Automorphisms of fine curve graphs match surface homeomorphisms for planar surfaces.
problem Understanding automorphisms of fine curve graphs on surfaces.
method Analyzing vertices and edges of fine curve graphs to match with surface homeomorphisms.
result Automorphism group of fine curve graphs is naturally isomorphic to the homeomorphism group of boundaryless planar surfaces with at least 7 punctures.
The paper classifies maximal translation surfaces in Lorentz-Minkowski space.
problem Classifying maximal translation surfaces in Lorentz-Minkowski space.
method Analyzing surfaces defined as the sum of two spatial curves, proving properties of generating curves, and classifying surfaces based on curve types.
result A full description of maximal translation surfaces, including new examples not found in Euclidean space.
Describes curves on surfaces with punctures and boundaries.
problem Representing multiple curves on surfaces with punctures and boundaries.
method Using geometric intersection numbers with embedded curves.
result Each multiple curve can be uniquely described.
In this study, we introduce a new type of surface curves called D-type curve. This curve is defined by the property that the unit Darboux vector W0 of a space curve r(s) and unit surface normal n along the curve r(s) satisfy the condition <n,W0>=constant. We point out that a D-type curve is a geodesic curve or an asymp…
The paper studies hyperbolic phenomena on closed surfaces using bicorn curves.
problem Understanding hyperbolic phenomena on curve graphs of closed surfaces.
method Using the theory of bicorn curves to analyze the curve graphs of closed surfaces.
result Proves that the curve graph of any closed surface is 15-hyperbolic with one exception.
Automorphisms of fine 1-curve graph linked to surface homeomorphisms.
problem Understanding automorphisms of fine 1-curve graphs.
method Isomorphic mapping to surface homeomorphisms.
result Automorphism group is isomorphic to homeomorphism group of a surface.
Automorphism group of nonorientable surface curve graph matches surface homeomorphisms.
problem Identifying automorphisms of nonorientable surface curve graphs.
method Using Bowden, Hensel, and Webb's fine curve graph and Long, Margalit, Pham, Verberne, and Yao's proof as a foundation.
result Automorphism group of nonorientable surface curve graph is isomorphic to the surface's homeomorphism group.
Surfaces and curves play an important role in geometric design. In recent years, problem of finding a surface passing through a given curve has attracted much interest. In the present paper, we propose a new method to construct a surface interpolating a given curve as the asymptotic curve of it. Also, we analyze the co…
Investigates Darboux rectifying curves on smooth surfaces.
problem Characterizing Darboux rectifying curves on smooth surfaces.
method Analyzes the position vector under isometry and finds conformal invariance conditions.
result Identifies sufficient conditions for conformal invariance of Darboux rectifying curves.
Study automorphisms of smooth curve graphs on surfaces.
problem Understanding automorphisms of fine curve graphs.
method Examined automorphisms of continuously differentiable curves on surfaces.
result Automorphisms on surfaces of genus ≥ 2 are induced by homeomorphisms.
Uniform hyperbolicity proved for nonorientable surface curve graphs.
problem Proving uniform hyperbolicity for nonorientable surface curve graphs.
method Using bicorn curves and arguments from orientable surfaces.
result Graph of nonseparating curves is uniformly hyperbolic.
Study minimizes crossing points of up to 12 curves on a genus 2 surface.
problem Minimizing intersection points of curves on a surface.
method Analyzes systems of up to 12 simple closed curves on a genus 2 surface to find the minimum crossing number.
result Determines the minimal crossing number of up to 12 curves on a genus 2 surface and proves the minimization systems are unique.
Paper studies surface knotting and its fold curves.
problem Understanding the knotting of surfaces and their fold curves.
method Examines the relationship between surface knotting and fold curves, defines a new invariant.
result Every surface can be isotoped so that its fold curves form an unlink.
In this paper, we study the computation of curvatures at the singular points of algebraic curves and surfaces. The idea is to convert the problem to compute the curvatures of the corresponding regular parametric curves and surfaces, which have intersections with the original curves and surfaces at the singular points. …
Study on focal surfaces and evolutes of framed curves in hyperbolic 3-space using Legendrian duality.
problem Investigate differential geometry properties of framed curves, including singular points.
method Use Legendrian dualities to analyze focal surfaces and evolutes of hyperbolic framed curves.
result Show the relationship among focal surfaces, evolutes, and dual surfaces of evolutes.
Study curves in non-orientable surfaces with specific intersection properties.
problem Enumerate and understand curves in non-orientable surfaces with intersection constraints.
method Generalized construction of Malestein-Rivin-Theran to non-orientable surfaces.
result Lower bound for maximum number of curves in generic non-orientable surface.
Characterizes when curves form bouquets in surfaces.
problem Understanding when simple closed curves form bouquets in surfaces.
method Characterization in terms of relations between Dehn twists.
result Characterization of bouquets of curves in surfaces.
Automorphisms of fine curve graph match surface homeomorphisms.
problem Understanding automorphisms of curve graphs for surfaces.
method Building on previous work, proving isomorphism to surface homeomorphisms.
result The group of automorphisms of the fine curve graph is isomorphic to the extended mapping class group of the surface.
Acyclicity proven for curve complex on surfaces.
problem Acyclicity of curve complex on surfaces.
method Analyzing homologous curves on surfaces of genus g.
result Complex is (g-3)--acyclic.
Study curve shortening flow on Riemann surfaces with conic singularities.
problem Analyzing curve shortening flow on surfaces with conic singularities.
method Generalized Huisken's comparison function to Riemann surfaces and surfaces with conic singularities. Reproofed Gage-Hamilton-Grayson theorem. Proved CSF can't touch conic singularities with cone angles ≤ π.
result CSF can't touch conic singularities with cone angles ≤ π for embedded simple closed curves.
Transforms curves and surfaces for efficient geometric analysis.
problem Efficiently analyzing and comparing curves and surfaces.
method Square root velocity transformation for curves and intrinsic comparison for surfaces.
result Fundamental geometric properties of curves under the transformation.
New examples of translation surfaces on hyperelliptic curves with many automorphisms.
problem Determining when translation surfaces are supported on the same algebraic curve.
method Analyzing eigenforms of automorphisms on hyperelliptic curves with many automorphisms.
result Presentation of infinitely many examples of translation surfaces on hyperelliptic curves.
We investigate the relationship among characteristic curves on developable surfaces. In case parameter curves coincide with these curves, we show that the base curve of a developable surface could be either a plane curve, a circular helix, a general helix or a slant helix.
Survey on rational curves on complex surfaces, highlighting different approaches.
problem Existence of rational curves on complex surfaces.
method Classification of complex surfaces and systematic study of rational curves in each class.
result Highlighting the different approaches to study rational curves on complex surfaces.
Curved flats linked to pairs of Lie applicable surfaces.
problem Understanding curved flats in Lie sphere geometry.
method One-to-one correspondence with pairs of Demoulin families of Lie applicable surfaces via Darboux transformation.
result Curved flats correspond to specific Lie applicable surface pairs.
New Finsler metrics derived from pedal curves.
problem Constructing new Finsler metrics.
method Using pedal curves or surfaces of other curves/surfaces.
result Generalization of slope metric.
Proves prime theta-curves for knots on minimal genus surfaces.
problem Prime knots and essential arcs on Seifert surfaces.
method Analyzes prime knot unions with essential arcs on minimal genus surfaces.
result Each prime knot union an essential arc on a minimal genus Seifert surface is a prime theta-curve.
Parabolic mapping class acts on curve graphs of infinite type surfaces.
problem Understanding parabolic isometries on curve graphs of infinite type surfaces.
method Fine curve graph tools to prove existence of parabolic isometries.
result Existence of parabolic isometries on graphs of curves of infinite type surfaces.
Investigates the vertex curve of smooth surfaces in 3D space, connecting geometry and image analysis.
problem Understanding the geometry of smooth surfaces in 3D space.
method Analyzes the vertex curve, related to differential geometry and symmetry sets of isophote curves.
result Establishes connections between the vertex curve and other geometric curves like parabolic and flecnodal curves.
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.
Defines and analyzes generalized normal ruled surfaces of curves in 3D space.
problem Understanding the geometry of generalized normal ruled surfaces.
method Calculates Gaussian and mean curvatures to determine surface properties and examines curve conditions.
result Determines when surfaces are flat or minimal and identifies specific curve types.
A directed curve is a possibly singular curve with well-defined tangent lines along the curve. Then the tangent surface to a directed curve is naturally defined as the ruled surface by tangent geodesics to the curve, whenever any affine connection is endowed with the ambient space. In this paper the local diffeomorphis…
Surfaces and curves play an important role in geometric design. In recent years, problem of finding a surface passing through a given curve have attracted much interest. In the present paper, we propose a new method to construct a surface interpolating a given curve as the geodesic curve of it. Also, we analyze the con…
There is a natural duality between line congruences in R3 and surfaces in R4 that sends principal lines into asymptotic lines. The same correspondence takes the discriminant curve of a line congruence into the parabolic curve of the dual surface. Moreover, it takes the ridge curves to the flat r…
Characterizes special curves on surface tangent bundles.
problem Understanding curves on surface tangent bundles.
method Characterization of Legendre and slant curves.
result Characterizations for N-Legendre and N-slant curves.
Study helicoidal surfaces with singular points using frontals.
problem Investigate helicoidal surfaces with singular points.
method Use frontals in the Euclidean plane to analyze helicoidal surfaces.
result Provide criteria for the singularities of helicoidal surfaces of frontals.
Study shortest non-separating curves on non-orientable surfaces, proving NP-hardness and tractability.
problem Computing shortest non-separating simple closed curves on non-orientable surfaces.
method Developed tools for computing shortest curves, proving NP-hardness and tractability.
result Proved NP-hardness and fixed-parameter tractability for computing shortest orienting curves, and polynomial-time algorithm for non-orienting curves.
The paper studies surfaces with spherical curvature lines and their generation by constrained elastic curves.
problem Understanding surfaces with spherical curvature lines and their generation mechanisms.
method The approach involves Lie sphere transformations, Legendre curves, and polynomial conserved quantities of connections.
result Lie applicable surfaces with exactly one family of spherical curvature lines are generated by the lift of constrained elastic curves.
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.
Sharp estimate shown to be rigid on curved surfaces.
problem Sharp Bezout estimate on nonnegatively curved Riemann surfaces.
method General three circle theorem applied.
result Rigidity of the sharp Bezout estimate.
New proof shows rationality of scl for non-filling curves.
problem Understanding stable commutator length in non-filling curves.
method New proof using extremal surfaces for scl.
result Rationality of stable commutator length for non-filling curves.
Hensel-Przytycki-Webb proved that all curve graphs of orientable surfaces are 17-hyperbolic. In this paper, we show that curve graphs of non-orientable surfaces are 17-hyperbolic by applying Hensel-Przytycki-Webb's argument. We also show that arc graphs of non-orientable surfaces are 7-hyperbolic, and arc-curve graphs …
The paper studies singularities on parallels of tangent developable surfaces of frontal curves.
problem Understanding singularities on parallels of tangent developable surfaces.
method Generalization of tangent developable surfaces and parallel deformations for frontal curves in arbitrary dimensions.
result Classification of generic singularities on parallels of tangent developable surfaces for frontal curves in 3 or 4 dimensional Euclidean spaces.
T-curves are piecewise linear curves which have been used with success since the beginning of the 1990's to construct new real algebraic curves with prescribed topology mainly on the real projective plane. In fact T-curves can be used on any real projective toric surface. We generalize here the construction of the latt…
Simplified proof classifies surfaces using normal curves.
problem Classifying surfaces using algebraic topology invariants.
method Using normal curves and analogy to 3-manifolds classification.
result Simple proof without using algebraic topology invariants.
We view conformal surfaces in the 4--sphere as quaternionic holomorphic curves in quaternionic projective space. By constructing enveloping and osculating curves, we obtain new holomorphic curves in quaternionic projective space and thus new conformal surfaces. Applying these constructions to Willmore surfaces, we show…
Determines the crossing number of polynomial curve systems on surfaces.
problem Calculating the crossing number of polynomial curve systems on surfaces.
method Determines the crossing number in terms of the genus for polynomial curve systems.
result High precision determination of crossing number for polynomial curve systems.
Study curve shortening flow on Riemann surfaces with conical singularities.
problem Evolution of curves on Riemann surfaces with singular points.
method Curve shortening flow governed by a degenerate quasilinear parabolic equation.
result Evolving curves stay fixed at singular points and show collapsing and convergence results.