arXiv research

The paper calculates expected distances on partially oriented flag manifolds.

We study $n$-dimensional area-minimizing currents $T$ in $\mathbb{R}^{n+1},$ with boundary $\partial T$ satisfying two properties: $\partial T$ is locally a finite sum of $(n-1)$-dimensional $C^{1,α}$ orientable submanifolds which only meet tangentially and with same orientation, for some $α\in (0,1]$; $\partial T$ has…

We introduce and study co-dimension one area-minimizing locally rectifiable currents $T$ with $C^{1,α}$ tangentially immersed boundary: $\partial T$ is locally a finite sum of orientable co-dimension two submanifolds which only intersect tangentially with equal orientation. We show that any such $T$ is supported in a s…

It is well known that a countable group admits a left-invariant total order if and only if it acts faithfully on R by orientation preserving homeomorphisms. Such group actions are special cases of group actions on simply connected 1-manifolds, or equivalently, actions on oriented order trees. We characterize a class of…

Rational homology ribbon cobordism defines a partial order on 3-manifolds.

Partial coverings of hyperbolic surfaces equidistribute with geodesics.

New method uses non-orientable surfaces to describe knots in 3-manifolds.

We prove a finiteness result for the $\partial$-patterned guts decomposition of all 3-manifolds obtained by splitting a given orientable, irreducible and $\partial$-irreducible 3-manifold along a closed incompressible surface. Then using the Thurston norm, we deduce that the JSJ-pieces of all 3-manifolds dominated by a…

In this paper, the general formulation for inextensible flows of curves on oriented surface in $\mathbb{R}^3$ is investigated. The necessary and sufficient conditions for inextensible curve flow lying an oriented surface are expressed as a partial differential equation involving the geodesic curvature and the geodesic…

The paper solves conditions for extending circle-valued Morse functions.

We introduce several algebraic structures related to handlebody-knots, including $G$-families of biquandles, partially multiplicative biquandles and group decomposable biquandles. These structures can be used to color the semiarcs in $Y$-oriented spatial trivalent graph diagrams representing $S^1$-oriented handlebody-k…

Branched covers are applied frequently in topology - most prominently in the construction of closed oriented PL d-manifolds. In particular, strong bounds for the number of sheets and the topology of the branching set are known for dimension d<=4. On the other hand, Izmestiev and Joswig described how to obtain a simplic…

Generalizes Rochlin's theorem to 4-manifolds with boundary and arbitrary 3-manifold boundaries.

New theorems on Hodge numbers and Kähler structures derived from complex differential forms.

Study the symmetries of smooth functions on Möbius bands.

We introduce \textit{Niebrzydowski algebras}, algebraic structures with a ternary operation and a partially defined multiplication, with axioms motivated by the Reidemeister moves for $Y$-oriented trivalent spatial graphs and handlebody-links. As part of this definition, we identify generating sets of $Y$-oriented Reid…

A 3-manifold is foliar if it supports a codimension-one co-oriented taut foliation. Suppose $M$ is an oriented 3-manifold with connected boundary a torus, and suppose $M$ contains a properly embedded, compact, oriented, surface $R$ with a single boundary component that is Thurston norm minimizing in $H_2(M, \partial M)…

A partial formula is provided to calculate the smallest number of vertices possible in a quadrangulation on the closed orientable 2-manifold of given genus. This extends the previously known partial formula due to N. Hartsfield and G. Ringel [J. Comb. Theory, Ser. B, 1989, 46, 84-95].

Researchers create a new compactification of character varieties using geometric and algebraic methods.

The paper proves conditions for positive scalar curvature and Yamabe constant on noncompact cylinders.

Geometric cohomology model uses co-oriented maps to define a product structure.

Let $(M^{n+1},\partial M,g)$ be a compact manifold with non-negative Ricci curvature, convex boundary and $2\leq n\leq 6$. We show that the min-max minimal hypersurface with respect to one-parameter families of hypersurfaces in $(M,\partial M)$ is orientable, of index one and multiplicity one.

Reconstruct flows and manifolds from their boundary actions on circles.

In a recent work of I.\,Dynnikov and M.\,Prasolov a new method of comparing Legendrian knots is proposed. In general, to apply the method requires a lot of technical work. In particular, one needs to search all rectangular diagrams of surfaces realizing certain dividing configurations. In this paper, it is shown that, …

We give partial boundary regularity for co-dimension one absolutely area-minimizing currents at points where the boundary consists of a sum of $C^{1,α}$ submanifolds, possibly with multiplicity, meeting tangentially, given that the current has a tangent cone supported in a hyperplane with constant orientation vector; t…

We give partial boundary regularity for co-dimension one absolutely area-minimizing currents at points where the boundary consists of a sum of $C^{1,α}$ submanifolds, possibly with multiplicity, meeting tangentially, given that the current has a tangent cone supported in a hyperplane with constant orientation vector; t…

The paper extends torus surgery results in 4-manifolds and shows diffeomorphisms.

Generalising previous results on classical braid groups by Artin and Lin, we determine the values of m, n $\in$ N for which there exists a surjection between the n-and m-string braid groups of an orientable surface without boundary. This result is essentially based on specific properties of their lower central series, …

Matveev introduced Borromean surgery on 3-manifolds, and proved that the equivalence relation on closed, oriented 3-manifolds generated by Borromean surgeries is characterized by the first homology group and the torsion linking pairing. Massuyeau generalized this result to closed, spin 3-manifolds, and the second autho…

We show that the notion of $3$-hyperconvexity on oriented flag manifolds defines a partial cyclic order. Using the notion of interval given by this partial cyclic order, we construct Schottky groups and show that they correspond to images of positive representations in the sense of Fock and Goncharov. We construct poly…

New corks found that are not strong and exotic.

Study Kauffman bracket skein modules of Seifert fibered spaces.

We construct complete Riemannian metrics to show that the total space of tangent bundles of orientable closed surfaces (except torus) admits complete uniformly PSC-metrics. It gives a partial positive answer to one of Gromov's question.

Let $M_1$ and $M_2$ be orientable irreducible 3--manifolds with connected boundary and suppose $\partial M_1\cong\partial M_2$. Let $M$ be a closed 3--manifold obtained by gluing $M_1$ to $M_2$ along the boundary. We show that if the gluing homeomorphism is sufficiently complicated, then $M$ is not homeomorphic to $S^3…

An incompressible surface $F$ on the boundary of a compact orientable 3-manifold $M$ is arc-extendible if there is an arc $γ$ on $\partial M -$ Int $F$ such that $F \cup N(γ)$ is incompressible, where $N(γ)$ is a regular neighborhood of $γ$ in $\partial M$. Suppose for simplicity that $M$ is irreducible, and $F$ has n…

We consider a general family of curves $Γ$ on a compact oriented Finsler surface $(M,F)$ with boundary $\partial M$. Let $\varphi\in C^{\infty}(M)$ and $ω$ a smooth 1-form on $M$. We show that $\int_{γ(t)}\{\varphi(γ(t))+ω_{γ(t)}(\dotγ(t))\}\,dt=0$ holds for every $γ\inΓ$ whose endpoints belong to $\partial M$, $γ(a)…

Example of group action on surface that can't extend to 3-manifold.

Study on 3-manifolds admitting pseudo-Anosov maps on subsurfaces.

Let $B$ be a Möbius band and $f:B \to \mathbb{R}$ be a Morse map taking a constant value on $\partial B$, and $\mathcal{S}(f,\partial B)$ be the group of diffeomorphisms $h$ of $B$ fixed on $\partial B$ and preserving $f$ in the sense that $f\circ h = f$. Under certain assumptions on $f$ we compute the group $π_0\mathc…

We give a complete proof of Thurston's celebrated hyperbolic Dehn filling theorem, following the ideal triangulation approach of Thurston and Neumann-Zagier. We avoid to assume that a genuine ideal triangulation always exists, using only a partially flat one, obtained by subdividing an Epstein-Penner decomposition. Thi…

The Borel Conjecture predicts that closed aspherical manifolds are topological rigid. We want to investigate when a non-aspherical oriented connected closed manifold M is topological rigid in the following sense. If f: N --> M is an orientation preserving homotopy equivalence with a closed oriented manifold as target, …

An optimal control problem associated with the dynamics of the orientation of a bipolar molecule in the plane can be understood by means of tools in differential geometry. For first time in the literature $k$-symplectic formalism is used to provide the optimal control problems associated to some families of partial dif…

OGRePy simplifies tensor calculations in general relativity.

The study proves a geometric inequality for surfaces with genus G.

We extend the notion of the symmetric signature $σ(\bar{M},r)$ in L^n(R) for a compact n-dimensional manifold M without boundary, a reference map r from M to BG and a homomorphism of rings with involutions from ZG to R to the case with boundary $\partial M$, where $(\bar{M},\bar{\partial M}) \to (M,\partial M)$ is the …

Torus decomposition shows foliation detected slopes for glued knot manifolds.

As neural networks revolutionize many applications, significant privacy conflicts between model users and providers emerge. The cryptography community developed a variety of techniques for secure computation to address such privacy issues. As generic techniques for secure computation are typically prohibitively ineffec…

Suppose M is a non-compact connected n-manifold without boundary, DD(M) is the group of C^\infty-diffeomorphisms of M endowed with the Whitney C^\infty-topology and DD_0(M) is the identity connected component of DD(M), which is an open subgroup in the group DD_c(M) \subset DD(M) of compactly supported diffeomorphisms o…