Study surfaces with constant ratio of principal curvatures in Euclidean and isotropic geometries.
problem Characterize surfaces with constant ratio of principal curvatures in different geometries.
method Differential geometry, line geometry, Lie sphere geometry, ordinary differential equations, algebraic geometry.
result Characterized various types of surfaces like rotational, channel, ruled, helical, and translational.
Minimal surfaces help compare geometric shapes.
problem Comparing different geometric shapes.
method Applications of minimal surfaces.
result New insights into geometric comparisons.
Research on surfaces in Laguerre geometry, focusing on L-isothermic, L-minimal, and generalized L-minimal surfaces.
problem Exploring properties of surfaces in Laguerre geometry.
method Using the quadric model of Lie sphere geometry and the method of moving frames.
result Application of the Cartan-Kaehler theorem to study L-minimal surfaces.
The study characterizes channel surfaces in Lie sphere geometry and their transformations.
problem Characterizing channel surfaces in Lie sphere geometry.
method Using Ω0-surfaces and Dupin cyclide congruences to study transformations and Ribaucour pairs. result Characterization of channel surfaces and their transformations.
Study local geometries of symmetric affine surfaces.
problem Characterize local geometries of locally symmetric affine surfaces.
method Classify 6 non-isomorphic local geometries using Opozda's result and classify them up to linear isomorphism.
result 6 non-isomorphic local geometries identified and classified.
The study connects surface geometry in 5D to 4D projections and umbilic curvatures.
problem Understanding the geometry of surfaces in 5D space.
method Relating surfaces in 5D to surfaces in 4D via projections and normal sections, analyzing asymptotic directions and umbilic curvatures.
result Relations between asymptotic directions and umbilic curvatures in 5D surfaces and their counterparts in 4D projections.
Study of surfaces with corank 1 singularities in 4D space.
problem Understanding the geometry of surfaces with specific singularities.
method Defining curvature parabolas, asymptotic and binormal directions, and relating to regular surfaces.
result Established a connection between singular and regular surfaces in 4D space.
Classifies surfaces in isotropic geometry with constant curvature.
problem Classifying surfaces with constant curvature in isotropic geometry.
method Classifies surfaces under the condition that at least one translating curve lies in a plane.
result Classification of surfaces in isotropic geometry with constant curvature.
The study characterizes surfaces with specific harmonic properties in pseudo-conformal geometry.
problem Characterizing surfaces with harmonic properties in pseudo-conformal geometry.
method Investigating sphere congruences, quasi-umbilical surfaces, and constant mean curvature surfaces.
result Generically, Bryant's quartic differential is divergence free if and only if the surface is superconformal or orthogonal to a harmonic congruence of spheres.
The study characterizes surface singularities in Lie sphere geometry.
problem Understanding singularities of surfaces in Lie sphere geometry.
method Analyzing conditions for cuspidal edges, swallowtails, and Lie sphere transformations.
result Conditions for various surface singularities in Lie sphere geometry.
Study on focal surfaces of lightcone framed surfaces in Lorentz-Minkowski 3-space.
problem Investigate differential geometry properties of focal surfaces of lightcone framed surfaces.
method Introduced lightcone frame to define lightcone framed surfaces, then investigated their differential geometry properties.
result Investigated differential geometry properties of focal surfaces of lightcone framed surfaces.
Study geometry of surfaces glued along a curve.
problem Geometry of surfaces glued along a curve.
method Moving frame along the curve, developable surfaces defined.
result Developed geometric properties of glued surfaces.
Explains curves and surfaces in differential geometry.
problem Understanding smooth curves and surfaces in differential geometry.
method Problem-centered, elementary, visual approach focusing on essential techniques.
result Provides a solid foundation for further study in differential geometry.
Simpler method derived for path geometries on surfaces, characterizing projective path geometries.
problem Characterizing projective path geometries on surfaces.
method Solving the equivalence problem of sub-Riemannian geometry of signature (1,1) on a contact 3-manifold.
result Characterization of projective path geometries in terms of their chains.
Explains how surfaces can have hyperbolic geometries and connects them to Higgs bundles.
problem Understanding hyperbolic structures on surfaces and their relation to Higgs bundles.
method Describes how to obtain and parametrize hyperbolic structures on surfaces, introduces Higgs bundles.
result Established a connection between hyperbolic surfaces and Higgs bundles.
These notes introduce key techniques in differential geometry for curves and surfaces.
problem Understanding the basics of differential geometry for curve and surface analysis.
method Problem-centered, elementary, visual approach to teaching essential techniques.
result Provides a solid foundation for further study in differential geometry.
SU(2) flat connections link to 3D geometry with cosmological constant.
problem Mapping flat connections on Riemann surfaces to 3D twisted geometry.
method Relating flat connection quantities to geometrical quantities in discrete 3D space.
result Moduli space of SU(2) flat connections generalizes phase space of twisted geometry.
Spinor representation in isotropic space via Laguerre geometry.
problem Representing conformal and constant mean curvature surfaces in isotropic space.
method Developing Laguerre geometry of isotropic space, defining spin transformations, and constructing Weierstrass and Kenmotsu representations.
result Explicit constructions of zero mean curvature and constant mean curvature surfaces.
Defines discrete channel surfaces in Lie sphere geometry.
problem Defining discrete channel surfaces in Lie sphere geometry.
method Definition and associated data sets for reconstruction.
result Proof of a discrete version of Vessiot's Theorem for isothermic discrete channel surfaces.
Expository notes on integrable geometry transformations.
problem Exploring transformations in integrable geometry.
method Gauge-theoretic approach to classical examples.
result Discussion of surfaces of constant negative Gauss curvature and isothermic surfaces.
Study large-scale geometry of infinite type surface mapping class groups.
problem Classify surfaces based on mapping class group properties.
method Coarse geometry, using Rosendal's framework.
result Classification of surfaces based on group properties.
We introduce the notion of manifolds of amalgamation geometry and its generalization, split geometry. We show that the limit set of any surface group of split geometry is locally connected, by constructing a natural Cannon-Thurston map.
Study CR geometry surface area elements and singular Yamabe problem solutions.
problem Solving the singular CR Yamabe problem in 3D CR geometry.
method Expressed CR invariant surface area elements, deduced Euler-Lagrange equations, provided solutions.
result One energy functional coefficient is shown to be proportional to the log term in volume renormalization.
We define a class of two dimensional surfaces conformally related to minimal surfaces in flat three dimensional geometries. By the utility of the metrics of such surfaces we give a construction of the metrics of 2N dimensional Ricci flat (pseudo-) Riemannian geometries.
Study of surfaces in space forms using Lie sphere geometry.
problem Investigate channel linear Weingarten surfaces in various space forms.
method Lie sphere geometric approach to uniform treatment of different ambient geometries.
result Any channel linear Weingarten surface is isothermic and a surface of revolution.
Investigates surface immersions in normed spaces using affine differential geometry.
problem Differential geometry of immersed surfaces in normed spaces.
method Endows surface with a Riemannian metric related to normal curvature, re-calculates curvatures in terms of ambient affine distance functions, and characterizes minimal surfaces.
result Characterizes minimal surfaces as solutions to a differential equation and identifies conditions for affine normal and Birkhoff normal vector fields to coincide.
New representations for discrete surfaces derived from dual transforms.
problem Constructing discrete surfaces in differential geometry.
method Using Ω-dual transform and lightlike Gauss maps in Laguerre geometry. result All discrete linear Weingarten surfaces arise via Weierstrass-type representations.
Abstract: Study of surface transitions and IDE inflections via contact geometry.
problem Understanding transitions on surfaces and implicit differential equations.
method Contact geometry and Legendrian properties of projections.
result List of unavoidable local phenomena on surfaces and IDE solutions.
New surfaces defy traditional pants decompositions.
problem Creating surfaces without bounded pants decompositions.
method Constructing quasiconformally homogeneous hyperbolic Riemann surfaces.
result Found surfaces with infinite type and compact boundary components that defy pants decompositions.
Study equiangular surfaces in 3D, extending plane spirals.
problem Understanding 3D surfaces with constant normal-vector angles.
method Investigates three-dimensional extensions of equiangular spirals.
result Identifies self-similar structures in sea shell geometry.
Study on surface geometry in Lie groups with CR structures.
problem Understanding surface curvature in Lie groups with CR structures.
method Defined Gauss and mean curvature in Tanaka-Webster geometry.
result Gave specific examples of surface curvature calculations.
The paper studies elliptical surfaces in 3D affine space, classifying them based on curvature.
problem Classifying regular elliptical surfaces in affine space A3 based on curvature. method Defined a moving frame of minimal order for regular elliptical surfaces and derived differential invariants.
result Classified regular elliptical surfaces of constant curvatures up to affine congruence.
New theory assesses smoothness of polyhedral surfaces.
problem Evaluate smoothness of polyhedral surfaces.
method Incorporates geometry of polyhedral surfaces, proposes new notions of smoothness.
result Seemingly mild conditions significantly limit polyhedral surface shapes.
Approximates surfaces using Laguerre geometry with spherical faces.
problem Approximating smooth surfaces using Laguerre geometry.
method Using Laguerre conjugate nets and spherical faces to approximate surfaces.
result Laguerre conjugate nets provide a method for surface approximation.
Develops Riemannian geometry for noncommutative super surfaces.
problem No specific problem stated; focuses on mathematical development.
method Introduces metric and connections on noncommutative super surfaces, showing compatibility and zero torsion under certain conditions.
result Noncommutative super surfaces have a well-defined Riemannian geometry with properties analogous to classical Riemannian geometry.
New minimal surfaces found in ball with boundary constraints.
problem Finding minimal surfaces with boundary conditions.
method Equivariant differential geometry approach.
result A family of free boundary minimal surfaces in the unit ball.
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.
The relation between differential geometry of surfaces and some Heisenberg ferromagnet models is considered.
We introduce two basic invariant forms which define generic surface in 3-space uniquely up to Lie sphere equivalence. Two particularly interesting classes of surfaces associated with these invariants are considered, namely, the Lie-minimal surfaces and the diagonally-cyclidic surfaces. For diagonally-cyclidic surfaces …
Paper relates curvature ellipse and parabola of surface projections.
problem Relating local geometry of surfaces in different dimensions.
method Analyzes curvature ellipses and parabolas in R4 and R3. result Relates curvature geometry of surfaces to their projections.
Long geodesics imply a special shape of convex bodies.
problem Understanding the geometry of convex surfaces.
method Intrinsic geometry of convex surfaces and proof by contradiction.
result Long geodesics on a convex surface imply the shape is an isosceles tetrahedron.
Study of minimal surfaces based on boundary geometry.
problem Understanding the shape of compact singular minimal surfaces from their boundaries.
method Estimates of area and height derived from boundary geometry; conditions for rotational surfaces; non-existence results for certain boundary configurations.
result Derived estimates and conditions for minimal surfaces based on boundary properties.
Extends mean curvature to surfaces in Riemann-Cartan geometry with torsion.
problem Addressing surfaces in Riemann-Cartan geometry with nontrivial torsion.
method Introducing a complex-valued 2-form associated with the torsion, which interacts with other geometric concepts.
result Complex-valued mean curvature quantity interacts with Hopf differential and Gauss map.
Study the spectral geometry of surfaces with curved conic singularities.
problem Understanding the spectral properties of surfaces with conic singularities.
method Using the heat trace expansion, express spectral geometry terms through the geometry and curvature of the singularities.
result The first few terms in the heat trace expansion are expressed through the geometry and curvature of the singularities.
Study of links on surfaces, generalizing Menasco's polyhedral decomposition.
problem Understanding the geometry of links on surfaces embedded in 3-manifolds.
method Decomposing the complement of links into simpler pieces and using hyperbolic geometry.
result Proved various properties of hyperbolic geometry of generalised alternating links.
We study surfaces with decorations and prove uniformization in non-Euclidean geometries.
problem Discrete conformal equivalence in non-Euclidean geometries.
method Variational principle and continuous deformation.
result One master theory of discrete conformal equivalence across different geometries.
Study uses crochet to visualize non-Euclidean geometry.
problem Understanding non-Euclidean surfaces through physical models.
method Parametrization of crochet models to represent Lobachevskian surface.
result Crochet models reflect non-Euclidean geometry characteristics.
The lightlike geometry of codimension two spacelike submanifolds in Lorentz-Minkowski space has been developed in [Izumiya, S. and Romero Fuster, M. C. Selecta Mathematica (NS), 13 23--55 (2007)] which is a natural Lorentzian analogue of the classical Euclidean differential geometry of hypersurfaces. In this paper we i…