arXiv research

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

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

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

Theorems on the existence of vector fields with given sets of Indexes of isolated Singular points are proved for the cases of closed manifolds, pairs of manifolds, manifolds with boundary, and gradient fields. It is proved that, on a two-dimensional manifold, an index of an isolated Singular point of the gradient field…

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

For a G-invariant holomorphic 1-form with an isolated singular point on a germ of a complex-analytic G-variety with an isolated singular point (G is a finite group) one has notions of the equivariant homological index and of the (reduced) equivariant radial index as elements of the ring of complex representations of th…

Given a compact Hermitian complex space with isolated singular points, we construct a Dolbeault-type Hilbert complex whose cohomology is isomorphic to the cohomology of the structure sheaf. We show that the corresponding K-homology class coincides with the one constructed by Baum-Fulton-MacPherson.

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

Let $(M,g_0)$ be a compact $n$-dimensional Riemannian manifold with a finite number of singular points, where the metric is asymptotic to a non-negatively curved cone over $(\mathbb{S}^{n-1},g)$. We show that there exists a smooth Ricci flow starting from such a metric with curvature decaying like C/t. The initial metr…

We study isolated singularities of two dimensional Yang-Mills-Higgs fields defined on a fiber bundle, where the fiber space is a compact Riemannian manifold and the structure group is a compact connected Lie group. In general the singularity can not be removed due to possibly non-vanishing limit holonomy around the sin…

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 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…

Degenerations of rank-two bundles on threefolds lead to isolated point singularities, with rigidity and bubbling properties.

We consider harmonic maps into pseudo-Riemannian manifolds. We show the removability of isolated singularities for continuous maps, i.e. that any continuous map from an open subset of R^m into a pseudo-Riemannian manifold which is two times continuously differentiable and harmonic everywhere outside an isolated point i…

We discuss the bi-Lipschitz geometry of an isolated singular point of a complex surface which particular emphasis on when it is metrically conical.

In this paper, we study existence, regularity, classification, and asymptotical behaviors of solutions of some Monge-Ampère equations with isolated and line singularities. We classify all solutions of $\det \nabla^2 u=1$ in $\R^n$ with one puncture point. This can be applied to characterize ellipsoids, in the same spir…

We characterize those closed $2k$-manifolds admitting smooth maps into $(k+1)$-manifolds with only finitely many critical points, for $k\in\{2,4\}$. We compute then the minimal number of critical points of such smooth maps for $k=2$ and, under some fundamental group restrictions, also for $k=4$. The main ingredients ar…

We survey what is known about singularities of special Lagrangian submanifolds (SL m-folds) in (almost) Calabi-Yau manifolds. The bulk of the paper summarizes the author's five papers math.DG/0211294, math.DG/0211295, math.DG/0302355, math.DG/0302356, math.DG/0303272 on SL m-folds X with isolated conical singularities.…

We study positive solutions of the Yamabe equation with isolated singularity and prove the existence of solutions with prescribed asymptotic expansions near singular points and an arbitrarily high order of approximation.

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 …

Study of singular metrics with negative scalar curvature on compact manifolds.

In this paper we study the short time existence problem for the (generalized) Lagrangian mean curvature flow in (almost) Calabi--Yau manifolds when the initial Lagrangian submanifold has isolated conical singularities modelled on stable special Lagrangian cones. Given a Lagrangian submanifold $F_0:L\rightarrow M$ in an…

For two complex vector bundles admitting a homomorphism between them, a Poincaré-Hopf formula for the difference of the Chern character numbers of these two vector bundles with isolated singularities is established by Huitao Feng, Weiping Li and Weiping Zhang. This article extend their reslut about Poincaré-Hopf type f…

Smooth approximations near singularities of constant mean curvature surfaces are found.

First we review the definition of a negative point mass singularity. Then we examine the gravitational lensing effects of these singularities in isolation and with shear and convergence from continuous matter. We review the Inverse Mean Curvature Flow and use this flow to prove some new results about the mass of a sing…

Proves smoothness of conical singularities in mean curvature flow.

Given $(M,g_0)$ a closed Riemannian manifold and a nonempty closed subset $X$ in $M$, the singular $σ_k-$Yamabe problem asks for a complete metric $g$ on $M\backslash X$ conformal to $g_0$ with constant $σ_k-$curvature. The $σ_k-$curvature is defined as the $k-$th elementary symmetric function of the eigenvalues of the…

Constructs minimal immersions with singularities.

The paper calculates critical configurations and Morse indices for polygons on circles or ellipses.

The study confirms a conjecture about critical points of smooth functions.

We study asymptotic behaviors of positive solutions to the Yamabe equation and the $σ$k-Yamabe equation near isolated singular points and establish expansions up to arbitrary orders. Such results generalize an earlier pioneering work by Caffarelli, Gidas, and Spruck, and a work by Korevaar, Mazzeo, Pacard, and Schoen, …

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

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.

The paper proves local isometric embeddings for singular metrics near a point.

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.

We show that every closed oriented smooth 4-manifold admits a complete singular Poisson structure in each homotopy class of maps to the 2-sphere. The rank of this structure is 2 outside a small singularity set, which consists of finitely many circles and isolated points. The Poisson bivector has rank 0 on the singulari…

Uniqueness proven for stable hypersurface tangent cones.

A maximal surface $\sb$ with isolated singularities in a complete flat Lorentzian 3-manifold $\N$ is said to be entire if it lifts to a (periodic) entire multigraph $\tilde{\sb}$ in $ł^3.$ In addition, $\sb$ is called of finite type if it has finite topology, finitely many singular points and $\tilde{\sb}$ is finitely …

Study shows persistence of singularities in minimal hypersurfaces is rare.

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

Paper studies metrics with constant Q-curvature near singular points.

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

Characterizes W-congruences to study their stable umbilical points.

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.