Study on special surfaces in 4D pseudo-Euclidean space with specific Gauss maps.
problem Characterizing rotational surfaces with pointwise 1-type Gauss maps in E4-2.
method Analyzing surfaces with spacelike profile curves in E4-2.
result Characterizations for surfaces to have pointwise 1-type Gauss maps.
In this paper, we study spacelike rotational surfaces which are called boost invariant surfaces in Minkowski 4-space E41. We give necessary and sufficient condition for flat spacelike rotational surface to have pointwise 1-type Gauss map. Also, we obtain a characterization for boost invariant marginally trapped surface…
The study classifies quasi-minimal Lorentz surfaces with specific Gauss maps.
problem Classifying quasi-minimal Lorentz surfaces with pointwise 1-type Gauss map.
method Analyzing Lorentz surfaces in a four-dimensional pseudo-Euclidean space with neutral metric.
result Complete classification of quasi-minimal Lorentz surfaces with pointwise 1-type Gauss map.
A marginally trapped surface in the four-dimensional Minkowski space is a spacelike surface whose mean curvature vector is lightlike at each point. In the present paper we find all marginally trapped surfaces with pointwise 1-type Gauss map. We prove that a marginally trapped surface is of pointwise 1-type Gauss map if…
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 show that a meridian surface has a harmonic Gauss map if and only if it is part of a plane. Fur…
In this work, we study some classes of rotational surfaces in the pseudo-Euclidean space Et4 with profile curves lying in 2-dimensional planes. First, we determine all such surfaces in the Minkowski 4-space E14 with pointwise 1-type Gauss map of the first kind and second kind. Then, we obtain …
In this paper, by the studying of the Gauss map, Laplacian operator, curvatures of surfaces in R13 and Bour's theorem, we are going to identify surfaces of revolution with pointwise 1-type Gauss map property in 3−dimensional Minkowski space.
In this paper we study general rotational surfaces in the 4- dimensional Euclidean space E4 and give a characterization of flat general rotation surface with pointwise 1-type Gauss map. Also, we show that a non-planar flat general rotation surface with pointwise 1-type Gauss map is a Lie group if and only if it is a Cl…
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.
In this work we firstly classify space-like surfaces in Minkowski space E14, de-Sitter space S13 and hyperbolic space H3 with harmonic Gauss map. Then we give a characterization and classification of space-like surfaces with pointwise 1-type Gauss map of the first kind. We also give s…
In this paper, we study the Gauss map of the surfaces in the de Sitter space-time S14(1). First, we prove that a space-like surface lying in the de Sitter space-time has pointwise 1-type Gauss map if and only if it has parallel mean curvature vector. Then, we obtain the complete classification of the quasi-…
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…
Study classifies hypersurfaces in 4D Lorentz-Minkowski space using specific operators.
problem Classifying tubular hypersurfaces in 4D Lorentz-Minkowski space.
method Analysis of Gauss map and linearized operators L1 and L2. result Classifications of hypersurfaces with specific types of Gauss maps.
In this paper, we study the Lorentzian minimal surfaces in the Minkowski space-time with finite type Gauss map. First, we obtain the classification of this type of surfaces with pointwise 1-type Gauss map. Then, we proved that there are no Lorentzian minimal surface in the Minkowski space-time with null 2-type Gauss ma…
Study pseudo-spherical submanifolds with 1-type Gauss map in pseudo-spheres.
problem Characterize and classify pseudo-Riemannian submanifolds with 1-type pseudo-spherical Gauss map.
method Classify Lorentzian surfaces and pseudo-Riemannian submanifolds in pseudo-spheres with 1-type pseudo-spherical Gauss map.
result Classification of submanifolds with 1-type pseudo-spherical Gauss map in pseudo-spheres.
Classifies surfaces in a pseudo-sphere with harmonic or 1-type pseudo-spherical Gauss maps.
problem Classifying surfaces in a pseudo-sphere with specific properties.
method Complete classification through Riemannian and Lorentzian surfaces, explicit systems of partial differential equations.
result Complete classification of surfaces with harmonic or 1-type pseudo-spherical Gauss maps.
Wintgen ideal submanifolds in space forms are those ones attaining equality pointwise in the so-called DDVV inequality which relates the scalar curvature, the mean curvature and the scalar normal curvature. They are Moebius invariant objects. The mean curvature sphere defines a conformal Gauss map into a Grassmann mani…
New curvature measures for 4D manifolds with corners defined and related to Gauss-Bonnet.
problem Defining curvature measures for 4D manifolds with corners.
method Defined two new extrinsic curvature quantities, one conformal invariant, and a new conformally invariant operator.
result Gauss-Bonnet theorem reformulated in terms of new curvature measures.
Study on curvature functions for compact manifolds with boundary.
problem Understanding curvature functions on compact manifolds with boundary.
method Proves necessary and sufficient conditions for geodesic and Gaussian curvature, solves problems in the pointwise conformal case.
result New existence and nonexistence results for metrics with prescribed curvature, depending on Euler characteristic.
The Gauss-Bonnet curvature of order 2k is a generalization to higher dimensions of the Gauss-Bonnet integrand in dimension 2k, as the usual scalar curvature generalizes the two dimensional Gauss-Bonnet integrand. In this paper, we evaluate the first variation of the integrals of these curvatures seen as functionals…
New forms calibrate minimal graphs in arbitrary dimensions.
problem Calibrating minimal graphs in arbitrary codimension.
method Constructing closed forms from minimal graphs and estimating their comass.
result Conditions ensuring minimal graphs are calibrated and area-minimizing.
Characterizes conformal Gauss maps into spheres.
problem Understanding conformal Gauss maps of surfaces.
method Invariant formulation and proof of characterisation of harmonic maps.
result Characterisation of harmonic maps as conformal Gauss maps of Willmore surfaces.
The paper studies Gauss maps of a Ricci-mean curvature flow.
problem Investigating the Gauss maps of a Ricci-mean curvature flow.
method Deduced the evolution equation for the Gauss maps of a Ricci-mean curvature flow and proved they satisfy the vertically harmonic map heat flow equation.
result The Gauss maps of a Ricci-mean curvature flow satisfy the vertically harmonic map heat flow equation.
The Gauss map of a special surface is studied, leading to symmetry conclusions.
problem Understanding the Gauss map of free boundary minimal surfaces.
method Analyzing eigenfunctions of the Jacobi-Steklov operator.
result Rotationally symmetric surfaces have components of their Gauss map as eigenfunctions of the Jacobi-Steklov operator.
Study Gauss maps of surfaces in Heisenberg group using hyperbolic geometry.
problem Understanding the geometry of surfaces in Heisenberg group.
method Using Gans model of hyperbolic plane, relate tension field of Gauss map to surface's mean curvature.
result Established a relationship between tension field and mean curvature.
Study on Gauss map of anisotropic minimal surfaces with Morse index estimates.
problem Estimating the Morse index of anisotropic minimal surfaces.
method Local analysis of Gauss map, conformal geometric techniques applied to the Gauss map.
result Upper and lower estimates for the Morse index of anisotropic minimal surfaces.
The paper studies the space of Gauss maps of complete minimal surfaces and their homotopy types.
problem Understanding the space of Gauss maps of complete minimal surfaces and their homotopy types.
method Proves the Gauss map assignment is a Serre fibration and determines the homotopy type of the space of meromorphic functions.
result The space of meromorphic functions on M that are the Gauss map of a complete full conformal minimal immersion has the same homotopy type as the space of all continuous maps from M to the 2-sphere. Study on Gauss maps of minimal surfaces in 4D space.
problem Understanding the Gauss map of minimal surfaces in Euclidean 4-space.
method Systematic study of Gauss map properties for complete minimal surfaces.
result Optimal results for the maximal number of exceptional values of the Gauss map.
Study on singularities of Gauss maps of wave fronts with specific properties.
problem Characterizing singularities of Gauss maps of wave fronts.
method Geometric properties and boundedness of Gaussian curvatures.
result Relation between boundedness of Gaussian curvatures and types of singularities of Gauss maps.
We study the Gauss map of minimal surfaces in the Heisenberg group Nil3 endowed with a left-invariant Riemannian metric. We prove that the Gauss map of a nowhere vertical minimal surface is harmonic into the hyperbolic plane H2. Conversely, any nowhere antiholomorphic harmonic map into $\mathbb{…
Study on Gauss map surfaces in 3D space, focusing on anchor rings.
problem Classifying finite type Gauss map surfaces in Euclidean 3-space.
method Investigating a subclass of tubes, anchor rings, and analyzing their Gauss map properties.
result Anchor rings are of infinite type Gauss map.
Paper derives a formula for renormalized area of minimal hypersurfaces in 5D Poincaré-Einstein spaces.
problem Calculating the renormalized area of minimal hypersurfaces in 5D Poincaré-Einstein spaces.
method Derives a Gauss-Bonnet formula for renormalized area in terms of scalar invariants and characterizes minimal hypersurfaces in terms of conformal geometry.
result Derives a formula expressing renormalized area in terms of integrals of scalar Riemannian invariants.
Study of conformal Gauss map for Willmore surfaces in model spaces.
problem Characterizing Willmore surfaces and their geometric properties.
method Detailed study of conformal Gauss map and its application to Willmore surfaces.
result New characterizations of minimal and CMC surfaces using conformal Gauss map.
The paper bounds the energy index of harmonic Gauss maps on surfaces.
problem Bounding the energy index of harmonic Gauss maps on surfaces.
method Analyzes closed and complete non-compact surfaces in Lie groups with bi-invariant metrics.
result The energy index of the Gauss map is bounded by the topological genus.
We study the biharmonic stress-energy tensor S2 of Gauss map. Adding few assumptions, the Gauss map with vanishing S2 would be harmonic.
Gauss map of complete minimal surfaces avoids certain hypersurfaces.
problem Characterizing the range of Gauss maps of minimal surfaces.
method Analyzing the degree of hypersurfaces omitted by the Gauss map.
result Gauss map can omit hypersurfaces of degree at most nn+2(n+1)n+2. Harmonic and minimal great circle fibrations have special Gauss maps.
problem Characterizing Gauss maps of harmonic and minimal great circle fibrations.
method Analyzing the relationship between the Gauss map and the generating unit vector field.
result The Gauss map of a great circle fibration is harmonic (minimal) if and only if the generating unit vector field is harmonic (minimal).
We prove that the distortion function of the Gauss map of a harmonic surface coincides with the distortion function of the surface. Consequently, Gauss map of a harmonic surface is K quasiregular if and only if the surface is K quasiregular, provided that the Gauss map is regular or what is …
The paper analyzes Gauss maps of Lorentzian surfaces in anti-de Sitter space.
problem Determining the type numbers of pseudo-hyperbolic Gauss maps of Lorentzian surfaces.
method Investigates the type numbers of pseudo-hyperbolic Gauss maps of oriented Lorentzian surfaces with constant curvatures in anti-de Sitter space.
result Investigates the behavior of type numbers of pseudo-hyperbolic Gauss maps along parallel families of surfaces.
Geometrically, a new obstruction is found for 4-manifold realizations.
problem Realizing normal 1-types of 4-manifolds with a given boundary.
method Three-stage obstruction theory, with a geometric tertiary obstruction.
result A new geometric tertiary obstruction for 4-manifold realizations.
The paper proves a Fenchel theorem for Gauss maps and shows circles and disks minimize certain energies.
problem Finding minimizers of nonlocal curvature energies.
method Combining Fenchel-type theorems with geometric analysis techniques.
result Circles and disks minimize specific energy functionals.
The study examines surfaces in isotropic space with specific Gauss map properties.
problem Understanding surfaces in simply isotropic space with degenerate metric.
method Investigates surfaces with Gauss map coordinates as eigenfunctions of the Laplace-Beltrami operator for minimal and parabolic normals.
result Identifies surfaces characterized by eigenfunction properties of the Gauss map.
Study the singularities of Gauss map components of surfaces in 4D.
problem Characterize singularities of Gauss map components of surfaces in R4. method Analyze the geometric properties and stability of singularities.
result Singularities of Gauss map components are generically stable and related to surface geometry and J-holomorphic curves. Study on tubes with specific Gauss map properties in 3D space.
problem Characterizing surfaces with a specific Gauss map property in 3D space.
method Analyzing tubes in Euclidean 3-space with Gauss map n satisfying ΔIn = Λn.
result Circular cylinders are the only surfaces of coordinate finite I-type Gauss map.
Study on Gauss images of specific minimal surfaces with finite curvature.
problem Characterizing Gauss images of minimal surfaces with finite total curvature.
method Analyzing the number and weight of omitted and totally ramified values of Gauss maps.
result Construction of new minimal surfaces with specific Gauss map properties.
Minimal hypersurfaces in Euclidean space are restricted to planes if their Gauss maps avoid a half-equator.
problem Characterizing minimal hypersurfaces in Euclidean space.
method Using the Gauss map to restrict the image of minimal hypersurfaces.
result Minimal hypersurfaces must be planes if their Gauss maps avoid a half-equator.
Study on unique generalized Gauss maps of minimal surfaces sharing hypersurfaces in projective varieties.
problem Uniqueness of generalized Gauss maps for minimal surfaces with shared hypersurfaces in projective varieties.
method Analysis of minimal surfaces in Rn+1 with inverse images of hypersurfaces in a projective subvariety. result Generalization and improvement of previous results on the uniqueness of generalized Gauss maps.
Study relates Gaussian curvature signs to cuspidal edge types and geometric invariants.
problem Understanding the relationship between Gaussian curvature and singularities of Gauss maps of cuspidal edges.
method Analyzes geometric invariants and types of singularities of Gauss maps to define and characterize positivity/negativity of cusps.
result Defines and characterizes positivity/negativity of cusps of Gauss maps by geometric invariants of cuspidal edges, and shows relation between sign of cusps and Gaussian curvature.