arXiv research

Paper improves proof of theorem on convex 2-disk homeomorphisms.

Even-dimensional simply connected manifolds that are rational homology spheres and double disk bundles are homeomorphic to spheres.

Paper shows braid groups can't be realized geometrically.

Harmonic extension of Weil-Petersson circle homeomorphisms

In this paper we study the homeomorphisms of the disk that are liftable with respect to a simple branched covering. Since any such homeomorphism maps the branch set of the covering onto itself and liftability is invariant up to isotopy fixing the branch set, we are dealing in fact with liftable braids. We prove that th…

New disks fill Legendrian knots without being smoothly isotopic.

Corks transform complex curves without changing topology.

In this paper we give a proof of the existence of an orthogonal geodesic chord on a Riemannian manifold homeomorphic to a closed disk and with concave boundary. This kind of study is motivated by the link of the multiplicity problem with the famous Seifert conjecture (formulated in 1948) about multiple brake orbits for…

Essential arcs on punctured disks intersect at most twice, proving a size limit.

Bi-Lipschitz proof for 2-varifolds near critical Allard condition.

We give a parametrization to the asymptotic Teichmuller space of the open unit disk through equivalent classes of shear functions induced by quasisymmetric homeomorphisms on the Farey tesselation of the unit disk. Then using the parametrization, we define a new metric on the asymptotic Teichmuller space. Two other rela…

Let ${\mathcal H}$ be the group of isotopy classes of orientation preserving homeomorphisms of $S^3$ that preserve a Heegaard splitting of genus two. In this paper, we use a tree in the barycentric subdivision of the disk complex of a handlebody of the splitting to obtain a finite presentation of ${\mathcal H}$.

The paper finds at least N orthogonal Finsler geodesic chords in a disk-like manifold.

Let $\{V_{i}\}_{i=1}^{n}$ be a finite family of closed subsets of a plane or a sphere $S^{2}$, each homeomorphic to the two-dimensional disk. In this paper we discuss the question how the boundary of connected components of a complement $\rr^{2} \setminus \bigcup_{i=1}^{n} V_{i}$ (accordingly, $S^{2} \setminus \bigcup_…

Study curves on surfaces with surgeries connecting them.

M Handel has proved in [Topology 38 (1999) 235--264] a fixed point theorem for an orientation preserving homeomorphism of the open unit disk, that may be extended to the closed disk and that satisfies a linking property of orbits. We give here a new proof of Handel's fixed point theorem, based on Brouwer theory and som…

Given a stabilized Heegaard splitting of a $3$-manifold, the primitive disk complex for the splitting is the subcomplex of the disk complex for a handlebody in the splitting spanned by the vertices of the primitive disks. In this work, we study the structure of the primitive disk complex for the genus two Heegaard spli…

Stable capillary surfaces in weighted balls are disks.

In a very influential paper Gehring and Palka introduced the notions of quasiconformally homogeneous and uniformly quasiconformally homogeneous subsets of Euclidean space. Their motivation was to provide a characterization of quasi-disks, i.e. domains which are quasiconformally homeomorphic to the unit disk. As a gener…

We develop tools to study the topology and geometry of self-affine fractals in dimension three and higher. We use the self-affine structure and obtain rather detailed information about the connectedness of interior and boundary sets, and on the dimensions and intersections of boundary sets. As an application, we descri…

The paper generalizes Thurston's earthquake map to cluster algebras of finite type.

For a smooth immersion $f$ from the punctured disk $D\backslash\{0\}$ into $\mathbb{R}^n$ extendable continuously at the puncture, if its mean curvature is square integrable and the measure of $f(D)\cap B_{r_k}=o(r_k)$ for a sequence $r_k\to 0$, we show that the Riemannian surface $(D_r\backslash\{0\},g)$ where $g$ is …

Minimal diffeomorphisms extend uniquely with $L^1$ Hopf differential.

Let B_n be the braid group on n > 3 strands. We prove that B_n modulo its center is co-Hopfian. We then show that any injective endomorphism of B_n is geometric in the sense that it is induced by a homeomorphism of a punctured disk. We further prove that any injection from B_n to B_n+1 is geometric. Additionally, we ob…

We prove a version of Smirnov type theorem and Charatheodory type theorem for a harmonic homeomorphism of the unit disk onto a Jordan surface with rectifiable boundary. Further we establish the classical isoperimetric inequality and Riesz--Zygmund inequality for Jordan harmonic surfaces without any smoothness assumptio…

We classify right-veering homeomorphisms of the once-punctured torus using the Burau representation of the 3-strand braid group. We show that reducible and periodic mapping classes in B_3 can be identified as right-veering by consideration of the reduced version of the Burau representation. Given any element beta in B_…

Fixed pseudo-Anosov homeomorphisms detect cinquefoil knot.

Geodesics on polygons in a unit disk are studied with unique metric properties.

Study $\mathbb{C}$-Fuchsian subgroups of non-arithmetic lattices.

Let $X$ be a topological space, $U$ -- opened subset of $X$. We will say that point $x \in \partial U$ is {\it accessible} from $U$ if there exists continuous injective mapping $φ: I \to \Cl D$ such that $φ(1)=x$, $φ([0,1)) \subset \Int U$. We proove the next main theorem. The following conditions are neccesary and suf…

We prove a global fixed point theorem for the centralizer of a homeomorphism of the two dimensional disk $D$ that has attractor-repeller dynamics on the boundary with at least two attractors and two repellers. As one application, we show that there is a finite index subgroup of the centralizer of a pseudo-Anosov homeom…

We prove that any minimal weak symplectic filling of the canonical contact structure on the unit cotangent bundle of a nonorientable closed surface other than the real projective plane is s-cobordant rel boundary to the disk cotangent bundle of the surface. If the nonorientable surface is the Klein bottle, then we show…

We study a recent general criterion for the injectivity of the conformal immersion of a Riemannian manifold into higher dimensional Euclidean space, and show how it gives rise to important conditions for Weierstrass-Ennerper lifts defined in the unit disk $\mathbb{D}$ endowed with a conformal metric. Among the corollar…

Dynnikov system maps Teichmüller space boundary, aiding pseudo-Anosov braid study.

We prove that in dimensions not equal to 4, 5, or 7, the homology and homotopy groups of the classifying space of the topological group of diffeomorphisms of a disk fixing the boundary are finitely generated in each degree. The proof uses homological stability, embedding calculus and the arithmeticity of mapping class …

This is a continuation of the joint paper with the same title by A.Belenkiy and Yu.Burago. It is proved here that two homeomorphic closed Alexandrov surfaces (of bounded integral curvature) are bi-Lipschitz with a constant depending only on upper bounds of their Euler number, diameters, negative integral curvatures, an…

Study infinite circle patterns in the Weil-Petersson class using discrete harmonic functions.

A Riemann surface $M$ is said to be $K$-quasiconformally homogeneous if for every two points $p,q \in M$, there exists a $K$-quasiconformal homeomorphism $f \colon M \rightarrow M$ such that $f(p) = q$. In this paper, we show there exists a universal constant $K_0 > 1$ such that if $M$ is a $K$-quasiconformally homogen…

The boundary of certain hyperbolic groups is like a Menger curve.

One can define the complexity of a smooth 4-manifold as the minimal sum of the number of disks, strands and crossings in a Kirby diagram. Martelli proved that the number of homeomorphism classes of complexity less than n grows as $n^2$. In this paper we prove that the number of diffeomorphism classes grows at least as …

Let $h:X \to Y$ be a homeomorphism between hyperbolic surfaces with finite topology. If $h$ is homotopic to a holomorphic map, then every closed geodesic in $X$ is at least as long as the corresponding geodesic in $Y$, by the Schwarz Lemma. The converse holds trivially when $X$ and $Y$ are disks or annuli, and it holds…

Optimal inequalities for metric surfaces derived from filling minimality.

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

The paper proves unique factorization of knotted handlebodies and examines handlebody-knot symmetry.

A mathematical theorem shows generalized dunce hats can't be split into two parts.

Fixed points found in Teichmüller space via anti-de Sitter geometry.

Decomposition theory explores topological spaces and their quotient spaces.

As in the case of the associahedron and cyclohedron, the permutohedron can also be defined as an appropriate compactification of a configuration space of points on an interval or on a circle. The construction of the compactification endows the permutohedron with a projection to the cyclohedron, and the cyclohedron with…