Related papers: On Laguerre Isotropic Hypersurfaces
We consider the inverse mean curvature flow by parallel hypersurfaces in space forms. We show that such a flow exists if and only if the initial hypersurface is isoparametric. The flow is characterized by an algebraic equation satisfied by…
The purpose of this paper is to investigate the geometric properties of real hypersurfaces of D'Angelo infinite type in ${\mathbb C}^n$. In order to understand the situation of flatness of these hypersurfaces, it is natural to ask whether…
We consider Einstein hypersurfaces of warped products $I\times_\omega\mathbb Q_\epsilon^n,$ where $I\subset\mathbb R$ is an open interval and $\mathbb Q_\epsilon^n$ is the simply connected space form of dimension $n\ge 2$ and constant…
The paper investigates higher dimensional analogues of Burago's inequality bounding the area of a closed surface by its total curvature. We obtain sufficient conditions for hypersurfaces in 4-space that involve the Ricci curvature. We get…
In this paper, we prove a Heintze-Karcher type inequality for capillary hypersurfaces supported on various hypersurfaces in the hyperbolic space. The equality case only occurs on capillary totally umbilical hypersurfaces. Then we apply this…
We consider isometric immersions of complete connected Riemannian manifolds into space forms of nonzero constant curvature. We prove that if such an immersion is compact and has semi-definite second fundamental form, then it is an embedding…
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…
In this paper we consider the equiform motion of a helix in Euclidean space $\mathbf{E}^7$. We study and analyze the corresponding kinematic three dimensional surface under the hypothesis that its scalar curvature $\mathbf{K}$ is constant.…
We present a method to construct a large family of Lagrangian surfaces in complex Euclidean plane by using Legendre curves in the 3-sphere and in the anti de Sitter 3-space or, equivalently, by using spherical and hyperbolic curves,…
We investigate 3-dimensional complete minimal hypersurfaces in the hyperbolic space $\mathbb{H}^{4}$ with Gauss-Kronecker curvature identically zero. More precisely, we give a classification of complete minimal hypersurfaces with…
The Dirichlet Laplacian between two parallel hypersurfaces in Euclidean spaces of any dimension in the presence of a magnetic field is considered in the limit when the distance between the hypersurfaces tends to zero. We show that the…
We prove that a hemisphere in the Euclidean space $R^{n+1}$, viewed as the graph of a function, admits no smooth perturbations as graphs with mean curvature $H\ge 1$ whose boundary equator is fixed up to $C^2$. This is an extension of the…
In this work we introduce the notion of constant angle null hypersurface of a Lorentzian manifold with respect to a given ambient vector field. We analyze the case in which the vector field is closed and conformal, thus finding that such…
We investigate helicoidal surfaces in three-dimensional Euclidean space whose profile curves are frontals. Using the framework of Legendre curves and framed surfaces, we establish conditions under which helicoidal surfaces generated by…
We study normal forms of germs of singular real-analytic Levi-flat hypersurfaces. We prove the existence of rigid normal forms for singular Levi-flat hypersurfaces which are defined by the vanishing of the real part of complex…
In this paper, we show the fundamental theorems for rotationally symmetric hypersurfaces, and thus, together with the earlier results in [3] and [4], provide a complete classification of umbilic hypersurfaces in the Heisenberg groups…
In this paper, we introduce the Laguerre bounded variation space and the Laguerre perimeter, thereby investigating their properties. Moreover, we prove the isoperimetric inequality and the Sobolev inequality in the Laguerre setting. As…
The theory of the isoptic curves is widely studied in the Euclidean plane $\bE^2$ (see \cite{CMM91} and \cite{Wi} and the references given there). The analogous question was investigated by the authors in the hyperbolic $\bH^2$ and elliptic…
In this paper, we are concerned with hypersurfaces in $H^n\times R$ with constant r-mean curvature, to be called $H_r$-hypersurfaces. We construct examples of complete $H_r$-hypersurfaces which are invariant by parabolic screw motion or by…
We prove that if u is a bounded smooth function in the kernel of a nonnegative Schrodinger operator $-L=-(\Delta +q)$ on a parabolic Riemannian manifold M, then u is either identically zero or it has no zeros on M, and the linear space of…