arXiv research

Study curves in 3-sphere using invariant geometric flows.

We introduce pseudoconformal structures on 4--dimensional manifolds and study their properties. Such structures are arising from two different complex operators which agree in a 2--dimensional subbundle of the tangent bundle; this subbundle thus forms a codimension 2 ${\rm CR}$ structure. A special case is that of a st…

Let $ X: M \hook S^5$ be a compact Legendrian surface in pseudoconformal(CR) 5-sphere. We introduce a pseudoconformally invariant Willmore type second order functional $ \W(X)$, and study its critical points called Willmore Legendrian surfaces. The fifth order structure equations show that Willmore dual can be defined …

This paper is devoted to the specific class of pseudoconformal mappings of quaternion and octonion variables. Normal families of functions are defined and investigated. Four criteria of a family being normal are proven. Then groups of pseudoconformal diffeomorphisms of quaternion and octonion manifolds are investigated…

The authors study the geometry of lightlike hypersurfaces on a four-dimensional manifold $(M, c)$ endowed with a pseudoconformal structure $c = CO (2, 2)$. They prove that a lightlike hypersurface $V \subset (M, c)$ bears a foliation formed by conformally invariant isotropic geodesics and two isotropic distributions ta…

We consider real isotropic geodesics on manifolds endowed with a pseudoconformal structure and their applications to the theory of lightlike hypersurfaces on such manifolds, the geometry of four-dimensional conformal structures of Lorentzian type, and a classification of the Einstein spaces.

There are three types of hypersurfaces in a pseudoconformal space C^n_1 of Lorentzian signature: spacelike, timelike, and lightlike. These three types of hypersurfaces are considered in parallel. Spacelike hypersurfaces are endowed with a proper conformal structure, and timelike hypersurfaces are endowed with a conform…

The authors study the geometry of lightlike hypersurfaces on manifolds $(M, c)$ endowed with a pseudoconformal structure $c = CO (n - 1, 1)$ of Lorentzian signature. Such hypersurfaces are of interest in general relativity since they can be models of different types of physical horizons. On a lightlike hypersurface, th…

This paper concerns the problem of existence of taut foliations among 3-manifolds. Since the contribution of David Gabai, we know that closed 3-manifolds with non-trivial second homology group admit a taut foliations. The essential part of this paper focuses on Seifert fibered homology 3-spheres. The result is quite di…

Stable 2-lobed Delaunay tori found in 3-sphere.

Characterizes a specific type of convex curves on a 3-sphere.

Real algebraic structures help classify overtwisted contact 3-spheres.

Disk complexes show 3-sphere surfaces are topologically minimal.

We show that a graph manifold which is a Z-homology 3-sphere not homeomorphic to either the 3-sphere or the Poincaré homology 3-sphere admits a horizontal foliation. This combines with known results to show that the conditions of not being an L-space, of having a left-orderable fundamental group, and of admitting a co-…

5 minimal tori found in 3-spheres with positive Ricci curvature.

The paper examines conditions for contact surgeries on rational homology 3-spheres.

For any knot, a 3-sphere triangulation exists with a knotted edge.

Minimal surfaces in spheres found for any genus.

The paper studies invariants of surfaces in the 3-sphere using handlebody-links.

Flat tori in 3-sphere have π extrinsic diameter under specific conditions.

For a hypersurface V of a conformal space, we introduce a conformal differential invariant I = h^2/g, where g and h are the first and the second fundamental forms of V connected by the apolarity condition. This invariant is called the conformal quadratic element of V. The solution of the problem of conformal rigidity i…

Classifies Legendrian Hopf links in 3-sphere.

Given a genus-g Heegaard splitting of a 3-sphere, the genus-g Goeritz group is defined to be the group of the isotopy classes of orientation preserving homeomorphism of the 3-sphere that preserve the splitting. In this paper, we determine the twisted first (co)homology group of the genus-2 Goeritz group of 3-sphere.

We prove Mayberry-Murasugi's formula for links in homology 3-spheres, which was proved before only for links in the 3-sphere. Our proof uses Franz-Reidemeister torsions.

The paper classifies decompositions of 3-sphere and lens spaces with handlebodies.

Diagrammatic method characterizes non-split surfaces in 3-sphere.

First examples of 3-spheres smoothly embedding in S^2 x S^2 but not S^4.

By the Fox's re-embedding theorem, any compact submanifold of the 3-sphere can be re-embedded in the 3-sphere so that it is unknotted. It is unknown whether the Fox's re-embedding can be replaced with twistings. In this paper, we will show that any closed 2-manifold embedded in the 3-sphere can be unknotted by twisting…

Perelman's proof confirmed, new method uses 4D topology.

Reduces weak reducing pairs to spheres in 3-sphere Heegaard surfaces.

Constructs fat, shellable 3-spheres with specific $f$-vectors.

Study invariants of $\mathbb{Z}/p$-homology 3-spheres from abelianization of mapping class groups.

Paper connects hypersurfaces in 4D to surfaces in 3-sphere.

New rack and multiple group rack cohomology for surfaces in 3-sphere.

New invariant fully describes finite type invariants of knots in homology 3-spheres.

Authors prove a conjecture about the Goeritz group of 3-sphere Heegaard splittings.

New steady Euler flows found on 3-sphere and Sasakian manifolds.

For a surface in 3-sphere, by identifying the conformal round 3-sphere as the projectivized positive light cone in Minkowski 5-spacetime, we use the conformal Gauss map and the conformal transform to construct the associate homogeneous 4-surface in Minkowski 5-spacetime. We then derive the local fundamental theorem for…

We show that on a Sasakian 3-sphere the Sasaki-Ricci flow initiating from a Sasakian metric of positive transverse scalar curvature converges to a gradient Sasaki- Ricci soliton. We also show the existence and uniqueness of gradient Sasaki-Ricci soliton on each Sasakian 3-sphere.

For each rational homology 3-sphere $Y$ which bounds simply connected definite 4-manifolds of both signs, we construct an infinite family of irreducible rational homology 3-spheres which are homology cobordant to $Y$ but cannot bound any simply connected definite 4-manifold. As a corollary, for any coprime integers $p,…

We prove the existence of a new algorithm for 3-sphere recognition based on Groebner basis methods applied to the variety of $\text{\em SL}(2,\C)$-representation of the fundamental group. An essential input is a recent result of the second author, stating that any integer homology 3-sphere different from the 3-sphere a…

We show that the perturbative ${\frak g}$ invariant of rational homology 3-spheres can be recovered from the LMO invariant for any simple Lie algebra ${\frak g}$, i.e, the LMO invariant is universal among the perturbative invariants. This universality was conjectured in [25]. Since the perturbative invariants dominate …

For all positive integers n we construct a 1-parameter family of conformal tori of revolution in the 3-sphere with n bulges. These tori arise by Darboux transformations of constant mean curvature tori but do not have constant mean curvature in the 3-sphere.

New mixed singularities help classify real algebraic links.

The paper details folding of branched covers of the 3-sphere over knots.

A steady flow on a 3-sphere connects to a Faddeev-Skyrme model.

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

Morse theory connects low energy submanifolds in 3-sphere.