arXiv research

Study localizes integrals at isolated degenerate zeros.

Analyzes Kähler blowups with isolated zeros using localization.

The paper proves uniform stable radius and Milnor number equality for specific mappings.

We generalize the Novikov inequalities for 1-forms in two different directions: first, we allow non-isolated critical points (assuming that they are non-degenerate in the sense of R.Bott), and, secondly, we strengthen the inequalities by means of twisting by an arbitrary flat bundle. The proof uses Bismut's modificatio…

Paper describes links of mixed polynomials with specific properties.

Study on singularities of specific polynomial functions.

We define two types of local indices of a vector field at an isolated zero on the boundary, and prove Poincare-Hopf-type index theorems for certain vector fields on a compact smooth manifold which have only isolated zeros.

Homotopy method removes isolated zeroes in continuous maps.

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

Polyhomogeneous expansions for Calabi-Yau metrics near singularities.

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

Formula for sections on complex manifolds with non-isolated components.

Unified interpretation of sub-Riemannian Gauss-Bonnet theorem for surfaces in 3D contact manifolds.

Proves certain Calabi-Yau varieties are projective.

Develops a formalism for studying general horizons and derives a near-horizon equation.

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

In this work we prove a residue formula for Morita-Futaki-Bott invariant with respect any holomorphic vector fields with isolated (possibly degenerated) singularities in terms of Grothendieck's residues.

In this article, we derive a topological obstruction to the removal of a isolated degenerate complex tangent to an embedding of a 3-manifold into $\mathbb{C}^3$ (without affecting the structure of the remaining complex tangents). We demonstrate how the vanishing of this obstruction is both a necessary and sufficient co…

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.

Two non-Morse-Bott Chern-Simons functions on homology 3-spheres.

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

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

The study shows that nonpositively curved 4-manifolds with zero Euler characteristic have degenerating Ricci curvature.

Integrability of mean curvature near degenerate points in Heisenberg group.

We establish sufficient conditions for existence of curves minimizing length as measured with respect to a degenerate metric on the plane while enclosing a specified amount of Euclidean area. Non-existence of minimizers can occur and examples are provided. This continues the investigation begun in [ABCDS] where the met…

We study critical points of holomorphic sections of $\ocal(m)$ on $\CP^n$. For quadrics, we give a complete discription of their critical points. When $n=1$, we prove a spherical Gauss-Lucas theorem. For general situation, we prove that a general section has all its critical points isolated and non-degenerate.

Improves Bernstein theorem for zero mean curvature hypersurfaces in Lorentz-Minkowski space.

Study describes how compact Ricci solitons degenerate as cone angles approach zero.

Sharp eigenvalue estimates on degenerating hyperbolic surfaces.

The abstract extends a Poincaré-Hopf formula to non-isolated singularities.

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…

Consider degenerations of Abelian differentials with prescribed number and multiplicity of zeros and poles. Motivated by the theory of limit linear series, we define twisted canonical divisors on pointed nodal curves to study degenerate differentials, give dimension bounds for their moduli spaces, and establish smootha…

The paper discusses polynomial convergence to conical Kähler-Einstein metrics.

In this paper we define a new type of 2-degenerate Cartan curves in Minkowski spacetime $R_{1}^{4}$. We prove that this type of curves contain only the polynomial functions as its components whose third derivative vanish completely. No curve with acceleration zero in $R_{1}^{4}$ is a 2-degenerate Cartan curve, therefor…

A self-dual harmonic 2-form on a 4-dimensional Riemannian manifold is symplectic where it does not vanish. Furthermore, away from the form's zero set, the metric with the 2-form give a compatible almost complex structure and thus pseudo-holomorphic subvarieties. Such a subvariety is said to have finite energy when the …

The paper studies umbilics on surfaces in Lorentz-Minkowski space.

For a conformal vector field $ξ$ on a Riemannian manifold, we say that a point is essential if there is no local metric in the conformal class for which $ξ$ is Killing. We show that the only essential points are isolated zeros of $ξ$. As an application, we show that every connected component of the zero set of $ξ$ is t…

The moduli space $\mathcal{M}_{g}$ of compact Riemann surfaces of genus $g$ has orbifold structure, and the set of singular points of such orbifold is the \textit{branch locus} $\mathcal{B}_{g}$. Given a prime number $p \ge 7$, $\mathcal{B}_{g}$ contains isolated strata consisting of $p$-gonal Riemann surfaces for gene…

We describe the local conformal geometry of a Lorentzian spin manifold $(M,g)$ admitting a twistor spinor $φ$ with zero. Moreover, we describe the shape of the zero set of $φ$. If $φ$ has isolated zeros then the metric $g$ is locally conformally equivalent to a static monopole. In the other case the zero set consists o…

We study the geodesic flow on the normal line congruence of a minimal surface in ${\Bbb{R}}^3$ induced by the neutral Kähler metric on the space of oriented lines. The metric is lorentz with isolated degenerate points and the flow is shown to be completely integrable. In addition, we give a new holomorphic description …

New Calabi-Yau metrics found on C^n for n>=3.

Study existence of harmonic and Dirac-harmonic maps from degenerating surfaces.

Study of special ODE subclasses using geometric structures.

Stability of Morse index for harmonic maps on degenerating surfaces analyzed.

New proof shows certain 4D metrics are non-degenerate if curvature is negative definite.

Space-like maximal surfaces and time-like minimal surfaces in Lorentz-Minkowski3-space are both characterized as zero mean curvature surfaces. We are interested in the case where the zero mean curvature surface changes type from space-like to time-like at a given non-degenerate null curve. We consider this phenomenon a…

Paper develops estimates for Lagrangian phase changes in 2D.

Given two Lagrangian spheres in an exact symplectic manifold, we find conditions under which the Dehn twists about them generate a free non-abelian subgroup of the symplectic mapping class group. This extends a result of Ishida for Riemann surfaces. The proof generalises the categorical version of Seidel's long exact s…