Related papers: Remarks on biharmonic hypersurfaces in space forms

We show the regularity of, and derive a-priori estimates for (weakly) harmonic maps from a Riemannian manifold into a Euclidean sphere under the assumption that the image avoids some neighborhood of a half-equator. The proofs combine…

We classify the biharmonic Legendre curves in a Sasakian space form, and obtain their explicit parametric equations in the $(2n+1)$-dimensional unit sphere endowed with the canonical and deformed Sasakian structures defined by Tanno. Then,…

We consider stable manifolds of a holomorphic diffeomorphism of a complex manifold. Using a conjugation of the dynamics to a (non-stationary) polynomial normal form, we show that typical stable manifolds are biholomorphic to complex…

We show that closed starshaped hypersurfaces of space forms with almost constant mean curvature or almost constant higher order mean curvature are closed to geodesic spheres.

We prove qualitative estimates on the total curvature of closed minimal hypersurfaces in closed Riemannian manifolds in terms of their index and area, restricting to the case where the hypersurface has dimension less than seven. In…

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…

In this paper we consider the complete biconservative surfaces in Euclidean space $\mathbb{R}^3$ and in the unit Euclidean sphere $\mathbb{S}^3$. Biconservative surfaces in 3-dimensional space forms are characterized by the fact that the…

A submanifold $M^n$ of a Euclidean space $\mathbb{E}^N$ is called biharmonic if $\Delta\vec{H}=0$, where $\vec{H}$ is the mean curvature vector of $M^n$. A well known conjecture of B.Y. Chen states that the only biharmonic submanifolds of…

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…

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…

In this paper, we prove a local rigidity of convex hypersurfaces in the spaces of constant curvature of dimension $n\ge4$. Namely, we show that two convex isometric hypersurfaces are congruent locally around their corresponding under the…

In this work, we extend the concepts of $p$-biharmonic maps and $p$-biharmonic hypersurfaces to provide a broader characterization of $(p,q)$-harmonic hypersurfaces and $(p,q)$-harmonic curves in Riemannian manifolds, including Einstein…

Hypersurfaces embedded in conformal manifolds appear frequently as boundary data in boundary-value problems in cosmology and string theory. Viewed as the non-null conformal infinity of a spacetime, we consider hypersurfaces embedded in a…

We revisit Allendoerfer-Weil's formula for the Euler characteristic of embedded hypersurfaces in constant sectional curvature manifolds, first taking some time to re-prove it while demonstrating techniques of [2] and then applying it to…

We derive the closed form solutions for the surface area, the capacitance and the demagnetizing factors of the ellipsoid immersed in the Euclidean space R^3. The exact solutions for the above geometrical and physical properties of the…

Within the framework of the well-known curvature models, a fluid lipid bilayer membrane is regarded as a surface embedded in the three-dimensional Euclidean space whose equilibrium shapes are described in terms of its mean and Gaussian…

Let $\Sigma$ be a smooth closed hypersurface with non-negative Ricci curvature, isometrically immersed in a space form. It has been proved in \cite{P}, \cite{CZ}, and \cite{C2} that there are some $L^2$ inequalities on $\Sigma$ which…

We prove that the problem of constructing biharmonic conformal maps on a $4$-dimensional Einstein manifold reduces to a Yamabe-type equation. This allows us to construct an infinite family of examples on the Euclidean 4-sphere. In addition,…

We classify all proper-biharmonic Legendre curves in a Sasakian space form and point out some of their geometric properties. Then we provide a method for constructing anti-invariant proper-biharmonic submanifolds in Sasakian space forms.…

We provide a general B\"ochner type formula which enables us to prove some rigidity results for $V$-static spaces. In particular, we show that an $n$-dimensional positive static triple with connected boundary and positive scalar curvature…