arXiv research

Alternative proof for ribbon surfaces in 3D space.

Ribbonness proven for slice knots, solving an old question.

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 …

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 characterizes Ricci solitons on the Poincaré upper half plane.

Geodesics on extended Siegel-Jacobi upper half-plane determined.

Inverse metric matrices on Siegel-Jacobi spaces are calculated for Berezin quantization.

Analyzes Berry phases and connection matrices on Siegel-Jacobi spaces.

Paper extends Enami-Ozeki-Yamaguchi's work on planar quadrangulations.

Study calculates mass of special polyhedra in hyperbolic space.

New geometry based on Siegel upper half-space with volume formula.

In this paper we classify the solutions to the geometric Neumann problem for the Liouville equation in the upper half-plane or an upper half-disk, with the energy condition given by finite area. As a result, we classify the conformal Riemannian metrics of constant curvature and finite area on a half-plane that have a f…

We introduce a method in differential geometry to study the derivative operators of Siegel modular forms. By determining the coefficients of the invariant Levi-Civita connection on a Siegel upper half plane, and further by calculating the expressions of the differential forms under this connection, we get a non-holomor…

Study on migrating elastic flows of curves across half-planes.

Three models are shown to be isometrically equivalent, with a gapless first eigenvalue.

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

We consider Bridgeland stability conditions for three-folds conjectured by Bayer-Macrì-Toda in the case of Picard rank one. We study the differential geometry of numerical walls, characterizing when they are bounded, discussing possible intersections, and showing that they are essentially regular. Next, we prove that w…

Study on non-classical generating sets in Fuchsian Schottky groups.

Constructs weight 1/2 multiplier systems for a specific group and relates to geometric edge paths.

Half grid diagrams prove every link can be represented by a special type of grid diagram.

The real Jacobi group $G^J_1(\mathbb{R})={\rm SL}(2,\mathbb{R})\ltimes {\rm H}_1$, where ${\rm H}_1$ denotes the 3-dimensional Heisenberg group, is parametrized by the $S$-coordinates $(x,y,θ,p,q,κ)$. We show that the parameter $η$ that appears passing from Perelomov's un-normalized coherent state vector based on the S…

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

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

This paper considers metric balls $B(p,R)$ in two dimensional Riemannian manifolds when $R$ is less than half the convexity radius. We prove that $Area(B(p,R)) \geq \frac{8}πR^2$. This inequality has long been conjectured for $R$ less than half the injectivity radius. This result also yields the upper bound $μ_2(B(p,R)…

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.

Paper defines end Khovanov homology to detect exotic planes.

Study proves unique compactification of hyperbolic space.

Every knot can be embedded in the union of finitely many half planes with a common boundary line in such a way that the portion of the knot in each half plane is a properly embedded arc. The minimal number of such half planes is called the arc index of the knot. We have identified all prime knots with arc index up to 1…

We consider a surface link in the 4-space which can be presented by a simple branched covering over the standard torus, which we call a torus-covering link. Torus-covering links include spun $T^2$-knots and turned spun $T^2$-knots. In this paper we braid a torus-covering link over the standard 2-sphere. This gives an u…

Extended metric defined on Siegel-Jacobi space using invariant forms.

An extended Kleinian group whose orientation-preserving half is a Schottky group is called an extended Schottky group. These groups correspond to the real points in the Schottky space. Their geometric structures is well known and it permits to provide information on the locus of fixed points of symmetries of handlebodi…

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.

Study on free-boundary CMC hypersurfaces in upper hemisphere, proving Morse index and eigenvalue bounds.

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 phase space of a compact, irreducible, simply connected, Riemannian symmetric space admits a natural family of Kähler polarizations parametrized by the upper half plane $S$. Using this family, geometric quantization, including the half-form correction, produces the field $H^{corr}\rightarrow S$ of quantum Hilbert s…

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

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.

Paper shows how to embed Möbius bands with many twists and small aspect ratios.

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…

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…

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.

A new method for describing surface-links in 4-space.