arXiv research

The study examines differential singularities in 3D Nijenhuis operators.

Generic 3D vector fields have singularly hyperbolic transitive sets.

New Heegaard Floer homology for orbifolds with cyclic singularities.

Paper finds singular solutions for a specific physics problem on a sphere.

Proves K-stability of cubic threefolds and calculates their Kähler-Einstein metrics.

We calculate the first and the second variation formula for the sub-Riemannian area in three dimensional pseudo-hermitian manifolds. We consider general variations that can move the singular set of a C^2 surface and non-singular variation for C_H^2 surfaces. These formulas enable us to construct a stability operator fo…

We show that a rescale limit at any degenerate singularity of Ricci flow in dimension 3 is a steady gradient soliton. In particular, we give a geometric description of type I and type II singularities.

Study of singularity formation in dHYM flow on a blown-up CP^3.

The study proves a key inequality for specific types of three-dimensional spaces.

Study on evolutes and focal surfaces of pseudo-spherical framed immersions in anti-de Sitter space.

We study three-dimensional Alexandrov spaces with a lower curvature bound, focusing on extending three classical results on three-dimensional manifolds: First, we show that a closed three-dimensional Alexandrov space of positive curvature, with at least one topological singularity, must be homeomorphic to the suspensio…

Construction of a semigroup with 15 generators and 84 relations is given. The center of this semigroup is in one-to-one correspondence with the set of all isotopy classes of non-oriented singular knots (links with finitely many double intersections in general position) in three-dimensional space.

Timelike minimal surfaces in Lorentzian Heisenberg group have singular points.

Three-dimensional N=2 superconformal field theories are constructed by compactifying M5-branes on three-manifolds. In the infrared the branes recombine, and the physics is captured by a single M5-brane on a branched cover of the original ultraviolet geometry. The branch locus is a tangle, a one-dimensional knotted subm…

We show that timelike maximal cylinders in $\RR^{1 + 2}$ always develop singularities in finite time and that, infinitesimally at a generic singularity, their time slices are evolved by a rigid motion or a self-similar motion. We also prove a mild generalization in non-flat backgrounds.

Modeling 3D continua with singular points using Yin sets.

Gurau argued in [arXiv:1006.0714] that the gluing spaces arising as Feynman diagrams of three-dimensional group field theory are not all pseudo-manifolds. I dispute this conclusion: albeit not properly triangulated, these spaces are genuine pseudo-manifolds, viz. their singular locus is of codimension at least two.

Gromov-Hausdorff convergence of time-slices of singular Ricci flows

The surface singular braid monoid corresponds to marked graph diagrams of knotted surfaces in braid form. In a quest to resolve linearity problem for this monoid, we will show that if it is defined on at least two or at least three strands, then its two or respectively three dimensional representations are not faithful…

As indicated by the third author in [19], there is a gap in the previous version of this paper by the first two authors [5]. We provide in this version an argument to fix the aforementioned gap. The main proposition, whose proof uses Perelman's techniques, is implied by Ding [9] and is covered by [19]. Our approach, ho…

Transforms instantons to monopoles for torus products.

Defines Milnor number for foliations and shows its topological invariance.

It is shown that every knot or link is the set of complex tangents of a 3-sphere smoothly embedded in the three-dimensional complex space. We show in fact that a one-dimensional submanifold of a closed orientable 3-manifold can be realised as the set of complex tangents of a smooth embedding of the 3-manifold into the …

Motivated by the hinge structure present in protein chains and other molecular conformations, we study the singularities of certain maps associated to body-and-hinge and panel-and-hinge chains. These are sequentially articulated systems where two consecutive rigid pieces are connected by a hinge, that is, a codimension…

Classifies left invariant Kundt structures on 3D Lie groups.

3D Ricci flows have bounded diameter before Type I singularities.

Survey classifies singularity models in 3D Ricci flow.

We provide a characterization of quotients of three-dimensional complex tori by finite groups that act freely in codimension one via a vanishing condition on the first and second orbifold Chern class. We also treat the case of actions free in codimension two, using instead the "birational" second Chern class, as we cal…

In this paper we classify the four dimensional gradient shrinking solitons under certain curvature conditions satisfied by all solitons arising from finite time singularities of Ricci flow on compact four manifolds with positive isotropic curvature. As a corollary we generalize a result of Perelman on three dimensional…

The paper defines and studies $d$-minimal surfaces in a degenerate semi-Euclidean space.

Study null surfaces of pseudo-spherical curves in anti-de Sitter space.

We introduce a geometric transition between two homogeneous three-dimensional geometries: hyperbolic geometry and anti de Sitter (AdS) geometry. Given a path of three-dimensional hyperbolic structures that collapse down onto a hyperbolic plane, we describe a method for constructing a natural continuation of this path i…

The paper studies singularities on parallels of tangent developable surfaces of frontal curves.

We prove a number of results relating various measures (volume, Legendrian index, stability index, and spectral curve genus) of the geometric complexity of special Lagrangian $T^2$-cones. We explain how these results fit into a program to understand the "most common" three-dimensional isolated singularities of special …

We construct stationary flat three-dimensional Lorentzian manifolds with singularities that are obtained from Euclidean surfaces with cone singularities and closed one-forms on these surfaces. In the application to (2+1)-gravity, these spacetimes correspond to models containing massive particles with spin. We analyse t…

Two single parameter families of polyhedra $P(ψ)$ are constructed in three dimensional spaces of constant curvature $C(ψ)$. Identification of the faces of the polyhedra via isometries results in cone manifolds $M(ψ)$ which are topologically $S^1\timesS^2$, $S^3$ or singular $S^2$. The singular set of $M(ψ)$ can have se…

The Sard conjecture is proven for smooth Martinet surfaces and under certain conditions for singular ones.

We use topological methods to prove a semicontinuity property of the Hodge spectra for analytic germs defined on an isolated surface singularity. For this we introduce an analogue of the Seifert matrix (the fractured Seifert matrix), and of the Levine--Tristram signatures associated with it, defined for null-homologous…

Study shows singular sets for certain fluid equations are negligible.

Study helicoidal surfaces from frontals, revealing geometric rigidity and stability of singularities.

A compact manifold is called Bieberbach if it carries a flat Riemannian metric. Bieberbach manifolds satisfy an isosystolic inequality by a general and fundamental result of M. Gromov. In dimension 3, there exist four classes of non-orientable Bieberbach manifolds up to an affine diffeomorphism. In this paper, We prove…

The homotopy theory of topological defects in ordered media fails to completely characterize systems with broken translational symmetry. We argue that the problem can be understood in terms of the lack of rotational Goldstone modes in such systems and provide an alternate approach that correctly accounts for the intera…

Study optimal trade execution in markets with price impact and resilience.

The authors study smooth lines on projective planes over the algebra C of complex numbers, the algebra C^1 of double numbers, and the algebra C^0 of dual numbers. In the space RP^5, to these smooth lines there correspond families of straight lines describing point three-dimensional tangentially degenerate submanifolds …

Study on Hausdorff dimension of singular CR Yamabe problem.

Study on bounded Gaussian curvature in mixed type surfaces of Lorentzian manifolds.

The paper explores actions on metric spaces similar to 3D manifolds, proving rigidity results.

Study of symmetries in 3D Lie groups, determining index and moduli space properties.