Biharmonic Submanifold Research Articles

On extended biharmonic hypersurfaces with three curvatures

The subject of harmonic and biharmonic submanifolds, with important role in mathematical physics and differential geometry, arises from the variation problems of ordinary mean curvature vector field. Generally, harmonic submanifolds are biharmonic, but not vice versa. Of course, many examples of biharmonic hypersurfaces are harmonic. A well-known conjecture of Bang-Yen Chen on Euclidean spaces says that every biharmonic submanifold is harmonic. Although the conjecture has not been proven (in general case), it has been affirmed in many cases, and this has led to its spread to various types of submanifolds. Inspired by the conjecture, we study the Lorentz submanifolds of the Lorentz-Minkowski spaces. We consider an advanced versión of the conjecture (namely, L1-conjecture) on Lorentz hypersurfaces of the pseudo-Euclidean 5-space L5 := E15 (i.e. the Minkowski 5-space). We confirm the extended conjecture on Lorentz hypersurfaces with three principal curvatures.

Biharmonic maps and biharmonic submanifolds with small curvature integral

In this article, we study biharmonic maps and biharmonic submanifolds with small curvature integral. Let φ:(Mn,g)→(Nm,h) be a biharmonic map from a complete noncompact Riemannian manifold (Mn,g) into a Riemannian manifold (Nm,h) satisfying that the L2-norm of the tension field of the map is finite. If the domain manifold of the map satisfies a Sobolev inequality and the Ln2-norm of the sectional curvature on the image φ(M) is sufficiently small, then we are able to prove the harmonicity of the biharmonic map. It turns out that the fundamental tone of M is sufficiently large, then such a biharmonic map φ must be harmonic. In case where the map is an isometric immersion, we prove that if M satisfies a Sobolev inequality, then M must be minimal under the assumption that the Ln2-norm of the Ricci curvature on M is sufficiently small. Moreover it is shown that if the fundamental tone of a biharmonic submanifold is sufficiently big, then it is minimal.

On Chen's biharmonic conjecture for hypersurfaces in [formula omitted

A longstanding conjecture on biharmonic submanifolds, proposed by Chen in 1991, is that any biharmonic submanifold in a Euclidean space is minimal. In the case of a hypersurface Mn in Rn+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 Mn remains open for n>3. In this paper, we settle Chen's conjecture for hypersurfaces in R5 for n=4.

Some classifications of biharmonic hypersurfaces with constant scalar curvature

We give some classifications of biharmonic hypersurfaces with constant scalar curvature. These include biharmonic Einstein hypersurfaces in space forms, compact biharmonic hypersurfaces with constant scalar curvature in a sphere, and some complete biharmonic hypersurfaces of constant scalar curvature in space forms and in a non-positively curved Einstein space. Our results provide additional cases (Theorem 2.3 and Proposition 2.8) that supports the conjecture that a biharmonic submanifold in a sphere has constant mean curvature, and two more cases that support Chen's conjecture on biharmonic hypersurfaces (Corollaries 2.2,2.7).

Chen’s Biharmonic Conjecture and Submanifolds with Parallel Normalized Mean Curvature Vector

The well known Chen’s conjecture on biharmonic submanifolds in Euclidean spaces states that every biharmonic submanifold in a Euclidean space is a minimal one. For hypersurfaces, we know from Chen and Jiang that the conjecture is true for biharmonic surfaces in E 3 . Also, Hasanis and Vlachos proved that biharmonic hypersurfaces in E 4 ; and Dimitric proved that biharmonic hypersurfaces in E m with at most two distinct principal curvatures. Chen and Munteanu showed that the conjecture is true for δ ( 2 ) -ideal and δ ( 3 ) -ideal hypersurfaces in E m . Further, Fu proved that the conjecture is true for hypersurfaces with three distinct principal curvatures in E m with arbitrary m. In this article, we provide another solution to the conjecture, namely, we prove that biharmonic surfaces do not exist in any Euclidean space with parallel normalized mean curvature vectors.

Biharmonic submanifolds in manifolds with bounded curvature

We consider a complete biharmonic submanifold [Formula: see text] in a Riemannian manifold with sectional curvature bounded from above by a non-negative constant [Formula: see text]. Assume that the mean curvature is bounded from below by [Formula: see text]. If (i) [Formula: see text], for some [Formula: see text], or (ii) the Ricci curvature of [Formula: see text] is bounded from below, then the mean curvature is [Formula: see text]. Furthermore, if [Formula: see text] is compact, then we obtain the same result without the assumption (i) or (ii). These are affirmative partial answers to Balmuş–Montaldo–Oniciuc conjecture.

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 curvature ([21]). The most recent work of Chen-Munteanu [18] shows that Chen's conjecture is true for $\delta(2)$-ideal hypersurfaces in $\mathbb E^m$, where a $\delta(2)$-ideal hypersurface is a hypersurface whose principal curvatures take three special values: $\lambda_1, \lambda_2$ and $\lambda_1+\lambda_2$. In this paper, we prove that Chen's conjecture is true for hypersurfaces with three distinct principal curvatures in $\mathbb E^m$ with arbitrary dimension, thus, extend all the above-mentioned results. As an application we also show that Chen's conjecture is true for $O(p)\times O(q)$-invariant hypersurfaces in Euclidean space $\mathbb E^{p+q}$.

$L_k$-BIHARMONIC HYPERSURFACES IN THE EUCLIDEAN SPACE

Chen conjecture states that every Euclidean biharmonic submanifold is minimal. In this paper we consider the Chen conjecture for $ L_k $-operators. The new conjecture ($ L_k $-conjecture) is formulated as follows: If $ L_k^{2}x=0 $ then $ H_{k+1}=0$ where $ x:M^{n}\rightarrow\Bbb{R}^{n+1} $ is an isometric immersion of a Riemannian manifold $ M^n $ into the Euclidean space $ \Bbb{R}^{n+1} $, $ H_{k+1} $ is the $ (k+1) $-th mean curvature of $ M $, and $ L_k $ is the linearized operator of the $ (k+1) $-th mean curvature of the Euclidean hypersurface $ M $. We prove the $ L_k $-conjecture for the hypersurface $ M $ with at most two principal curvatures.

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 Ou and Tang (2012). 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) which satisfies an integral condition: for some p∈(0,+∞), ∫M|H→|pdμg<+∞, where H→ is the mean curvature vector field of M↪N, must be minimal. This generalizes the recent results due to N. Nakauchi and H. Urakawa in Nakauchi and Urakawa (2013, 2011). 2. Any complete biharmonic submanifold (resp. hypersurface) in a Riemannian manifold of at most polynomial volume growth whose sectional curvature (resp. Ricci curvature) is non-positive must be minimal. 3. Any complete biharmonic submanifold (resp. hypersurface) in a non-positively curved manifold whose sectional curvature (resp. Ricci curvature) is smaller than −ϵ for some ϵ>0 which satisfies that ∫Bρ(x0)|H→|p+2dμg(p≥0) is of at most polynomial growth of ρ, must be minimal. We also consider ε-superbiharmonic submanifolds defined recently in Wheeler (2013) by G. Wheeler and prove similar results for ε-superbiharmonic submanifolds, which generalize the result in Wheeler (2013).

Biharmonic maps from a complete Riemannian manifold into a non-positively curved manifold

We consider biharmonic maps $$\phi :(M,g)\rightarrow (N,h)$$ from a complete Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. Assume that $$ p $$ satisfies $$ 2\le p <\infty $$ . If for such a $$ p $$ , $$\int _M|\tau (\phi )|^{ p }\,\mathrm{d}v_g<\infty $$ and $$\int _M|\,\mathrm{d}\phi |^2\,\mathrm{d}v_g<\infty ,$$ where $$\tau (\phi )$$ is the tension field of $$\phi $$ , then we show that $$\phi $$ is harmonic. For a biharmonic submanifold, we obtain that the above assumption $$\int _M|\,\mathrm{d}\phi |^2\,\mathrm{d}v_g<\infty $$ is not necessary. These results give affirmative partial answers to the global version of generalized Chen’s conjecture.

Biharmonic PNMC submanifolds in spheres

We obtain several rigidity results for biharmonic submanifolds in $\mathbb{S}^{n}$ with parallel normalized mean curvature vector field. We classify biharmonic submanifolds in $\mathbb{S}^{n}$ with parallel normalized mean curvature vector field and with at most two distinct principal curvatures. In particular, we determine all biharmonic surfaces with parallel normalized mean curvature vector field in $\mathbb{S}^n$. Then we investigate, for (not necessarily compact) proper biharmonic submanifolds in $\mathbb{S}^n$, their type in the sense of B-Y. Chen. We prove: (i) a proper biharmonic submanifold in $\mathbb{S}^n$ is of 1-type or 2-type if and only if it has constant mean curvature ${\mcf}=1$ or ${\mcf}\in(0,1)$, respectively; (ii) there are no proper biharmonic 3-type submanifolds with parallel normalized mean curvature vector field in $\mathbb{S}^n$.

On the generalized Chen's conjecture on biharmonic submanifolds

The generalized Chen's conjecture on biharmonic submanifolds asserts that any biharmonic submanifold of a non-positively curved manifold is minimal (see e.g., [CMO1], [MO], [BMO1], [BMO2], [BMO3], [Ba1], [Ba2], [Ou1], [Ou2], [IIU]). In this paper, we prove that this conjecture is false by constructing foliations of proper biharmonic hyperplanes in a 5-dimensional conformally flat space with negative sectional curvature. Many examples of proper biharmonic submanifolds of non-positively curved spaces are also given.

Biharmonic Submanifolds in a Riemannian Manifold with Non-Positive Curvature

In this paper, we show that, for every biharmonic submanifold (M, g) of a Riemannian manifold (N, h) with non-positive sectional curvature, if $${\int_M\vert \eta \vert^2 v_g < \infty}$$ , then (M, g) is minimal in (N, h), i.e., $${\eta\equiv0}$$ , where η is the mean curvature tensor field of (M, g) in (N, h). This result gives an affirmative answer under the condition $${\int_M\vert \eta \vert^2 v_g < \infty}$$ to the following generalized Chen’s conjecture: every biharmonic submanifold of a Riemannian manifold with non-positive sectional curvature must be minimal. The conjecture turned out false in case of an incomplete Riemannian manifold (M, g) by a counter example of Ou and Tang (in The generalized Chen’s conjecture on biharmonic sub-manifolds is false, a preprint, 2010).

Some classification theorems for submanifolds in Minkowski space-time

if its mean curvature vector H is an eigenvector of A with eigenvalue 0. The study of submanifolds whose mean curvature vector is an eigenvector of A was initiated in [3] in which the author proved that a submanifold M of a Euclidean space satisfies the condi- tion AH = 2H for some constant 2 if and only if either M is biharmonic (2 = 0), or M is of l-type or of null 2-type. The complete classification of surfaces in E 3 satisfy the condition AH = 2H was done in t988 by the author. In this paper we investigate submanifolds of hyperbolic spaces satisfying the condition AH = 2H. More precisely, we obtain the following results. Theorem 1. Let M be an n-dimensional submanifoM of the hyperbolic space H "~ -t (_ 1), imbedded standardly in the Minkowski space-time E"~. Then the mean curvature vector H of M in E'~ is an eigenvector of A if and only if either M is a minimal submanifold of Hm-1 (_ 1) or M is contained in a totally umbilical hypersurface of t-1 ~- I (_ 1) as a minimal submanifold. Theorem 2. Let M be an n-dimensional, non-minimal submanifoM of the hyperbolic' space H m- 1 (_ 1), imbedded standardly in the Minkowski space-time E'~. Then the Jbllowing statements are equivalent: (a) M is a biharmonic submanifold of E";; (b) the mean curvature vector of M in E'; is a light-like constant vector; (c) M is a pseudo-umbilical submanifold with unit parallel mean curvature vector in

BIHARMONIC W-SURFACES IN 4-DIMENSIONAL PSEUDO-EUCLIDEAN SPACE

In this article, W-surface in a 4-dimensional pseudo-Euclidean space Et4 (t = l, 2) is defind and a classification problem of biharmonic W-surfaces in Et4 is investigated. The study of biharmonic submanifold in a pseudo-Euclidean space is an off-spring of the study of a problem “ Classify all of finite type submanifolds in a Euclidean space Em” proposed by B. Y. Chen (see [5],[6] and [11]). As a result, one classification theorem for biharmonic W-surfaces with flat normal connection in Et4 is obtained. It is a generalization of main theorem in [7].