The paper defines and studies d-minimal surfaces in a degenerate semi-Euclidean space.
problem Investigating surfaces in a degenerate semi-Euclidean space with a degenerate metric.
method Defining d-minimal surfaces and proving a representation formula of Weierstrass type. result The correspondence between d-minimal surfaces in R0,2,1 and spacelike flat zero mean curvature surfaces in R14. We define the notions of St1×Ss1-valued lightcone Gauss maps, lightcone pedal surface and Lorentzian lightcone height function of Lorentzian surface in semi-Euclidean 4-space and established the relationships between singularities of these objects and geometric invariants of the surface as applications of s…
Discussing rigidity in codimension 2, extending rigidity concepts.
problem Local isometric rigidity problem in codimension 2.
method Extending rigidity concepts to include genuine and honest rigidity, studying isometric immersions in semi-Euclidean spaces.
result Necessity of natural singularity in inner product for transitivity.
In this study, we try to semi-real quaternionic curves in the semi-Euclidean space E_2^4. Firstly, we introduce algebraic properties of semi-real quaternions. And then, we give some characterizations of semi-real quaternionic involute-evolute curves in the semi-Euclidean space E_2^4. Lastly, we illustrate some examples…
Study on rotating surfaces in 4D space with matrices.
problem Understanding rotational surfaces in pseudo-Euclidean 4-space.
method Defined hyperbolic and elliptic rotational surfaces using curves and matrices in 4D semi-Euclidean space.
result Generated rotated surfaces using specific rotation matrices.
The paper explores null hypersurfaces in Lorentzian manifolds using geometric immersions.
problem Understanding the geometry of null hypersurfaces in Lorentzian manifolds.
method The approach involves isometric immersions of leafs of the screen distribution into semi-Euclidean spheres or hyperbolic spaces.
result Null hypersurfaces are shown to be umbilic and screen totally umbilic under certain geometric conditions.
In this note, we extend the notion of a Monge hypersurface from its roots in semi-Euclidean space to more general spaces. For the degenerate case, the geometry of these structures is studied using the Bejancu-Duggal method of screen distributions.
In this paper, we construct smooth isometric embeddings of multiple warped product manifolds in quadrics of semi-Euclidean spaces. Our main theorem generalizes previous results as given by Blanusa, Rozendorn, Henke and Azov.
The paper applies Clairaut's theorem to rotational surfaces in pseudo-Euclidean 4-space.
problem Exploring geodesic curves on rotational surfaces in pseudo-Euclidean 4-space.
method Expressing Clairaut's theorem and deriving equations for geodesic curves.
result Characterization of geodesic curves on hyperbolic and elliptic surfaces of rotation.
The notion of rectifying curve in the Euclidean space is introduced by Chen as a curve whose position vector always lies in its rectifying plane spanned by the tangent and the binormal vector field t and n_2 of the curve. In this study, we have obtained some characterizations of semi-real spatial quaternionic rectifyin…
In this paper we study two types of fibrations associated with a 3-dimensional unital associative irreducible algebra and their basic properties. We investigate trivial principal fibrations of degenerate semi-Euclidean sphere and their semi-conformal and projective models. We use Norden normalization method for constru…
In this paper, it is proved that, no special timelike Frenet curve is a Bertrand curve in E24 and also, in Eνn+1 (n≥3), such that the notion of Bertrand curve is definite only in E12 and E13. Therefore, a generalization of timelike Bertrand curve is defined…
Study gradient Yamabe solitons on warped product manifolds.
problem Characterize and provide examples of gradient Yamabe solitons on warped product manifolds.
method Prove triviality results and provide nontrivial examples using a conformal base and translation group action.
result Provide infinitely many explicit examples of complete steady gradient Yamabe solitons.
The first goal of this note is to study the Almansi property on an m-dimensional model in the sense of Greene and Wu and, more generally, in a Riemannian geometric setting. In particular, we shall prove that the only model on which the Almansi property is verified is the Euclidean space R^m. In the second part of the p…
The paper studies gradient Ricci-Harmonic solitons on warped product manifolds.
problem Characterizing gradient Ricci-Harmonic solitons on warped product manifolds.
method Warped product structure, potential function, warping function, harmonic map analysis.
result Nontrivial examples of warped product gradient Ricci-harmonic solitons are provided.
In this paper we investigate the problem of non-analytic embeddings of Lorentzian manifolds in Ricci-flat semi-Riemannian spaces. In order to do this, we first review some relevant results in the area, and then motivate both the mathematical and physical interest in this problem. We show that any n-dimensional compac…
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.
The paper proves weak continuity of Cartan structural system on semi-Riemannian manifolds with lower regularity.
problem Weak continuity of the Cartan structural system on semi-Riemannian manifolds with lower regularity.
method Formulated and proved a geometric compensated compactness theorem, deduced Lp weak continuity of the Cartan structural system. result Weak continuity of the Cartan structural system and Gauss-Codazzi-Ricci system on semi-Riemannian manifolds with lower regularity.
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.
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.
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…
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.
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.
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 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.
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.
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.
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 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 \(N-5\).
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 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.
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.