Related papers: Hyperbolic triangles of the maximum area with two …
Dogru and Tabachnikov in 2003 explored the polygonal outer billiard map in the hyperbolic plane and introduced a class of convex polygons called 'large'. They particularly sought conditions for a triangle to be classified as large. For a…
We consider hyperbolic manifolds with boundary, which admit an ideal triangulation with n ideal triangles and one edge. We prove that the number of these manifolds is $\exp(n\ln(n)+O(n))$.
We study hyperbolized versions of cohomological equations that appear with cocycles by isometries of the euclidean space. These (hyperbolized versions of) equations have a unique continuous solution. We concentrate in to know whether or not…
We establish a connection between two previously unrelated topics: a particular discrete version of conformal geometry for triangulated surfaces, and the geometry of ideal polyhedra in hyperbolic three-space. Two triangulated surfaces are…
In this paper, the isodiametric problem for centrally symmetric convex bodies in the Euclidean d-space R^d containing no interior non-zero point of a lattice L is studied. It is shown that the intersection of a suitable ball with the…
We describe first steps towards a system for automated triangle constructions in absolute and hyperbolic geometry. We discuss key differences between constructions in Euclidean, absolute and hyperbolic geometry, compile a list of primitive…
The sphere packing problem asks for the greatest density of a packing of congruent balls in Euclidean space. The current best upper bound in all sufficiently high dimensions is due to Kabatiansky and Levenshtein in 1978. We revisit their…
In a previous paper we have introduced the ortho-homological triangles, which are triangles that are orthological and homological simultaneously. In this article we call attention to two remarkable ortho-homological triangles (the given…
We prove that the (n-2)-dimensional surface area (perimeter) of central hyperplane sections of the n-dimensional unit cube is maximal for the hyperplane perpendicular to the vector (1,1,0,...,0). This gives a positive answer to a question…
We show that the third eigenvalue of the Neumann Laplacian in hyperbolic space is maximal for the disjoint union of two geodesic balls, among domains of given volume. This extends a recent result by Bucur and Henrot in Euclidean space,…
We study hyperbolic polyhedral surfaces with faces isometric to regular hyperbolic polygons satisfying that the total angles at vertices are at least $2\pi.$ The combinatorial information of these surfaces is shown to be identified with…
The reverse isoperimetric problem asks for existence and properties of bounded convex sets in a Riemannian manifold which maximise the perimeter under all those sets of fixed volume which roll freely in a ball of some given radius. If the…
Let $M$ be a compact oriented 3-manifold with non-empty boundary consisting of surfaces of genii $>1$ such that the interior of $M$ is hyperbolizable. We show that for each spherical cone-metric $d$ on $\partial M$ such that all cone-angles…
Using the method of C. V\"or\"os, we establish results on hyperbolic plane geometry, related to triangles. In this note we investigate the orthocenter, the concept of isogonal conjugate and some further center as of the symmedian of a…
We introduce a new technique for proving the classical Stable Manifold theorem for hyperbolic fixed points. This method is much more geometrical than the standard approaches which rely on abstract fixed point theorems. It is based on the…
For large $R$, we consider measurable sets $A\subseteq [0,R]^2$ that avoid triples of points of the form $(x,y)$, $(x+t,y)$, $(x,y+1/t)$ with $x,y\in\mathbb{R}$ and $t>0$, i.e., the vertices of upward-oriented, axis-aligned right triangles…
On a complete manifold, such as Euclidean 3-space or hyperbolic 3-space, the limit at infinity of the norm of the Higgs field is called the mass of the monopole. We show the existence, on hypebolic 3-space, of monopoles with given magnetic…
We construct a hyperbolic attractor of renormalization of bi-cubic circle maps with bounded combinatorics, with a codimension-two stable foliation.
In this paper we construct an example of a weakly complete maximal surface in the Lorentz-Minkowski space L^3, which is bounded by a hyperboloid. Moreover, all the singularities of our example are of lightlike type.
Gradients of the perimeter and area of a polygon have straightforward geometric interpretations. The use of optimality conditions for constrained problems and basic ideas in triangle geometry show that polygons with prescribed area…