In this paper we study the r-stability of closed spacelike hypersurfaces with constant $r$-th mean curvature in conformally stationary spacetimes of constant sectional curvature. In this setting, we obtain a characterization of $r-$stability through the analysis of the first eigenvalue of an operator naturally attached to the $r$-th mean curvature. As an application, we treat the case in which the spacetime is the de Sitter 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.
The aim of this paper is to verify that the study of generic conformally flat hypersurfaces in 4-dimensional space forms is reduced to a surface theory in the standard 3-sphere. The conformal structure of generic conformally flat (local-)hypersurfaces is characterized as conformally flat (local-)3-metrics with the Guichard condition. Then, there is a certain class of orthogonal analytic (local-)Riemannian 2-metrics with constant Gauss curvature -1 such that any 2-metric of the class gives rise to a one-parameter family of conformally flat 3-metrics with the Guichard condition. In this paper, w…
In this article we classify the conformally flat Euclidean hypersurfaces of dimension three with three distinct principal curvatures of $\mathbb{R}^4$, $\mathbb{S}^3\times \mathbb{R}$ and $\mathbb{H}^3\times \mathbb{R}$ with the property that the tangent component of the vector field $\partial/\partial t$ is a principal direction at any point. Here $\partial/\partial t$ stands for either a constant unit vector field in $\mathbb{R}^4$ or the unit vector field tangent to the factor $\mathbb{R}$ in the product spaces $\mathbb{S}^3\times \mathbb{R}$ and $\mathbb{H}^3\times \mathbb{R}$, respectivel…
Conformally flat submanifolds in spheres and integrable systems
E. Cartan证明了S^{n+1}（n>3）中的共形平坦超曲面至多有两个不同的主曲率，并且局部包络一个单参数的（n-1）-球面族.我们证明了S^4中共形平坦超曲面的Gauss-Codazzi方程是一个孤子方程，并利用孤子理论中的一个Dressing作用构造了这些超曲面的几何Ribaucour变换。我们描述了这些超曲面在S^4中的模及其环群对称性。我们还将这些结果推广到具有平坦法丛和常数重数的（2n-2）-球面中的共形平坦n-浸入。
In this paper, we study the relationship between isoparametric hypersurfaces and hypersurfaces with constant principal curvatures in Finsler spaces. We give some examples of isoparametric hypersurfaces with (non)constant principal curvatures on Randers manifolds with nonconstant flag curvatures. Furthermore, we construct an example of a conformally flat Randers manifold which admits a family of nonisoparametric hyperplanes with constant principal curvatures.
Let $M^n$ be an $n$-dimensional complete and locally conformally flat hypersurface in the unit sphere $\mathbb{S}^{n+1}$ with constant scalar curvature $n(n-1)$. We show that if the total curvature $\left( \int _ { M } | H | ^ { n } d v \right) ^ { \frac { 1 } { n } }$ of $M$ is sufficiently small, then $M^n$ is totally geodesic.
An important problem is to determine under which circumstances a metric on a conformally compact manifold is conformal to a Poincaré--Einstein metric. Such conformal rescalings are in general obstructed by conformal invariants of the boundary hypersurface embedding, the first of which is the trace-free second fundamental form and then, at the next order, the trace-free Fialkow tensor. We show that these tensors are the lowest order examples in a sequence of conformally invariant higher fundamental forms determined by the data of a conformal hypersurface embedding. We give a construction of the…
In a 4-manifold, the composition of a Riemannian Einstein metric with an almost paracomplex structure that is isometric and parallel, defines a neutral metric that is conformally flat and scalar flat. In this paper, we study hypersurfaces that are null with respect to this neutral metric and in particular we study their geometric properties with respect to the Einstein metric. Firstly, we show that all totally geodesic null hypersurfaces are scalar flat and their existence implies that the Einstein metric in the ambient manifold must be Ricci-flat. Then, we find a necessary condition for the e…
In this paper, we study stability and instability problem for type-II partitioning problem. First, we make a complete classification of stable type-II stationary hypersurfaces in a ball in a space form as totally geodesic $n$-balls. Second, for general ambient spaces and convex domains, we give some topological restriction for type-II stable stationary immersed surfaces in two dimension. Third, we give a lower bound for the Morse index for type-II stationary hypersurfaces in terms of their topology.
