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We prove that every Jordan curve in $$\mathbb {R}^2$$ inscribes uncountably many rhombi. No regularity condition is assumed on the Jordan curve.

This paper is concerned with the Riemann problem for bianalytic functions on a closed Jordan curve with a certain degree of fractality. We derive a solvability condition under which an explicit sol...

We consider paths of steepest descent, in the complex plane, for the norm of a non-constant one variable polynomial $f$. We show that such paths, starting from a zero of the logarithmic derivative of $f$ and ending in a root of $f$, draw a tree in the complex plane, and we give an upper bound estimate on their lengths. In some cases, we obtain a finer estimate that depends only on the set of roots of $f$, not on their multiplicity, and we wonder if this can be done in general. We also extend this question to finite Blaschke products for the unit disk., Comment: Several typos corrected, addition of a section on Blaschke products, 7 pages, 2 figures

In this paper an explicit numerical procedure is proposed to determine the non-fragility of parameters of controllers used to stabilize a class of time delay systems. The proposed procedure finds the largest possible circle inscribed in a stability subregion delimited by a simple closed curve in the plane of the controller parameters. The center of this circle is declared to be the controller’s least fragile parameters. The obtained theoretical results are applied and compared with some of the most relevant examples found in the literature. This illustrates that the proposed procedure improves previous results found in the literature so far.

We study the geometry of simply connected wandering domains for entire functions and we prove that every bounded connected regular open set, whose closure has a connected complement, is a wandering domain of some entire function. In particular such domain can be realized as an escaping or an oscillating wandering domain. As a consequence we obtain that every Jordan curve is the boundary of a wandering Fatou component of some entire function., Minor changes

We fix a Finsler norm F and, using the anisotropic curvature flow, we prove that in the plane the anisotropic maximum curvature $$k^F_{\max }$$ of a smooth Jordan curve is such that $$ k^F_{\max }(\gamma )\ge \sqrt{\kappa /A}$$ , where A is the area enclosed by $$\gamma $$ and $$\kappa $$ the area of the unitary Wulff shape associated to the anisotropy F.

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 moreover obtain continuity up to the boundary and interior Hölder regularity of solutions. Our results generalize corresponding results of Jost and Tomi-Tromba from the setting of Riemannian manifolds to that of proper metric spaces with a local quadratic isoperimetric inequality. The special case of a disc-type surface spanning a single Jordan curve corresponds to the classical problem of Plateau, in proper metric spaces recently solved by Lytchak and the second author.

Let $\Gamma$ be a closed smooth Jordan curve in the complex plane $\mathbb{C}$ ,G be a bounded domain in $\mathbb{C}$ with the boundary $\Gamma$ , and let $\overline{G}=G\cup\Gamma$ . We study functions that are continuous in $\mathbb{C}\setminus G$ and harmonic in $\mathbb{C}\setminus\overline{G}$ that grow more slowly than the function $|z|^2$ at $z\to\infty$ . It is shown that, if in the class of such functions there exists a solution to the overdetermined Neumann boundary value problem in which the function equals zero on $\Gamma$ and $\mu$ -almost everywhere on $\Gamma$ there exists and equals one the normal derivative of this function, then the domain G is a disk (Theorem 1). In this case, the solution is unique and it coincides up to a constant with the fundamental solution for the Laplace operator in $\mathbb{C}$ with a singularity in the center of the disk G. The proof of Theorem 1 is based on the application of the conformal mapping of the exterior of the unit disk onto the domain $\mathbb{C}\setminus \overline{G}$ . This mapping allows us to reduce the original problem for the domain $\mathbb{C}\setminus \overline{G}$ to overdetermined boundary value problem for the exterior of the disk in which the main difficulty is the heterogeneity of the boundary condition for the normal derivative. To study this condition some subtle results were required on the boundary properties of a function that performs the indicated conformal mapping as well as some properties of the Hardy classes Hp in the unit disk. Theorem 2 of the paper shows that in the general case the conditions in Theorem 1 cannot be relaxed. It states the existence of a bounded domain $G\subset\mathbb{C}$ different from a disk with a smooth Jordan boundary $\Gamma$ and functions $f_1,f_2,f_3\in C(\mathbb{C}\setminus G)$ harmonic in $\mathbb{C}\setminus\overline{G}$ for each of which exactly one of the conditions of Theorem 1 is not satisfied.

The classical Riemann mapping theorem implies that there exists a so-called Riemann mapping which takes the upper half plane onto the left domain bounded by a Jordan curve in the extended complex plane. The primary purpose of the paper is to study the basic problem: how does a Riemann mapping depend on the corresponding Jordan curve? We are mainly concerned with those Jordan curves in the Weil–Petersson class, namely, the corresponding Riemann mappings can be quasiconformally extended to the whole plane with Beltrami coefficients being square integrable under the Poincare metric. After giving a geometric characterization of a Weil–Petersson curve, we endow the space of all normalized Weil–Petersson curves with a new real Hilbert manifold structure in a geometric manner and show that this new structure is topologically equivalent to the standard complex Hilbert manifold structure, which implies that an appropriately chosen Riemann mapping depends continuously on a Weil–Petersson curve (and vice versa). This can be considered as the first result about the continuous dependence of Riemann mappings on non-smooth Jordan curves.

We study the following separation problem: Given a collection of pairwise disjoint coloured objects in the plane withkdifferent colours, compute a shortest “fence”F, i.e., a union of curves of minimum total length, that separates every pair of objects of different colours. Two objects are separated ifFcontains a simple closed curve that has one object in the interior and the other in the exterior. We refer to the problem asgeometrick-cut, as it is a geometric analog to the well-studied multicut problem on graphs. We first give an$$O(n^4\log ^3\!n)$$O(n4log3n)-time algorithm that computes an optimal fence for the case where the input consists of polygons of two colours withncorners in total. We then show that the problem is NP-hard for the case of three colours. Finally, we give a randomised$$4/3\cdot 1.2965$$4/3·1.2965-approximation algorithm for polygons and any number of colours.

Herein, the solution of partial differential equations (PDEs) using spectral methods is developed for irregular domains, which preserves their accuracy. Previously, to solve these problems, the finite differences method or the embedded domains method was typically applied. The approach presented in this article can be used for any boundary described by a Jordan curve, and the solution behavior outside the domain need not to considered. The computational process has low cost and generality because the map constructions and changing variables are unnecessary. In addition, by using the presented parametrization process, boundary conditions (boundary bordering) can be implemented conveniently, where the rectangular domains can be considered as an asymptotical case. The structure is numerically oriented, which facilitates the application of any algorithm related to spectral methods.

For embedded 2-spheres in a 4-manifold sharing the same embedded transverse sphere homotopy implies isotopy, provided the ambient 4-manifold has no $\BZ_2$-torsion in the fundamental group. This gives a generalization of the classical light bulb trick to 4-dimensions, the uniqueness of spanning discs for a simple closed curve in $S^4$ and $\pi_0(\Diff_0(S^2\times D^2)/\Diff_0(B^4))=1$. In manifolds with $\BZ_2$-torsion, one surface can be put into a normal form relative to the other.

Let $R$ be a compact surface and let $\Gamma$ be a Jordan curve which separates $R$ into two connected components $\Sigma_1$ and $\Sigma_2$. A harmonic function $h_1$ on $\Sigma_1$ of bounded Dirichlet norm has boundary values $H$ in a certain conformally invariant non-tangential sense on $\Gamma$. We show that if $\Gamma$ is a quasicircle, then there is a unique harmonic function $h_2$ of bounded Dirichlet norm on $\Sigma_2$ whose boundary values agree with those of $h_1$. Furthermore, the resulting map from the Dirichlet space of $\Sigma_1$ into $\Sigma_2$ is bounded with respect to the Dirichlet semi-norm.

Given a simple closed curve $$\gamma $$ on a connected, oriented, closed surface S of negative Euler characteristic, Mirzakhani showed that the set of points in the moduli space of hyperbolic structures on S having a simple closed geodesic of length L of the same topological type as $$\gamma $$ equidistributes with respect to a natural probability measure as $$L \rightarrow \infty $$ . We prove several generalizations of Mirzakhani’s result and discuss some of the technical aspects ommited in her original work. The dynamics of the earthquake flow play a fundamental role in the arguments in this paper.

As an application of the curve shortening flow, this paper will show an inequality for the maximum curvature of a smooth simple closed curve on surfaces.

The Loewner energy of a Jordan curve is the Dirichlet energy of its Loewner driving term. It is finite if and only if the curve is a Weil–Petersson quasicircle. In this paper, we describe cutting and welding operations on finite Dirichlet energy functions defined in the plane, allowing expression of the Loewner energy in terms of Dirichlet energy dissipation. We show that the Loewner energy of a unit vector field flow-line is equal to the Dirichlet energy of the harmonically extended winding. We also give an identity involving a complex-valued function of finite Dirichlet energy that expresses the welding and flow-line identities simultaneously. As applications, we prove that arclength isometric welding of two domains is sub-additive in the energy, and that the energy of equipotentials in a simply connected domain is monotone. Our main identities can be viewed as action functional analogs of both the welding and flow-line couplings of Schramm–Loewner evolution curves with the Gaussian free field., Geometric and Functional Analysis, 30 (1), ISSN:1016-443X, ISSN:1420-8970

In this chapter we introduce elements of the differential geometry of curves in an Euclidean space with three dimensions. We begin with the Serret-Frenet equations and their consequences and we devote a section on results related to closed curves.