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 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 examines geometric invariants near a specific type of singular point.
problem The behavior of geometric invariants near a singular point of a surface or curve.
method Analysis of geometric invariants for surfaces and curves that are suspensions of singular curves.
result Evaluation of the orders of Gaussian and mean curvatures for the studied surfaces and curves.
Geometric study of cuspidal S1 singularities using diffeomorphisms and isometries.
problem Understanding geometric properties of cuspidal S1 singularities. method Form representing deformation using diffeomorphisms and isometries, necessary and sufficient condition for frontal maps.
result Investigation of geometric properties and cuspidal cross caps in deformations.
Study geometric singular solutions of generalized Monge-Ampère equations.
problem Solving generalized Monge-Ampère equations on a plane.
method Using exterior differential systems and Cauchy characteristics.
result Criteria for geometric singular solutions to be equivalent to specific types.
Study wall singularities in spaces with upper curvature bounds.
problem Understanding singularities in spaces with curvature constraints.
method Geometric structure theorem and geometric characterization for codimension one and two.
result Necessary and sufficient conditions for singular sets to be of codimension at least two.
Study geometric properties of S1 singularities and their deformations.
problem Understanding differential geometric properties of S1 singularities and deformations.
method Representing deformation using diffeomorphisms and isometries, studying geometric properties.
result Differential geometric properties of S1 singularities and Whitney umbrellas in deformations.
Geometric models for Lie algebras from simple singularities.
problem Classifying simply-laced simple Lie algebras.
method Using polygonal wheels derived from Milnor fibers of simple singularities.
result Geometric root systems are isomorphic to Lie algebras.
The geometric monodromy of a plane curve singularity is a quasi-finite diffeomorphism. In this paper we locate the reduction curves of the geometric monodromy and the quadratic vanishing cycles of the singularity. An application to the geometric monodromy group is given.
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 paper establishes pressure gaps for manifolds with flat subtori singularities.
problem Understanding phase transitions in nonpositively curved manifolds with flat subtori.
method Derives a pressure gap criterion for closed rank 1 manifolds with specific singular sets and proves Hölder continuity of geometric potentials.
result Geometric potentials have pressure gaps and no phase transitions under certain curvature constraints.
Extends geometric quantization to singular spaces.
problem Handling singular symplectic spaces in geometric quantization.
method Developed stratified pseudobundles to replace auxiliary information.
result Provided results for singular quotients of toric manifolds and cotangent bundles.
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.
We prove that the introduction of the class of geometrically atomic bundle maps by Harvey and Lawson in their theory of singular connections is not necessary because an arbitrary map satisfies the conditions of geometric atomicity.
The paper describes the geometric properties of line congruences' singularities.
problem Understanding the singularities of generic line congruences.
method Use of an equiaffine pair to define generic line congruences.
result Geometric description of folds, cusps, and swallowtails as singularities of generic line congruences.
We simplify D4+-front singularities and apply to geometric invariants.
problem Simplifying D4+-front singularities in R3. method Coordinate transformation on source and isometry on target.
result Gauss-Bonnet type theorem for fronts with D4+-singularity. The abstract proves spherical surface decompositions with conical singularities.
problem Decomposing surfaces with spherical metrics and conical singularities.
method Geometric triangulations and irreducible components of standard shapes.
result Spherical polygons, including half-spherical concave polygons, can be arbitrarily complicated.
In this paper we prove geometric residue theorems for bundle maps over a compact manifold. The theory developed associates residues to the singularity submanifolds of the map for any invariant polynomial. The theory is then applied to a variety of settings: smooth maps between equidimensional manifolds, CR-singularitie…
Study geometric equations on cohomogeneity one manifolds near singular orbits.
problem Solving geometric equations like Ricci, Einstein, and soliton near singular orbits.
method Special assumption simplifies proof; general case solved in Part II.
result Existence and uniqueness of solutions near singular orbits.
The study reveals chaos in geometric objects embedded in higher dimensions.
problem Understanding chaos in higher-dimensional geometries.
method Analyzing the embedding of chaos in geometric objects of varying dimensions.
result Chaos in higher dimensions is a one-dimensional geometrical object embedded in a higher-dimensional object.
Criteria for sharksfin and deltoid singularities from plane to plane, with applications.
problem Identifying and understanding singularities in plane-to-plane mappings.
method Providing criteria and geometric meanings for singularities.
result Geometric meanings and criteria for sharksfin and deltoid singularities.
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 give criteria for which a principal curvature becomes a bounded C∞-function at non-degenerate singular points of wave fronts by using geometric invariants. As applications, we study singularities of parallel surfaces and extended distance squared functions of wave fronts. Moreover, we relate these singularit…
Constructs singular Yamabe solutions via equivariant reduction.
problem Constructs non-trivial geometric examples for the Yamabe equation.
method Reduces the problem to an equivariant setting for simpler analysis.
result Provides a non-trivial weak solution to the Yamabe problem.
The paper extends a geometric model using singular curves.
problem Understanding abnormal extremals in sub-Riemannian geometry.
method Analysis of singular curves and construction of a graded Lie algebra.
result A nilpotent graded Lie algebra is constructed isomorphic to F4. New method identifies vanishing arcs for curve singularities.
problem Characterizing arcs sent to geometric vanishing cycles.
method Introducing geometric variation operator and vanishing arcsets.
result Existence of topological exceptional collections of arcsets.
Positive braids linked to knot invariants and geometric monodromy groups.
problem Understanding knot invariants and geometric monodromy groups for positive braids.
method Associate braid monodromy groups to positive braids, identify these groups with framed mapping class groups for knots, and use these to determine knot invariants.
result Geometric monodromy groups of irreducible singularities are determined by genus and Arf invariant of associated knots.
Study geodesic paths on flat surfaces, comparing length and singularity counts.
problem Comparing geometric length and singularity counts on geodesic paths.
method Apply counting limit laws to infinite graphs and then to flat surfaces.
result Statistical comparison of geometric length and singularity counts on geodesic paths.
In this paper, we prove that any two birational projective varieties with finite quotient singularities can be realized as two geometric GIT quotients of a non-singular projective variety by a reductive algebraic group. Then, by applying the theory of Variation of Geometric Invariant Theory Quotients ([3]), we show tha…
In this paper, we investigate the geometric propagation and diffraction of singularities of solutions to the wave equation on manifolds with edge singularities.
Develops Chern-Weil theory for singular foliations.
problem Chern-Weil theory for Haefliger-singular foliations.
method Constructs explicit forms representing characteristic classes in de Rham cohomology.
result Theory applies to general smooth Haefliger structures up to homotopy.
Logic approach finds real singularities in differential equations.
problem Finding geometric singularities of implicit ODEs over the reals.
method Vessiot theory, parametric Gaussian elimination, heuristic simplification, real quantifier elimination.
result Effective computation of geometric singularities using logic methods.
Any ruled surface in Euclidean 3-space is described as a curve of unit dual vectors in the algebra of dual quaternions (=the even Clifford algebra of type (0,3,1)). Combining this classical framework and Singularity Theory, we characterize local diffeomorphic types of singular ruled surfaces in terms of geometric invar…
We study the local symplectic algebra of curves. We use the method of algebraic restrictions to classify symplectic T7 singularities. We define discrete symplectic invariants - the Lagrangian tangency orders. We use these invariants to distinguish symplectic singularities of classical A−D−E singularities of planar…
We study surfaces of constant positive Gauss curvature in Euclidean 3-space via the harmonicity of the Gauss map. Using the loop group representation, we solve the regular and the singular geometric Cauchy problems for these surfaces, and use these solutions to compute several new examples. We give the criteria on the …
In this paper we study a special case of the completion of cusp Kähler-Einstein metric on the regular part of varieties by taking the continuity method proposed by La Nave and Tian. The differential geometric and algebro-geometric properties of the noncollapsing limit in the continuity method with cusp singularities wi…
Smooth bundles with rough data maintain Hodge kernel isomorphism.
problem Maintaining Hodge kernel isomorphism for smooth bundles with non-smooth geometric data.
method Analyzing nilpotent differential operators and Hodge-Dirac-type operators under perturbations of geometric data.
result Kernels of Hodge-Dirac operators remain isomorphic under uniform perturbations of geometric data.
We show that a rescale limit at any degenerate singularity of Ricci flow in dimension 3 is a steady gradient soliton. In particular, we give a geometric description of type I and type II singularities.
Study shows continuity and geometric regularity of Kähler-Ricci flow blow-up limits.
problem Geometric regularity of blow-up limits of the Kähler-Ricci flow.
method Established geometric regularity for Type I blow-up limits based on sequences of Ricci vertices.
result The limiting flow is continuous in time in Gromov-Hausdorff and Gromov-W1 distance. Unified view on big bang singularities from initial data.
problem Understanding quiescent big bang singularities from initial data.
method Unified geometric perspective on various results.
result Solutions of Oude Groeniger et al. induce data on the singularity.
In this present paper, we study geometric structures of rank two prolongations of implicit second-order partial differential equations (PDEs) for two independent and one dependent variables and characterize the type of these PDEs by the topology of fibers of the rank two prolongations. Moreover, by using properties of …
We study 3-manifolds in R5 with corank 1 singularities. At the singular point we define the curvature locus using the first and second fundamental forms, which contains all the local second order geometrical information about the manifold.
Constructs a Lie groupoid integrating singular foliations.
problem Integrating singular foliations into higher Lie groupoids.
method Recursive use of bi-submersions and geometric resolutions.
result Finite-dimensional Lie groupoid integrating singular foliations.
We give a topological and geometrical description of focus-focus singularities of integrable Hamiltonian systems. In particular, we explain why the monodromy around these singularities is non-trivial, a result obtained before by J.J. Duistermaat and others for some concrete systems.
Study of holomorphic curves and surfaces using singularity theory.
problem Understanding the geometry of holomorphic curves and complex surfaces.
method Application of singularity theory to holomorphic curves and surfaces.
result Definition of geometric invariants for curves and surfaces.
We prove that monotonicity of density and energy inequality imply the rectifiability of the singular sets for Yang-Mills flow.
Global singularities propagate in magnetic mechanical systems on Riemannian manifolds.
problem Propagation of singularities in magnetic mechanical systems.
method Combines reduction from magnetic to Riemannian systems, analysis of reparameterized flows, and regularization techniques.
result Invariant singular set under generalized gradient flow dynamics.
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.