 2012, Chen and Garray [11] proved that δ(2)-ideal null 2-type hypersurfaces in Euclidean space are spherical cylinders. Recently, the third author extended Hasanis and Vlachos’s results and proved in [19] that null 2-type hypersurfaces with at most 123 Linear Weingarten λ-biharmonic hypersurfaces... 1535 three distinct principal curvatures in a Euclidean space have constant mean curvature and constant scalar curvature. Furthermore, Chen and the third author [10] made further progress by proving δ(3)-ideal null 2-type hypersurfaces have constant mean curvature and constant scalar curvature. The theory of null 2-type submanifolds with codimension ≥ 2 has been studied by some authors, among others, in [12,15,16]. For the most recent surveys in this field, we refer the readers to [7–9]. A hypersurface in a space form is said to be linear Weingarten if its normalized scalar curvature R and mean curvature H satisfy aR+bH = c for some constants a, b ∈ R.Inves- tigating the geometry and rigidity of submanifolds is an important and interesting problem in differential geometry. According to the definition, linear Weingarten hypersurfaces reduce to hypersurfaces with constant mean curvature if a = 0 and hypersurfaces with constant scalar curvature if b = 0. In [23], Li, Suh and Wei give the first rigidity result for linear Weingarten hypersurfaces in Sn+1 under the assumption that the hypersurface is compact. Recently, Aquino, De Lima and Velasquez [1] established a new characterization theorem concerning complete linear Weingarten hypersurfaces immersed in real space forms using the generalized maximum principle. There are also some interesting results concerning linear Weingarten hypersurfaces, for instance, see ([2,24]). In this paper, we will investigate linear Weingarten null 2-type or λ-biharmonic hypersurfaces in a Euclidean space. Without the assumptions compactness or completeness, by careful analysis of the Codazzi equation and Gauss equation, we will prove that

Theorem 1.1 Every linear Weingarten null 2-type hypersurfaces in a Euclidean space Rn+1 (n < 7) has constant mean curvature and constant scalar curvature.

Corollary 1.2 Every linear Weingarten biharmonic hypersurfaces in a Euclidean space Rn+1 (n < 7) has to be minimal.

Remark 1.3 Our results give further support evidence to Chen’s problem [5, Problem 12]. Note that the assumption linear Weingarten is much weaker than the general geometric assumptions constant scalar curvature or constant mean curvature. Hence, Theorem 1.1 is a generalization of Corollary 1.4 in [20].

The paper is organized as follows. In Sect. 2, we recall some necessary background for hypersurfaces and equivalent conditions for null 2-type hypersurfaces. In Sect. 3, we provide some useful lemmas and useful computations. At last, in Sect. 4, we give a proof of Theorem 1.1.

2 Preliminaries

We first recall some basic material in differential geometry. Let x : Mn → Rn+1 be an isometric immersion of a hypersurface Mn into a Euclidean space Rn+1. Denote the Levi-Civita connections of Mn and Rn+1 by ∇ and ∇¯ , respectively. Letting X and Y be vector fields tangent to Mn and ξ be a unite normal vector field, then the Gauss and Weingarten formulas ([8,9]) are given, respectively, by ¯ ∇X Y =∇X Y + h(X, Y ), ¯ ∇X ξ =−AX, 123 1536 D. Yang et al. where h is the second fundamental form and A is the Weingarten operator. It is well known that the second fundamental form h and the Weingarten operator A are related by the following

h(X, Y ), ξ=AX, Y . −→ And the mean curvature vector ﬁeld H is given by

−→ 1 H = trace h. (2.1) n The Gauss equations are given by

R(X, Y )Z =AY, ZAX −AX, ZAY, (2.2) where the curvature tensor is deﬁned by

R(X, Y )Z =∇X ∇Y Z −∇Y ∇X Z −∇[X,Y ] Z, (2.3)

The Codazzi equation is given in the following

(∇X A)Y = (∇Y A)X, (2.4) where (∇X A)Y is deﬁned by

(∇X A)Y =∇X (AY) − A(∇X Y ), for all X, Y , Z tangent to Mn. From Gauss equation (2.2), the scalar curvature R, the squared length of the second fundamental form B and the mean curvature H are related by

R = n2 H 2 − B. (2.5)

The Laplacian operator on Mn actingonasmoothfunction f is given by

n n =− v(∇ ) =− < ∇ (∇ ), >=− ( −∇ ) . f di f ei f ei ei ei ei ei f (2.6) i=1 i=1 By identifying the tangent and the normal parts of equation (1.1), the following necessary and sufﬁcient conditions for Mn to be of null 2-type in Rn+1 (cf. [9,10]) are obtained.

Proposition 2.1 The immersion φ : Mn −→ Rn+1 of a hypersurface Mn in a Euclidean space is null 2-type if and only if H + HtraceA2 = λH, (2.7) 2A grad H + nHgrad H = 0.

Recall that the third author in [19] studied the null 2-type hypersurfaces with at most three distinct principal curvatures in n+1-dimensional Euclidean space and obtained the following result:

Theorem 2.2 ([19]) Every null 2-type hypersurface with at most three distinct principal curvatures in a Euclidean space Rn+1 must have constant mean curvature. 123 Linear Weingarten λ-biharmonic hypersurfaces... 1537

3 Some results on linear Weingarten null 2-type hypesurfaces

In this section, we will consider the orientable null 2-type hypersurfaces Mn immersed in Rn+1 provided n < 7. We assume that Mn is not of 1-type, and hence Mn is not minimal. According to Theorem 2.2, it is necessary to consider the cases n = 5andn = 6. Since the technique is extremely similar, without loss of generality we need only to deal with the case n = 6. If the mean curvature H is constant, the ﬁrst equation of (2.7) implies that the length of the second fundamental form is also constant. Combining these with (2.5) gives that the scalar curvature R is constant as well. Now suppose that the mean curvature H is not constant, we will derive a contradiction. Since the mean curvature H is not constant, there exists a point x0 such that grad H(x0) = 0. In the following, we will work on an neighborhood U(x0) of x0 such that grad H(x0) = 0at any point. In view of the second equation of (2.7), grad H is an eigenvector of the Weingarten operator A with the corresponding principal curvature −3H. Hence, we choose e1 such that e1 is parallel to grad H. Under the assumption that the shape operator of the hypersurface immersed in E7 is diagonalizable, the Weingarten operator A takes the following form with respect to a suitable orthonormal frame {e1,...,e6} as

A = diag (λ1,...,λ5,λ6), where λi are the principal curvatures with λ6 =−3H. Therefore, it follows from (2.1)that

5 λi + λ6 = 6H, (3.1) i=1 that is, 5 λi = 9H =−3λ6. (3.2) i=1 Denote by B the squared length of the second fundamental form h of M,then

5 = 2 = λ2 + λ2. B trace A i 6 (3.3) i=1 From (2.5)and(3.3), we have

5 = 2 − = λ2 − λ2. R 36H B 3 6 i (3.4) i=1 In the following, we will consider the linear Weingarten hypersurface with aH+ bR = c for constant a, b, c. Since we have assumed that the mean curvature is not constant, so b = 0 and a c a c R =− H + = λ + . (3.5) b b 3b 6 b It follows from (3.4)and(3.5)that

5 λ2 = λ2 + λ − , i 3 6 c1 6 c2 (3.6) i=1 123 1538 D. Yang et al. where c1 =−a/3b, c2 = c/b.Sincee6 is parallel to grad H, it follows that

e6(H) = 0, ei (H) = 0, i = 1,...,5. (3.7) λ =− λ ∇ = 6 ωk Note that 6 3H implies that 6 depends only on one variable t.Let ei e j k=1 ijek. From (3.7), we have

[ei , e j ](H) = 0, which yields ω6 = ω6 , ij ji (3.8) for distinct i, j = 1, 2,...,5. A direct computation concerning the compatibility conditions ∇ , = ∇ , = ek ei ei 0and ek ei e j 0gives ωi = ,ωj + ωi = ki 0 ki kj 0 (3.9) for distinct i, j and i, j, k = 1, 2,...,6. Furthermore, by Codazzi equation (2.4), we can deduce that (λ ) = (λ − λ )ω j , ei j i j ji (3.10) (λ − λ )ω j = (λ − λ )ω j i j ki k j ik (3.11) for distinct i, j, k = 1, 2,...,6. From Gauss equation (2.2) and the deﬁnition of Gauss curvature tensor in (2.3), by considering R(ei , e j )ek, e6 and R(e j , ek )ei , e6, we can obtain ωk ω6 − ω6 = ωk ω6 − ω6 , ij jj kk ji ii kk ωi ω6 − ω6 = ωi ω6 − ω6 . jk kk ii kj jj ii Combining this with (3.9), we get ωk ω6 − ω6 = ωk ω6 − ω6 = ωi ω6 − ω6 . ij jj kk ji ii kk kj jj ii (3.12) On the other hand, taking into account the second expression of (3.9)and(3.11)forthree distinct principal curvatures λi ,λj and λk for i, j, k = 1,...,5, we obtain ωk (λ − λ ) = ωk (λ − λ ) = ωi (λ − λ ), ij j k ji i k kj j i (3.13) ωk ωk + ωi ωi + ω j ω j = . ij ji jk kj ik ki 0 (3.14) Next, we give an important proposition and its the proof for later use.

Proposition 3.1 Assume the mean curvature H is not constant at point x0.AtU(x0),the principle curvatures λi (i = 1,...,6) are one variable functions with respect to t.

Proof Substituting H =−λ6/3 into the ﬁrst equation of (2.7), using (2.6)and(3.3)–(3.7), we get 5 (λ ) = (λ ) ω6 + λ ( λ2 + λ − − λ). e6e6 6 e6 6 ii 6 4 6 c1 6 c2 (3.15) i=1

Since e6(λ6) = 0andλ6 = λ6(t),Eq.(3.15) becomes 5 1 ω6 = f (t) = e e (λ ) − λ (4λ2 + c λ − c − λ) . (3.16) ii 1 e (λ ) 6 6 6 6 6 1 6 2 i=1 6 6 123 Linear Weingarten λ-biharmonic hypersurfaces... 1539

ωi = = ,..., ω6 = By a direct computation in (3.9)and(3.10), we get 66 0foralli 1 6and ij 0 for distinct i, j = 1,...,5. On one hand, according to the deﬁnition of the Gauss curvature tensor (2.3), we have 6 ( , ) , = ωi + ω j ωi − ωi R e6 ei e6 ei e6 i6 i6 6 j ei 66 k=1 6 6 6 − ω j ωi − ω j ωi + ω j ωi 66 ij 6i j6 i6 j6 = = = k 1 k 1 k 1 =− ω6 + (ω6 )2. e6 ii ii (3.17) On the other hand, it follows from Gauss equation that

R(e6, ei )e6, ei =−λ6λi . (3.18) Combining (3.17) with (3.18)gives ω6 = ω6 2 + λ λ e6 ii ii 6 i (3.19) for distinct i, j = 1,...,5. Taking the sum of (3.19)fori = 1,...,5, using (3.2)and(3.16), we have 5 ω6 2 = ( ) = λ2 + ( ). ii f2 t 3 6 e6 f1 (3.20) i=1 Putting i = 6in(3.10) and from (3.9), we get (λ ) = λ ω6 − λ ω6 . e6 i i ii 6 ii (3.21) Taking the sum of (3.21)fori = 1,...,5, using (3.2)and(3.16), we have 5 λ ω6 = ( ) = λ − (λ ). i ii g1 t 6 f1 3e6 6 (3.22) i=1 ω6 = ,..., Multiplying by ii on both sides of (3.19) and taking the sum for i 1 5, then using (3.20)and(3.22), we obtain 5 3 1 ω6 = f (t) = e ( f ) − λ g . (3.23) ii 3 2 6 2 6 1 i=1

Differentiating (3.22) with to e6 and using (3.21)and(3.19), we have 5 5 5 ( ) = λ ω6 2 − λ ω6 2 + λ λ2. e6 g1 2 i ii 6 ii 6 i i=1 i=1 i=1 Equations (3.6)and(3.20) yield 5 6 2 1 2 λi ω = g (t) = e (g ) + λ f − λ (3λ + c λ − c ) . (3.24) ii 2 2 6 1 6 2 6 6 1 6 2 i=1 ω6 2 = ,..., Multiplying by ii on both sides of (3.19) and taking the sum for i 1 5, we have 5 5 5 1 6 3 6 4 6 2 e ω = ω + λ λi ω . 3 6 ii ii 6 ii i=1 i=1 i=1 123 1540 D. Yang et al.

Applying (3.23)and(3.24), we obtain 5 4 1 ω6 = f (t) = e ( f ) − λ g . (3.25) ii 4 3 6 3 6 2 i=1 Similar to the above computations completely, we can get 5 5 1 ω6 = f (t) = e ( f ) − λ g . (3.26) ii 5 4 6 4 6 3 i=1 (λ ) = (λ − λ )ω j λ = λ On the other hand, it follows from (3.10)thate6 j 6 j j6,so j 6 for j = 1,...,5. There are at most ﬁve distinct principal curvatures except for λ6. If every principal curvature is single, according to (3.21), we have (λ ) ω6 = e6 i . ii λi − λ6 λ = λ ω6 = ω6 , = ,..., Equations (3.12)and(3.13) imply that i j is equivalent to ii jj for i j 1 5. ω6 = ,..., Thus, ii are mutually different for i 1 5. The polynomial equations (3.16), (3.20), (3.23), (3.25)and(3.26) reduce to a system of equations: 5 (ω6 )k = = , ,..., . jj fk for k 1 2 5 (3.27) j=1

Differentiating the system of equations (3.27) with respect to ei for any i = 1,...,5, from ei ( fk ) = 0, we can get a new system of homogeneous equations: 5 (ω6 )k−1 (ω6 ) = = , ,..., . jj ei jj 0fork 1 2 5 (3.28) j=1 Since the determinant of the coefﬁcient matrix in the system of equations (3.28) is not (ω6 ) = , = ,..., vanishing, we have ei jj 0foranyi j 1 5. Moreover, 5 ω6 − ω6 =[ , ] ω6 = ωk − ωk ω6 , ei e6 jj e6ei jj ei e6 jj i6 6i ek jj k=1 so (ω6 ) = , ei e6 jj 0 for i, j = 1,...,5.Accordingto(3.19), we have λ (λ ) = ω6 − ω6 ω6 = . 6ei j ei e6 jj 2 jjei jj 0 (3.29)

Hence, ei (λ j ) = 0foranyi, j = 1,...,5, and all principal curvatures λi depend only on one variable t. If there exist multiple principal curvatures, i.e., the number m of distinct principle curvatures satisﬁes m ≤ 5, then there are at most four distinct principal curvatures for λi except for λ ω6 = ,..., 6. Then, the number of different ii is less than or equal to 4 for i 1 5. If four of the ω6 = , , , ii are mutually different, it is only necessary to consider the system (3.27)fork 1 2 3 4. ω6 A similar discussion yields the conclusion. If three or less of the ii are mutually different, then the conclusion follows by similar argument. Thus, we complete the proof of Proposition 3.1. 123 Linear Weingarten λ-biharmonic hypersurfaces... 1541

4 The proof of main Theorem

In this section, we will prove Theorem 1.1 by deriving a contradiction. We assume that the mean curvature H is not constant. Firstly, we compute R(ei , e j )ei , e j according to the deﬁnition of the Gauss curvature tensor (2.3) and the Gauss equation (2.2) respectively. One could get 6 6 6 ω j + ωk ω j − ω j − ωk ω j − ωk − ωk ω j =−λ λ ei ji ji ik e j ii ii jk ij ji ki i j (4.1) k=1 k=1 k=1 for distinct i, j = 1,...,5. Since λi = λ j , it follows from (3.9), (3.10) and Proposition 3.1 that 5 ω j = ωi = , ωk ω j = . ji jj 0 ii jk 0 k=1 Then, (4.1) becomes 5 5 ω6 ω6 + ωk ω j − ωk − ωk ω j =−λ λ , ii jj ji ik ij ji ki i j (4.2) k=1,k=i, j k=1,k=i, j which together with (3.9)and(3.14)gives 5 ω6 ω6 − ωk ωk =−λ λ , ii jj 2 ij ji i j (4.3) k=1,k=i, j for distinct i, j = 1,...,5. According to the number of the distinct principle curvatures, we distinguish the following three cases Case A. m = 6. In this case, λ1,...,λ6 are mutually different. According to (3.12)–(3.14), we only consider the following cases: ωk , , Case A.1: There are three of the ij that are not zero for distinct i j k. Without loss of ω3 = ,ω4 = ω5 = generality, we suppose 12 0 12 0and 12 0. In this case, by considering equations (3.12)and(3.13), we have ω6 − ω6 ω6 − ω6 ω6 − ω6 11 22 = 11 jj = 22 jj λ1 − λ2 λ1 − λ j λ2 − λ j = , , λ ω6 for j 3 4 5. Combining this with Proposition 3.1, we know that 6 and ii depend only on one variable t. Thus, there exist two smooth functions χ and ω depending on t such that ω6 = χλ + ω, ii i (4.4) for i = 1, 2,...,5. Differentiating (4.4) with respect to e6 and using (3.19)and(3.21), we get 2 e6(χ) = λ6(χ + 1) + χω, (4.5)

e6(ω) = ω(λ6χ + ω). (4.6)

Differentiating (3.2) with respect to e6 and using (3.2), (3.6), (3.21)and(4.4), we obtain 5 (λ ) = (λ − λ )ω6 = χ(− λ2 − λ + ) + λ ω. 3e6 6 6 i ii 6 6 c1 6 c2 8 6 (4.7) i=1 123 1542 D. Yang et al.

Taking into account (4.4) and using (3.2), we get

5 ω6 =− λ χ + ω. ii 3 6 5 (4.8) i=1

Then, Eq. (3.15) becomes (λ ) = (λ )(− λ χ + ω) + λ λ2 + λ − − λ . e6e6 6 e6 6 3 6 5 6 4 6 c1 6 c2 (4.9)

Differentiating (4.7) with respect to e6 and using (4.5), (4.6)and(4.9), we may eliminate e6e6(λ6) and get ( ω + χ) (λ ) = λ − λ2 − λ + + λ . 4 c1 e6 6 6 18 6 4c1 6 4c2 3 (4.10)

Eliminating e6(λ6) between (4.7)and(4.10)gives ( ω + χ) − λ2 − λ + χ + λ ω = λ − λ2 − λ + + λ . 4 c1 6 6 c1 6 c2 8 6 3 6 18 6 4c1 6 4c2 3 (4.11) Furthermore, differentiating (4.10) with respect to e6,by(4.7), (4.9)and(4.10), we have 4 3 2 2 216λ + 72c1λ + −54c2 + 6c λ − 10c1c2λ6 + (4c2 + 3λ)c2 χ 6 6 1 6 (4.12) − λ3 + λ2 + ( + λ)λ ω = . 156 6 12c1 6 28c2 18 6 0

Differentiating (4.12) with respect to e6 and using (4.5)and(4.6), we get 864λ3 + 216c λ2 + −108c + 12c2 λ − 10c c e (λ )χ 6 1 6 2 1 6 1 2 1 6 + 216λ4 + 72c λ3 + −54c + 6c2 λ2 − 10c c λ + (4c + 3λ)c 6 1 6 2 1 6 1 2 6 2 2 × λ χ 2 + 1 + χω − 468λ2 + 24c λ + 28c + 18λ e (λ )ω 6 6 1 6 2 1 6 + λ3 + λ2 + ( + λ) λ ω(ω + χλ ) = . 156 6 12c1 6 28c2 18 6 6 0

Multiplying by 3 on both sides of the above equation and using (4.7)and(4.10), we have 864λ3 + 216c λ2 + −108c + 12c2 λ − 10c c ) χ 6 1 6 2 1 6 1 2 × χ −6λ2 − c λ + c + 8λ ω 6 1 6 2 6 4 3 2 2 + 3 216λ + 72c1λ + −54c2 + 6c λ − 10c1c2λ6 + (4c2 + 3λ) c2 6 6 1 6 (4.13) × λ χ 2 + 1 + χω 6 − 468λ2 + 24c λ + 28c + 18λ n χ −6λ2 − c λ + c + 8λ ω 6 1 6 2 6 1 6 2 6 + λ3 + λ2 + ( + λ) λ ω(ω + χλ ) = . 3 156 6 12c1 6 28c2 18 6 6 0

Equation (4.13) could be rewritten as

2 2 q1(λ6)χ + q2(λ6)χω + q3(λ6)ω + q4(λ6) = 0, (4.14) 123 Linear Weingarten λ-biharmonic hypersurfaces... 1543 where qi are nontrivial polynomials concerning the function λ6 and given by: q (λ ) = 864λ3 + 216c λ2 + −108c + 12c2 λ − 10c c ) −6λ2 − c λ + c 1 6 6 1 6 2 1 6 1 2 6 1 6 2 + 3 216λ4 + 72c λ3 + −54c + 6c2 λ2 − 10c c λ + (4c + 3λ)c λ , 6 1 6 2 1 6 1 2 6 2 2 6 q (λ ) = 864λ3 + 216c λ2 + −108c + 12c2 λ − 10c c 8λ 2 6 6 1 6 2 1 6 1 2 6 + 3 216λ4 + 72c λ3 + −54c + 6c2 λ2 − 10c c λ + (4c + 3λ) c 6 1 6 2 1 6 1 2 6 2 2 − 468λ2 + 24c λ + 28c + 18λ −6λ2 − c λ + c 6 1 6 2 6 1 6 2 + 3 156λ3 + 12c λ2 + (28c + 18λ) λ λ , 6 1 6 2 6 6 q (λ ) =− 468λ2 + 24c λ + 28c + 18λ 8λ 3 6 6 1 6 2 6 + 3 156λ3 + 12c λ2 + (28c + 18λ) λ , 6 1 6 2 6 (λ ) = λ4 + λ3 + − + 2 λ2 − λ + ( + λ) λ . q4 6 3 216 6 72c1 6 54c2 6c1 6 10c1c2 6 4c2 3 c2 6 (4.15) Inthesameway,(4.11)and(4.12) could be rewritten as: 2 2 p1(λ6)χ + p2(λ6)χω + p3(λ6)ω + p4(λ6) = 0, (4.16)

h1(λ6)χ + h2(λ6)ω = 0, (4.17) where pi , hi (i = 1, 2) are nontrivial polynomials concerning the function λ6 given by: p (λ ) = c −6λ2 − c λ + c , 1 6 1 6 1 6 2 (λ ) = − λ2 + λ + , p2 6 4 6 6 c1 6 c2 p3(λ6) = 32λ6, (4.18) p (λ ) =−3λ −18λ2 − 4c λ + 4c + 3λ , 4 6 6 6 1 6 2 h (λ ) = 216λ4 + 72c λ3 + −54c + 6c2 λ2 − 10c c λ + (4c + 3λ)c , 1 6 6 1 6 2 1 6 1 2 6 2 2 (λ ) =− λ3 + λ2 + ( + λ)λ . h2 6 156 6 12c1 6 28c2 18 6 Combining (4.14), (4.16) with (4.17) to eliminate χ,weget 2 P1ω + P2 = 0, (4.19) 2 Q1ω + Q2 = 0. (4.20) Moreover, combining (4.19) with (4.20) to eliminate ω2,weobtain

P2 Q1 − P1 Q2 = 0, (4.21) where Pi , Qi (i = 1, 2) are nontrivial polynomials concerning the function λ6 given by: = 2 − + 2 = λ11 +··· , P1 h2q1 h1h2q2 h1q3 101896704 6 = 2 = λ13 +··· , P2 h1q4 30233088 6 = 2 − + 2 = λ9 +··· , Q1 h2 p1 h1h2 p2 h1 p3 684288 6 = 2 = λ11 +··· , Q2 h1 p4 11664 6 where we only need to write the highest order terms of λ6. By substituting Pi and Qi into (4.21), we get a polynomial equation concerning λ6 with constant coefficients ci = ci (c1, c2,λ): 22 λi = , ci 6 0 (4.22) i=0 123 1544 D. Yang et al. where the coefficient c22 of the highest order term satisfies

c22 = 30233088 × 684288 − 101896704 × 11664 = 0.

Therefore, λ6 has to be constant and H =−λ6/3 is a constant, which contradicts to the assumption. ωk Case A.2: Two ones of ij are not zero, and the others are zero. Without loss of generality, ω3 = ω4 = ω5 = we assume that 12 0, 12 0and 12 0. In this case, it follows from (4.3)that ω6 ω6 =−λ λ , ii 55 i 5 (4.23) for i = 1,...,4. From (3.12)and(3.13), similar to Case A.1, there exist two smooth functions χ and ω depending on t such that ω6 = χλ + ω, ii i (4.24) for i = 1,...,4, where χ and ω satisfy the differential equations (4.5)and(4.6). Substituting (4.24)into(4.23), we obtain χω6 =−λ . 55 5 (4.25) Substituting (4.25)and(4.24)into(4.23), we obtain

ωλ5 = 0, (4.26) ω6 λ χ ω which means that 55 and 5 are determined completely by and . Substitute (4.24)–(4.26) into (4.7), and then differentiate it with respect to e6.Byusing(3.19), (3.21), (4.5)and(4.6), a similar discussion as Case A.1 gives λ6 has to be constant, which is a contradiction as well. . : ωk , , CaseA 3 Only one of ij is not zero for distinct i j k. Without loss of generality, we ω3 = ω4 = ,ω5 = let 12 0and 12 0 12 0. In this case, according to (4.3), we have ω6 ω6 =−λ λ , ii jj i j (4.27) for i = 1, 2, 3and j = 4, 5. Similar to Case A.2, we obtain λ6 has to be constant, which yields a contradiction as well. Case B. m = 5, that is, the hypersurface Mn have ﬁve distinct principle curvatures λ ,λ ,λ ,λ = λ ,λ ωk = ωk = = , , 1 2 3 4 5 6.From(3.13), we have 45 54 0fork 1 2 3. In this case, we only consider the following two subcases: ωk ω3 = Case B.1: Two ones of ij are not zero. Without loss of generalization, we let 12 0 ω4 = and 12 0. In this case, by considering equations (3.12)and(3.13), we have ω6 − ω6 ω61 − ω6 ω6 − ω6 11 22 = 11 33 = 22 33 λ1 − λ2 λ1 − λ3 λ2 − λ3 ω6 − ω6 ω6 − ω6 = 11 uu = 22 uu , λ1 − λu λ2 − λu = , λ ω6 for u 4 5. Combining this with Proposition 3.1, we know that i and ii depend only on one variable t. Thus, there exist two smooth functions χ and ω depending on t such that ω6 = χλ + ω, ii i (4.28) 123 Linear Weingarten λ-biharmonic hypersurfaces... 1545 for i = 1, 2,...,5. And the next steps are similar to Case A.1. Therefore, λ6 has to be constant, which contradicts to the assumption. ωk , , Case B.2: Only one of ij for distinct i j k is not zero. Without loss of generality, we ω3 = ω4 = let 12 0and 12 0. In this case, Eq. (4.3) implies that ω6 ω6 =−λ λ , ii 44 i 4 (4.29) for i = 1, 2, 3. And the next steps are similar to the previous Case A.2. We get a contradiction. Case C. m = 4, that is the hypersurface have four distinct principal curvatures λ1,λ2,λ3 = λ = λ ,λ ωk = ωk = ωk = ωk = ωk = ωk = = , 4 5 6.From(3.13), we have 34 43 35 53 45 54 0fork 1 2. ωk , , In this case, we only consider one of ij for distinct i j k is not zero. Without loss of ω3 = generality, we let 12 0. By considering equations (3.12)and(3.13), we have ω6 − ω6 ω6 − ω6 ω6 − ω6 11 22 = 11 uu = 22 uu , λ1 − λ2 λ1 − λu λ2 − λu = , , λ ω6 for u 3 4 5. Combining this with Proposition 3.1, we know that i and ii depend only on one variable t. Thus, there exist two smooth functions χ and ω depending on t such that ω6 = χλ + ω, ii i (4.30) for i = 1, 2,...,5. And the next steps are similar to the previous Case A.1. Therefore, λ6 has to be constant, which means that H is constant. Hence, we get a contradiction and the proof of Theorem 1.1 is completed.

Acknowledgements The ﬁrst two authors are supported by the NSFC (No. 11801246) and the General Project for Department of Liaoning Education (No. LJC201901), and the third author is supported by the NSFC (No. 11601068) and the General Project for Department of Liaoning Education (No. LN2019J05).

References

1. Aquino, C., de Lima, H., Velasquez, M.: A new characterization of complete linear Weingarten hypersurfaces in real space forms. Pac. J. Math. 261(1), 33–43 (2013) 2. Alencar, H., do Carmo, M.: Hypersurfaces with constant mean curvature in spheres. Proc. Am. Math. Soc. 120(4), 1223–1229 (1994) 3. Chen, B.Y.: Null two-type surfaces in E3 are circular cylinders. Kodai Math. J. 11, 295–299 (1988) 4. Chen, B. Y.: Null two-type surfaces in Euclidean space. In: Proceedings of the Symposium in Honor of Cheng-Sung Hsu and Kung-Sing Shih: Algebra, Analysis, and Geometry (National Taiwan Univ. 1988). World Scientific Publ., Teaneck, NJ, pp. 1–18 (1988) 5. Chen, B.Y.:Some open problems and conjectures on submanifolds of finite type. Soochow J. Math. 17(2), 169–188 (1991) 6. Chen, B.Y.: Some pinching and classification theorems for minimal submanifolds. Arch. Math. 60, 568– 578 (1993) 7. Chen, B.Y.: Pseudo-Riemannian Geometry, δ-invariants and Applications. Word Scientific, Hackensack (2011) 8. Chen, B.Y.: Some open problems and conjectures on submanifolds of finite type: recent development. Tamkang J. Math. 45(1), 87–108 (2014) 9. Chen, B.Y.: Total Mean Curvature and Submanifolds of Finite Type, 2nd edn. World Scientific, Hacken- sack (2014) 10. Chen, B.Y., Fu, Y.: δ(3)-ideal null 2-type hypersurfaces in Euclidean space. Differ. Geom. Appl. 40, 43–56 (2015) 11. Chen, B.Y., Garray, O.J.: δ(2)-ideal null 2-type hypersurfaces of Euclidean space are spherical cylinders. Kodai Math. J. 35, 382–391 (2012) 123 1546 D. Yang et al.

12. Chen, B.Y., Lue, H.S.: Some 2-type submanifolds and applications. Ann. Fac. Sci. Toulouse Math. Sér. V 9, 121–131 (1988) 13. Chen, B.Y., Munteanu, M.I.: Biharmonic ideal hypersurfaces in Euclidean spaces. Differ. Geom. Appl. 31 , 1–16 (2013) −→ −→ 14. Defever, F.: Hypersurfaces of E4 satisfying H = λ H . Mich. Math. J. 44, 355–363 (1997) 15. Dursun, U.: Null 2-type submanifolds of the Euclidean space E5 with parallel normalized mean curvature vector. Kodai Math. J. 28(1), 191–198 (2005) 16. Dursun, U.: Null 2-type submanifolds of the Euclidean space E5 with non-parallel mean curvature vector. J. Geom. 86(1), 73–80 (2007) 17. Ferrândez, A., Lucas, P.: Null ﬁnite type hypersurfaces in space forms. Kodai Math. J. 14, 406–419 (1991) 18. Fetcu, D., Loubeau, E., Montaldo, S., Oniciuc, C.: Biharmonic submanifolds of CPn. Math. Z. 266, 505–531 (2010) 19. Fu, Y.: Null 2-type hypersurfaces with at most three distinct principal curvatures in Euclidean space. Taiwan. J. Math. 19(2), 519–533 (2015) 20. Fu, Y., Hong, M.C.: Biharmonic hypersurfaces with constant scalar curvature in space forms. Pac. J. Math. 294(2), 329–350 (2018) 21. Hasanis, T., Vlachos, T.: Hypersurfaces with constant scalar curvature and constant mean curvature. Ann. Glob. Anal. Geom. 13, 69–77 (1995) 22. Lemaire, L.: Minima and critical points of the energy in dimension two. In: Global Differential Geometry and Global Analysis, Lecture Notes in Mathematics, vol. 838, pp. 187–193 . Springer (1981) 23. Li, H.Z., Suh, Y.J., Wei, G.X.: Linear Weingarten hypersurfaces in a unit sphere. Bull. Korean Math. Soc. 46(2), 321–329 (2009) 24. Yang, D.: Linear Weingarten spacelike hypersurfaces in locally symmetric Lorentz space. Bull. Korean Math. Soc. 49(2), 271–284 (2012)

Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional afﬁliations.

123