Related papers: On some classical constructions extended to hyperb…
The framework of universal geometry allows us to consider metrical properties of affine views of elliptic curves, even over finite fields. We show how the Neuberg cubic of triangle geometry extends to the finite field situation and provides…
In this paper I present a kind of proof for classical Euclidean geometric problems which relies on both synthetic and analytic geometry. Using the elementary tools of polynomial algebra and multivariate calculus we manage to reduce the…
Germs of tubular neighborhood embeddings for submanifolds N of manifolds M are in one-one correspondence with germs of Euler-like vector fields near N. In many contexts, this reduces the proof of `normal forms results' for geometric…
By employing certain extended classical summation theorems, several surprising \pi and other formulae are displayed.
Noether's theorem on the equivalence of symmetry and conservation laws has applications to geometric problems on symmetric spaces. We remind the reader of the theorem and give an application to a variational problem on hyperbolic surfaces.
We describe the holonomy algebras of all canonical connections and their action on complex hyperbolic spaces $\mathbb{C}\mathrm{H}(n)$ in all dimensions ($n\in\mathbb{N}$). This thorough investigation yields a formula for all Kahler…
In the Euclidean setting, Napoleon's Theorem states that if one constructs an equilateral triangle on either the outside or the inside of each side of a given triangle and then connects the barycenters of those three new triangles, the…
This notes explores angle structures on ideally triangulated compact $3$-manifolds with high genus boundary. We show that the existence of angle structures implies the existence of a hyperbolic metric with totally geodesic boundary, and…
We study the Dirichlet problem for harmonic maps between hyperbolic planes, under the assumption that the Euclidean harmonic extension of the boundary map is quasiconformal.
It is known that any tame hyperbolic 3-manifold with infinite volume and a single end is the geometric limit of a sequence of finite volume hyperbolic knot complements. Purcell and Souto showed that if the original manifold embeds in the…
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…
Markov's theorem classifies the worst irrational numbers with respect to rational approximation and the indefinite binary quadratic forms whose values for integer arguments stay farthest away from zero. The main purpose of this paper is to…
We make observations about constant mean curvature surfaces in Euclidean 3-space and their dual surfaces, and the resulting pairs of surfaces in hyperbolic 3-space under the Lawson correspondence.
We present a variety of geometrical and combinatorial tools that are used in the study of geometric structures on surfaces: volume, contact, symplectic, complex and almost complex structures. We start with a series of local rigidity results…
This work makes a parallel construction for curves on threefolds to a ``current-theoretic'' proof of Abel's theorem giving the rational equivalence of divisors P and Q on a Riemann surface when Q - P is (equivalent to) zero in the Jacobian…
In this paper we relate some classical normal forms for complex elliptic curves in terms of 4-point sets in the Riemann sphere. Our main result is an alternative proof that every elliptic curve is isomorphic as a Riemann surface to one in…
Zamolodchikov's recursion relations are used to analyze the existence and approximations to the classical conformal block in the case of four parabolic weights. Strong numerical evidence is found that the saddle point momenta arising in the…
We show that any universal quasigeodesic cone of uniformly coarse median spaces admits a canonical coarse median structure. As an application, we recover a result of Bowditch which states that any hierarchically hyperbolic space admits a…
The paper initiates a systematic study of Moebius structures and Ptolemy spaces. We conjecture that every compact Ptolemy space with circles and many space inversions is Moebius equivalent to the boundary at infinity of a rank one symmetric…
We study geometry, topology and deformation spaces of noncompact complex hyperbolic manifolds (geometrically finite, with variable negative curvature), whose properties make them surprisingly different from real hyperbolic manifolds with…