arXiv research

We define general rotational surfaces of elliptic and hyperbolic type in the pseudo-Euclidean 4-space with neutral metric which are analogous to the general rotational surfaces of C. Moore in the Euclidean 4-space. We study Lorentz general rotational surfaces with plane meridian curves and give the complete classificat…

The paper applies Clairaut's theorem to rotational surfaces in pseudo-Euclidean 4-space.

In the present paper we consider a special class of spacelike surfaces in the Minkowski 4-space which are one-parameter systems of meridians of the rotational hypersurface with timelike or spacelike axis. They are called meridian surfaces of elliptic or hyperbolic type, respectively. We study these surfaces with respec…

Proper superminimal surfaces in hyperbolic 4-space can be approximated by conformal immersions.

We describe the relationship between complex-valued harmonic morphisms from Minkowski 4-space} and the shear-free ray congruences of mathematical physics. Then we show how a horizontally conformal submersion on a domain of Euclidean 3-space gives the boundary values at infinity of a complex-valued harmonic morphism on …

4 flat 3-manifolds realized in hyperbolic 4-space.

Equivalences between conformal foliations on Euclidean $3$-space, Hermitian structures on Euclidean $4$-space, shear-free ray congruences on Minkowski $4$-space, and holomorphic foliations on complex $4$-space are explained geometrically and twistorially; these are used to show that 1) any real-analytic complex-valued …

The study classifies timelike meridian surfaces in Minkowski 4-space.

The study classifies points on ruled surfaces in 4-space based on geometric properties.

Minimal discs count as knot invariants in hyperbolic 4-space.

In this paper, for each finite group $G$, we construct explicitly a non-compact complete finite-volume arithmetic hyperbolic $4$-manifold $M$ such that $\mathrm{Isom}\,M \cong G$, or $\mathrm{Isom}^{+}\,M \cong G$. In order to do so, we use essentially the geometry of Coxeter polytopes in the hyperbolic $4$-space, on o…

We give the classification of constant mean curvature rotational surfaces of elliptic, hyperbolic, and parabolic type in the four-dimensional pseudo-Euclidean space with neutral metric.

In this paper we develop a complete theory of factorization for isometries of hyperbolic 4-space. Of special interest is the case where a pair of isometries is linked, that is, when a pair of isometries can be expressed each as compositions of two involutions, one of which is common to both isometries. Here we develop …

Given a discrete subgroup of the isometries of n-dimensional hyperbolic space there is always a region kept precisely invariant under the stabilizer of a parabolic fixed point, called the Margulis region. While in dimensions 2 and 3 this region is a horoball, it has in general a more complicated shape due to the existe…

The paper finds the maximum scalar curvature for 3D minimal hypersurfaces in hyperbolic 4-space.

We describe a family of 4-dimensional hyperbolic orbifolds, constructed by deforming an infinite volume orbifold obtained from the ideal, hyperbolic 24-cell by removing two walls. This family provides an infinite number of infinitesimally rigid, infinite covolume, geometrically finite discrete subgroups of the isometry…

Investigate the local geometry of smooth surfaces in 4-space via contact with 2-planes and apparent contours.

The paper studies canal hypersurfaces in Lorentz-Minkowski 4-space.

A Laguerre geometric local characterization is given of L-minimal surfaces and Laguerre deformations (T-transforms) of L-minimal isothermic surfaces in terms of the holomorphicity of a quartic and a quadratic differential. This is used to prove that, via their Laguerre Gauss maps, the T-transforms of L-minimal isotherm…

The paper proves a Penrose inequality for 4-discs with hyperbolic ends and conic singularities.

We study the geometry of oriented right-angled hexagons in H^4, the hyperbolic 4-space, via Clifford numbers or quaternions. We show how to augment alternate sides of such a hexagon so that for the non-augmented sides, we can define quaternion half side-lengths whose angular parts are obtained from half the Euler angle…

The paper studies canal hypersurfaces in Lorentz-Minkowski 4-space.

We consider a special class of spacelike surfaces in the Minkowski 4-space which are one-parameter systems of meridians of the rotational hypersurface with timelike or spacelike axis. We call these surfaces meridian surfaces of elliptic or hyperbolic type, respectively. On the base of our invariant theory of surfaces w…

The paper characterizes biconservative surfaces in hyperbolic 4-space.

New homotopy 4-spheres and real projective 4-spaces created.

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 charge and arbitrary mass. Previously, aside from charge one monopoles, existenc…

We say that a topologically embedded 3-sphere in a smoothing of Euclidean 4-space is a barrier provided, roughly, no diffeomorphism of the 4-manifold moves the 3-sphere off itself. In this paper we construct infinitely many one parameter families of distinct smoothings of 4-space with barrier 3-spheres. \par The existe…

The paper generalizes Bour's theorem to Minkowski 4-space.

Submanifolds in Lorentz-Minkowski space are investigated from various mathematical viewpoints and are of interest also in relativity theory. We define the hyperbolic surface and the de Sitter surface of a curve in the spacelike hypersurface M in the Minkowski 4-space. These surfaces are respectively located in the hype…

Paper defines end Khovanov homology to detect exotic planes.

Study character varieties of a Coxeter group in hyperbolic and Anti-de Sitter spaces.

We introduce a generalization of the Euclidean algorithm for rings equipped with an involution, and completely enumerate all isomorphism classes of orders over definite, rational quaternion algebras equipped with an orthogonal involution that admit such an algorithm. We give two applications: first, any order that admi…

We give a conformal representation in terms of meromorphic data for a certain class of spacelike surfaces in the Lorentz-Minkowski 4-space L^4 whose mean curvature vector is either lightlike or zero at each point. This representation extends simultaneously the Weierstrass representation for minimal surfaces in Euclidea…

The group of bordism classes of unoriented surfaces in 4-space is determined. The bordism classes are characterized by normal Euler numbers,double linking numbers, and triple linking numbers.

Alternative proof for ribbon surfaces in 3D space.

Study timelike meridian surfaces in Minkowski 4-space with specific properties.

Ribbonness proven for slice knots, solving an old question.

In this paper, we give the characterizations of spacelike B2-slant helix by means of curvatures of the spacelike curve in Minkowski 4-space. Furthermore, we give the integral characterization of the spacelike B2-slant helix.

In this paper, we study generic conformally flat hypersurfaces in the Euclidean $4$-space $\mathbb{R}^4$ using the framework of Möbius geometry. First, we classify locally the generic conformally flat hypersurfaces with closed Möbius form under the Möbius transformation group of $\mathbb{R}^4$. Such examples come from …

No minimal charts with exactly seven white vertices found.

Paper proves uniqueness of bridge multisections for surfaces in 4-space.

We consider computational complexity of problems related to the fundamental group and the first homology group of (embeddable) $2$-complexes. We show, as an extension of an earlier work, that computing first homology of $2$-complexes is equivalent in computational complexity to matrix diagonalization. That is, the usua…

We construct a special class of spacelike surfaces in the Minkowski 4-space which are one-parameter systems of meridians of the rotational hypersurface with lightlike axis and call these surfaces meridian surfaces of parabolic type. They are analogous to the meridian surfaces of elliptic or hyperbolic type. Using the i…

The hexabasic book is the cone of the 1-dimensional skeleton of the union of two tetrahedra glued along a common face. The universal 3-dimensional polyhedron UP is the product of a segment and the hexabasic book. We show that any 2-dimensional link in 4-space is isotopic to a surface in UP. The proof is based on a repr…

This paper studies minimal charts of a specific type to understand embedded surfaces in 4-space.

No minimal chart of type (4,3) exists in 4-space.

In the present study we consider knotted spheres in Euclidean $4$-space $\mathbb{E}^{4}$. Firstly, we give some basic curvature properties of knotted spheres in $\mathbb{E}^{4}$. Further, we obtained some results related with the conjugate nets and Laplace transforms of these kind of surfaces.

Machine learning predicts minimal surfaces for knots, supporting a conjecture.