Related papers: Blaschke's problem for hypersurfaces

We address the problem of determining the hypersurfaces $f\colon M^{n} \to \mathbb{Q}_s^{n+1}(c)$ with dimension $n\geq 3$ of a pseudo-Riemannian space form of dimension $n+1$, constant curvature $c$ and index $s\in \{0, 1\}$ for which…

Let $x: \mathbf{M} \rightarrow {\mathbb Q}^{n+1}_1$ be a regular space-like hypersurface in the conformal space ${\mathbb Q}^{n+1}_1$. We classify all those hypersurfaces with parallel Blaschke tensor in the conformal space up to the…

We consider the generalization of classical Blaschke's Problem to higher codimension case, characterizing Darboux pair of isothermic surfaces and dual S-Willmore surfaces as the only non-trivial surface pairs that envelop a 2-sphere…

Bernstein problem for affine maximal type equation has been a core problem in affine geometry. A conjecture proposed firstly by Chern for entire graph and then extended by Trudinger-Wang to its fully generality asserts that any Euclidean…

In this paper we classify Euclidean hypersurfaces $f\colon M^n \rightarrow \mathbb{R}^{n+1}$ with a principal curvature of multiplicity $n-2$ that admit a genuine conformal deformation $\tilde{f}\colon M^n \rightarrow \mathbb{R}^{n+2}$.…

Let $M^n$ be an $n$-dimensional umbilic-free hypersurface in the $(n+1)$-dimensional Lorentzian space form $M^{n+1}_1(c)$. Three basic invariants of $M^n$ under the conformal transformation group of $M^{n+1}_1(c)$ are a $1$-form $C$, called…

We consider the problem $F=f(\nu)$ for strictly convex, closed hypersurfaces in $S^{n+1}$ and solve it for curvature functions $F$ the inverses of which are of class $(K)$.

The width of a closed convex subset of Euclidean space is the distance between two parallel supporting planes. The Blaschke-Lebesgue problem consists of minimizing the volume in the class of convex sets of fixed constant width and is still…

Our first objective in this paper is to give a natural formulation of the Christoffel problem for hypersurfaces in $H^{n+1}$, by means of the hyperbolic Gauss map and the notion of hyperbolic curvature radii for hypersurfaces. Our second…

In this paper, we investigate the rigidity problems of complete hypersurfaces with constant mean curvature and constant scalar curvature in Euclidean spaces. Firstly, under some conditions of Gaussian-Kronecker curvature, we provide…

Conditions, related to the so-called bending problem are considered for hypersurfaces of a pseudo-Euclidean space. Corresponding theorems are proved.

Biharmonic hypersurfaces in a generic conformally flat space are studied in this paper. The equation of such hypersurfaces is derived and is used to determine the conformally flat metric $f^{-2}\delta_{ij}$ on the Euclidean space…

We study the problem posed by F. Burstall of developing a theory of isothermic Euclidean submanifolds of dimension greater than or equal to three. As a natural extension of the definition in the surface case, we call a Euclidean submanifold…

In this article we provide a general construction when $n\ge3$ for immersed in Euclidean $(n+1)$-space, complete, smooth, constant mean curvature hypersurfaces of finite topological type (in short CMC $n$-hypersurfaces). More precisely our…

In [2] we have classified the Blaschke quasi-umbilical submanifolds in the conformal space ${\mathbb Q}^n_s$. In this paper we shall classify the Blaschke para-umbilical hypersurfaces in the conformal space ${\mathbb Q}^n_s$. That may be…

In 3-dimensional Euclidean space, Scherk second surfaces are singly periodic embedded minimal surfaces with four planar ends. In this paper, we obtain a natural generalization of these minimal surfaces in any higher dimensional Euclidean…

In this paper, we use two conformal non-homogeneous coordinate systems, modeled on the de Sitter space ${\mathbb S}^{m+1}_1$, to cover the conformal space ${\mathbb Q}^{m+1}_1$, so that the conformal geometry of regular space-like…

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…

We proved that any complete hypersurface in the Euclidean space $\mathbb{R}^{n+1}$ whose Gauss image is contained in an open hemisphere has to be proper. As applications, we derive a counterpart of Hoffman-Osserman-Schoen's result for…

We develop the basic formulae of hyperspherical trigonometry in multidimensional Euclidean space, using multidimensional vector products, and their conversion to identities for elliptic functions. We show that the basic addition formulae…