$L_1$-Biharmonic Hypersurfaces in Euclidean Spaces with Three Distinct Principal Curvatures
Authors
Abstract:
Chen's biharmonic conjecture is well-known and stays open: The only biharmonic submanifolds of Euclidean spaces are the minimal ones. In this paper, we consider an advanced version of the conjecture, replacing $Delta$ by its extension, $L_1$-operator ($L_1$-conjecture). The $L_1$-conjecture states that any $L_1$-biharmonic Euclidean hypersurface is 1-minimal. We prove that the $L_1$-conjecture is true for $L_1$-biharmonic hypersurfaces with three distinct principal curvatures and constant mean curvature of a Euclidean space of arbitrary dimension.
similar resources
Real Hypersurfaces with Constant Principal Curvatures in Complex Hyperbolic Spaces
We present the classification of all real hypersurfaces in complex hyperbolic space CHn, n ≥ 3, with three distinct constant principal curvatures.
full textLk-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 Lk-operators. The new conjecture (Lk-conjecture) is formulated as follows: If Lkx = 0 then Hk+1 = 0 where x : M → R is an isometric immersion of a Riemannian manifold M into the Euclidean space R, Hk+1 is the (k+1)-th mean curvature of M , and Lk is the linearized ope...
full textPrincipal Curvatures of Isoparametric Hypersurfaces in Cp
Let M be an isoparametric hypersurface in CPn, and M the inverse image of M under the Hopf map. By using the relationship between the eigenvalues of the shape operators of M and M , we prove that M is homogeneous if and only if either g or l is constant, where g is the number of distinct principal curvatures of M and l is the number of non-horizontal eigenspaces of the shape operator on M .
full text$L_k$-biharmonic spacelike hypersurfaces in Minkowski $4$-space $mathbb{E}_1^4$
Biharmonic surfaces in Euclidean space $mathbb{E}^3$ are firstly studied from a differential geometric point of view by Bang-Yen Chen, who showed that the only biharmonic surfaces are minimal ones. A surface $x : M^2rightarrowmathbb{E}^{3}$ is called biharmonic if $Delta^2x=0$, where $Delta$ is the Laplace operator of $M^2$. We study the $L_k$-biharmonic spacelike hypersurfaces in the $4$-dimen...
full textRigidity of minimal hypersurfaces of spheres with two principal curvatures
Let ν be a unit normal vector field along M . Notice that ν : M −→ S satisfies that 〈ν(m),m〉 = 0. For any tangent vector v ∈ TmM , m ∈ M , the shape operator A is given by A(v) = −∇̄vν, where ∇̄ denotes the Levi Civita connection in S. For every m ∈ M , A(m) defines a linear symmetric transformation from TmM to TmM ; the eigenvalues of this transformation are known as the principal curvatures of ...
full textConvex Hypersurfaces with Pinched Principal Curvatures and Flow of Convex Hypersurfaces by High Powers of Curvature
We consider convex hypersurfaces for which the ratio of principal curvatures at each point is bounded by a function of the maximum principal curvature with limit 1 at infinity. We prove that the ratio of circumradius to inradius is bounded by a function of the circumradius with limit 1 at zero. We apply this result to the motion of hypersurfaces by arbitrary speeds which are smooth homogeneous ...
full textMy Resources
Journal title
volume 13 issue 2
pages 59- 70
publication date 2018-10
By following a journal you will be notified via email when a new issue of this journal is published.
Hosted on Doprax cloud platform doprax.com
copyright © 2015-2023