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.
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.
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.
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.
Paper constructs hyperbolic metrics using circle packings and curvature parameters.
problem Creating polyhedral metrics for surfaces of various topologies.
method Using circle packings and curvature parameters, the paper constructs hyperbolic polyhedral metrics.
result Unified approach to producing polyhedral metrics for surfaces of broader topological types.
Paper solves degenerated circle pattern metric problem in spherical geometry.
problem Existence and rigidity of (degenerated) circle pattern metrics with prescribed total geodesic curvatures.
method Defined prescribed combinatorial Ricci flows and studied their convergence.
result First degenerated result for total geodesic curvatures in spherical background geometry.
A 5-manifold fibers over a circle with nonpositive curvature.
problem Finding nonpositively curved manifolds that fiber over a circle.
method Exhibited a specific 5-manifold with nonpositive curvature.
result The fiber is a 4-manifold with nonhyperbolic fundamental group.
Constructs infinitely many examples of large manifolds with circle bundles of positive scalar curvature.
problem Existence of circle bundles over large manifolds with positive scalar curvature metrics.
method Symplectic geometry techniques.
result Infinitely many examples of macroscopically large manifolds with circle bundles of positive scalar curvature.
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.
Thurston's circle packing approximation of the Riemann Mapping (proven to give the Riemann Mapping in the limit by Rodin-Sullivan) is largely based on the theorem that any topological disk with a circle packing metric can be deformed into a circle packing metric in the disk with boundary circles internally tangent to t…
Paper resolves spherical curvature flow problem.
problem Existence of ideal circle patterns in spherical background geometry.
method Introduces a combinatorial geodesic curvature flow in spherical background geometry.
result Characterizes sufficient and necessary conditions for flow convergence.
Study free circle actions on specific 7-manifolds with positive Ricci curvature.
problem Classify and construct metrics on manifolds with a specific cohomology ring.
method Classify manifolds, construct free circle actions, and find positive Ricci curvature metrics.
result Almost all manifolds admit infinitely many non-equivalent free circle actions with positive Ricci curvature.
We obtain an asymptotic formula for the number of circles of curvature at most T in any given bounded Apollonian circle packing. For an integral packing, we obtain the upper bounds for the number of circles with prime curvature as well as of pairs of circles with prime curvatures, which are sharp up constant multiples.…
New method finds metrics on surfaces with prescribed curvatures using circle packings and surgery.
problem Finding piecewise Euclidean metrics on surfaces with prescribed combinatorial curvatures.
method Combinatorial curvature flows with surgery for inversive distance circle packings.
result Longtime existence and global convergence of combinatorial curvature flows with surgery.
The study finds surfaces with constant anisotropic mean curvature foliated by circles in Euclidean space.
problem Existence and geometric description of surfaces with constant anisotropic mean curvature.
method Analyzes surfaces with constant anisotropic mean curvature of the Dirichlet energy, proving existence and classifying them.
result Existence and geometric description of surfaces foliated by circles with zero anisotropic mean curvature.
Study on curve diffusion flows with scale-critical curvature term.
problem Analyzing stability of curve diffusion flows with scale-critical curvature.
method Introduced and studied a one-parameter family of curve diffusion flows with a scale-critical cubic curvature term. Analyzed dynamical stability of homothetic circles using variational methods.
result Established that any small perturbation of an ω-fold circle monotonically approaches the unit ω-circle after rescaling, translation, and reparametrisation. Sharp convergence rate for curvature stability in planar free elastic flow.
problem Stability of ω-circles under the planar free elastic flow. method Improved closeness measurement via curvature scalar, leading to a sharp convergence rate.
result Sharp convergence rate for curvature stability in planar free elastic flow.
Study verifies Joyce's conjectures for circle-invariant Lagrangian surfaces.
problem Verifying Joyce's conjectures for specific Lagrangian surfaces.
method Continuation of Lagrangian mean curvature flow through finite time neck pinches.
result Flow converges to a chain of special Lagrangians, verifying conjectures.
The paper proves three circles theorems and Liouville type theorems for subharmonic and holomorphic functions.
problem Establishing theorems for subharmonic and holomorphic functions on specific geometric structures.
method Using subharmonic and holomorphic functions on Riemannian manifolds and gradient shrinking Ricci solitons.
result Proves Liouville type theorems as applications of the established theorems.
Paper studies degenerated circle packings in hyperbolic geometry and finds conditions for their existence.
problem Whether a prescribed total geodesic curvature can be realized by a degenerated circle packing.
method Introduced combinatorial Ricci flow to find the desired degenerated circle packed surface, analogous to Chow-Luo and Takatsu methods.
result Fully characterized sufficient and necessary conditions for the existence of degenerated circle packings and showed their uniqueness.
The macroscopic version of Urysohn width for scalar curvature is disproven in high dimensions.
problem Disproving the macroscopic version of Gromov's Urysohn width conjecture for scalar curvature.
method Novel estimate on Urysohn width of circle bundles and a new notion of ruling for Riemannian manifolds.
result The macroscopic version of Gromov's Urysohn width conjecture for scalar curvature is false in dimensions four and above.
Riemann zero mean curvature examples in the Lorentz-Minkowski space are surfaces with zero mean curvature foliated by circles contained in parallel planes. In contrast to the Euclidean case, this family of surfaces presents new and rich features because of the variety of types of circles. In this paper, we give a geome…
Proves rigidity of certain transformations on specific geometric manifolds.
problem Rigidity of conformal circle-preserving transformations on Berwaldian manifolds.
method Analyzes properties of Berwaldian manifolds and flag curvatures.
result Rigidity condition for nontrivial conformal circle-preserving transformations.
Given a triangulated surface M, we use Ge-Xu's α-flow \cite{Ge-Xu1} to deform any initial inversive distance circle packing metric to a metric with constant α-curvature. More precisely, we prove that the inversive distance circle packing with constant α-curvature is unique if αχ(M)≤0, which generalize And…
The paper finds hyperbolic metrics on surfaces with boundary using combinatorial curvature flows.
problem Finding hyperbolic metrics on surfaces with totally geodesic boundaries of prescribed lengths.
method Introducing combinatorial Ricci flow and combinatorial Calabi flow for generalized circle packings.
result Proves longtime existence and global convergence of combinatorial curvature flows.
New method finds ideal circle patterns on spheres.
problem Finding ideal circle patterns on spheres with prescribed curvatures.
method Combinatorial Calabi flow in spherical geometry.
result Existence and convergence of the flow for ideal circle patterns.
We prove that a compact 4-manifold which supports a circle-invariant fat SO(3)-bundle is diffeomorphic to either S^4 or CP^2-bar. The proof involves studying the resulting Hamiltonian circle action on an associated symplectic 6-manifold. Applying our result to the twistor bundle of Riemannian 4-manifolds shows that S^4…
Study spherical curves with curvature dependent on distance to a great circle.
problem Understanding spherical curves with curvature dependent on distance to a great circle.
method Introducing spherical angular momentum, characterizing known curves, finding new families, and obtaining arc length parametrizations.
result New families of spherical curves with intrinsic equations in elementary or Jacobi elliptic functions.
The paper classifies manifolds with free torus actions and positive Ricci curvature.
problem Classifying manifolds with specific geometric properties.
method Using circle bundles and free torus actions on connected sums of products of spheres.
result Infinitely-many manifolds with positive Ricci curvature and torus actions are classified.
In this work we find all helicoidal surfaces in Minkowski space with constant mean curvature whose generating curve is a the graph of a polynomial or a Lorentzian circle. In the first case, we prove that the degree of the polynomial is 0 or 1 and that the surface is ruled. If the generating curve is a Lorentzian ci…
Study of harmonic maps to the circle with applications to hyperbolic 3-manifolds.
problem Understanding harmonic maps and their properties.
method Analyzing maps from 3-manifolds to the circle, proving dihedral rigidity.
result Special case of dihedral rigidity of three dimensional cubes with π/2 angles. 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.
Study curvature and torsion from cross-ratios in discrete curves.
problem Define curvature and torsion for discrete curves using cross-ratios.
method Use Möbius invariant point-insertion-rule to construct circles and express torsion using cross-ratio.
result Discrete curvature and torsion defined using cross-ratios converge to smooth curvature and torsion as sampling density increases.
Proves existence of unique circle packings on polyhedral surfaces.
problem Existence of unique circle packings on polyhedral surfaces with specified discrete curvature.
method Constructs diffeomorphism between fiber bundles, uses discrete Ricci flow and edge flipping.
result Proves existence of unique inversive distance circle packings.
Survey on discrete minimal surfaces and their properties.
problem Discretizing minimal surfaces in Euclidean space.
method Polyhedral surfaces with parallel face offsets and circle patterns.
result All simply connected discrete minimal surfaces can be constructed from circle patterns.
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.
For a circle packing P on the sphere invariant under a geometrically finite Kleinian group, we compute the asymptotic of the number of circles in P of spherical curvature at most T which are contained in any given region.
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.
Explains examples of Lagrangian flow with circle symmetry.
problem Understanding Lagrangian flow with symmetry.
method Examining specific examples in C2. result Shows various types of flow, including compact and non-compact.
Study curvature flows on pinched Hadamard surfaces, proving convexity preservation and convergence.
problem Preserving convexity and convergence of curves under curvature flows on pinched Hadamard surfaces.
method Area- and length-preserving curvature flows, refined comparison arguments, delicate curvature estimates.
result Convexity is preserved and curves converge to a geodesic circle under certain conditions.
The paper solves circle packings on surfaces with boundaries.
problem Circle packing on surfaces with boundaries and finite genus.
method Using Thurston's algorithm and discrete Schwarz-Pick lemma.
result A unique solution to the boundary value problem exists.
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.
Study a flow preserving area of plane curves, ending in a circle.
problem Preserving area while deforming plane curves.
method Non-local flow of convex closed plane curves.
result Limiting curve is a circle in the C∞ metric. We prove that a closed immersed plane curve with total curvature 2πm has entropy at least m times the entropy of the embedded circle, as long as it generates a type I singularity under the curve shortening flow (CSF). We construct closed immersed plane curves of total curvature 2πm whose entropy is less than m …
The classical Hadamard three circle theorem is generalized to complete Kähler manifolds. More precisely, we show that the nonnegativity of the holomorphic sectional curvature is a necessary and sufficient condition for the three circle theorem. As corollaries, two sharp monotonicity formulae for holomorphic functions a…
In this work, we study spacelike surfaces in Minkowski space E13 foliated by pieces of circles and that satisfy a linear Weingarten condition of type aH+bK=c, where a,b and c are constant and H and K denote the mean curvature and the Gauss curvature respectively. We show that such surfaces must be surfa…
This paper investigates circle patterns with obtuse exterior intersection angles on surfaces of finite topological type. We characterise the images of the curvature maps and establish several equivalent conditions regarding long time behaviors of Chow-Luo's combinatorial Ricci flows for these patterns. As consequences,…
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.