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…
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Basic Classes of Timelike General Rotational Surfaces in the Four-dimensional Minkowski Spacemath.DG
Study of timelike surfaces in Minkowski space with specific geometric properties.
problem Characterizing geometric properties of timelike surfaces in Minkowski space.
method Analytical study of two types of timelike general rotational surfaces.
result Explicit descriptions of minimal and surfaces with specific curvature 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…
The study characterizes loxodromes on specific rotational surfaces in 3D space.
problem Characterizing loxodromes on rotational surfaces with special geometric properties.
method Parametrizations and curvature/torsion calculations for loxodromes on various rotational surfaces.
result The loxodrome on a flat rotational surface is a general helix.
Study on rotating surfaces in 4D space with matrices.
problem Understanding rotational surfaces in pseudo-Euclidean 4-space.
method Defined hyperbolic and elliptic rotational surfaces using curves and matrices in 4D semi-Euclidean space.
result Generated rotated surfaces using specific rotation matrices.
Rotation systems can't always be drawn in surfaces.
problem Rotation systems and simple drawings in surfaces.
method Extended the plane result to all fixed surfaces.
result Existence of rotation systems not arising from simple drawings in any fixed surface.
The paper applies Clairaut's theorem to rotational surfaces in pseudo-Euclidean 4-space.
problem Exploring geodesic curves on rotational surfaces in pseudo-Euclidean 4-space.
method Expressing Clairaut's theorem and deriving equations for geodesic curves.
result Characterization of geodesic curves on hyperbolic and elliptic surfaces of rotation.
In-plane drill rotations are impossible for smooth shells.
problem In-plane drill rotations on smooth shells are impossible.
method Analyzing the differential geometry of surfaces and isometries.
result Any isometry that coincides with the given surface at a portion of the boundary is the identity.
The study explores special surfaces in a normed space.
problem Constant Gaussian and mean curvature surfaces in normed spaces.
method Analyzes rotational surfaces with specific curvature properties.
result Generalizes catenoid, pseudo-sphere, and Delaunay surfaces.
In this work, we study a class of rotational surfaces in the pseudo-Euclidean space whose profile curves lie in two-dimensional planes. We solve the differential equation that characterizes the rotational surfaces with zero mean curvature to determine the profile curves of such rotational surfaces. The…
We investigate the duality between minimal surfaces in Euclidean space and maximal surfaces in Lorentz-Minkowski space in the family of rotational surfaces. We study if the dual surfaces of two congruent rotational minimal (or maximal) surfaces are congruent. We show that in the duality process by means of a one-parame…
New approach to rotational Weingarten surfaces using geometric momentum.
problem Classifying and characterizing rotational Weingarten surfaces.
method Introducing geometric linear momentum of a plane curve to reduce Weingarten conditions to differential equations.
result Classification of non-degenerate quadric surfaces and elasticoids.
In the four-dimensional pseudo-Euclidean space with neutral metric there are three types of rotational surfaces with two-dimensional axis - rotational surfaces of elliptic, hyperbolic or parabolic type. A surface whose mean curvature vector field is lightlike is said to be quasi-minimal. In this paper we classify all q…
Positive factorization found for a specific map on surfaces.
problem Balanced superelliptic rotation on surfaces.
method Positive factorization approach.
result Positive factorization for balanced superelliptic rotation.
Study rotational surfaces with prescribed Gauss curvature in 3D space.
problem Classify and analyze rotational surfaces with prescribed Gauss curvature.
method Phase plane analysis and mild assumptions on the prescribed function.
result Existence of singular radial solutions intersecting orthogonally the axis of rotation.
The paper classifies and constructs rotational surfaces with constant astigmatism in space forms.
problem Classifying surfaces with constant astigmatism in space forms.
method Classification and construction of surfaces using variational problems and binormal evolution.
result Locally constructed all rotational surfaces of constant astigmatism.
Study on rotational surfaces in de Sitter space with specific curvature conditions.
problem Characterizing rotational surfaces in de Sitter space with Weingarten conditions.
method Analyzing spacelike and timelike rotational surfaces in 3D de Sitter space, determining profile curves, and classifying surfaces based on curvature relations.
result Classification of Weingarten rotational surfaces in de Sitter space with specific curvature relations.
The paper defines and analyzes homotopic rotation sets for surfaces of higher genus.
problem Defining and analyzing homotopic rotation sets for surfaces of higher genus.
method Developed a definition and proved several results using the theory of Le Calvez and Tal.
result Found that the homotopic rotation set can imply the existence of infinitely many periodic orbits under certain conditions.
Rotational Ricci surfacesmath.DG
The study classifies surfaces with specific curvature properties.
problem Classifying surfaces with a particular curvature equation.
method Analyzing surfaces in 3D Euclidean space with a specific curvature equation.
result A one-parameter family of surfaces meeting the unit ball orthogonally.
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…
Classifies surfaces with special curvature properties.
problem Rotational surfaces with specific curvature conditions.
method Classifies surfaces with rotationally symmetric norms and linear curvature relations.
result Rotational surfaces with linearly related curvatures are classified.
Solves curvature prescription on rotational surfaces.
problem Prescribing different types of curvatures on rotational surfaces.
method Using arbitrary continuous functions of distance from axis of revolution.
result Complete classification of surfaces with specific curvature relationships.
New approach classifies rotational Weingarten surfaces in Lorentz-Minkowski space.
problem Classifying rotational Weingarten surfaces in Lorentz-Minkowski space.
method Using geometric linear momentum of generatrix curves with respect to axes of revolution.
result Unified framework for three causal types of rotation axes.
Characterizes rotational solitons for curve shortening flow on revolution surfaces.
problem Understanding the behavior of curves under curve shortening flow on revolution surfaces.
method Characterization and asymptotic behavior analysis.
result Asymptotic behavior of rotational solitons to parallel geodesics.
Helicoidal surfaces rotate and translate under mean curvature flow.
problem Existence of helicoidal surfaces under mean curvature flow.
method One-parameter families of helicoidal surfaces rotating and translating.
result Existence of helicoidal surfaces under mean curvature flow.
New surfaces in Lorentz-Minkowski space with constant mean curvature identified.
problem Identifying surfaces with constant mean curvature in Lorentz-Minkowski space.
method Using a specific coordinate system and properties of the Weingarten endomorphism, the mean curvature is shown to be constant under certain conditions.
result Constant mean curvature surfaces identified, including spacelike and timelike Enneper surfaces.
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.
Study on surfaces with constant curvature under a specific connection.
problem Classifying surfaces with constant sectional curvature under a semi-symmetric non-metric connection.
method Analyzing surfaces in with a canonical semi-symmetric non-metric connection determined by a vector field.
result Classification of surfaces under various conditions (cylindrical, rotational) with constant sectional curvature.
Investigates physical properties on surfaces of rotation using Clairaut's theorem.
problem Understanding specific energy and angular momentum on surfaces of rotation.
method Used Clairaut's theorem with geodesic conditions to derive specific energy and angular momentum.
result Physical expressions for specific energy and angular momentum on surfaces of rotation were derived.
Study curve shortening flows on specific surfaces, proving properties and existence.
problem Analyzing curve shortening flows on rotational surfaces with negative Gauss curvatures.
method Assume negative Gauss curvatures and conditions on Gauss curvature and curve curvature. Prove curve remains a graph and establish flow properties.
result Prove the curve remains a graph over parallels and establish long-time existence of the flow.
Researchers classify cmc surfaces using Jacobi elliptic functions.
problem Classifying rotational cmc surfaces in non-Euclidean space forms.
method Lie sphere geometric description of rotational linear Weingarten surfaces.
result Explicit parametrizations of cmc surfaces in hyperbolic space.
The paper classifies surfaces with constant skew curvature in 3-space forms.
problem Classifying surfaces with constant skew curvature in 3-space forms.
method Variational characterization and flow of binormal vector field.
result Classification of rotational surfaces with constant skew curvature.
In this paper, we study rotational surfaces of elliptic, hyperbolic and parabolic type with pointwise 1-type Gauss map which have spacelike profile curve in four dimensional pseudo Euclidean space E4-2 and obtain some characterizations for these rotational surfaces to have pointwise 1-type Gauss map.
We obtain an infinite family of complete non embedded rotational surfaces in whose second fundamental forms have length equal to one at any point. Also we prove that a complete rotational surface with second fundamental form of constant length is either a round sphere, a circular cylinder or, up to a homo…
The paper classifies rotational K^α-translators in Minkowski space.
problem Classifying rotational K^α-translators in Minkowski space.
method Analyzing the properties of spacelike and timelike surfaces in Minkowski space.
result Classification of all rotational K^α-translators depending on the causal character of the rotation axis.
Study axisymmetric surfaces in Euclidean space for energy minimization.
problem Finding surfaces in Euclidean space that minimize energy.
method Phase plane analysis and maximum principle.
result Helicoidal stationary surfaces must be rotational.
Minimal surfaces found in 4D space.
problem Minimal surfaces in 4D space.
method Reduced biharmonic equation to ODEs, excluded non-minimal solutions.
result Biharmonic rotational surfaces in 4D are minimal.
Finite translation surfaces can be classified by the order of their singularities. When generalizing to infinite translation surfaces, however, the notion of order of a singularity is no longer well-defined and has to be replaced by new concepts. This article discusses the nature of two such concepts, recently introduc…
Derives a new first order differential equation for smooth surfaces.
problem Finding new equations to describe smooth surfaces.
method Derives a linear differential equation of the first order.
result Proves the maximum principle for Darboux rotation fields.
Lie minimal surfaces are characterized by differential equations of principal curvatures.
problem Characterizing Lie minimal surfaces in Riemannian space forms.
method Using Euler-Lagrange equations and differential equations of principal curvatures.
result Rotational surfaces are found for certain relationships between principal curvatures.
Study on unfolding maps of surfaces in 3D space, proving versality conditions.
problem Investigating the versality of rotation unfolding of folding maps for surfaces in .
method Introducing and analyzing the rotation unfolding of folding maps, proving versality conditions in terms of geometry.
result Proved conditions for the rotation unfolding to be versal, showing diffeomorphic type of tangent plane locus.
Study irrational rotations and construct 2-filling rays on infinite type surfaces.
problem Understanding dense orbits in skew product transformations.
method Skew product transformation with continued fraction analysis.
result Existence of infinite cliques of 2-filling rays on infinite type surfaces.
We apply the invariant theory of surfaces in the four-dimensional Euclidean space to the class of general rotational surfaces with meridians lying in two-dimensional planes. We find all minimal super-conformal surfaces of this class.
We study the third Laplace-Beltrami operator of timelike rotational surface of (T,L)-type in three dimensional Lorentz-Minkowski space.
Study of 17 surface behaviors and singularities for elliptic Weingarten equations.
problem Characterizing and understanding elliptic Weingarten surfaces and their singularities.
method Phase space analysis and classification of surface behaviors.
result Classification of 17 possible qualitative behaviors for rotational surfaces.
Researchers found multiple spherical Ricci metrics on tori with rotational symmetry.
problem Constructing and analyzing spherical Ricci metrics with rotational symmetry.
method Explicitly constructed a two-parameter family of metrics with rotational symmetry and showed their existence on tori.
result Infinitely many non-isometric spherical Ricci metrics can be realized on the same torus.
Solves surface problem in 3D light cone.
problem Björling problem for zero mean curvature surfaces in the three-dimensional light cone.
method Solves the Björling problem for zero mean curvature surfaces in the three-dimensional light cone.
result Constructs and classifies all rotational zero mean curvature surfaces.
New Bäcklund transformations for discrete pseudospherical surfaces of revolution are found.
problem Constructing new non-rotational discrete pseudospherical surfaces.
method Explicit parametrizations and Bäcklund transformations for discrete constant negative Gaussian curvature surfaces of revolution.
result Conditions for Bäcklund transformations to preserve periodicity are provided.