arXiv research

Study on Coxeter groups' boundary planarity, finding exceptions.

Compact Special Weingarten surfaces with planar convex boundaries are disks.

Estimates the index of the Laplace operator on planar domains with Robin boundary condition.

Study constructs disks with curved boundaries in a 3D ball.

Study the boundaries of ε-neighborhoods of planar sets, showing their structure and curvature.

Proves existence of non-planar minimal disks in ellipsoids.

Determine lens spaces as closures of homology cobordisms over planar surfaces.

We show that an immersed minimal annulus, with two planar boundary curves along which the surface meets these planes with constant contact angle, is part of the catenoid.

Framework for isometric immersions of planar regions from framed curves.

This paper defines ribbons and ribbon complexes in CW spaces and analyzes their topological properties.

Using open book foliations we show that an overtwisted disc in a planar open book can be put in a topologically nice position. As a corollary, we prove that a planar open book whose fractional Dehn twist coefficients grater than one for all the boundary components supports a tight contact structure.

We prove that maximal annuli in $\mathbb{L}^{3}$ bounded by circles, straight lines or cone points in a pair of parallel spacelike planes are part of either a Lorentzian catenoid or a Lorentzian Riemann's example. We show that under the same boundary condition, the same conclusion holds even when the maximal annuli hav…

Study of group boundaries and subgroup properties.

Study on planar graphs in Poincare model of hyperbolic geometry.

We give a necessary and sufficient condition for a hyperbolic Coxeter group with planar nerve to have Sierpiński curve as its Gromov boundary.

The Blaschke rolling disk theorem is extended to non-convex domains.

We present a grid diagram analogue of Carter, Rieger and Saito's smooth movie theorem. Specifically, we give definitions for grid movies, grid movie isotopies and present a definition of grid planar isotopy as a particular subset of the grid diagram moves: stabilization, destabilization and commutation. We show that gr…

We extend our discrete uniformization theorems for planar, $m$-connected, Jordan domains [Journal für die reine und angewandte Mathematik 670 (2012), 65--92] to closed surfaces of non-positive genus.

Proves convergence groups on a 2-sphere are Kleinian groups.

The paper classifies vertices in planar polygons formed by convex domains.

We give a simple, combinatorial construction of a unital, spherical, non-degenerate $\ast$-planar algebra over the ring $\mathbb{Z}[q^{1/2},q^{-1/2}]$. This planar algebra is similar in spirit to the Temperley-Lieb planar algebra, but computations show that they are different. The construction comes from the combinator…

Consider a planar, bounded, $m$-connected region $Ω$, and let $\bordΩ$ be its boundary. Let $\mathcal{T}$ be a cellular decomposition of $Ω\cup\bordΩ$, where each 2-cell is either a triangle or a quadrilateral. From these data and a conductance function we construct a canonical pair $(S,f)$ where $S$ is a genus $(m-1)$…

The study finds counterexamples to a conjecture about incompressible planar surfaces in hyperbolic link exteriors.

It is shown that there exist alternating non-Montesinos knots whose essential spanning surfaces with maximal and minimal boundary slopes are not realised by the checkerboard surfaces coming from a reduced alternating planar diagram.

Nontrivial infinitesimal bendings for a class of two-dimensional surfaces are constructed. The surfaces considered here are orientable; compact; with boundary; have positive curvature everywhere except at finitely many planar points; and have vanishing first homology group.As a consequence, a nonrigidity result for suc…

Defines W-volume for planar domains with circular boundaries, relating to Laplacian determinant and Schottky uniformization.

Strict convexity of graphs with constant mean curvature is proven under certain conditions.

Explains hyperbolic group boundaries using amalgams.

We describe some of the algebra underlying the decomposition of planar grid diagrams. This provides a useful toy model for an extension of Heegaard Floer homology to 3-manifolds with parametrized boundary. This paper is meant to serve as a gentle introduction to the subject, and does not itself have immediate topologic…

Spatial graphs of non-Eulerian or proper Eulerian planar graphs are unknottable by region crossing changes.

Reconstructing a planar domain from its Dirichlet-to-Neumann data

We consider compact connected minimal surfaces, with a pair of boundary curves (not necessarily convex) in distinct planes, that have least-area amongst all orientable surfaces with the same boundary. When the planes containing these two boundary curves are either parallel or sufficiently close to parallel, and when th…

In this article, we find the complete list of all contact structures (up to isotopy) on closed three-manifolds which are supported by an open book decomposition having planar pages with three (but not less) boundary components. We distinguish them by computing their first Chern classes and three dimensional invariants …

In this note we introduce the (homologically essential) arc complex of a surface as a tool for studying properties of open book decompositions and contact structures. After characterizing destabilizability in terms of the essential translation distance of the monodromy of an open book we given an application of this re…

Rigidity theorem for ideal surfaces with flat boundary conditions.

The study explores planar Cayley graphs and their connection to Kleinian groups.

In this paper we construct some invariants of spatial graphs by disk-summing the constituent knots and show the delta edge-homotopy invariance of them. As an application, we show that there exist infinitely many slice spatial embeddings of a planar graph up to delta edge-homotopy, and there exist infinitely many bounda…

We prove that the curvature flow of an embedded planar network of three curves connected through a triple junction, with fixed endpoints on the boundary of a given strictly convex domain, exists smooth until the lengths of the three curves stay far from zero. If this is the case for all times, then the evolution exists…

It is shown that given any link-manifold, there is an algorithm to decide if the manifold contains an embedded, essential planar surface; if it does, the algorithm will construct one. If a slope on the boundary of the link-manifold is given, there is an algorithm to determine if the slope bounds an embedded punctured-d…

Simple closed curves in ε-boundaries separate sets in the plane.

This paper is the third in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed 3-manifold. In [CM3]-[CM5] we describe the case where the surfaces are topologically disks on any fixed small scale. To describe general planar domains (in [CM6]) we need in …

Study on curve shortening flow with boundary conditions, proving convergence or contraction.

The study finds lower bounds for the first eigenvalue of the Laplacian in planar domains with magnetic fields.

A graph G is intrinsically S^1-linked if for every embedding of the vertices of G into S^1, vertices that form the endpoints of two disjoint edges in G form a non-split link in the embedding. We show that a graph is intrinsically S^1-linked if and only if it is not outer-planar. A graph is outer-flat if it can be embed…

Generalizes Seifert algorithm to integral homology spheres.

Study on elastic curves pinned at the boundary, focusing on minimizers and their interaction with obstacles.

In [20], Ros and Vergasta proved that an immersed orientable compact stable constant mean curvature surface $Σ$ with free boundary in a closed ball $B\subset\mathbb{R}^3$ must be a planar equator, a spherical cap or a surface of genus 1 with at most two boundary components. In this article, by using a modified Hersch t…

Study stable capillary hypersurfaces with planar boundaries in half-spaces and domains.