arXiv research

We study three-dimensional generalized Ricci solitons, both in Riemannian and Lorentzian settings. We shall determine their homogeneous models, classifying left-invariant generalized Ricci solitons on three-dimensional Lie groups.

Three elements generate balanced superelliptic mapping class groups.

Extends three circle theorem to almost Hermitian manifolds.

Investigates smoothness of specific algebra structures.

Three involutions generate the mapping class group for surfaces of genus 6 or more.

Extended mapping class group can be generated by three involutions for certain surfaces.

Kashaev's invariants for a knot in a three sphere are generalized to invariants of a knot in a three manifold. A relation between the newly constructed invariants and the hyperbolic volume of the knot complement is observed for some knots in lens spaces.

We prove that the extended mapping class group is generated by three orientation reversing involutions.

We construct a three-dimensional topological sigma model which is induced from a generalized complex structure on a target generalized complex manifold. This model is constructed from maps from a three-dimensional manifold $X$ to an arbitrary generalized complex manifold $M$. The theory is invariant under the diffeomor…

We solve the metrisability problem for generic three-dimensional projective structures.

We show that for generic Riemannian metrics on a closed spin manifold of dimension three the Dirac operator has only simple eigenvalues.

We prove that the mapping class group of a closed connected orientable surface of genus at least eight is generated by three involutions.

We explicitly compute the lower algebraic K-theory of the split three-dimensional crystallographic groups; i.e., the groups G that act properly and cocompactly on three-dimensional Euclidean space by isometries, such that the natural map from G to O(3) is a split injection onto its image. There are 73 split three-dimen…

Proof that specific groups are quasi-isometrically rigid.

Proves mapping class group of nonorientable surfaces can be generated by three torsions.

We study the generalization of quasipositive links from the three-sphere to arbitrary closed, orientable three-manifolds. Our main result shows that the boundary of any smooth, properly embedded complex curve in a Stein domain is a quasipositive link. This generalizes a result due to Boileau and Orevkov, and it provide…

The paper proves a generalized Bour's theorem for invariant surfaces.

The notion of a braid is generalized into two and three dimensions. Two-dimensional braids are described by braid monodromies or graphics called charts. In this paper we introduce the notion of curtains, and show that three-dimensional braids are described by braid monodromies or curtains.

Measure contraction property is one of the possible generalizations of Ricci curvature bound to more general metric measure spaces. In this paper, we discover sufficient conditions for a three dimensional contact subriemannian manifold to satisfy this property.

The mapping class group of a Heegaard splitting is the group of connected components in the set of automorphisms of the ambient manifold that map the Heegaard surface onto itself. For the genus three Heegaard splitting of the 3-torus, we find an eight element generating set for this group. Six of these generators induc…

Develops methods for constructing exact, non-stationary solutions to Euler equations.

A three dimensional supergravity theory which generalizes the super IG theory of Witten and resembles the model discussed recently by Mann and Papadopoulos is displayed. The partition function is computed, and is shown to be a three-manifold invariant generalizing the Casson invariant.

Let S be a triangulated 2-sphere with fixed triangulation T. We apply the methods of thin position from knot theory to obtain a simple version of the three geodesics theorem for the 2-sphere [5]. In general these three geodesics may be unstable, corresponding, for example, to the three equators of an ellipsoid. Using a…

We show that the mapping class group of any closed connected orientable surface of genus at least five is generated by only two commutators, and if the genus is three or four, by three commutators.

We construct four-dimensional symplectic cobordisms between contact three-manifolds generalizing an example of Eliashberg. One key feature is that any handlebody decomposition of one of these cobordisms must involve three-handles. The other key feature is that these cobordisms contain chains of symplectically embedded …

Following Riemann's idea, we prove the existence of a minimal disk in Euclidean space bounded by three lines in generic position and with three helicoidal ends of angles less than $π$. In the case of general angles, we prove that there exist at most four such minimal disks, we give a sufficient condition of existence i…

Researchers found that the twist subgroup can be generated by two elements for certain surface genera.

We provide the twisted Alexander polynomials of finite abelian covers over three-dimensional manifolds whose boundary is a finite union of tori. This is a generalization of a well-known formula for the usual Alexander polynomial of knots in finite cyclic branched covers over the three-dimensional sphere.

We introduce canonical principal parameters on any strongly regular minimal surface in the three dimensional sphere and prove that any such a surface is determined up to a motion by its normal curvature function satisfying the Sinh-Poisson equation. We obtain a classification theorem for bi-umbilical hypersurfaces of t…

This paper introduces two virtual knot theory ``analogues'' of a well-known family of invariants for knots in thickened surfaces: the Grishanov-Vassiliev finite-type invariants of order two. The first, called the three loop isotopy invariant, is an invariant of virtual knots while the second, called the three loop fram…

The paper characterizes stable cohomotopy groups in codimensions two and three, linking algebraic and geometric perspectives.

In the paper, we show that for a generic $C^1$ vector field $X$ on a closed three dimensional manifold $M$, any isolated transitive set of $X$ is singular hyperbolic. It is a partial answer of the conjecture in \cite{MP}.

This paper adresses the following problem: Given a closed orientable three-manifold M, are there at most finitely many closed orientable three-manifolds 1-dominated by M? We solve this question for the class of closed orientable graph manifolds. More presisely the main result of this paper asserts that any closed orien…

Symmetry-breaking in three differential geometry conjectures.

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.

The paper studies invariant metrics with positive scalar curvature on 3-manifolds.

Extended dual Coxeter and Artin groups theory to rank-three systems.

In this short note, we prove that the only simply connected noncompact three-dimensional Type I $κ$-solution to the Ricci flow is the shrinking cylinder. This work can be regarded as a generalization of Cao and Chow, and a complement of Ding and Ni. Up to this point, three-dimensional $κ$-solutions of Type I are comple…

In this paper we show how to place Michael Berry's discovery of knotted zeros in the quantum states of hydrogen in the context of general knot theory and in the context of our formulations for quantum knots. Berry gave a time independent wave function for hydrogen, as a map from three space to the complex plane and suc…

Three-layer networks learn more complex features than two-layer networks.

The main goal of our paper is the study of several classes of submanifolds of generalized complex manifolds. Along with the generalized complex submanifolds defined by Gualtieri and Hitchin (we call these ``generalized Lagrangian submanifolds'' in our paper), we introduce and study three other classes of submanifolds. …

Manolescu correction terms are numerical invariants of homology three-spheres arising from $\mathrm{Pin}(2)$-equivariant Seiberg-Witten theory that contain information about homology cobordism. We discuss several constraints on these invariants for homology spheres obtained by Dehn surgery on a knot in the three-sphere…

R-circles in general three dimensional CR manifolds (of contact type) are the analogues to traces of Lagrangian totally geodesic planes on the sphere viewed as the boundary of two dimensional complex hyperbolic space. They form a family of certain legendrian curves on the manifold. We prove that a diffeomorphism betwee…

Novel flow modifies Ricci flow with string theory correction.

I calculate optimistically asymptotic behaviors of the WRT SU(2) invariants for the three-manifolds obtained from the figure-eight knot by p-surgeries with p=0,1,2,...,10, from which one can extract volumes and the Chern-Simons invariants of these closed manifolds. I conjecture that this also holds for general closed t…

The first group of differentiable cohomology of $\Diff(S^1)$, vanishing on the Möbius subgroup $PSL(2,R)\subset\Diff(S^1)$, with coefficients in modules of linear differential operators on $S^1$ is calculated. We introduce three non-trivial $PSL(2,R)$-invariant 1-cocycles on $\Diff(S^1)$ generalizing the Schwarzian der…

A classification and examples of four-dimensional isoclinic three-webs of codimension two are given. The examples considered prove the existence theorem for many classes of webs for which the general existence theorems are not proved yet.

A classification and examples of four-dimensional isoclinic three-webs of codimension two are given. The examples considered prove the existence theorem for many classes of webs for which the general existence theorems are not proved yet.