Related papers: Embedded constant curvature curves on convex surfa…
We present a collection of easily stated open problems in the theory of compact constant mean curvature surfaces with boundary. We also survey the current status of answering them.
We say that a simple, closed curve $\gamma$ in the plane has bounded convex curvature if for every point $x$ on $\gamma$, there is an open unit disk $U_x$ and $\varepsilon_x>0$ such that $x\in\partial U_x$ and $B_{\varepsilon_x}(x)\cap…
We study discrete curvatures computed from nets of curvature lines on a given smooth surface, and prove their uniform convergence to smooth principal curvatures. We provide explicit error bounds, with constants depending only on properties…
We investigate the existence, convergence and uniqueness of modified general curvature flow of convex hypersurfaces in hyperbolic space with a prescribed asymptotic boundary.
We classify all real hypersurfaces with constant principal curvatures in the complex hyperbolic plane.
The existence of essential closed surfaces surfaces is proven for finite coverings of 3-manifolds that are triangulated by finitely many topological ideal tetrahedra and admit a regular, negatively curved, ideal structure.
We consider surfaces with parallel mean curvature vector field and finite total curvature in product spaces of type $\mathbb{M}^n(c)\times\mathbb{R}$, where $\mathbb{M}^n(c)$ is a space form, and characterize certain of these surfaces. When…
Constant curvature surfaces are constructed from the finite action solutions of the supersymmetric $\mathbb{C}P^{N-1}$ sigma model. It is shown that there is a unique holomorphic solution which leads to constant curvature surfaces: the…
On one hand, we study the class of graphs on surfaces, satisfying tessellation properties, with positive Forman curvature on each edge. Via medial graphs, we provide a new proof for the finiteness of the class, and give a complete…
This paper deals with a generalized length-preserving flow for convex curves in the plane. It is shown that the flow exists globally and deforms convex curves into circles as time tends to infinity.
We announce the classification of complete, almost embedded surfaces of constant mean curvature, with three ends and genus zero: they are classified by triples of points on the sphere whose distances are the asymptotic necksizes of the…
In this note, we investigate the existence of smooth complete hypersurfaces in hyperbolic space with constant $(n-2)$-curvature and a prescribed asymptotic boundary at infinity. Previously, the existence was known only for a restricted…
Given two closed curves in a surface, we propose an algorithm to detect whether they are of the same type or not.
The aim of this paper is to investigate the sufficient condition for the invariance of a normal curve on a smooth immersed surface under isometry. We also find the the deviations of the tangential and normal components of the curve with…
The aim of this paper is to extend classic results of the theory of CMC surfaces in the product spaces to the class of immersed surfaces in $\mathbb{M}^2(\kappa)\times\mathbb{R}$ whose mean curvature is given as a $C^1$ function depending…
In this paper, we obtain some results about the mean curvature integrals of the outer parallel convex body of constant width.
It is shown that various questions about the existence of simple closed curves in normal subgroups of surface groups are undecidable.
Let $S$ be a complete flat surface, such as the Euclidean plane. We obtain direct characterizations of the connected components of the space of all curves on $S$ which start and end at given points in given directions, and whose curvatures…
We prove the existence of closed hypersurfaces of prescribed scalar curvature in globally hyperbolic Lorentzian manifolds provided there are barriers.
We consider constant mean curvature surfaces of finite topology, properly embedded in three-space in the sense of Alexandrov. Such surfaces with three ends and genus zero were constructed and completely classified by the authors in…