In 3-dimensional Lorentz-Minkowski space we determine the number of catenoids connecting two coaxial circles in parallel planes. This study is separated according to the types of circles and the causal character (spacelike and timelike) of the catenoid.
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Harmonic Enneper Immersion in math.DG
Method constructs harmonic immersions in R^3 using Enneper-type representation.
problem Constructing harmonic immersions in R^3.
method Enneper-type representation approach.
result Any harmonic immersion in R^3 can be obtained using this approach.
Annular type surfaces with fixed boundary and with prescribed, almost constant mean curvaturemath.DG
Study proves existence and nonexistence for annular surfaces with specific curvature and boundary.
problem Existence and nonexistence of annular surfaces with prescribed mean curvature.
method Proves existence and nonexistence results for normal graphs of unduloids or nodoids.
result Existence and nonexistence results for annular type surfaces with prescribed mean curvature.
Study smoothings of singular intersections of ellipsoids.
problem Smooth singular intersections of ellipsoids.
method Generic singularities of coaxial intersections of ellipsoids studied.
result Special attention to 3D case.
The Brylinski beta function is extended for coaxial layers on submanifolds.
problem Extending the Brylinski beta function to coaxial layers on submanifolds.
method Analytic continuation and computation of residues for the function.
result The Brylinski beta function has an analytic continuation with simple poles.
Wright showed that, if a 1-ended simply connected locally compact ANR Y with pro-monomorphic fundamental group at infinity admits a proper Z-action, then that fundamental group at infinity can be represented by an inverse sequence of finitely generated free groups. Geoghegan and Guilbault strengthened that result, prov…
We construct a new family of high genus examples of free boundary minimal surfaces in the Euclidean unit 3-ball by desingularizing the intersection of a coaxial pair of a critical catenoid and an equatorial disk. The surfaces are constructed by singular perturbation methods and have three boundary components. They are …
Rotational symmetry of self-expanders to the inverse mean curvature flow with cylindrical endsmath.DG
We show that any complete, immersed self-expander to the inverse mean curvature flow, which has one end asymptotic to a cylinder, or has two ends asymptotic to two coaxial cylinders, must be rotationally symmetric.
In this article we give a criterion for the existence of a metric of curvature on a -sphere with conical singularities of prescribed angles and non-coaxial holonomy. Such a necessary and sufficient condition is expressed in terms of linear inequalities in $\vartheta_1,\dot…
Study smoothings of ellipsoid intersections with singularities.
problem Smooth singular intersections of ellipsoids.
method Study 3D manifolds with singularities as small covers of Coxeter polyhedral orbifolds.
result Introduce -pyramitoid to generalize -pyramids.
The idea of end-to-end learning of communication systems through neural network-based autoencoders has the shortcoming that it requires a differentiable channel model. We present in this paper a novel learning algorithm which alleviates this problem. The algorithm enables training of communication systems with an unkno…
This study examines geometric properties and offsets of slant timelike-ruled surfaces.
problem Geometric properties and offsets of slant timelike-ruled surfaces in Minkowski 3-space.
method Derivation of parametric formulation, conditions for coaxial alignment, examination through Blaschke and Darboux frames.
result Conditions ensuring the coaxial alignment of the central normal with the ruling direction of the offset surface.
Study local features of decorated representation spaces for spherical surfaces.
problem Local structure of moduli space of spherical surfaces with conical points.
method Analysis of decorated representation spaces of fundamental groups in SU(2).
result Smooth locus of decorated representation spaces is dense and connected.
Combinatorial p-th Calabi Flows for Total Geodesic Curvatures in hyperbolic background geometrymath.DG
The paper finds circle packings with specific curvatures in hyperbolic geometry.
problem Finding circle packings with prescribed total geodesic curvatures and discrete Gaussian curvatures.
method Established existence and rigidity via variational principle, introduced combinatorial p-th Calabi flows.
result Introduced combinatorial p-th Calabi flows to find circle packings with prescribed curvatures.
A game of packingsmath.MG
Study generates infinite circle packings with a specific property.
problem Generating infinite circle packings with a unique property.
method Investigates an infinite family of circle packings and uses them to create Apollonian packings.
result Created an infinite set of circle packings with the Apollonian property.
The paper extends Descartes' circle theorem to n-flower configurations using hyperbolic geometry.
problem Extending Descartes' circle theorem to n-flower configurations.
method Spinorial description of horospheres in hyperbolic geometry.
result An explicit equation satisfied by the curvatures of n-flower configurations.
The class of traveling wave solutions of the sine-Gordon equation is known to be in 1-1 correspondence with the class of (necessarily singular) pseudospherical surfaces in Euclidean space with screw-motion symmetry: the pseudospherical helicoids. We explicitly describe all pseudospherical helicoids in terms of elliptic…
Solves Apollonius' problem using oriented circles and inversive geometry.
problem Constructing a circle tangent to three given circles.
method Using oriented circles and inversive invariants, reversing each given circle to find solutions.
result The problem has 0, 1, or 2 solutions, depending on the configuration of given circles.
The paper explores universal circles for Anosov foliations and their uniqueness.
problem Exploring the uniqueness of universal circles for Anosov foliations.
method Using the flow space of an Anosov flow to parameterize the circle bundle at infinity of the foliations.
result Several constructions of a universal circle are typically distinct and not conjugate.
Link projections with the same circle arrangement can be transformed by specific moves.
problem Characterizing link projections based on their circle arrangements.
method Local moves to transform link projections and analyze their circle arrangements.
result Two link projections have the same circle arrangement if and only if they can be transformed into each other by certain local moves.
A Steiner chain of length k consists of k circles, tangent to two given non-intersecting circles (the parent circles) and tangent to each other in a cyclic pattern. The Steiner porism states that once a chain of k circles exists, there exists a 1-parameter family of such chains with the same parent circles that can be …
Proves rigidity of circle packings in the plane, generalizing previous work.
problem Rigidity of infinite inversive distance circle packings in the plane.
method Maximal principle for generic weighted Delaunay inversive distance circle packings and ring lemma for inversive distance circle packings in hexagonal triangulated plane.
result Proves Bowers-Stephenson's conjecture for inversive distance circle packings.
We consider circle packings and, more generally, Delaunay circle patterns - arrangements of circles arising from a Delaunay decomposition of a finite set of points - on surfaces equipped with a complex projective structure. Motivated by a conjecture of Kojima, Mizushima and Tan, we prove that the forgetful map sending …
Classifies surfaces with great and small circles through each point.
problem Identifying surfaces with specific circle properties.
method Topological classification of surfaces in 3D unit sphere.
result Surfaces are homeomorphic to five normal forms.
Study local diffeomorphisms of conformal circles in pseudo-Riemannian manifolds.
problem Understanding local diffeomorphisms of conformal circles.
method Variations of conformal circles and pseudo-Riemannian manifolds.
result Local diffeomorphisms of conformal circles are conformal local diffeomorphisms.
Boundary Value Problem and Discrete Schwarz-Pick Lemma for Generalized Hyperbolic Circle Packingsmath.DG
Paper proves a discrete Schwarz-Pick lemma for generalized circle packings.
problem Comparing geometric quantities of circle packings with different boundary values.
method Combinatorial Calabi flows and maximum principle.
result Discrete Schwarz-Pick lemma proven for generalized circle packings.
Hyperideal circle patternsmath.GT
A ``hyperideal circle pattern'' in is a finite family of oriented circles, similar to the ``usual'' circle patterns but such that the closed disks bounded by the circles do not cover the whole sphere. Hyperideal circle patterns are directly related to hyperideal hyperbolic polyhedra, and also to circle packings. …
Circle graph automorphisms match circle's and are strongly universal.
problem Identifying the automorphism group of the circle.
method Proving the circle graph's automorphism group coincides with the circle's and showing the circle graph's rational chords form a strongly universal element.
result The circle graph's automorphism group is strongly universal.
The paper studies circle packings using renormalization and subdivision rules.
problem Characterizing and proving properties of circle packings with specific subdivision rules.
method Iterations of skinning maps on Teichmüller spaces, renormalization theory, subdivision rules.
result Uniformly contracting renormalization operator and geometric inflexibility of circle packings.
Paper proves circle packings converge to Riemann mapping for Jordan domains.
problem Proving discrete conformal maps converge to Riemann mapping.
method Establishing solvability theorem for inversive distance circle packings.
result Bowers-Stephenson's conjecture for Jordan domains is proven.
Paper introduces new flows to find circle packings with specific curvature.
problem Finding circle packings with prescribed total geodesic curvatures.
method Introduces combinatorial Calabi flow, fractional combinatorial Calabi flow, and combinatorial p-th Calabi flow.
result Establishes conditions for the longtime behaviors of these flows.
Study of combinatorial Calabi flow on ideal circle patterns.
problem Finding ideal circle patterns with prescribed curvatures.
method Combinatorial Calabi flow in hyperbolic and Euclidean geometry.
result Flow converges exponentially to ideal circle patterns.
Proves existence of circle patterns on surfaces with cusps.
problem Existence of circle patterns with prescribed angles on surfaces with cusps.
method Introduced combinatorial Ricci and Calabi flows to prove longtime existence and convergence.
result Existence of generalized circle patterns with prescribed angles on surfaces with cusps.
Unique circle patterns on spheres found for spherical conical metrics.
problem Non-uniqueness in circle packing for spherical metrics.
method Prescribed geodesic total curvature instead of cone angles.
result Unique existence of circle patterns for spherical conical metrics.
Zippersmath.GT
Paper introduces 'zippers' for constructing universal circles.
problem Constructing universal circles for hyperbolic 3-manifolds.
method Introduces zippers to directly construct universal circles.
result New and direct way to construct universal circles.
Perron's method and spherical ideal circle patterns with prescribed total geodesic curvaturesmath.GT
Paper proves existence and uniqueness of circle patterns on surfaces with assigned geodesic curvatures.
problem Existence and uniqueness of circle patterns on surfaces with prescribed geodesic curvatures.
method Applied Perron's method and Thurston's algorithm to prove existence and convergence.
result Existence and uniqueness of circle patterns on surfaces with prescribed geodesic curvatures.
Study circle patterns on tori, linking symplectic forms and homeomorphisms.
problem Understanding circle patterns on tori and their symplectic properties.
method Investigates the space of circle patterns on closed tori with complex projective structures, embedding it into Teichmüller spaces and analyzing symplectic forms.
result Non-degeneracy of the pulled-back Weil-Petersson symplectic form and homeomorphism between circle patterns and Teichmüller spaces.
Study of circle arrangements related to Morse-Bott functions.
problem Understanding the geometry and singularity theory of Morse-Bott functions.
method Systematic construction of circle arrangements centered at existing circles, studying local changes in Reeb graphs.
result Reeb graphs of Morse-Bott functions are spaces of all components of preimages of single points.
The paper studies circle packings on surfaces with boundary and their total geodesic curvatures.
problem Existence and rigidity of circle packings with conical singularities.
method Variational principle and combinatorial Ricci flow.
result Existence and rigidity of circle packings with prescribed total geodesic curvature.
Large hyperbolic circlesmath.DS
Asymptotics for equidistribution of circles on hyperbolic surfaces.
problem Equidistribution of circles on hyperbolic surfaces.
method Spectral method and statistical limit theorems.
result Precise asymptotics for the rate of equidistribution of circles.
A Delaunay cell decomposition of a surface with constant curvature gives rise to a circle pattern, consisting of the circles which are circumscribed to the facets. We treat the problem whether there exists a Delaunay cell decomposition for a given (topological) cell decomposition and given intersection angles of the ci…
The paper extends the Discrete Schwarz-Pick Lemma to circle packings with obtuse intersections and disjoint packings.
problem Proving the Discrete Schwarz-Pick Lemma for circle packings with various inversive distances.
method Using a variational principle for circle packings with inversive distances, the paper extends the lemma to a broader range of packings.
result The Discrete Schwarz-Pick Lemma holds for circle packings with inversive distances in , provided an additional condition on triangle weights.
New surface class defined using osculating circles.
problem Defining a new surface class in Euclidean space.
method Using osculating circles of curves and classification of specific types.
result Classification of canal and Weingarten surfaces.
Paper extends circle pattern theory to obtuse angles.
problem Circle patterns with obtuse angles not previously covered.
method Using topological degree theory, extends Koebe-Andreev-Thurston Theorem.
result Generalized Andreev's Theorem for obtuse dihedral angles.
Study circles to understand dynamics and rigidity in homogeneous spaces.
problem Understanding dynamics and rigidity in infinite-volume homogeneous spaces.
method Addressing four questions about circle packings.
result Highlighting the interplay between dynamics, geometry, and rigidity.
Study circle families' envelopes and related curves.
problem Understanding relationships between circle families and special curves.
method Investigate envelopes of circle families and their connections to evolutes, pedals, evolutoids, and pedaloids.
result Characterized relationships between circle families and related curves.
The paper examines circle graphs of Gauss diagrams and finds counterexamples to previous descriptions.
problem Problems with previous descriptions of realizable Gauss diagrams.
method Experimental checking and formulation of new descriptions of realizable circle graphs.
result New descriptions of realizable circle graphs and an algorithm for checking realizability.
Deformation of semi-circle law for the correlated time series and Phase transitioncond-mat.stat-mech
Study on eigenvalue distribution of correlated time series deforming the semi-circle law.
problem Eigenvalue distribution of correlated time series differs from the semi-circle law.
method Analysis of Wigner random matrix with temporal correlation.
result Eigenvalue distribution converges to a deformed semi-circle law with longer tail and higher peak.