Stability of Wasserstein spaces under various convergence types.
problem Stability and finiteness of Wasserstein spaces over singular and non-singular spaces.
method Gromov--Hausdorff convergence and equivariant Gromov--Hausdorff convergence.
result Analogue of Perelman's stability theorem on Wasserstein spaces.
Survey on Ricci flow on spaces with conical singularities.
problem Analyzing Ricci flow on spaces with isolated conical singularities.
method Ricci de Turck flow preserving conical singularities, stability of Ricci flat metrics, preservation of positive scalar curvature under certain conditions.
result Ricci flat metrics with isolated conical singularities are stable and positive scalar curvature is preserved under the flow.
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 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.
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. Study of symplectic manifolds degenerating into singular spaces.
problem Understanding degenerations of symplectic manifolds into singular spaces.
method Topological framework focusing on Lagrangian manifolds and holomorphic membranes.
result Degenerations into singular toric varieties yield exotic Lagrangian tori.
The study calculates best Sobolev constants with sharp Hardy terms in Euclidean and hyperbolic spaces.
problem Computing best Sobolev constants with sharp Hardy terms in different environments.
method Analyzes constants in Euclidean and hyperbolic spaces with interior and boundary point singularities.
result Computed best Sobolev constants for Hardy-Sobolev inequalities with sharp terms.
Study classifies non-removable singularities in Minkowski 3-space equations.
problem Classifying non-removable singularities in Minkowski 3-space equations.
method Classification through real analytic solutions of the prescribed mean curvature equation.
result Classification of non-removable isolated singularities.
Two constructions of classifying spaces for singular maps are connected.
problem Understanding the classifying spaces of singular maps.
method Homotopy theoretical connection between two constructions.
result Some classifying spaces have a simple product structure.
Positive mass theorem for singular hyperbolic spaces proved.
problem Positive mass theorem for singular asymptotically hyperbolic manifolds.
method Careful analysis of the Yamabe equation.
result Positive mass theorem established for singular asymptotically hyperbolic manifolds.
We prove a conjecture due to M. Kazarian, connecting two classifying spaces in singularity theory. These spaces are: - Kazarian's space (generalizing Vassiliev's algebraic complex and) showing which cohomology classes are represented by singularity strata. - Author's space Xτ giving homotopy representation of cobord…
Extends space theory to singular spaces.
problem Analyzing Ricci flows of bounded curvature.
method Generalizes Cheeger, Colding, and Naber's theory.
result Theory applies to singular space limits of manifolds.
New findings on Kähler-Einstein currents dominating Kähler forms on compact spaces.
problem Understanding Kähler-Einstein currents on singular spaces.
method Analyzing currents' properties and applying them to compact spaces with log terminal singularities.
result Kähler-Einstein currents dominate Kähler forms on compact spaces with log terminal singularities.
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.
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.
Compact RCD spaces derived from singular Kahler metrics on 3D projective varieties.
problem Understanding geometric structures of singular Kahler spaces.
method Proving RCD spaces homeomorphic to 3D projective varieties with bounded Nash entropy and Ricci curvature.
result Compact RCD spaces are equivalent to underlying projective varieties.
Study lightlike hypersurfaces in anti-de Sitter space.
problem Characterize singularities of lightlike hypersurfaces.
method Investigate lightlike hypersurfaces along spacelike submanifolds in anti-de Sitter space.
result Characterize singularities of lightlike hypersurfaces.
Constructs graphs with singularities in a special space.
problem Creating graphs with specific singularities in a unique space.
method Using Weierstrass representation for minimal surfaces.
result Constructs entire singly periodic graphs with isolated cone-like singularities.
Analyzes singularities of convex hypersurfaces in hyperbolic space.
problem Understanding singularities of convex hypersurfaces with constant curvature.
method Analyzes the structure of singular sets using convex curvature functions.
result Describes the structure of singular sets in hyperbolic space.
The study examines conditions for scalar curvature and Dirac operators on singular spaces.
problem Existence of scalar curvature measures and Dirac operators on singular spaces.
method Investigation of smooth manifolds with singular Riemannian metrics.
result Sufficient conditions for the existence of scalar curvature measures and Dirac operators.
Complex surface singularities linked to L-spaces and left-orderable groups.
problem Characterizing links of complex surface singularities.
method Using recent results on L-spaces and left-orderable groups.
result Rational singularities have rational link, non-rational links have left-orderable fundamental groups.
Establishes Hermite-Einstein metrics on complex spaces with singularities.
problem Existence of Hermite-Einstein metrics on complex spaces with singularities.
method Established existence of estimable Hermite-Einstein metrics for stable reflexive coherent sheaves on compact normal Kähler spaces with klt singularities.
result Obtained precise results for varieties with klt singularities.
The paper defines function spaces on manifolds with bounded or singular geometries.
problem Defining function spaces on manifolds with various geometries.
method Introduces and analyzes Sobolev, Besov, and Bessel potential spaces on uniformly regular and singular Riemannian manifolds.
result Demonstrates maximal regularity for a linear parabolic problem on singular manifolds.
The paper classifies singularities of line congruences in 4D space.
problem Classifying singularities of line congruences in 4D space.
method Generic classification approach for 3-parameter line congruences and Blaschke normal congruences.
result Generic classification of singularities of 3-parameter line congruences in R4. Study flat spacetimes with BTZ singularities using BTZ-extension.
problem Understanding singularities in flat spacetimes.
method Use BTZ-extension to describe Moduli spaces and construct cauchy-surface.
result Construct convex polyhedral cauchy-surface in Cauchy-compact flat spacetimes with BTZ.
Study two types of singular Kähler-Einstein metrics on complex varieties.
problem Understanding different types of singular Kähler-Einstein metrics.
method Analyzing Ricci flat Kähler cone metrics and applying to more general spaces.
result A weaker notion of singular Kähler-Einstein metrics is equivalent to a stronger one under certain conditions.
Study special Lagrangian submanifolds with edge singularities using elliptic theory.
problem Characterize the moduli space of deformations of special Lagrangian submanifolds with edge singularities.
method Use elliptic theory for edge-degenerate differential operators on singular manifolds.
result Obtain a general theorem describing the local structure of the moduli space.
New theorem shows no complete flat fronts with singularities in higher dimensions.
problem Existence of complete flat fronts with singularities in higher-dimensional Euclidean spaces.
method Used a representation formula and theorem from Murata and Umehara.
result Proved the non-existence of complete flat fronts with non-empty singular set in Euclidean (n+1)-space for n ≥ 3.
Continuity of Kähler-Einstein potentials at singularities proven.
problem Regularity of solutions to degenerate complex Monge-Ampère equations on singular spaces.
method Investigation of Dirichlet problem and global continuity of solutions.
result Kähler-Einstein potentials are continuous at isolated singularities.
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.
Two singular links are cobordant if one can be obtained from the other by singular link isotopy together with a combination of births or deaths of simple unknotted curves, and saddle point transformations. A movie description of a singular link cobordism in 4-space is a sequence of singular link diagrams obtained from …
Study on surfaces with unbounded mean curvature and singularities.
problem Understanding surfaces with unbounded mean curvature and singularities.
method Analyzing surfaces given by Kenmotsu-type formula.
result Characterization of singularities in surfaces with unbounded mean curvature.
Study Kähler-Einstein metrics on singular varieties, proving metric completion properties.
problem Kähler-Einstein metrics on singular projective varieties.
method Approximation with constant scalar curvature metrics, RCD space analysis.
result Metric completion of smooth part is non-collapsed RCD space and homeomorphic to original variety.
Perimeter minimizers in curved spaces have a singular set no more than 5 dimensions.
problem Understanding the structure of minimizers in spaces with bounded Ricci curvature.
method Analysis of non-collapsed Ricci limit spaces with two-sided curvature bounds.
result The Hausdorff dimension of the singular set is at most \(N-5\).
Study on monopoles with Dirac type singularities in 3D space.
problem Characterizing singularities of monopoles.
method Characterizations via growth orders of norms of sections and Higgs fields.
result Two characterizations of Dirac type singularities of monopoles.
We study the topological and differentiable singularities of the configuration space C(Γ) of a mechanical linkage Γin d-dimensional Euclidean space, defining an inductive sufficient condition to determine when a configuration is singular. We show that this condition holds for generic singularities, provide a mechanical…
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 on stable maps from 3-sphere to 3-space, focusing on singularities and invariants.
problem Understanding the behavior of stable maps from S3 to R3. method Analysis of codimension-one transitions, singular set behavior, and global invariants.
result Effects of decompositions on global invariants with prescribed branch sets.
Classifies tangent cones of Kähler metrics with cone singularities.
problem Understanding the behavior of Kähler metrics near singular points.
method Constructs and classifies tangent cones for non-collapsed sequences of Kähler-Einstein metrics.
result Classifies possible tangent cones in two complex dimensions.
Study on moduli spaces of sextic curves with simple singularities and their compactifications.
problem Understanding moduli spaces of sextic curves with simple singularities.
method Using period maps of K3 surfaces with ADE singularities, algebraic open embeddings into arithmetic quotients of type IV domains, and GIT and Looijenga compactifications.
result Identifications of GIT and Looijenga compactifications for all cases.
LCK metrics extended to spaces with quotient singularities, preserving key properties.
problem Extending LCK metrics to complex spaces with singularities.
method Generalization to complex analytic spaces with quotient singularities, proving conditions for existence.
result Spaces with quotient singularities admit LCK metrics if their universal cover has a specific Kaehler metric.
Study curves in a spacelike hypersurface using geometric and singularity theory.
problem Investigate the geometry of curves in spacelike hypersurfaces.
method Use techniques of the theory of singularities to describe the shapes and singularities of hyperbolic and de Sitter surfaces.
result Contribute to the study of extrinsic geometry of curves in different ambient spaces.
This paper studies Gorenstein singularities and their applications in moduli spaces of holomorphic differentials.
problem Understanding Gorenstein singularities and their moduli spaces.
method Construction of Gorenstein curve singularities via test configurations and miniversal deformation spaces.
result Classification of Gorenstein singularities and compactification of nonvarying strata.
Tian's theorem applies to Moishezon spaces with singular metrics.
problem Distribution of currents on Moishezon spaces with singular metrics.
method Proving asymptotic distribution of Fubini-Study currents.
result Curvature currents of metrics on singular Hermitian line bundles.
Let X be the moduli space of SL(n,C), SU(n), GL(n,C), or U(n)-valued representations of a rank r free group. We classify the algebraic singular stratification of X. This comes down to showing that the singular locus corresponds exactly to reducible representations if there exist singularities at all. Then by relating a…
Investigates singular Finsler foliations on (α,β)-spaces and their relation to Riemannian foliations.
problem Understanding conditions for singular Finsler foliations to be singular Riemannian foliations.
method Analyzes (α,β)-spaces and verifies conditions for SFFs to be SRFs, extending Molino's conjecture. result Equifocality of regular leaves for SFFs under certain conditions.
New cochain algebra for diffeological spaces connects de Rham and singular cohomologies.
problem Incompatibility of de Rham and singular cohomologies in diffeology.
method Introduces a new singular de Rham complex and proves it quasi-isomorphic to the original de Rham complex for manifolds and spaces with singularities.
result The new cochain complex resolves the incompatibility issue in diffeology.
Formula for foliations' singularities in complex projective spaces.
problem Counting singularities of foliations on complex projective spaces.
method Global residue formula for logarithmic indices of foliations with isolated singularities.
result Formula for the number of singularities in the complement of the invariant divisor on complex projective spaces.