Related papers: Fundamental domains in the Einstein Universe
Crooked planes are piecewise linear surfaces that were introduced by Drumm in the early 1990s to construct fundamental domains for properly discontinuous actions of free groups on Minkowski 3-space. In a previous paper, we introduced…
In hyperbolic space, the angle of intersection and distance classify pairs of totally geodesic hyperplanes. A similar algebraic invariant classifies pairs of hyperplanes in the Einstein universe. In dimension 3, symplectic splittings of a…
Crooked planes were defined by Drumm to bound fundamental polyhedra in Minkowski space for Margulis spacetimes. They were extended by Frances to closed polyhedral surfaces in the conformal compactification of Minkowski space (Einstein…
We construct open domains in Euclidean 3-space which do not admit complete properly immersed minimal surfaces with an annular end. These domains can not be smooth by a recent result of Martin and Morales
In the present paper, we propose a new discrete surface theory on 3-valent embedded graphs in the 3-dimensional Euclidean space which are not necessarily discretization or approximation of smooth surfaces. The Gauss curvature and the mean…
This paper is the third in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed 3-manifold. In [CM3]-[CM5] we describe the case where the surfaces are topologically disks on…
The Einstein universe $\mathbf{Ein}^{p,q}$ of signature $(p,q)$ is a pseudo-Riemannian analogue of the conformal sphere; it is the conformal compactification of the pseudo-Riemannian Minkowski space. For $p,q \geq 1$, we show that, up to a…
The Einstein universe is the conformal compactification of Minkowski space. It also arises as the ideal boundary of anti-de Sitter space. The purpose of this article is to develop the synthetic geometry of the Einstein universe in terms of…
We describe a fundamental domain for the punctured Riemann surface $V_{3,m}$ which parametrises (up to M\"obius conjugacy) the set of quadratic rational maps with numbered critical points, such that the first critical point has period…
In this paper, the first family of conforming discrete three dimensional Gradgrad-complexes consisting of finite element spaces is constructed. These discrete complexes are exact in the sense that the range of each discrete map is the…
Main aim of this work is to give a suitable explanation of present accelerating universe through an acceptable interactive dynamical cosmological model. A three-fluid cosmological model is introduced in the background of…
We give a diameter bound for fundamental domains for isometric actions of the fundamental group of a closed hyperbolic surface on a delta-hyperbolic space, where the bound depends on the hyperbolicity constant delta, the genus of the…
We show that a vector field non-minimally coupled to gravity reproduces exactly the dynamics of an Einstein cluster -- a large ensemble of non-interacting particles moving on circular geodesics under their collective gravitational field.…
We study the coupled Einstein-Klein-Gordon equations for a complex scalar field with and without a quartic self-interaction in a curvatureless Friedman-Lema\^{\i}\-tre Universe. The equations can be written as a set of four coupled first…
We study the dynamics of domain walls in Einstein-Born-Infeld-dilaton theory. Dilaton is non-trivially coupled with the Born-Infeld electromagnetic field. We find three different types of solutions consistent with the dynamic domain walls.…
A triangulated piecewise-linear minimal surface in Euclidean 3-space defined using a variational characterization is critical for area amongst all continuous piecewise-linear variations with compact support that preserve the simplicial…
In this paper, we prove a similar result to the fundamental theorem of regular surfaces in classical differential geometry, which extends the classical theorem to the entire class of singular surfaces in Euclidean 3-space known as frontals.…
This paper considers some fundamental questions concerning marginally trapped surfaces, or apparent horizons, in Cauchy data sets for the Einstein equation. An area estimate for outermost marginally trapped surfaces is proved. The proof…
Generic relevant deformations of Einstein's gravity theory contain additional degrees of freedom that have a multi-facetted stabilization dynamics on curved spacetimes. We show that these relevant degrees of freedom are self-protected…
An algebraic system is proposed that represent surface cobordisms in thickened surfaces. Module and comodule structures over Frobenius algebras are used for representing essential curves. The proposed structure gives a unified algebraic…