Study describes singularities of height functions on specific singular surfaces.
problem Analyzing singularities of height functions on singular surfaces.
method Using geometric language and blowing-ups, investigate singularities of height functions and dual surfaces.
result Characterized singularities of height functions and dual surfaces on specific singular surfaces.
The study examines singularities and geometric properties of surfaces derived from frontals with specific singular points.
problem Characterizing and understanding the singularities and geometric properties of surfaces formed by the singular loci of normal congruences of frontals with pure-frontal singular points.
method Characterizations of singularities in terms of geometric invariants of the initial frontal are provided for the normal ruled surface. Relations between certain singularities of focal surfaces and geometric properties of the frontal are also explored.
result Behavior of Gaussian curvature of focal surfaces of frontals with a 5/2-cuspidal edge is considered. Study proves surfaces with constant curvature are simple shapes.
problem Characterizing singular minimal surfaces with constant curvature.
method Proved geometric properties of surfaces with constant curvature.
result Singular minimal surfaces with constant curvature are planes, spheres, and cylindrical surfaces.
Formula proves Euler characteristic of singularized surfaces.
problem Calculating the Euler characteristic of singularized surfaces.
method Three operations: collapsing, zipping, and double loop identification.
result Formula for Euler characteristic of singularized surfaces.
Fold singular points play important roles in the theory of maximal surfaces. For example, if a maximal surface admits fold singular points, it can be extended to a timelike minimal surface analytically. Moreover, there is a duality between conelike singular points and folds. In this paper, we investigate fold singular …
Study on singular points of translation surfaces under linearly dependent conditions.
problem Investigate singular points of translation surfaces under linearly dependent conditions.
method Use theories of generalised framed surfaces and framed surfaces.
result Introduce translation generalised framed surfaces and investigate their singular points.
Simple criteria for codimension two surface singularities.
problem Identifying singularities in surfaces of codimension two.
method Provided criteria for singularities in surfaces of codimension less than or equal to two.
result Conditions for codimension two singularities in ruled surfaces and center maps.
Paper studies singularities of timelike minimal surfaces in Minkowski 3-space.
problem Exploring singularities of timelike minimal surfaces in Minkowski 3-space.
method Existence and non-existence theorems, criteria for specific singularities.
result Various singularities unique to timelike minimal surfaces, including cuspidal butterfly and (2,5)-cuspidal edge. The paper proves ideal triangulations and disk unfolding for singular flat surfaces.
problem Proving ideal triangulations and disk unfolding for singular flat surfaces.
method Using geodesic triangulation and finite geodesic connections.
result Each singular flat surface has an ideal triangulation and can be unfolded into a flat disk.
Study singularities of CMC 1 surfaces in de Sitter space.
problem Characterize singularities of spacelike CMC 1 surfaces in de Sitter space.
method Proved duality between singular points and introduced invariants.
result Classified non-degenerate singular points on spacelike CMC 1 surfaces.
Study describes singularities of distance squared functions on singular surfaces.
problem Characterizing singularities of distance squared functions on singular surfaces.
method Using smooth map-germs Sk, Bk, Ck, and F4 singularities, the study describes singularities via blowing-ups. result Characterization of singularities of wave-fronts and caustics of singular surfaces.
Study geometric properties of surfaces with specific formulae.
problem Calculate curvature of surfaces with singular points.
method Investigate singular curvature and limiting normal curvature of surfaces.
result Determine geometric properties of surfaces with singular points.
The paper proves cylindrical nature of singular minimal ruled surfaces.
problem Understanding minimal potential energy surfaces under gravitational forces.
method Analyzing singular minimal ruled surfaces in Euclidean and Lorentz-Minkowski 3-spaces.
result Singular minimal ruled surfaces are cylindrical, including as α-catenary cylinders.
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.
We use integrable systems techniques to study the singularities of timelike non-minimal constant mean curvature (CMC) surfaces in the Lorentz-Minkowski 3-space. The singularities arise at the boundary of the Birkhoff big cell of the loop group involved. We examine the behaviour of the surfaces at the big cell boundary,…
Extends Frohman and Rannard's result to Seifert fiber spaces with singular surfaces.
problem Characterize essential surfaces in Seifert fiber spaces with singular surfaces.
method Extends Frohman and Rannard's approach to handle surfaces with singular fibers.
result Characterizes essential surfaces in Seifert fiber spaces with singular surfaces.
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.
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 D4± singularities. We give criteria for these singularities type…
We shall investigate maximal surfaces in Minkowski 3-space with singularities. Although the plane is the only complete maximal surface without singular points, there are many other complete maximal surfaces with singularities and we show that they satisfy an Osserman-type inequality.
We prove that finite area isolated singularities of surfaces with constant positive curvature in R^3 are removable singularities, branch points or immersed conical singularities. We describe the space of immersed conical singularities of such surfaces in terms of the class of real analytic closed locally convex curves …
Optimizes bounds for multiple T-singularities on surfaces.
problem Bounding T-singularities on non-rational projective surfaces with many singularities.
method Analyzes combinatorial configurations and classifies them to find optimal bounds.
result Classifies all combinatorial configurations leading to high bounds, proving their non-existence gives optimal bounds.
Mean curvature flow shows singularities on smooth surfaces.
problem Understanding singularities in mean curvature flow.
method Analyzing spherical or nondegenerate neck pinches.
result First singular time has isolated singularities.
The study examines Riemannian surfaces with simple singularities.
problem Geometry of Riemannian surfaces with discrete singular points.
method Local and global description using divisors, Gauss-Bonnet formula, classifications theorem.
result Classification of flat metrics with simple singularities on compact surfaces.
Study of cuspidal edges on focal surfaces of regular surfaces.
problem Clarifying the sign of singular curvature at cuspidal edges.
method Investigation using singularities of parallel surfaces.
result Clarification of the sign of singular curvature at cuspidal edges.
Constructs surfaces with conical singularities using variational methods.
problem Creating Hamiltonian Stationary Surfaces with specific singularities.
method Variational methods and convergence process similar to Ginzburg-Landau analysis.
result Obtained surfaces with prescribed conical singularities related to optimal Wente constants.
Characterizes Wahl singularities in del Pezzo surface degenerations.
problem Classifying Wahl singularities in degenerations of del Pezzo surfaces.
method Introducing del Pezzo Wahl chains with markings, proving degenerations to toric surfaces, establishing correspondences, and using Hacking's exceptional collections.
result Established a one-to-one correspondence between marked del Pezzo surfaces and fake weighted projective planes.
We give an explicit formula for singular surfaces of revolution with prescribed unbounded mean curvature. Using it, we give conditions for singularities of that surfaces. Periodicity of that surface is also discussed.
We study global log canonical thresholds of cubic surfaces with canonical singularities, and we prove the existence of a Kahler-Einstein metric on two singular cubic surfaces.
In this paper we define and analyze singularities of discrete linear Weingarten surfaces with Weierstrass-type representations in 3-dimensional Riemannian and Lorentzian spaceforms. In particular, we discuss singularities of discrete surfaces with non-zero constant Gaussian curvature, and parallel surfaces of discret…
Study focal surfaces of wave fronts with unbounded curvatures.
problem Characterizing singularities of focal surfaces near non-degenerate singular points.
method Characterizations based on types of singularities and geometrical properties of initial fronts.
result Investigation of Gaussian curvature behavior of focal surfaces.
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.
Study on helicoidal singular minimal surfaces with specific properties.
problem Characterizing singular minimal surfaces invariant by helicoidal motions.
method Analyzing surfaces with mean curvature defined by a specific formula and studying their invariance under helicoidal motions.
result Helicoidal singular minimal surfaces have a specific geometric configuration.
This paper calculates interaction strength for translation surfaces with multiple singularities.
problem Computing the interaction strength of translation surfaces with multiple singularities is challenging.
method The authors study interaction strength of specific families of translation surfaces, including regular polygons and Bouw-Möller surfaces.
result The paper provides exact computations of KVol on translation surfaces with multiple singularities.
In this paper we study the problem of existence of orbifold Kaehler-Einstein metrics on del Pezzo surfaces of degree 1 with Du Val singular points. Moreover we compute global log canonical thresholds of del Pezzo surfaces of degree 1 with Du Val singularities and of del Pezzo surfaces of Picard rank 1 with Du Val singu…
Study helicoidal surfaces from frontals, revealing geometric rigidity and stability of singularities.
problem Investigate helicoidal surfaces of frontals in Euclidean space.
method Using Legendre curves and framed surfaces, derive curvature expressions and analyze deformations.
result Singularities of curves persist under deformations, revealing geometric rigidity and stability.
Study K-R flow on Hirzebruch surfaces, showing tangent flows are K-R flows with orbifold singularities.
problem Finite time singularities in Kähler-Ricci flow on Hirzebruch surfaces.
method Analyze tangent flows based at singular points.
result Tangent flows are K-R flows with orbifold singularities.
The paper studies singularities in discrete indefinite affine minimal surfaces.
problem Characterizing singularities in discrete indefinite affine minimal surfaces.
method Discretizing smooth curves and applying discrete Lelieuvre's formulas to study the resulting surfaces.
result The definition of singular edges and vertices in discrete asymptotic nets mirrors properties of smooth surfaces.
Study ruled surfaces with finite multiplicity, focusing on their curves and singularities.
problem Understanding ruled surfaces with finite multiplicity.
method Analyzing striction curves and singularities of ruled surfaces.
result Geometric meanings of invariants related to ruled surfaces.
We calculate finite volumes of moduli spaces of flat surfaces with conical singularities.
problem Calculating volumes of moduli spaces of flat surfaces with prescribed conical singularities.
method Induction on the Euler characteristics of the punctured surface for almost all orders of the singularities.
result Explicit computation of volumes is possible.
We consider smoothings of a complex surface with singularities of class T and no nontrivial holomorphic vector field. Under an hypothesis of non degeneracy of the smoothing at each singular point, we prove that if the singular surface admits an extremal metric, then the smoothings also admit extremal metrics in nearby …
The surface singular braid monoid corresponds to marked graph diagrams of knotted surfaces in braid form. In a quest to resolve linearity problem for this monoid, we will show that if it is defined on at least two or at least three strands, then its two or respectively three dimensional representations are not faithful…
Study on minimal surfaces with Y-singularities, proving rigidity for Morse index one.
problem Geometric constraints on minimal surfaces with Y-singularities.
method Investigation of surfaces with low Morse index, focusing on Morse index one.
result Partial uniqueness theorem for Y-catenoid with Morse index one.
Paper solves curvature assignment on surfaces with sharp points and edges.
problem Prescribing curvatures on surfaces with conical singularities and corners.
method New variational formulation for surfaces with singularities.
result First results for prescribed curvatures on surfaces with singularities.
Paper provides a formula for translating solitons and singular minimal surfaces.
problem Representing translating solitons and singular minimal surfaces in 3D space.
method Develops a Weierstrass representation formula.
result Solves a general Cauchy problem for the class of surfaces.
The paper derives Gauss-Bonnet formulas for mappings between surfaces with boundary.
problem Calculating topological invariants for mappings between surfaces with boundaries.
method Defining singular points, constructing coherent tangent bundles, and applying Gauss-Bonnet formulas.
result Derives two Gauss-Bonnet type formulas for mappings between surfaces with boundaries.
The Gauss-Bonnet formula for classical translation surfaces relates the cone angle of the singularities (geometry) to the genus of the surface (topology). When considering more general translation surfaces, we observe so-called wild singularities for which the notion of cone angle is not applicable any more. We study w…
We study symplectic deformation types of minimal symplectic fillings of links of quotient surface singularities. In particular, there are only finitely many symplectic deformation types for each quotient surface singularity.
Proves existence and uniqueness of metrics with negative curvature and singularities on compact surfaces.
problem Existence and uniqueness of conformal metrics with negative curvature and singularities.
method Proves existence and uniqueness of conformal metrics with negative curvature and singularities on compact surfaces.
result Existence and uniqueness of conformal metrics with negative curvature and singularities on compact surfaces.