Hot spots conjecture proven for small eigenvalue domains.
problem Hot spots conjecture for hyperbolic planar domains with small eigenvalues.
method Proved a variant of Rauch's hot spots conjecture.
result Second Neumann Laplace eigenfunctions have no interior critical points on large convex domains.
Study visibility properties of Kobayashi distance on unbounded domains.
problem Understanding visibility properties of Kobayashi distance on unbounded domains.
method Analyzing visibility properties in the context of Kobayashi hyperbolic domains, focusing on unbounded domains and their boundary behavior.
result Carathéodory-type extension theorem for biholomorphisms between planar domains, including infinitely-connected domains.
Sharp bounds for spanning tree entropy in planar lattices.
problem Estimating spanning tree entropy in planar lattice graphs.
method Using hyperbolic geometry and polyhedra volumes.
result Proved bounds are easy to compute and provide excellent estimates.
New inequalities for planar convex domains' Laplacian eigenvalues.
problem Neumann eigenvalues of the Laplacian on planar convex domains.
method Established two new universal inequalities.
result New inequalities for Laplacian eigenvalues on convex domains.
One-parameter hyperbolic planar motion was first studied by S. Yu¨ce and N. Kuruog~lu. Moreover, they analyzed the relationships between the absolute, relative and sliding velocities of one-parameter hyperbolic planar motion as well as the related pole curves, \cite{Yuc}. One-paramete…
The aim of this paper is to clarify the relationship between Gromov-hyperbolicity and amenability for planar maps.
We define and study a Möbius invariant energy associated to planar domains, as well its generalization to space curves. This generalization is a Möbius version of Banchoff-Pohl's notion of area enclosed by a space curve. A relation with Gauss-Bonnet theorems for complete surfaces in hyperbolic space is also described.
Study on hyperbolic knotoids, proving their volumes add and providing tables.
problem Defining and studying hyperbolic knotoids.
method Definitions and proofs for hyperbolicity of spherical and planar knotoids, including volume calculations.
result Volumes of hyperbolic spherical knotoids add and rational knotoids have least volume.
In this paper we give two examples of sequences of embedded minimal planar domains in R3 which converge to singular laminations of R3. In contrast with the situation for embedded minimal disks, these examples do not arise from complete embedded minimal planar domains and highlight some of the su…
Study on planar graphs in Poincare model of hyperbolic geometry.
problem Investigating Morse flows on a 2-disk using planar graphs.
method Using planar graphs and spherical graphs to describe topological structures.
result Listed all planar graphs with at least 3 edges and described those with 4 edges.
Paper extends tree bijection for hyperbolic surfaces without requiring cusps.
problem Computing volumes and distances on hyperbolic surfaces without cusps.
method Extend tree bijection to half-tight cylinders, using Busemann function.
result Tree bijection can now be applied to surfaces without cusps.
Estimates the index of the Laplace operator on planar domains with Robin boundary condition.
problem Index estimates for planar domains with Robin boundary condition
method Combines conformal and spectral techniques with topology of the domain.
result Lower bounds for the index in terms of the number of boundary components.
We initiate the study of the higher-order Escobar constants Ik(M), k≥3, on bounded planar domains M. The Escobar constants Ik of the unit disk and a family of polygons are provided.
In this paper, by the method of moving planes, we establish the monotonicity and symmetry properties of convex solutions for Monge-Ampere systems on bounded smooth planar domains.
We establish a sharp geometric constant for the upper bound on the resonance counting function for surfaces with hyperbolic ends. An arbitrary metric is allowed within some compact core, and the ends may be of hyperbolic planar, funnel, or cusp type. The constant in the upper bound depends only on the volume of the cor…
Study of pursuit-evasion game on sphere and its relation to planar Apollonius circle.
problem Analyzing pursuit-evasion game on a sphere and its properties.
method Extending classical planar pursuit-evasion game to spherical geometry, studying equilibrium intercept points and their relation to Apollonius domain.
result Condition for intercept point to belong to Apollonius domain on sphere, analogous to planar game.
We study the curvature flow of planar nonconvex lens-shaped domains, considered as special symmetric networks with two triple junctions. We show that the evolving domain becomes convex in finite time; then it shrinks homothetically to a point. Our theorem is the analog of the result of Grayson for curvature flow of clo…
In \cite{Mul} one-parameter planar motion was first introduced and the relations between absolute, relative, sliding velocities (and accelerations) in the Euclidean plane E2 were obtained. Moreover, the relations between the Complex velocities one-parameter motion in the Complex plane were provided by \cite…
We prove that any non-simply connected planar domain can be properly and minimally embedded in H^2 x R. The examples that we produce are vertical bi-graphs, and they are obtained from the conjugate surface of a Jenkins-Serrin graph.
We give a necessary and sufficient condition for a hyperbolic Coxeter group with planar nerve to have Sierpiński curve as its Gromov boundary.
Isothermic tori with one planar curvature line found and characterized.
problem Classifying isothermic tori with specific curvature lines.
method Complex analytic methods and explicit theta function formulas.
result Explicit formulas for family of plane curves and their relation to hyperbolic elastica.
The Blaschke rolling disk theorem is extended to non-convex domains.
problem Classical inclusion principle for non-convex domains.
method Geometric conditions based on curvature, algorithm for decomposition.
result Necessary and sufficient conditions for rolling disks in non-convex domains.
The study finds lower bounds for the first eigenvalue of the Laplacian in planar domains with magnetic fields.
problem Finding lower bounds for the first eigenvalue of the Laplacian in planar domains with magnetic fields.
method Analyzing the spectrum of the Laplacian with magnetic Neumann boundary conditions, focusing on multiply connected domains with convex curves. Lower bounds are derived based on geometric invariants such as area, perimeter, diameter, and fluxes around inner holes.
result Sharp lower bounds for the first eigenvalue are derived for doubly connected domains and domains with an arbitrary number of holes, and a lower bound is obtained for Aharonov-Bohm operators with an arbitrary number of poles when holes shrink to points.
In this note, we exhibit infinite families of tight non-fillable contact manifolds supported by planar open books with vanishing Heegaard Floer contact invariants. Moreover, we also exhibit an infinite such family where the supported manifold is hyperbolic.
The study finds counterexamples to a conjecture about incompressible planar surfaces in hyperbolic link exteriors.
problem Finding counterexamples to a conjecture about incompressible planar surfaces in hyperbolic link exteriors.
method Constructing examples of hyperbolic links and analyzing their exteriors to find incompressible spanning planar surfaces.
result Examples of 3-component hyperbolic links with exterior containing incompressible spanning planar surfaces with nonmeridional and nonintegral boundary slopes.
The paper discusses fractional Sobolev immersions of flat domains into 3D space.
problem Developing C1 regularity and isometric immersions of flat domains with fractional Sobolev regularity. method Analysis of weak Codazzi-Mainardi equations, study of $W^{2, rac2s}$ planar deformations, and properties of the distributional Jacobian determinant.
result Generalization of isometric immersions with local fractional Sobolev regularity.
Proves convergence groups on a 2-sphere are Kleinian groups.
problem Proving convergence groups on a 2-sphere are Kleinian groups.
method Analyzing relatively hyperbolic groups with planar boundaries and applying to various versions of the Cannon conjecture.
result Proves relatively hyperbolic groups with planar boundaries are virtually Kleinian.
Upper bounds for magnetic Laplacian eigenvalues on planar domains.
problem Estimating the ground state energy of magnetic Laplacian on planar domains.
method Gauge invariance, flux analysis, and Cheeger-type constants.
result Upper bounds on the ground state energy depending on the ratio of holes to area, with sharpness and optimality conditions.
Infinite-genus surfaces have many isospectral hyperbolic structures.
problem Finding many isospectral hyperbolic structures on infinite-genus surfaces.
method Constructing families of isospectral hyperbolic structures on infinite-type surfaces without planar ends.
result Uncountable families of isospectral and quasiconformally distinct hyperbolic structures on infinite-genus surfaces with self-similar end spaces.
Reconstructing a planar domain from its Dirichlet-to-Neumann data
problem Reconstructing a planar domain from its Dirichlet-to-Neumann data
method Using the Hilbert transform of the boundary curve
result Reconstructing a simply connected planar domain from the DN data
The paper finds conditions for graphs with constant mean curvature in hyperbolic space.
problem Finding conditions for graphs with constant mean curvature in hyperbolic space.
method Analyzing geodesic curvature and bounding conditions for graphs in hyperbolic space.
result Conditions for existence of H-graphs with constant mean curvature in hyperbolic space. In 1997, Collin proved that any properly embedded minimal surface in R3 with finite topology and more than one end has finite total Gaussian curvature. Hence, by an earlier result of Lopez and Ros, catenoids are the only non-planar, non-simply connected, properly embedded, minimal planar domains in $\mathbb…
The paper classifies vertices in planar polygons formed by convex domains.
problem Classifying vertices in planar polygons formed by convex domains.
method Analyzing polygons formed by homothets and translates of a convex domain.
result The number of singular boundary points in a C-polygon is between n and 2(n−1)+m for a strictly convex domain with m singular boundary points. Study of group boundaries and subgroup properties.
problem Characterizing boundaries of relatively hyperbolic group pairs.
method Analyzing Bowditch boundaries and convergence group actions.
result Rigidity of group pairs leads to specific subgroup properties.
The Fridman function is bounded by the injectivity radius for certain hyperbolic manifolds.
problem Bounding the Fridman function for hyperbolic manifolds.
method Analyzing the relationship between the Fridman function and the injectivity radius function.
result The Fridman function is bounded above by the injectivity radius function for certain hyperbolic manifolds.
In this expository note, we illustrate phenomena and conjectures about boundaries of hyperbolic groups by considering the special cases of certain amalgams of hyperbolic groups. While doing so, we describe fundamental results on hyperbolic groups and their boundaries by Bowditch and Haissinsky, along with special treat…
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.
An ideal triangulation T of a hyperbolic 3-manifold M with one cusp is non-peripheral if no edge of T is homotopic to a curve in the boundary torus of M. For such a triangulation, the gluing and completeness equations can be solved to recover the hyperbolic structure of M. A planar project…
Defines W-volume for planar domains with circular boundaries, relating to Laplacian determinant and Schottky uniformization.
problem Determining the volume of conformal metrics on planar domains with circular boundaries.
method Extending Epstein maps to conformal metrics, defining W-volume, using Schottky uniformization and Loewner energy.
result Shows a bound on the renormalized volume of Schottky uniformization and provides a realization of Loewner energy.
New bounds on inscribed triangles in arbitrary planar domains.
problem Finding inscribed triangles in arbitrary planar domains with specific angle constraints.
method Proving the existence of uniformly fat triangles and not-too-fat triangles in bounded open sets.
result Existence of a maximal number Θ (between 0 and 60) for inscribed triangles with angles ≥ Θ degrees.
We deduce from a rooted tree in the disk a slalom divide and a slalom knot. A slalom knot is either the local link of a simple plane curve singularity of type A_2n, E_6, E_8 or a fibered hyperbolic knot with very special monodromy.
We study an explicit construction of planar open books with four binding components on any three-manifold which is given by integral surgery on three component pure braid closures. This construction is general, indeed any planar open book with four binding components is given this way. Using this construction and resul…
New proof shows not all Salem numbers are growth rates of Coxeter groups.
problem Identifying growth rates of Coxeter groups using Salem numbers.
method New proof using spectral radii and Coxeter transformations.
result Not every Salem number is a growth rate of hyperbolic Coxeter groups.
Study stable capillary hypersurfaces with planar boundaries in half-spaces and domains.
problem Characterize stable capillary hypersurfaces with planar boundaries in bounded domains.
method Analyzes hypersurfaces in half-spaces and domains bounded by hyperplanes, proving conditions for stability and shape.
result Stable hypersurfaces in certain domains are spherical caps or pieces of spheres.
These notes outline recent developments in classical minimal surface theory that are essential in classifying the properly embedded minimal planar domains M in R^3 with infinite topology (equivalently, with an infinite number of ends). This final classification result by Meeks, Perez, and Ros states that such an M must…
The paper characterizes sets with infinite hyperbolic convex hull volume.
problem Characterizing sets with infinite hyperbolic convex hull volume.
method Geometric conditions and self-similar sets.
result Characterizes continua and planar self-similar sets with infinite hyperbolic convex hull volume.
Random hyperbolic surfaces with punctures converge to the Brownian sphere.
problem Understanding the geometry of random hyperbolic surfaces with punctures.
method Rescaling and encoding via plane trees with continuous labels.
result Rescaled random hyperbolic surfaces converge to the Brownian sphere.
Fix a finite set of points in Euclidean n-space $\euc^n$, thought of as a point-cloud sampling of a certain domain $D\subset\euc^n$. The Rips complex is a combinatorial simplicial complex based on proximity of neighbors that serves as an easily-computed but high-dimensional approximation to the homotopy type of D. …