Related papers: $f$-Biharmonic hypersurfaces into a conformally fl…

We use functions of a bicomplex variable to unify the existing constructions of harmonic morphisms from a 3-dimensional Euclidean or pseudo-Euclidean space to a Riemannian or Lorentzian surface. This is done by using the notion of…

We study conformally flat hypersurfaces $f\colon M^{3} \to \Q^{4}(c)$ with three distinct principal curvatures and constant mean curvature $H$ in a space form with constant sectional curvature $c$. First we extend a theorem due to Defever…

This paper studies conformal biharmonic immersions. We first study the transformations of Jacobi operator and the bitension field under conformal change of metrics. We then obtain an invariant equation for a conformal biharmonic immersion…

We study $p$-harmonic maps, $p$-harmonic morphisms, biharmonic maps, and quasiregular mappings into submanifolds of warped product Riemannian manifolds ${I}\times_f S^{m-1}(k)\, $ of an open interval and a complete simply-connecteded…

We study f-biharmonic and bi-f-harmonic submanifolds in both generalized complex and Sasakian space forms. We prove necessary and sufficient condition for f-biharmonicity and bi-f-harmonicity in the general case and many particular cases.…

A $k$-harmonic map is a critical point of the $k$-energy in the space of smooth maps between two Riemannian manifolds. In this paper, we prove that if $M^{n} (n\ge 3)$ is a CMC proper triharmonic hypersurface with at most three distinct…

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 Balmu\c{s}-Montaldo-Oniciuc, we…

An isometric immersion $X: \Sigma^n \longrightarrow \mathbb{E}^{n+1}$ is biharmonic if $\Delta^2 X = 0$, i.e. if $\Delta H =0$, where $\Delta$ and $H$ are the metric Laplacian and the mean curvature vector field of $\Sigma^n$ respectively.…

We consider a non-negative biminimal properly immersed submanifold $M$ (that is, a biminimal properly immersed submanifold with $\lambda\geq0$) in a complete Riemannian manifold $N$ with non-positive sectional curvature. Assume that the…

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…

We show that an asymptotically flat Riemannian three-manifold with non-negative scalar curvature is isometric to flat $\mathbb{R}^3$ if it admits an unbounded area-minimizing surface. This answers a question of R. Schoen.

We consider a complete biharmonic submanifold $\phi:(M,g)\rightarrow (N,h)$ in a Riemannian manifold with sectional curvature bounded from above by a non-negative constant $c$. Assume that the mean curvature is bounded from below by $\sqrt…

In this note, we investigate conformally flat submanifolds of Euclidean space with positive index of relative nullity. Let $M^n$ be a complete conformally flat manifold and let $f\colon M^n\to \R^m$ be an isometric immersion. We prove the…

We first prove that, unlike the biharmonic case, there exist triharmonic curves with nonconstant curvature in a suitable Riemannian manifold of arbitrary dimension. We then give the complete classification of triharmonic curves in surfaces…

In this note we prove that a constant mean curvature surface is proper-biharmonic in the unit Euclidean sphere $\mathbb{S}^4$ if and only if it is minimal in a hypersphere $\mathbb{S}^3(\frac{1}{\sqrt{2}})$.

Liebmann's Theorem asserts that a compact, connected, convex surface with constant mean curvature (CMC) in the Euclidean space must be a totally umbilical sphere. In this article we extend Liebmann's result to hypersurfaces with boundary.…

In this paper we will show the following result: Let $\mathcal{N} $ be a complete (noncompact) connected orientable Riemannian three-manifold with nonnegative scalar curvature $S \geq 0$ and bounded sectional curvature $ K_{s} \leq K $.…

In this paper we study triharmonic hypersurfaces immersed in a space form $N^{n+1}(c)$. We prove that any proper CMC triharmonic hypersurface in the sphere $\mathbb S^{n+1}$ has constant scalar curvature; any CMC triharmonic hypersurface in…

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…

We compute a Simons' type formula for the stress-energy tensor of biharmonic maps from surfaces. Specializing to Riemannian immersions, we prove several rigidity results for biharmonic CMC surfaces, putting in evidence the influence of the…