Paper extends theorem on covering spaces and Jordan curves.
problem Covering and extending theorems for Alexandrov spaces.
method Introduces proximal homotopic cycles to extend the Mitsuishi-Yamaguchi theorem.
result Extensions of the Mitsuishi-Yamaguchi Good Covering Theorem and Jordan curve theorem.
Two curves in hyperbolic space share a bounded distance, leading to a minimizing surface.
problem Finding a surface minimizing area between two disjoint curves in hyperbolic space.
method Analyzing the asymptotic boundary of hyperbolic 3-space, applying Definition 1.8 for distance bounds, and proving Theorems 1.7 and 1.11.
result Existence of an area-minimizing surface between two disjoint curves with bounded distance.
Theorem converse to Jordan's curve theorem says that {\it if a compact set K has two complementary domains in R2, 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…
This paper extends the Good Covering Theorem and Jordan Curve Theorem for proximal Alexandrov spaces.
problem Extending the Good Covering Theorem and Jordan Curve Theorem to proximal Alexandrov spaces.
method Introducing path cycles and using them to extend the Good Covering Theorem and Jordan Curve Theorem.
result Extensions of the Mitsuishi-Yamaguchi Good Covering Theorem and Jordan Curve Theorem for proximal Alexandrov spaces.
Similarity maps cyclic quadrilaterals onto smooth curves.
problem Mapping cyclic quadrilaterals onto smooth Jordan curves.
method Uses the theorem of Polterovich and Viterbo.
result Existence of orientation-preserving similarity.
Two proofs show that removing a loop from a plane circuit splits the plane.
problem Proving the Weak Jordan Theorem about plane circuits.
method Detailed presentation of Thomassen's and Filippov's proofs.
result The complement of any loop in a plane circuit is disconnected.
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.
Rectangles can fit on smooth curves, proving a theorem about Klein bottles.
problem Fitting rectangles on smooth curves.
method Shevchishin's theorem about Klein bottle embeddings.
result Similar rectangles can be placed on smooth Jordan curves.
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…
Every curve can fit countless rhombuses.
problem Finding many rhombi within any curve.
method No curve regularity assumed.
result Uncountably many rhombi fit every curve.
New Jordan Floer homology shows every curve inscribes every isosceles trapezoid.
problem Understanding inscriptions of isosceles trapezoids in Jordan curves.
method Constructing a new Lagrangian Floer homology chain complex.
result Establishes new cases of non-smooth Jordan curves inscribing isosceles trapezoids.
The study extends inscription problems to non-Euclidean geometries.
problem Generalizing inscription problems to non-Euclidean geometries.
method Symplectic and Riemannian geometry techniques.
result Proved generalized inscription theorems for hyperbolic and spherical surfaces.
The study shows that inscribed rectangles in smooth curves cover at least one third of all possible aspect ratios.
problem Determining the coverage of inscribed rectangles in smooth Jordan curves.
method Analyzing sets of disjoint homologically nontrivial projective planes and applying Kemperman's theorem.
result The Lebesgue measure of the set of aspect ratios is at least 1/3.
Polynomially inscribe 6 concyclic points into any smooth curve.
problem Inscribing points into smooth curves using polynomials.
method Combines Fukaya-Irie theorem and Floer homology computation.
result Supports conjecture about polynomial inscriptions.
We study a notion of "width" for Jordan curves in CP1, 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.
problem Finding inscribed squares in curves formed by graph functions.
method Analysis of spectral invariants of Jordan Floer homology under curve perturbations.
result Existence of inscribed squares in curves with specific Lipschitz constants.
Curves inscribe rectangles with positive area.
problem Finding angles for inscribing rectangles within Jordan curves.
method Proving existence of a subset of angles with measure at least A/R^2.
result Angles inscribing rectangles have measure at least A/R^2.
Let R be an open Riemann surface. In this paper we prove that every continuous function M→Rn, n≥3, defined on a divergent Jordan arc M⊂R can be approximated in the Carleman sense by conformal minimal immersions; thus providing a new generalization of Carleman's theor…
Floer homology applied to inscribing rectangles into curves.
problem Determining if a Jordan curve can inscribe a square.
method Constructing Floer homology from inscribed rectangles and using spectral invariants.
result A Jordan curve inscribes a square if its enclosed area exceeds half a circle's area.
Rectangular peg problem solved for many curves.
problem Rectangular peg problem for continuous Jordan curves.
method Microlocal sheaf theory and recent work of Greene and Lobb.
result Affirmative answer for a large class of rectifiable curves.
Let C be a real-analytic Jordan curve in R3. Then C cannot bound infinitely many disk-type minimal surfaces which provide relative minima of area.
Normal forms and moduli stacks for flat connections on complex manifolds.
problem Understanding singular flat connections on complex manifolds.
method Introducing homogeneous Lie groupoids and studying their representation theory to prove normal form theorems and moduli space structures.
result Moduli spaces of singular flat connections admit the structure of algebraic quotient stacks.
Two optimization problems for Loewner energy curves and their symmetries.
problem Optimizing Jordan curves and positive curves on boundary spaces.
method Using conformal welding and Möbius transformations.
result Symmetries between boundary spaces and pleated planes.
Paper extends Brouwer Fixed Point Theorem with amiable and almost amiable fixed sets.
problem Extending the Brouwer Fixed Point Theorem to approximate fixed sets.
method Introducing shape boundary regions in CW spaces as amiable and almost amiable fixed subsets of dpc maps.
result Variation of Jordan Curve Theorem and Fixed Cell Complex Theorem.
Study shows continuous evolution of curves in Fréchet distance.
problem Continuous evolution of curves under curvature flow.
method Curvature flow and level-set flow, analyzed in Fréchet distance.
result Evolution of curves depends continuously on initial curve.
Square can fit inside curves close to smooth ones.
problem Finding inscribed squares in nearly smooth curves.
method Using curvature and a map to relate curves, proving existence of inscribed squares.
result Curves close to smooth ones contain inscribed squares.
The paper finds a unique curve minimizing Loewner energy among piecewise geodesic Jordan curves.
problem Finding a unique curve minimizing Loewner energy among piecewise geodesic Jordan curves.
method Defining a complex projective structure and using accessory parameters to characterize the curve.
result The accessory parameters are the residues of the quadratic differential comparing the projective structure to the trivial one.
Study Schiffer operators on Riemann surfaces, linking conformal and topological invariants.
problem Investigate Schiffer operators on Riemann surfaces and their connections to conformal and topological invariants.
method Develop calculus for Schiffer and Cauchy operators, derive index theorems, and characterize kernels and images.
result Derive index theorems for Schiffer operators, connecting conformal invariants to topological invariants.
We construct embedded minimal surfaces which are n-periodic in Rn. They are new for codimension n−2≥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.
problem Square-peg problem in space curves.
method Local curvature analysis and limiting argument on approximating curves.
result Every embedded curve with finite total curvature has an inscribed square-like quadrilateral.
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.
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…
The problem of which Gauss diagram can be realized by knots is an old one and has been solved in several ways. In this paper, we present a direct approach to this problem. We show that the needed conditions for realizability of a Gauss diagram can be interpreted as follows "the number of exits = the number of entrances…
The problem of which Gauss diagram can be realized by plane curves is an old one and has been solved in several ways. In this paper, we present a direct approach to this problem. We show that needed conditions for realizability of a Gauss diagram can be interpreted as follows "the number of exits = the number of entran…
Study the boundaries of ε-neighborhoods of planar sets, showing their structure and curvature.
problem Understanding the structure and smoothness of boundaries of ε-neighborhoods of planar sets.
method Analyzing the global topological structure and smoothness of boundaries of ε-neighborhoods of compact planar sets.
result The boundary of ε-neighborhoods can be expressed as a disjoint union of Jordan curves and singularities.
In this paper we prove that if γ is a Jordan curve on S2 then there is a smooth curve shortening flow defined on (0,T) which converges to γ in C0 as t→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.
problem Characterizing self-affine Jordan arcs without parabolic segments.
method Analyzing the weak separation property and proving implications for arc types.
result Self-affine Jordan arcs without parabolic segments are attractors of multizippers.
The abstract proves polygon inscriptions in curves with specific edge ratios.
problem Proving the existence of polygons inscribed in Jordan curves with prescribed edge ratios.
method Using the properties of differentiable curves and proportional side lengths.
result Existence of polygons inscribed in Jordan curves with prescribed 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…
Motivated by the ideas and methods used by Naitoh in the consideration of parallel totally real submanifolds in complex space forms, the author of the present paper successfully makes use of the so called Jordan triple and (restricted) structure Lie algebra associated with a given Jordan algebra to establish a one-to-o…
Maps continuous Riemann surfaces to complex space with specific properties.
problem Embedding Riemann surfaces into complex space with controlled poles and boundaries.
method Continuous map with specified properties, including effective poles and Hausdorff dimension constraints.
result Existence of Jordan curves in the image of the map, each of Hausdorff dimension one.
The study examines uniqueness and non-uniqueness of minimal surfaces in hyperbolic space.
problem Uniqueness and non-uniqueness of minimal surfaces in hyperbolic space.
method Analyzes criteria for uniqueness and constructs examples of non-uniqueness.
result Uniqueness of minimal surfaces is equivalent to uniqueness in a smaller class of stable minimal disks.
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 2N such minimal surfaces. As a consequence, there exists so…
The paper proves that any smooth curve can have two similar inscribed rectangles.
problem Finding two similar inscribed rectangles in a smooth Jordan curve.
method Lagrangian Floer homology and differential topological computation.
result Generic doubling of inscribed rectangles in smooth Jordan curves.
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…
Simple closed curves in ε-boundaries separate sets in the plane.
problem Separating sets with simple closed curves in ε-boundaries.
method Analyzing ε-boundaries of planar sets and proving the existence of simple closed curves.
result Simple closed curves in ε-boundaries separate sets in the plane.
Study on links formed by pseudocircle arrangements, focusing on three unavoidable cases.
problem Counting non-equivalent positive oriented links with pseudocircle arrangements as shadows.
method Analyzing three unavoidable arrangements of pseudocircles to estimate the number of non-equivalent links.
result Sharp estimates on the number of non-equivalent positive oriented links for the three unavoidable arrangements.
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…