arXiv research

Constructs a family of genus three minimal surfaces with parallel ends.

The paper studies singularities on parallels of tangent developable surfaces of frontal curves.

We construct a special class of Lorentz surfaces in the pseudo-Euclidean 4-space with neutral metric which are one-parameter systems of meridians of rotational hypersurfaces with timelike or spacelike axis and call them meridian surfaces. We give the complete classification of the meridian surfaces with parallel mean c…

We study parallel surfaces and dual surfaces of cuspidal edges. We give concrete forms of principal curvature and principal direction for cuspidal edges. Moreover, we define ridge points for cuspidal edges by using those. We clarify relations between singularities of parallel and dual surfaces and differential geometri…

The study proves a theorem for surfaces using Codazzi operators and investigates parallel mean curvature surfaces.

We construct a special class of Lorentz surfaces in the pseudo-Euclidean 4-space with neutral metric which are one-parameter systems of meridians of rotational hypersurfaces with lightlike axis and call them meridian surfaces. We give the complete classification of the meridian surfaces with constant Gauss curvature an…

We investigate singularities of all parallel surfaces to a given regular surface. In generic context, the types of singularities of parallel surfaces are cuspidal edge, swallowtail, cuspidal lips, cuspidal beaks, cuspidal butterfly and 3-dimensional $D_4^\pm$ singularities. We give criteria for these singularities type…

The paper explores $K^α$-translators on parallel and canal surfaces in 3D space.

Geometrically describes surfaces with parallel mean curvature in warped product spaces.

We study surfaces with parallel normalized mean curvature vector field in Euclidean or Minkowski 4-space. On any such surface we introduce special isothermal parameters (canonical parameters) and describe these surfaces in terms of three invariant functions. We prove that any surface with parallel normalized mean curva…

The study classifies parallel mean curvature spheres in a sphere-hyperbolic product space.

We present a reduction of codimension theorem for surfaces with parallel mean curvature in symmetric spaces.

Two holomorphic Hopf differentials for surfaces of non-null parallel mean curvature vector in S^2xS^2 and H^2xH^2 are constructed. A 1:1 correspondence between these surfaces and pairs of constant mean curvature surfaces of S^2xR and H^2xR is established. Using that, surfaces with vanishing Hopf differentials (in parti…

Defines surfaces with specific contour patterns.

Study timelike surfaces with parallel mean curvature in Minkowski 4-space.

Study of cuspidal edges on focal surfaces of regular surfaces.

We study affine immersions as introduced by Nomizu and Pinkall. We classify those affine immersions of a surface in 4-space which are degenerate and have vanishing cubic form (i.e. parallel second fundamental form). This completes the classification of parallel surfaces of which the first results were obtained in the b…

The paper examines soliton surfaces using a parallel transport frame field in 4D space.

We classify complete biharmonic surfaces with parallel mean curvature vector field and non-negative Gaussian curvature in complex space forms.

The paper studies parallel surfaces of cuspidal cross caps and their degeneracy.

In the present article we study a special class of surfaces in the four-dimensional Euclidean space, which are one-parameter systems of meridians of the standard rotational hypersurface. They are called meridian surfaces. We classified semi-parallel meridian surface in 4-dimensional Euclidean space $E^4$.

The purpose of this article is to determine explicitly the complete surfaces with parallel mean curvature vector, both in the complex projective plane and the complex hyperbolic plane. The main results are as follows: When the curvature of the ambient space is positive, there exists a unique such surface up to rigid mo…

Unified view of integrable systems linking CMC, isothermic, and Willmore surfaces.

Simple proof for special surface classification.

In this study, we consider canal surfaces according to parallel transport frame in Euclidean space $\mathbb{E}^{4}$. The curvature properties of these surfaces are investigated with respect to $k_{1}$, $k_{2}$ and $k_{3}$ which are principal curvature functions according to parallel transport frame. We also give an exa…

We consider a quadratic form defined on the surfaces with parallel mean curvature vector of an any dimensional complex space form and prove that its $(2,0)$-part is holomorphic. When the complex dimension of the ambient space is equal to $2$ we define a second quadratic form with the same property and then determine th…

We survey different classification results for surfaces with parallel mean curvature immersed into some Riemannian homogeneous four-manifolds, including real and complex space forms, and product spaces. We provide a common framework for this problem, with special attention to the existence of holomorphic quadratic diff…

Using Traizet's regeneration method, we prove that for each positive integer n there is a family of embedded, doubly periodic minimal surfaces with parallel ends in Euclidean space of genus 2n-1 and 4 ends in the quotient by the maximal group of translations. The genus 2n-1 family converges smoothly to 2n copies of Sch…

We prove a Simons type equation for non-minimal surfaces with parallel mean curvature vector (pmc surfaces) in $M^n(c)\times\mathbb{R}$, where $M^n(c)$ is an $n$-dimensional space form. Then, we use this equation in order to characterize complete non-minimal pmc surfaces with non-negative Gaussian curvature.

The paper introduces surface signatures for irregular surfaces and rough surfaces.

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

The paper examines stable CMC hypersurfaces with boundaries on parallel hyperplanes.

Study of curves and surfaces in Riemannian spaces making a constant angle with a parallel transported direction.

We prove a Simons type equation for non-minimal surfaces with parallel mean curvature vector (pmc surfaces) in $M^3(c)\times\mathbb{R}$, where $M^3(c)$ is a 3-dimensional space form. Then, we use this equation in order to characterize certain complete non-minimal pmc surfaces.

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

In this paper we proof that the Holomorphic angle for compact minimal surfaces in the sphere $S^5$ with constant Contact angle and with a parallel normal vector field must be constant.

A nice differential-geometric framework for (non-abelian) higher gauge theory is provided by principal 2-bundles, i.e. categorified principal bundles. Their total spaces are Lie groupoids, local trivializations are kinds of Morita equivalences, and connections are Lie-2-algebra-valued 1-forms. In this article, we const…

One can embed arbitrarily many disjoint, non-parallel, non-boundary parallel, incompressible surfaces in any three manifold with at least one boundary component of genus two or greater [4]. This paper proves the contrasting, but not contradictory result that although one can sometimes embed arbitrarily many surfaces in…

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

We study the global geometry of surfaces in Sasakian space forms whose mean curvature vector is parallel in the normal bundle (these include the Riemannian Heisenberg space of dimension $2n+1$). We prove a codimension reduction theorem. We introduce two holomorphic quadratic differentials on anti-invariant such surface…

We prove that the natural principal parameters on a given Weingarten surface are also natural principal parameters for the parallel surfaces of the given one. As a consequence of this result we obtain that the natural PDE of any Weingarten surface is the natural PDE of its parallel surfaces. We show that the linear fra…

New classifications of totally real surfaces in nearly Kähler C⁴.

In [dLMu05], DeLellis and Müller proved a quantitative version of Codazzi's theorem, namely for a smooth embedded surface $\ Σ\subseteq \mathbb{R}^3\$ with area normalized to $\ {\cal H}^2(Σ) = 4 π$, it was shown that $\ \parallel A_Σ- id \parallel_{L^2(Σ)} \leq C \parallel A^0_Σ\parallel_{L^2(Σ)}\$, and building on…

Study of complex surfaces in a specific pseudo-Riemannian space.

We establish what semi-discrete linear Weingarten surfaces with Weierstrass-type representations in $3$-dimensional Riemannian and Lorentzian spaceforms are, confirming their required properties regarding curvatures and parallel surfaces, and then classify them. We then define and analyze their singularities. In partic…

We give a full classification of higher order parallel surfaces in three-dimensional homogeneous spaces with four-dimensional isometry group, i.e. in the so-called Bianchi-Cartan-Vranceanu family. This gives a positive answer to a conjecture formulated in 2002. As a partial result, we prove that totally umbilical surfa…

We consider a general theory of curvatures of discrete surfaces equipped with edgewise parallel Gauss images, and where mean and Gaussian curvatures of faces are derived from the faces' areas and mixed areas. Remarkably these notions are capable of unifying notable previously defined classes of surfaces, such as discre…

We study symmetric affine surfaces which have non-vanishing torsion tensor. We give a complete classification of the local geometries possible if the torsion is assumed parallel. This generalizes a previous result of Opozda in the torsion free setting; these geometries are all locally homogeneous. If the torsion is not…