Related papers: A note on circle patterns on surfaces

The deformation theory of hyperbolic and Euclidean cone-manifolds with all cone angles less then 2{\pi} plays an important role in many problems in low dimensional topology and in the geometrization of 3-manifolds. Furthermore, various old…

In this paper we study topological surfaces as gridded surfaces in the 2-dimensional scaffolding of cubic honeycombs in Euclidean and hyperbolic spaces.

The Epstein-Baer theory of curve isotopies is basic to the remarkable theorem that homotopic homeomorphisms of surfaces are isotopic. The groundbreaking work of R. Baer was carried out on closed, orientable surfaces and extended by D. B. A.…

We develop the deformation theory of hyperbolic cone-3-manifolds with cone-angles less than $2\pi$, i.e. contained in the interval $(0,2\pi)$. In the present paper we focus on deformations keeping the topological type of the cone-manifold…

We prove an existence theorem for convex hypersurfaces of prescribed Gauss curvature in the complement of a compact set in Euclidean space which are close to a cone.

We prove the following: 1. Let epsilon>0 and let S_1,S_2 be two closed hyperbolic surfaces. Then there exists locally-isometric covers S'_i of S_i (for i=1,2) such that there is a (1+\epsilon) bi-Lipschitz homeomorphism between S'_1 and…

The theory of surfaces in Euclidean space can be naturally formulated in the more general context of Legendre surfaces into the space of contact elements. We address the question of deformability of Legendre surfaces with respect to the…

We study the geometry of hyperbolic cone surfaces, possibly with cusps or geodesic boundaries. We prove that any hyperbolic cone structure on a surface of non-exceptional type is determined up to isotopy by the geodesic lengths of a finite…

Surfaces with concentric $K$-contours and parallel $K$-contours in Euclidean $3$-space are defined. Crucial examples are presented and characterization of them are given.

In Euclidean geometry, the Pythagorean theorem is presented as an equation involving three squares. This paper explores how analogous expressions may be identified in spherical and hyperbolic geometries.

Coning off a collection of uniformly quasiconvex subsets of a Gromov hyperbolic space leaves a new space, called the cone-off. Kapovich and Rafi generalized work of Bowditch to show this space is still Gromov hyperbolic. We show that the…

A finite subset S of a closed hyperbolic surface F canonically determines a "centered dual decomposition" of F: a cell structure with vertex set S, geodesic edges, and 2-cells that are unions of the corresponding Delaunay polygons. Unlike a…

There are multiple generalisations of the Pythagorean theorem to spherical and hyperbolic geometry. A natural one, involving areas of disks with radii equal to the sides of a proper triangle, was discovered in the hyperbolic case by Maria…

We show that for every $\epsilon>0$, there exists a compact lamination by $\epsilon$-holomorphic surfaces in the complex projective plane, minimal, and that carries hyperbolic holonomy. We call $\epsilon$-holomorphic a real 2-dimensional…

We prove the existence of a minimal diffeomorphism isotopic to the identity between two hyperbolic cone surfaces $(\Sigma,g_1)$ and $(\Sigma,g_2)$ when the cone angles of $g_1$ and $g_2$ are different and smaller than $\pi$. When the cone…

The inscribed angle theorem, a famous result about the angle subtended by a chord within a circle, is well known and commonly taught in school curricula. In this paper, we present a generalisation of this result (and other related circle…

We deal with the restriction phenomenon for the Fourier transform. We prove that each of the restriction conjectures for the sphere, the paraboloid, the elliptic hyperboloid in $\mathbb{R}^n$ implies that for the cone in $\mathbb{R}^{n+1}$.…

The existence of closed hypersurfaces of prescribed curvature in globally hyperbolic Lorentzian manifolds is proved provided there are barriers.

Let $N$ be a connected finite type nonorientable surface with or without boundary components and punctures. We prove that the graph of nonseparating curves of $N$ is connected and Gromov hyperbolic with a constant which does not depend on…

The aim of this paper is to investigate the fractional combinatorial Calabi flow for hyperbolic bordered surfaces. By Lyapunov theory, it is proved that the flow exists for all time and converges exponentially to a conformal factor that…