arXiv research

Introduces hyperbolic generalized framed surfaces and their properties.

New surfaces generalize Dini surfaces in 4D.

Study of timelike surfaces in Minkowski space with specific geometric properties.

General rotational surfaces as a source of examples of surfaces in the four-dimensional Euclidean space have been introduced by C. Moore. In this paper we consider the analogue of these surfaces in the Minkowski 4-space. On the base of our invariant theory of spacelike surfaces we study general rotational surfaces with…

Researchers generalize Ribaucour-type surfaces with new mathematical representation.

This paper studies symplectic critical surfaces in Hermite surfaces.

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…

In the present study we consider the generalized rotational surfaces in Euclidean spaces. Firstly, we consider generalized spherical curves in Euclidean $(n+1)-$space $\mathbb{E}^{n+1}$. Further, we introduce some kind of generalized spherical surfaces in Euclidean spaces $\mathbb{E}^{3}$ and $$ respect…

Paper proves stable minimal surfaces in 3D are flat.

We present methods to construct interesting surfaces of general type via $\mathbb{Q}$-Gorenstein smoothing of a singular surface obtained from an elliptic surface. By applying our methods to special Enriques surfaces, we construct new examples of a minimal surface of general type with $p_g=0$, $π_1=\mathbb{Z}/2\mathbb{…

Proves existence of circle patterns on surfaces with cusps.

New surfaces in 4-ball constructed from knits, described by charts.

The Enneper surface and helix surfaces are unique in their geometric properties.

Defines and analyzes generalized normal ruled surfaces of curves in 3D space.

The paper classifies maximal translation surfaces in Lorentz-Minkowski space.

Paper generalizes discrete CMC surfaces and shows how they can be derived.

Generators found for nonorientable surfaces with many punctures.

Simply-connected surfaces of general type for n≥5.

We prove that the Gauss curvature and the curvature of the normal connection of any minimal surface in the four dimensional Euclidean space satisfy an inequality, which generates two classes of minimal surfaces: minimal surfaces of general type and minimal super-conformal surfaces. We prove a Bonnet-type theorem for st…

It is shown that any handle-irreducible summand of every stable-ribbon surface-link is a unique ribbon surface-link up to equivalences, so that every stable-ribbon surface-link is a ribbon surface-link. This is a generalization of a previously observed result for a stably trivial surface-link. Two observations are give…

This paper finds minimal sets of generators for mapping class groups of specific surfaces.

Two families of general affine surface areas are introduced. Basic properties and affine isoperimetric inequalities for these new affine surface areas as well as for $L_φ$ affine surface areas are established.

A novel class of integrable surfaces is recorded. This class of O surfaces is shown to include and generalize classical surfaces such as isothermic, constant mean curvature, minimal, `linear' Weingarten, Guichard and Petot surfaces and surfaces of constant Gaussian curvature. It is demonstrated that the construction of…

Study finds conditions for stationary spacelike surfaces in a generalized Robertson-Walker spacetime.

The paper describes a specific type of Kähler surfaces.

Study classifies ruled surfaces from mean curvature flow solutions.

The Weierstrass representation for minimal surfaces in $\mathbb{R}^3$ provides a flexible method for constructing minimal surfaces of arbitrary genus. The topological limitations of minimal surfaces interfere with this providing a more general geometric modeling tool. Minimal surfaces lie in the larger class of harmoni…

Extensions of the generalized Weierstrass representation to generic surfaces in 4D Euclidean and pseudo-Euclidean spaces are given. Geometric characteristics of surfaces are calculated. It is shown that integrable deformations of such induced surfaces are generated by the Davey -Stewartson hierarchy. Geometrically thes…

This paper classifies discrete conformal structures on surfaces with boundary.

Given a 3 manifold M with torus boundary and an ideal triangulation, Yoshida and Tillmann give different methods to construct surfaces embedded in M from ideal points of the deformation variety. Yoshida builds a surface from twisted squares whereas Tillmann produces a spun-normal surface. We investigate the relation be…

Stable generalized complex structures on certain surfaces are constant.

New algorithms compute points on generalized Bolza surfaces.

The study examines minimal surfaces in Riemannian products of surfaces.

Study helicoidal surfaces from frontals, revealing geometric rigidity and stability of singularities.

The notion of a generalized harmonic inverse mean curvature surface in the Euclidean four-space is introduced. A backward Bäcklund transform of a generalized harmonic inverse mean curvature surface is defined. A Darboux transform of a generalized harmonic inverse mean curvature surface is constructed by a backward Bäck…

In this paper, we study general rotational surfaces in the 4- dimensional pseudo-Euclidean space E4-2 and obtain a characterization of flat general rotation surfaces with pointwise 1-type Gauss map in E4-2 and give an example of such surfaces.

We examine some common features of minimal surfaces, nonzero constant mean curvature surfaces and marginally outer trapped surfaces, concerning their stability and rigidity, and consider some applications to Riemannian geometry and general relativity.

New representations of surface groups into higher-dimensional PSL generalize pleated surfaces.

The paper counts geodesic loops on surfaces without conjugate points.

We describe a method for generating minimal hard prime surface-link diagrams. We extend the known examples of minimal hard prime classical unknot and unlink diagrams up to three components and generate figures of all minimal hard prime surface-unknot and surface-unlink diagrams with prime base surface components up to …

We use integrable systems techniques to study the singularities of timelike non-minimal constant mean curvature (CMC) surfaces in the Lorentz-Minkowski 3-space. The singularities arise at the boundary of the Birkhoff big cell of the loop group involved. We examine the behaviour of the surfaces at the big cell boundary,…

Study complex surfaces fibered over Teichmüller curves with Veech fibers.

In this paper, we generalize the polar transforms of spacelike isothermic surfaces in $Q^4_1$ to n-dimensional pseudo-Riemannian space forms $Q^n_r$. We show that there exist $c-$polar spacelike isothermic surfaces derived from a spacelike isothermic surface in $Q^n_r$, which are into $S^{n+1}_r(c)$, $H^{n+1}_{r-1}(c)$…

Minimal moves for surfaces in 4D discovered, linking planar and spatial moves.

A space-like surface in Minkowski space-time is minimal if its mean curvature vector field is zero. Any minimal space-like surface of general type admits special isothermal parameters - canonical parameters. For any minimal surface of general type parameterized by canonical parameters we obtain Weierstrass representati…

Extends Goldberg's result for string links over surfaces.

The study of special Lagrangian classes and semistable Mukai vectors on K3 surfaces.

This research solves Plateau's problem for CRPC surfaces.