Related papers: Surfaces in Sol$_3$ space foliated by circles
We study the classification of area-stationary and stable $C^2$ regular surfaces in the space of the rigid motions of the Minkowski plane E(1,1), equipped with its sub-Riemannian structure. We construct examples of area-stationary surfaces…
We obtain a complete classification of ruled zero mean curvature surfaces in the three-dimensional light cone. En route, we examine geodesics and screw motions in the space form, allowing us to discover helicoids. We also consider their…
We show that a complete non-compact 3-manifold with scalar curvature bounded below by a positive constant admits a singular foliation by surfaces of controlled area and diameter.
We classify rotational surfaces in a normed 3-space with rotationally symmetric norm whose principal curvatures satisfy a linear relation.
We elucidate the geometric background of function-theoretic properties for the Gauss maps of several classes of immersed surfaces in three-dimensional space forms, for example, minimal surfaces in Euclidean three-space, improper affine…
We give a short proof of the contractibility of the space of geodesic triangulations with fixed combinatorial type of a convex polygon in the Euclidean plane. Moreover, for any $n>0$, we show that there exists a space of geodesic…
In this paper, we study the rotational surfaces in the isotropic 3-space I^3. satisfying Weingarten conditions in terms of the relative curvature K (analogue of the Gaussian curvature) and the isotropic mean curvature H. In particular, we…
In this work, we consider spacelike surfaces in Minkowski space $\hbox{\bf E}%_{1}^{3}$ that satisfy a linear Weingarten condition of type $\kappa_{1}=m\kappa_{2}+n$, where $m$ and $n$ are constant and $\kappa_{1}$ and $\kappa_{2}$ denote…
In this paper, we study polar foliations on simply connected symmetric spaces with non-negative curvature. We will prove that all such foliations are isoparametric as defined by Heintze, Liu and Olmos. We will also prove a splitting theorem…
Hypersurfaces are studied and classified under multiple additional assumptions in any Riemannian homogeneous space $(\mathbb{C}P^3, g_a)$, including nearly K\"ahler $\mathbb{C}P^3$. Notably, all extrinsically homogeneous hypersurfaces are…
A very interesting problem in the classical theory of minimal surfaces consists of the classification of such surfaces under some geometrical and topological constraints. In this short paper, we give a brief summary of the known…
In this article parametric versions of Wilson's plug and Kuperberg's plug are discussed. We show that there is a weak homotopy equivalence induced by the inclusion between the space of non-singular vector fields tangent to a foliation and…
In this work we find all helicoidal surfaces in Minkowski space with constant mean curvature whose generating curve is a the graph of a polynomial or a Lorentzian circle. In the first case, we prove that the degree of the polynomial is $0$…
In this study, we define some new types of non-null ruled surfaces called slant ruled surfaces in the Minkowski 3-space E_1^3. We introduce some characterizations for a non-null ruled surface to be a slant ruled surface in E_1^3. Moreover,…
Nikulin and Vinberg proved that there are only a finite number of lattices of rank $\geq 3$ that are the N\'eron-Severi group of projective K3 surfaces with a finite automorphism group. The aim of this paper is to provide a more geometric…
We study rotational surfaces with constant Minkowski Gaussian curvature and rotational surfaces with constant Minkowski mean curvature in a $3$-dimensional normed space with rotationally symmetric norm. We have a generalization of the…
It is classically known that the only zero mean curvature entire graphs in the Euclidean 3-space are planes, by Bernstein's theorem. A surface in Lorentz-Minkowski 3-space $\boldsymbol{R}^3_1$ is called of mixed type if it changes causal…
We use the solution space of a pair of ODEs of at least second order to construct a smooth surface in Euclidean space. We describe when this surface is a proper embedding which is geodesically complete with finite total Gauss curvature. If…
Topology and geometry are deeply intertwined in the study of surfaces, though their interaction manifests differently in smooth and discrete settings. In the smooth category, a classical result asserts that any closed smooth surface…
In this paper are studied the nets of principal curvature lines on surfaces embedded in Euclidean $3-$space near their end points, at which the surfaces tend to infinity. This is a natural complement and extension to smooth surfaces of the…