Related papers: Algebraic cmc hypersurface of order 3 in Euclidean…

We prove that there are no regular algebraic hypersurfaces with non-zero constant mean curvature in the Euclidean space $\mathbb R^{n+1}$, $n\geq 2$, defined by polynomials of odd degree. Also we prove that the hyperspheres and the round…

In this paper we prove that the only algebraic constant mean curvature (cmc) surfaces in R^3 of order less than four are the planes, the spheres and the cylinders. The method used heavily depends on the efficiency of algorithms to compute…

We obtain some nonexistence results for complete noncompact stable hyppersurfaces with nonnegative constant scalar curvature in Euclidean spaces. As a special case we prove that there is no complete noncompact strongly stable hypersurface…

We prove that a complete hyperbolic 3-manifold of finite volume does not admit a properly embedded noncompact surface of finite topology with constant mean curvature greater than or equal to 1.

In the theory of finite type submanifolds, null 2-type submanifolds are the most simple ones, besides 1-type submanifolds (cf. e.g., [3, 12]). In particular, the classification problems of null 2-type hypersurfaces are quite interesting and…

Three classes of new, algebraic, zero-mean-curvature hypersurfaces in pseudo-Euclidean spaces are given.

Let $M^n$ be a biharmonic hypersurface with constant scalar curvature in a space form $\mathbb M^{n+1}(c)$. We show that $M^n$ has constant mean curvature if $c>0$ and $M^n$ is minimal if $c\leq0$, provided that the number of distinct…

In hyperbolic 3-space $\mathbb{H}^3$ surfaces of constant mean curvature $H$ come in three types, corresponding to the cases $0 \leq H < 1$, $H = 1$, $H > 1$. Via the Lawson correspondence the latter two cases correspond to constant mean…

We prove that proper biharmonic hypersurfaces with constant scalar curvature in Euclidean sphere $\mathbb S^5$ must have constant mean curvature. Moreover, we also show that there exist no proper biharmonic hypersurfaces with constant…

We show that closed hypersurfaces in Euclidean space with nonnegative scalar curvature are weakly mean convex. In contrast, the statement is no longer true if the scalar curvature is replaced by the k-th mean curvature, for k greater than…

In this paper, we give a full classification of the separable hypersurfaces of constant sectional curvature in the Euclidean $n$-space $\mathbb{R}^n$. In dimension $n=3$, this classification was solved by Hasanis and L\'opez [Manuscripta…

In this paper we prove some results concerning stability of hypersurfaces in the four dimensional Euclidean space with zero scalar curvature. First we prove there is no complete stable hypersurface with zero scalar curvature, polynomial…

We prove that the only surfaces in $3$-dimensional Euclidean space $\R^3$ with constant Gaussian curvature $K$ and constructed by the sum of two space curves are cylindrical surfaces, in particular, $K=0$.

We describe local similarities and global differences between minimal surfaces in Euclidean 3-space and constant mean curvature 1 surfaces in hyperbolic 3-space. We also describe how to solve global period problems for constant mean…

We will prove that \emph{there are no stable complete hypersurfaces of $\mathbb{R}^4$ with zero scalar curvature, polynomial volume growth and such that $\dfrac{(-K)}{H^3}\geq c>0$ everywhere, for some constant $c>0$}, where $K$ denotes the…

Generalizing a theorem of Huang, Cheng and Wan classified the complete hypersurfaces of $\mathbb R^4$ with non-zero constant mean curvature and constant scalar curvature. In our work, we obtain results of this nature in higher dimensions.…

We consider regular surfaces $M$ that are given as the zeros of a polynomial function $p:R^3\rightarrow R$, where the gradient of $p$ vanishes nowhere. We assume that $M$ has non-zero mean curvature and prove that there exist only two…

We investigate complete minimal hypersurfaces in the Euclidean space $% \ {R}^{4}$, with Gauss-Kronecker curvature identically zero. We prove that, if $f:M^{3}\to {R}^{4}$ is a complete minimal hypersurface with Gauss-Kronecker curvature…

We prove a Simons type equation for non-minimal surfaces with parallel mean curvature vector (pmc surfaces) in $M^3(c)\times\mathbb{R}$, where $M^3(c)$ is a 3-dimensional space form. Then, we use this equation in order to characterize…

We make observations about constant mean curvature surfaces in Euclidean 3-space and their dual surfaces, and the resulting pairs of surfaces in hyperbolic 3-space under the Lawson correspondence.