arXiv research

Every curve can fit countless rhombuses.

New Jordan Floer homology shows every curve inscribes every isosceles trapezoid.

We study a notion of "width" for Jordan curves in $\mathbb{CP}^1$, paying special attention to the class of quasicircles. The width of a Jordan curve is defined in terms of the geometry of its convex hull in hyperbolic three-space. A similar invariant in the setting of anti de Sitter geometry was used by Bonsante-Schle…

Square inscribed in a curve made of two graph functions.

Paper extends theorem on covering spaces and Jordan curves.

Similarity maps cyclic quadrilaterals onto smooth curves.

Curves inscribe rectangles with positive area.

For two disjoint rectifiable star-shaped Jordan curves (including round circles) in the asymptotic boundary of hyperbolic 3-space, if the distance (see Definition 1.8) between these two Jordan curves are bounded from above by some constant, then there exists an annulus-type area minimizing (or equivalently least area) …

Floer homology applied to inscribing rectangles into curves.

Rectangular peg problem solved for many curves.

Let C be a real-analytic Jordan curve in $R^3$. Then C cannot bound infinitely many disk-type minimal surfaces which provide relative minima of area.

Two optimization problems for Loewner energy curves and their symmetries.

Study shows continuous evolution of curves in Fréchet distance.

Square can fit inside curves close to smooth ones.

The paper finds a unique curve minimizing Loewner energy among piecewise geodesic Jordan curves.

We construct embedded minimal surfaces which are $n$-periodic in $\mathbb{R}^n$. They are new for codimension $n-2\ge 2$. We start with a Jordan curve of edges of the $n$-dimensional cube. It bounds a Plateau minimal disk which Schwarz reflection extends to a complete minimal surface. Studying the group of Schwarz refl…

Square-like quadrilaterals inscribed in space curves proven for finite total curvature.

Rectangles can fit on smooth curves, proving a theorem about Klein bottles.

This paper extends the Good Covering Theorem and Jordan Curve Theorem for proximal Alexandrov spaces.

Theorem converse to Jordan's curve theorem says that {\it if a compact set $K$ has two complementary domains in $R^{2}$, from each of which it is at every point accessible, it is a simple closed curve}. We show that the requirement of this theorem that {\it all} points of $K$ were accessible from {\it both} complementa…

For a Jordan domain in the plane the length metric space of points connected to an interior point by a curve of finite length is a CAT(0)space and Gromov hyperbolic. With respect to the cone topology, that space plus its boundary at infinity is topologically the same as the original Jordan domain.

Two proofs show that removing a loop from a plane circuit splits the plane.

We show that both, the Jordan curve theorem and the Schoenflies theorem extend to non-metric manifolds (at least in the two-dimensional context), and conclude by some dynamical applications à la Poincaré-Bendixson.

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

Polynomially inscribe 6 concyclic points into any smooth curve.

In this paper we prove that if $γ$ is a Jordan curve on $\mathbb{S}^2$ then there is a smooth curve shortening flow defined on $(0,T)$ which converges to $γ$ in $\mathcal{C}^0$ as $t\to 0^+$. Another perspective is that the level-set flow of $γ$ is smooth. This is a generalization of the author's previous work where t…

Self-affine arcs without inner weak separation are parabolic segments.

The fixed-point index of a homeomorphism of Jordan curves measures the number of fixed-points, with multiplicity, of the extension of the homeomorphism to the full Jordan domains in question. The now-classical Circle Index Lemma says that the fixed-point index of a positive-orientation-preserving homeomorphism of round…

The abstract proves polygon inscriptions in curves with specific edge ratios.

We study compact stable embedded minimal surfaces whose boundary is given by two collections of closed smooth Jordan curves in close planes of Euclidean 3-space. Our main result is a classification of these minimal surfaces, under certain natural geometric asymptotic constraints, in terms of certain associated varifold…

The study extends inscription problems to non-Euclidean geometries.

Maps continuous Riemann surfaces to complex space with specific properties.

The study examines uniqueness and non-uniqueness of minimal surfaces in hyperbolic space.

It is well known that every quasi-Fuchsian manifold admits at least one closed incompressible minimal surface, and at most finitely many of them. In this paper, for any prescribed integer $N>0$, we construct a quasi-Fuchsian manifold which contains at least $2^N$ such minimal surfaces. As a consequence, there exists so…

The paper proves that any smooth curve can have two similar inscribed rectangles.

We introduce a topological object, called hairy Cantor set, which in many ways enjoys the universal features of objects like Jordan curve, Cantor set, Cantor bouquet, hairy Jordan curve, etc. We give an axiomatic characterisation of hairy Cantor sets, and prove that any two such objects in the plane are ambiently homeo…

Study on links formed by pseudocircle arrangements, focusing on three unavoidable cases.

The Plateau-Douglas problem asks to find an area minimizing surface of fixed or bounded genus spanning a given finite collection of Jordan curves in Euclidean space. In the present paper we solve this problem in the setting of proper metric spaces admitting a local quadratic isoperimetric inequality for curves. We more…

In this paper we introduce a geometric quantity, the $r$-multiplicity, that controls the length of a smooth curve as it evolves by curve shortening flow. The length estimates we obtain are used to prove results about the level set flow in the plane. If $K$ is locally-connected, connected and compact, then the level set…

Let $\mathcal{R}$ be an open Riemann surface. In this paper we prove that every continuous function $M \to \mathbb{R}^n$, $n\ge 3$, defined on a divergent Jordan arc $M \subset \mathcal{R}$ can be approximated in the Carleman sense by conformal minimal immersions; thus providing a new generalization of Carleman's theor…

We show that for any C^0 Jordan curve C in the sphere at infinity of H^3, there exists an embedded $H$-plane P_H in H^3 with asymptotic boundary C for any H in (-1,1). As a corollary, we proved that any quasi-Fuchsian hyperbolic 3-manifold M=SxR contains an H-surface S_H in the homotopy class of the core surface S for …

We prove existence and a.e. regularity of an area minimizing soap film with a bound on energy spanning a given Jordan curve in R^3. The energy of a film is defined to be the sum of its surface area and the length of its singular branched set. The class of surfaces over which area is minimized includes images of disks, …

In this paper we find approximate solutions of certain Riemann-Hilbert boundary value problems for minimal surfaces in $\mathbb{R}^n$ and null holomorphic curves in $\mathbb{C}^n$ for any $n\ge 3$. With this tool in hand we construct complete conformally immersed minimal surfaces in $\mathbb{R}^n$ which are normalized …

The paper defines Benoist-Hulin groups and explores their properties.

Let $α$ be a polygonal Jordan curve in $\bfR^3$. We show that if $α$ satisfies certain conditions, then the least-area Douglas-Radó disk in $\bfR^3$ with boundary $α$ is unique and is a smooth graph. As our conditions on $α$ are not included amongst previously known conditions for embeddedness, we are enlarging the set…

We provide a new proof of the following inequality: the maximum curvature $k_\mathrm{max}$ and the enclosed area $A$ of a smooth Jordan curve satisfy $k_\mathrm{max}\ge \sqrt{π/A}$. The feature of our proof is the use of the curve shortening flow.

We construct Peano curves $γ: [0,\infty) \to \mathbb{R}^2$ whose "footprints" $γ([0,t])$, $t>0$, have $C^\infty$ boundaries and are tangent to a common continuous line field on the punctured plane $\mathbb{R}^2 \setminus \{γ(0)\}$. Moreover, these boundaries can be taken $C^\infty$-close to any prescribed smooth family…

Let $U(\boldsymbol r),\boldsymbol r\inΩ\subset \mathbb R^2$ be a harmonic function that solves an exterior Dirichlet problem. If all the level sets of $U(\boldsymbol r),\boldsymbol r\inΩ$ are smooth Jordan curves, then there are several geometric inequalities that correlate the curvature $κ(\boldsymbol r)$ with the ma…