Related papers: Singularities of parallels to tangent developable …

We consider developable surfaces along the singular set of a swallowtail which are considered to be flat approximations of the swallowtail. For the study of singularities of such developable surfaces, we introduce the notion of Darboux…

We study singularities of surfaces which are given by Kenmotsu-type formula with prescribed unbounded mean curvature.

We prove that any piece of a rotational hypersurface with prescribed mean curvature function in a Euclidean space can be uniquely extended infinitely, which generalizes the results by Euler and Delaunay for surfaces of revolution with…

In this paper we introduce the notion of the singular evolutoid set which is the set of all singular points of all evolutoids of a fixed smooth planar curve with at most cusp singularities. By the Gauss-Bonnet Theorem for Coherent Tangent…

We investigate necessary and sufficient conditions for the extendibility and boundedness of Gaussian curvature, Mean curvature and principal curvatures near all types of singularities on fronts. We also study the convergence to infinite…

In this study, we define some new types of ruled surfaces called slant ruled surfaces. We give some characterizations for a regular ruled surface to be a slant ruled surface in Euclidean 3- space. We show that if the slant ruled surface is…

In this paper, we consider a regular curve on an oriented surface in Euclidean 3-space with the Darboux frame $\{\mathsf{T},\mathsf{V},\mathsf{U}\}$ along the curve, where $\mathsf{T}$ is the unit tangent vector field of the curve,…

In Euclidean space, we investigate surfaces whose mean curvature $H$ satisfies the equation $H=\alpha\langle N,\mathbf{x}\rangle+\lambda$, where $N$ is the Gauss map, $\mathbf{x}$ is the position vector and $\alpha$ and $\lambda$ are two…

Consider an orientable compact surface in three dimensional Euclidean space with minimum total absolute curvature. If the Gaussian curvature changes sign to finite order and satisfies a nondegeneracy condition along closed asymptotic…

The Legendre curve in the unit tangent bundle over Euclidean plane is a plane curve with a moving frame. We have the (Legendre) curvature of the Legendre curve, and the existence and uniqueness theorems for the curvature are valid. In this…

We consider a general theory of curvatures of discrete surfaces equipped with edgewise parallel Gauss images, and where mean and Gaussian curvatures of faces are derived from the faces' areas and mixed areas. Remarkably these notions are…

Motivated by a number of recent investigations, we define and investigate the various properties of the ruled surfaces depend on three dimensional Lie groups with a bi-variant metric. We give useful results involving the characterizations…

In this paper we define and analyze singularities of discrete linear Weingarten surfaces with Weierstrass-type representations in $3$-dimensional Riemannian and Lorentzian spaceforms. In particular, we discuss singularities of discrete…

Minimal surfaces with planar curvature lines in the Euclidean space have been studied since the late 19th century. On the other hand, the classification of maximal surfaces with planar curvature lines in the Lorentz-Minkowski space has only…

We survey different classification results for surfaces with parallel mean curvature immersed into some Riemannian homogeneous four-manifolds, including real and complex space forms, and product spaces. We provide a common framework for…

For singular corank 1 surfaces in $\mathbb R^3$ we introduce a distinguished normal vector called the axial vector. Using this vector and the curvature parabola we define a new type of curvature called the axial curvature, which generalizes…

In this paper we consider developable surfaces which are isometric to planar domains and which are piecewise differentiable, exhibiting folds along curves. The paper revolves around the longstanding problem of existence of the so-called…

We recall two basic conjectures on the developables of convex projective curves, prove one of them and disprove the other in the firdt nontrivial case of curves in RP^3. Namely, we show that i) the tangent developable surface of any convex…

We consider surfaces immersed in three-dimensional pseudohermitian manifolds. We define the notion of (p-)mean curvature and of the associated (p-)minimal surfaces, extending some concepts previously given for the (flat) Heisenberg group.…

We study equisingular deformation problems for curves and surfaces in algebraic families, with particular emphasis on situations where nodal behavior is no longer generic. Extending classical Severi theory, we develop deformation--theoretic…