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Linear Weingarten -Biharmonic Hypersurfaces in Euclidean Space

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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 sub- manifold 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 hyper- surfaces 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.
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