关于黎曼流形中扭形成向量场与双调和超曲面

p-Biharmonic hypersurfaces in Einstein space and conformally flat space

In this paper, we present some new properties for p-biharmonic hypersurfaces in Riemannian manifold. We also characterize the p-biharmonic submanifolds in an Einstein space. We construct a new example of proper p-biharmonic hypersurfaces. We present some open problems.

Biharmonic space-like hypersurfaces in pseudo-Riemannian space

We classify the space-like biharmonic surfaces in 3-dimension pseudo-Riemannian space form, and construct explicit examples of proper biharmonic hypersurfaces in general ADS space.

Classification theorems for biharmonic real hypersurfaces in a complex projective space

First, we classify proper biharmonic Hopf real hypersurfaces in $\mathbb{C}P^2$. Next, we classify proper biharmonic real hypersurfaces with two distinct principal curvatures in $\mathbb{C}P^n$, where $n\geq 2$. Finally, we prove that biharmonic ruled real hypersurfaces in $\mathbb{C}P^n$ are minimal, where $n\geq 2$.

Remarks on biharmonic hypersurfaces in space forms

We consider closed biharmonic hypersurfaces in the Euclidean sphere and prove a rigidity result under a suitable condition on the scalar curvature. Moreover, we establish an integral formula involving the position vector for biharmonic hypersurfaces in space forms.

Biharmonic homogeneous hypersurfaces in compact symmetric spaces

In this paper, we study biharmonic hypersurfaces in Einstein manifolds. Then, we determine all the biharmonic hypersurfaces in irreducible symmetric spaces of compact type which are regular orbits of commutative Hermann actions of cohomogeneity one.

Unique Continuation Property for Biharmonic Hypersurfaces in Spheres

We study properties of non-minimal biharmonic hypersurfaces of spheres. The main result is a CMC Unique Continuation Theorem for biharmonic hypersurfaces of spheres. We then deduce new rigidity theorems to support the Conjecture that biharmonic submanifolds of Euclidean spheres must be of constant mean curvature.

Stability and the index of biharmonic hypersurfaces in a Riemannian manifold

In this paper, we give an explicit second variation formula for a biharmonic hypersurface in a Riamannian manifold similar to that of a minimal hypersurface. We then use the second variation formula to compute the stability index of the known biharmonic hypersurfaces in a Euclidean sphere, and to prove the non-existence of unstable proper biharmonic hypersurface in a Euclidean space or a hyperbolic space, which adds another special case to support Chen's conjecture on biharmonic submanifolds.

关于双调和超曲面和双调和曲线的推广

On the generalization of biharmonic hypersurfaces and biharmonic curves

本文扩展了p-双调和映射和p-双调和超曲面的概念，以更全面地描述黎曼流形中的(p,q)-调和超曲面和(p,q)-调和曲线，包括爱因斯坦空间。同时，提供了新的显式例子。

Totally Biharmonic Hypersurfaces in Space Forms and 3-Dimensional BCV Spaces

A hypersurface is said to be totally biharmonic if all its geodesics are biharmonic curves in the ambient space. We prove that a totally biharmonic hypersurface into a space form is an isoparametric biharmonic hypersurface, which allows us to give the full classification of totally biharmonic hypersurfaces in these spaces. Moreover, restricting ourselves to the 3-dimensional case, we show that totally biharmonic surfaces into Bianchi-Cartan-Vranceanu spaces are isoparametric surfaces and we give their full classification. In particular, we show that, leaving aside surfaces in the 3-dimensional…

Tanaka-Webster biharmonic hypersurfaces in the Sasakian space form

In this article we consider the concept of biharmonicity about the hypersurfaces in the Sasakian space form which is equipped with the Tanaka-Webster connection.

Survey on the Biharmonic Hypersurfaces in Terms of the Induced Metric of Tensor Ricci

In this article, we study the biharmonic hypersurfaces in the Sasakian space form with the induced metric of tensor Ricci. We find the existence necessary and sufficient condition of the biharmonic hypersurfaces there. We show that the biharmonic Hopf hypersurfaces are minimal where gradient of the mean curvature is in direction of the structural vector fields. Furthermore, we prove that does not exist any biharmonic Hopf hypersurfaces when the gradient of the mean curvature is a principal direction.

Biharmonic hypersurfaces in a product space $L^m\times \mathbb{R}$

In this paper, we study biharmonic hypersurfaces in a product of an Einstein space and a real line. We prove that a biharmonic hypersurface with constant mean curvature in such a product is either minimal or a vertical cylinder generalizing a result of \cite{OW} and \cite{FOR}. We derived the biharmonic equation for hypersurfaces in $S^m\times \mathbb{R}$ and $H^m\times \mathbb{R}$ in terms of the angle function of the hypersurface, and use it to obtain some classifications of biharmonic hypersurfaces in such spaces. These include classifications of biharmonic hypersurfaces which are totally u…

A new gap for $CMC$ biharmonic hypersurfaces in Euclidean spheres

In this note we improve a gap result concerning the range of the mean curvature of complete $CMC$ proper-biharmonic hypersurfaces in unit Euclidean spheres.

Biharmonic hypersurfaces in a conformally flat space

Biharmonic hypersurfaces in a generic conformally flat space are studied in this paper. The equation of such hypersurfaces is derived and is used to determine the conformally flat metric $f^{-2}δ_{ij}$ on the Euclidean space $\mathbb{R}^{m+1}$ so that a minimal hypersurface $M^m\longrightarrow (\mathbb{R}^{m+1}, δ_{ij})$ in a Euclidean space becomes a biharmonic hypersurface $M^m\longrightarrow (\mathbb{R}^{m+1}, f^{-2}δ_{ij})$ in the conformally flat space. Our examples include all biharmonic hypersurfaces found in [Ou1] and [OT] as special cases.

Biharmonic hypersurfaces in space forms with three distinct principal curvatures

In this paper, we have studied biharmonic hypersurfaces in space form $\bar{M}^{n+1}(c)$ with constant sectional curvature $c$. We have obtained that biharmonic hypersurfaces $M^{n}$ with at most three distinct principal curvatures in $\bar{M}^{n+1}(c)$ has constant mean curvature. We also obtain the full classification of biharmonic hypersurfaces with at most three distinct principal curvatures in arbitrary dimension space form $\bar{M}^{n+1}(c)$.

Biharmonic Hypersurfaces With The Recurrent Operators In The Euclidean Space

We show how some of well-known recurrent operators such as recurrent curvature operator, recurrent Ricci operator, recurrent Jacobi operator, recurrent shape and Weyl operators have the significant role for biharmonic hypersurfaces to be minimal in the Euclidean space.

Four dimensional biharmonic hypersurfaces in nonzero space form have constant mean curvature

In this paper, through making careful analysis of Gauss and Codazzi equations, we prove that four dimensional biharmonic hypersurfaces in nonzero space form have constant mean curvature. Our result gives the positive answer to the conjecture proposed by Balmus-Montaldo-Oniciuc in 2008 for four dimensional hypersurfaces.

关于双调和共形超曲面

On biharmonic conformal hypersurfaces

本文研究了共形超曲面的双调和性质，证明了在特定条件下共形因子必须为等参数函数，并展示了某些超曲面在特定空间形式中的双调和浸入性。

Biharmonic and biconservative hypersurfaces in space forms

We present some general properties of biharmonic and biconservative submanifolds and then survey recent results on such hypersurfaces in space forms. We also propose an alternative version for a well-known result of Nomizu and Smyth for hypersurfaces by replacing the CMC hypothesis with the more general condition of biconservativity.

Biharmonic hypersurfaces in hemispheres

In this paper we consider the Balmuş-Montaldo-Oniciuc's conjecture in the case of hemispheres. We prove that a compact non-minimal biharmonic hypersurface in a hemisphere of $S^{n+1}$ must be the small hypersphere $S^{n}\left(1/\sqrt{2}\right)$, provided that $n^{2}-H^{2}$ does not change sign.

Some classifications of biharmonic Lorentzian hypersurfaces in Minkowski 5-space $\mathbb E^5_1$

In this paper, we study Lorentzian hypersurfaces in Minkowski 5-space with non-diagonalizable shape operator whose characteristic polinomial is $(t-k_1)^2(t-k_3)(t-k_4)$ or $(t-k_1)^3(t-k_4)$. We proved that in these cases, a hypersurface is biharmonic if and only if it is minimal.

Biharmonic hypersurfaces with three distinct principal curvatures in Euclidean space

The well known Chen's conjecture on biharmonic submanifolds states that a biharmonic submanifold in a Euclidean space is a minimal one ([10-13, 16, 18-21, 8]). For the case of hypersurfaces, we know that Chen's conjecture is true for biharmonic surfaces in $\mathbb E^3$ ([10], [24]), biharmonic hypersurfaces in $\mathbb E^4$ ([23]), and biharmonic hypersurfaces in $\mathbb E^m$ with at most two distinct principal curvatures ([21]). The most recent work of Chen-Munteanu [18] shows that Chen's conjecture is true for $δ(2)$-ideal hypersurfaces in $\mathbb E^m$, where a $δ(2)$-ideal hypersurface i…

On biharmonic hypersurfaces with constant scalar curvatures in $\mathbb E^5(c)$

We prove that proper biharmonic hypersurfaces with constant scalar curvature in Euclidean sphere $\mathbb S^5$ must have constant mean curvature. Moreover, we also show that there exist no proper biharmonic hypersurfaces with constant scalar curvature in Euclidean space $\mathbb E^5$ or hyperbolic space $\mathbb H^5$, which give affirmative partial answers to Chen's conjecture and Generalized Chen's conjecture.

Biharmonic hypersurfaces with bounded mean curvature

We consider a complete biharmonic hypersurface with nowhere zero mean curvature vector field $ϕ:(M^m,g)\rightarrow (S^{m+1},h)$ in a sphere. If the squared norm of the second fundamental form $B$ is bounded from above by m, and $\int_M H^{- p }dv_g<\infty$, for some $0<p<\infty$, then the mean curvature is constant.

Biharmonic hypersurfaces with three distinct principal curvatures in spheres

We obtain a complete classification of proper biharmonic hypersurfaces with at most three distinct principal curvatures in sphere spaces with arbitrary dimension. Precisely, together with known results of Balmuş-Montaldo-Oniciuc, we prove that compact orientable proper biharmonic hypersurfaces with at most three distinct principal curvatures in sphere spaces $\mathbb S^{n+1}$ are either the hypersphere $\mathbb S^n(1/\sqrt2)$ or the Clifford hypersurface $\mathbb S^{n_1}(1/\sqrt2)\times\mathbb S^{n_2}(1/\sqrt2)$ with $n_1+n_2=n$ and $n_1\neq n_2$. Moreover, we also show that there does not exi…

On Chen's biharmonic conjecture for hypersurfaces in $\mathbb R^5$

A longstanding conjecture on biharmonic submanifolds, proposed by Chen in 1991, is that {\it any biharmonic submanifold in a Euclidean space is minimal}. In the case of a hypersurface $M^n$ in $\mathbb R^{n+1}$, Chen's conjecture was settled in the case of $n=2$ by Chen and Jiang around 1987 independently. Hasanis and Vlachos in 1995 settled Chen's conjecture for a hypersurface with $n=3$. However, the general Chen's conjecture on a hypersurface $M^n$ remains open for $n> 3$. In this paper, we settle Chen's conjecture for hypersurfaces in $\mathbb R^{5}$ for $n=4$.

进入共形平坦空间的 $f$-双调和超曲面

$f$-Biharmonic hypersurfaces into a conformally flat space

本文研究共形平坦及非正曲率流形中的 $f$-双调和超曲面，证明全脐情形的非紧性与极小性，并构造、分类相关曲面及高维真 $f$-双调和子流形。

f-Biharmonic maps and f-biharmonic submanifolds II

We continue our study [Ou4] of f-biharmonic maps and f-biharmonic submanifolds by exploring the applications of f-biharmonic maps and the relationships among biharmonicity, f-biharmonicity and conformality of maps between Riemannian manifolds. We are able to characterize harmonic maps and minimal submanifolds by using the concept of f-biharmonic maps and prove that the set of all f-biharmonic maps from 2-dimensional domain is invariant under the conformal change of the metric on the domain. We give an improved equation for f-biharmonic hypersurfaces and use it to prove some rigidity theorems a…

关于共形双调和映射与超曲面

On conformal biharmonic maps and hypersurfaces

本文引入共形双能量泛函，研究其临界点共形双调和映射，分类了球面中共形双调和超球面和广义Clifford环面，并发现双曲空间也存在共形双调和超曲面，最后计算了超球面的稳定性指标。

乘积空间 L^{m}\times \mathbb{R} 中的 λ-双调和超曲面

λ-Biharmonic hypersurfaces in the product space L^{m}\times \mathbb{R}

本文研究乘积空间中的λ-双调和超曲面，证明常平均曲率情形下其为极小或垂直圆柱面，并分类了空间形式中的全脐或半平行超曲面。

Biharmonic hypersurfaces with constant scalar curvature in space forms

Let $M^n$ be a biharmonic hypersurface with constant scalar curvature in a space form $\mathbb M^{n+1}(c)$. We show that $M^n$ has constant mean curvature if $c>0$ and $M^n$ is minimal if $c\leq0$, provided that the number of distinct principal curvatures is no more than 6. This partially confirms Chen's conjecture and Generalized Chen's conjecture. As a consequence, we prove that there exist no proper biharmonic hypersurfaces with constant scalar curvature in Euclidean space $\mathbb E^{n+1}$ or hyperbolic space $\mathbb H^{n+1}$ for $n<7$.

Biharmonic hypersurfaces in a Riemannian manifold with non-positive Ricci curvature

In this paper, we show that, for a biharmonic hypersurface $(M,g)$ of a Riemannian manifold $(N,h)$ of non-positive Ricci curvature, if $\int_M|H|^2 v_g<\infty$, where $H$ is the mean curvature of $(M,g)$ in $(N,h)$, then $(M,g)$ is minimal in $(N,h)$. Thus, for a counter example $(M,g)$ in the case of hypersurfaces to the generalized Chen's conjecture (cf. Sect.1), it holds that $\int_M|H|^2 v_g=\infty$.

欧氏空间中的双调和超曲面

Biharmonic Hypersurfaces in Euclidean Spaces

本文研究欧氏空间中双调和超曲面，构造了主曲率方向可积的双守恒超曲面，并证明完整双调和超曲面当且仅当为极小超曲面。

On cohomogeneity one biharmonic hypersurfaces into the Euclidean space

The aim of this paper is to prove that there exists no cohomogeneity one $G-$invariant proper biharmonic hypersurface into the Euclidean space ${\mathbb R}^n$, where $G$ denotes a tranformation group which acts on ${\mathbb R}^n$ by isometries, with codimension two principal orbits. This result may be considered in the context of the Chen conjecture, since this family of hypersurfaces includes examples with up to seven distinct principal curvatures. The paper uses the methods of equivariant differential geometry. In particular, the technique of proof provides a unified treatment for all these …

四阶超曲面流与高维Willmore流

The biharmonic hypersurface flow and the Willmore flow in higher dimensions

本文通过建立新的Gagliardo-Nirenberg不等式，证明了高维四阶超曲面流和Willmore流的全局存在性。

Biharmonic submanifolds of pseudo-Riemannian manifolds

In this paper, we derived biharmonic equations for pseudo-Riemannian submanifolds of pseudo-Riemannian manifolds which includes the biharmonic equations for submanifolds of Riemannian manifolds as a special case. As applications, we proved that a pseudo-umbilical biharmonic pseudo-Riemannian submanifold of a pseudo-Riemannian manifold has constant mean curvature, we completed the classifications of biharmonic pseudo-Riemannian hypersurfaces with at most two distinct principal curvatures, which were used to give four construction methods to produce proper biharmonic pseudo-Riemannian submanifol…

On the biharmonic hypersurfaces with three distinct principal curvatures in space forms

In [16] there was proved that any biharmonic hypersurface with at most three distinct principal curvatures in space forms has constant mean curvature. At the very last step of the proof, the argument relied on the fact that the resultant of two polynomials is a non-zero polynomial. In this paper we point out that, in fact, there is a case, and only one, when this resultant is the zero polynomial and therefore the original proof is not fully complete. Further, we prove that in this special case we still obtain that the hypersurface has constant mean curvature.

空间形式中的双调和超曲面

Biharmonic hypersurfaces in space forms

本文证明了非正截面曲率空间形式中任意维数的双调和超曲面均为极小，解决了双曲空间中广义陈省身猜想的超曲面情形；对单位球面中的双调和超曲面，推导出定量限制、逐点准则、数量曲率界及局部分类结果，为BMO猜想提供了证据。

Weakly convex biharmonic hypersurfaces in nonpositive curvature space forms are minimal

A submanifold $M^m$ of a Euclidean space $R^{m+p}$ is said to have harmonic mean curvature vector field if $Δ\vec{H}=0$, where $\vec{H}$ is the mean curvature vector field of $M\hookrightarrow R^{m+p}$ and $Δ$ is the rough Laplacian on $M$. There is a conjecture named after Bangyen Chen which states that submanifolds of Euclidean spaces with harmonic mean curvature vector fields are minimal. In this paper we prove that weakly convex hypersurfaces (i.e. hypersurfaces whose principle curvatures are nonnegative) with harmonic mean curvature vector fields in Euclidean spaces are minimal. Furthermo…

New results toward the classification of Biharmonic submanifolds in $\mathbb{S}^{n}$

We prove some new rigidity results for proper biharmonic immersions in ${\mathbb S}^n$ of the following types: Dupin hypersurfaces; hypersurfaces, both compact and non-compact, with bounded norm of the second fundamental form; hypersurfaces satisfying intrinsic properties; PMC submanifolds; parallel submanifolds.

乘积空间中的双守恒与双调和超曲面

Biconservative and Biharmonic Hypersurfaces in Product Spaces

本文研究乘积空间中的双守恒与双调和超曲面，给出等参与常角情形的完整描述，并证明分类、刚性及全局不存在性定理。

球面和乘积空间中的共形双能超曲面

Conformal-biharmonic hypersurfaces in spheres and product spaces

研究了球面和乘积空间中具有常主曲率的共形双曲面的性质及分类

Biharmonic submanifolds in a Riemannian manifold

In this paper, we solve affirmatively B.-Y. Chen's conjecture for hypersurfaces in the Euclidean space, under a generic condition. More precisely, every biharmonic hypersurface of the Euclidean space must be minimal if their principal curvatures are simple, and the associated frame field is irreducible.

Classification results for biharmonic submanifolds in spheres

We classify biharmonic submanifolds with certain geometric properties in Euclidean spheres. For codimension 1, we determine the biharmonic hypersurfaces with at most two distinct principal curvatures and the conformally flat biharmonic hypersurfaces. We obtain some rigidity results for pseudo-umbilical biharmonic submanifolds of codimension 2 and for biharmonic surfaces with parallel mean curvature vector field. We also study the type, in the sense of B-Y. Chen, of compact proper biharmonic submanifolds with constant mean curvature in spheres.

Some constructions of biharmonic maps and Chen's conjecture on biharmonic hypersurfaces

We give several construction methods and use them to produce many examples of proper biharmonic maps including biharmonic tori of any dimension in Euclidean spheres (Theorem 2.2, Corollaries 2.3, 2.4, and 2.6), biharmonic maps between spheres (Theorem 2.9) and into spheres (Theorem 2.10) via orthogonal multiplications and eigenmaps. We also study biharmonic graphs of maps, derive the equation for a function whose graph is a biharmonic hypersurface in a Euclidean space, and give an equivalent formulation of Chen's conjecture on biharmonic hypersurfaces by using the biahrmonic graph equation (Th…

欧氏空间中的双调和$\delta(r)$-理想超曲面是极小的

Biharmonic $δ(\lowercase{r})$-ideal hypersurfaces in Euclidean spaces are minimal

本文研究欧氏空间双调和子流形的陈邦彦猜想，证明n≥3时欧氏空间中δ(r)-理想双调和超曲面必为极小超曲面，推广了已有结果，还得到该条件下双保守超曲面为常平均曲率的结论。

Biconservative Lorentz hypersurfaces in $\mathbb{E}_{1}^{\lowercase{n}+1}$ with complex eigenvalues

Our paper is an attempt to to verify the Chen's conjecture on biharmonic submanifolds and to classify biconservative submanifolds. In doing so we provide an affirmative answer to Chen's conjecture on biharmonic submanifolds. We prove that every biconservative Lorentz hypersurface $M_{1}^{n}$ in $\mathbb{E}_{1}^{n+1}$ having complex eigenvalues has constant mean curvature. Moreover, every biharmonic Lorentz hypersurface $M_{1}^{n}$ having complex eigenvalues in $\mathbb{E}_{1}^{n+1}$ must be minimal.

Biharmonic hypersurfaces in Riemannian manifolds

We study biharmonic hypersurfaces in a generic Riemannian manifold. We first derive an invariant equation for such hypersurfaces generalizing the biharmonic hypersurface equation in space forms studied in \cite{Ji2}, \cite{CH}, \cite{CMO1}, \cite{CMO2}. We then apply the equation to show that the generalized Chen's conjecture is true for totally umbilical biharmonic hypersurfaces in an Einstein space, and construct a (2-parameter) family of conformally flat metrics and a (4-parameter) family of multiply warped product metrics each of which turns the foliation of an upper-half space of $\mathhb…

New Examples of Biharmonic Submanifolds in $CP^n$ and $S^{2n+1}$

We construct biharmonic real hypersurfaces and Lagrangian submanifolds of Clifford torus type in $CP^n$ via the Hopf fibration; and get new examples of biharmonic submanifolds in $S^{2n+1}$ as byproducts .

Classifications and Isolation Phenomena of Bi-Harmonic Maps and Bi-Yang-Mills Fields

Classifications of all biharmonic isoparametric hypersurfaces in the unit sphere, and all biharmonic homogeneous real hypersurfaces in the complex or quaternionic projective spaces are shown. Answers in case of bounded geometry to Chen's conjecture or Caddeo, Montaldo and Piu's one on biharmonic maps into a manifold of non positive curvature are given. Gauge field analogue is shown, and the isolation phenomena of bi-Yang-Mills fields are obtained.

沿曲线的超曲面的平坦逼近

Flat approximations of hypersurfaces along curves

本文研究超曲面沿给定曲线的平坦逼近问题，利用平坦曲面作为可展曲面的经典刻画，给出了具有相同沿曲线法向量场的平坦曲面的显式参数构造，并讨论其存在性与唯一性。

On biharmonic submanifolds in non-positively curved manifolds

In the biharmonic submanifolds theory there is a generalized Chen's conjecture which states that biharmonic submanifolds in a Riemannian manifold with non-positive sectional curvature must be minimal. This conjecture turned out false by a counter example of Y. L. Ou and L. Tang in \cite{Ou-Ta}. However it remains interesting to find out sufficient conditions which guarantee this conjecture to be true. In this note we prove that: 1. Any complete biharmonic submanifold (resp. hypersurface) $(M, g)$ in a Riemannian manifold $(N, h)$ with non-positive sectional curvature (resp. Ricci curvature) wh…

欧氏空间中的双守恒超曲面

Biconservative hypersurfaces in Euclidean Spaces

本文研究欧氏空间中的双守恒超曲面，先描述其定义，再构造这类仅双调和方程切向部分消失的等距浸入。

Chen's conjecture on biharmonic submanifolds in Riemannian manifolds

We study biharmonic hypersurfaces and biharmonic submanifolds in a Riemannian manifold. One of interesting problems in this direction is Chen's conjecture which says that any biharmonic submanifold in a Euclidean space is minimal. From the invariant equation for biharmonic submanifolds, we derive a fundamental identity involving the mean curvature vector field, and using this, we prove Chen's conjecture on biharmonic submanifolds in a Euclidean space. More generally, it is proved that any biharmonic submanifold in a space form of nonpositively sectional curvature is minimal. Furthermore we pro…