arXiv research

Paper describes links of mixed polynomials with specific properties.

All link types arise from semiholomorphic polynomials.

We prove that any weakly triholomorphic map from a compact hyperkähler surface to an algebraic K3 surface defined by a homogeneous polynomial of degree 4 in $\mathbb{C}P^3$ has only isolated singularities.

New method detects essential tori in mixed singularity links.

Study on singularities of specific polynomial functions.

Research on mixed polynomials, extending non-degeneracy concepts to complex variables.

Defines Perelman's functionals on manifolds with non-isolated conical singularities.

Study of $G_2$-structures with isolated singularities and bounded torsion.

Proves conditions for positive scalar curvature on certain manifolds with conical singularities.

The Poincaré-Hopf theorem is extended to projective varieties with isolated singularities.

Let $M$ be a smooth manifold and let $\F$ be a codimension one, $C^\infty$ foliation on $M$, with isolated singularities of Morse type. The study and classification of pairs $(M,\F)$ is a challenging (and difficult) problem. In this setting, a classical result due to Reeb \cite{Reeb} states that a manifold admitting a …

The Poincaré-Hopf theorem is extended to projective varieties with isolated singularities.

We study the Euler obstruction of essentially isolated determinantal singularities (EIDS). The EIDS were defined by W. Ebeling and S. Gusein-Zade, as a generalization of isolated singularity. We obtain some formulas to calculate the Euler obstruction for the determinantal varieties with singular set an ICIS.

Study reveals GAGA phenomenon in Poisson cohomology for plane structures with isolated singularities.

Robert Bryant (Theorie des varietes minimales et applications, 1988, 154: 321-347) proved that an isolated singularity of a conformal metric of positive constant curvature on a Riemann surface is a conical one. Using Complex Analysis, we find all of the local models for an isolated singularity of a flat metric whose ar…

We investigate the face numbers of simplicial complexes with Buchsbaum vertex links, especially pseudomanifolds with isolated singularities. This includes deriving Dehn-Sommerville relations for pseudomanifolds with isolated singularities and establishing lower bound theorems when the singularities are also homological…

Constructs Kahler-Einstein metrics near isolated log canonical singularities.

Study on solutions near isolated singularities in 6D Yamabe equation.

Generic singularities found in spacetimes with weakly trapped submanifolds.

Formula calculates homology groups of Milnor fibres for real hypersurface singularities.

Wavelet analysis reveals limitations in detecting multifractality in signals with isolated singularities.

In the present paper, we deform isolated singularities of a certain class of polar weighted homogeneous mixed polynomials, and show that there exists a deformation which has only definite fold singularities and mixed Morse singularities.

We give an explicit construction of complex maps whose nodal line have the form of lemniscate knots. We review the properties of lemniscate knots, defined as closures of braids where all strands follow the same transverse (1, $\ell$) Lissajous figure, and are therefore a subfamily of spiral knots generalising the torus…

The paper extends deformation theory to Calabi-Yau varieties with isolated log canonical singularities.

The paper builds complex hyperbolic 2-manifolds with isolated singularities.

New families of weakly symmetric nilmanifolds discovered.

Proves heat expansion for Laplacian on a singularity.

We give a simple direct proof of uniqueness of tangent cones for singular projectively Hermitian Yang-Mills connections on reflexive sheaves at isolated singularities modelled on $μ$-polystable holomorphic bundles over $\mathbf{P}^{n-1}$.

We study the local symplectic algebra of the 0-dimensional isolated complete intersection singularities. We use the method of algebraic restrictions to classify these symplectic singularities. We show that there are non-trivial symplectic invariants in this classification.

We prove that there are just two types of isolated singularities of special Kähler metrics in real dimension two provided the associated holomorphic cubic form does not have essential singularities. We also construct examples of such metrics.

For isolated complex hypersurface singularities with real defining equation we show the existence of a monodromy vector field such that complex conjugation intertwines the local monodromy diffeomorphism with its inverse. In particular, it follows that the geometric monodromy is the composition of the involution induced…

We give a classification of non-removable isolated singularities for real analytic solutions of the prescribed mean curvature equation in Minkowski $3$-space.

Constructs graphs with singularities in a special space.

Given a real analytic function $f$ from $\mathbb{R}^4$ to $\mathbb{R}^2$ with isolated critical point at the origin, the link $L_f$ of the singularity is a real fibred knot in $\mathbb{S}^{3}$. From this singularities, we construct a family of real isolated suspension singularities from $\mathbb{R}^6$ to $\mathbb{R}^2$…

We show that in cohomogeneity 3 there are G-manifolds with any given number of isolated singular orbits and an invariant metric of positive Ricci curvature. We show that the corresponding result is also true in cohomogeneity 5 provided the number of singular orbits is even.

In this paper, we develop the theory of Perelman's $W$-functional on manifolds with isolated conical singularities. In particular, we show that the infimum of $W$-functional over a certain weighted Sobolev space on manifolds with isolated conical singularities is finite, and the minimizer exists, if the scalar curvatur…

Khimshiashvili proved a topological degree formula for the Eu-ler characteristic of the Milnor fibres of a real function-germ with an isolated singularity. We give two generalizations of this result for non-isolated singularities. As corollaries we obtain an algebraic formula for the Euler characteristic of the fibres …

Analytic lattice cohomology defined for isolated singularities, linking to Heegaard Floer cohomology.

Study Euler obstruction of 1-forms on determinantal singularities.

We study graphs of positive extrinsic curvature with a non-removable isolated singularity in 3-dimensional warped product spaces, and describe their behavior at the singularity in several natural situations. We use Monge-Ampère equations to give a classification of the surfaces in 3-dimensional space forms which are em…

Study differentiable maps on hypersurface links, finding fold maps with circle singular value sets.

We extend the results of Hardt and Simon on area-minimizing cones to prove that isolated singularities of stationary one-sided area-minimizing hypersurfaces can be locally perturbed away on the side that they are minimizing.

We construct local models of isolated singularities for special Kähler structures in real dimension two assuming that the associated holomorphic cubic form does not have essential singularities. As an application we compute the holonomy of the flat symplectic connection, which is a part of the special Kähler structure.

We study solutions to conformally invariant equations with isolated singularties.

Mean curvature flow shows singularities on smooth surfaces.

The study reveals chaos in geometric objects embedded in higher dimensions.

Study rigidity in Penrose's singularity theorem with weakly trapped surfaces.

We establish a correspondence on a Riemann surface between hyperbolic metrics with isolated singularities and bounded projective functions whose Schwarzian derivatives have at most double poles and whose monodromies lie in ${\rm PSU}(1,\,1)$. As an application, we construct explicitly a new class of hyperbolic metrics …