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.
Paper solves long-standing problem of infinite ideal polyhedra in hyperbolic space.
problem Characterize infinite ideal polyhedra in hyperbolic 3-space.
method Introduced combinatorial Ricci flow for infinite ideal circle patterns.
result Proved characterization of infinite ideal circle patterns under specific conditions.
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.
The flow converges without Kähler-Einstein and develops ideal sheaves.
problem Analyzing convergence of inverse Monge-Ampere flow without Kähler-Einstein metrics.
method Generalizing the flow and providing conditions for convergence and ideal sheaves development.
result The flow converges without Kähler-Einstein metrics and develops Nadel multiplier ideal sheaves.
New method simplifies ideal curve flow with length constraint.
problem Analyzing ideal curve flow with length constraint.
method Introduced length constraint to simplify sixth order curvature flow.
result Flow exists for all time and converges to a round circle.
New jellyfish found in various flows.
problem Existence of geometrically distinct shapes in flows.
method Analyzing elastic, curve diffusion, and ideal flows.
result Infinitely many distinct shapes discovered.
Given a general pseudo-Anosov flow in a three manifold, the orbit space of the lifted flow to the universal cover is homeomorphic to an open disk. We compactify this orbit space with an ideal circle boundary. If there are no perfect fits between stable and unstable leaves and the flow is not topologically conjugate to …
The paper extends circle pattern flows to hyperbolic and Euclidean geometry.
problem Extending circle pattern flows to hyperbolic and Euclidean geometry.
method Proving the existence and exponential convergence of combinatorial Calabi flows for ideal circle patterns.
result The solution to combinatorial Calabi flows converges exponentially fast to a flat cone metric.
In this article we obtain a simple topological and dynamical systems condition which is necessary and sufficient for an arbitrary pseudo-Anosov flow in a closed, hyperbolic three manifold to be quasigeodesic. Quasigeodesic means that orbits are efficient in measuring length up to a bounded multiplicative distortion whe…
Study shows universal circle isomorphic to flow space ideal boundary.
problem Understanding foliations and pseudo-Anosov flows in 3-manifolds.
method Analyzing depth-one foliations and pseudo-Anosov flows in atoroidal 3-manifolds.
result Universal circle is isomorphic to flow space ideal boundary.
Multiplier ideal sheaves are constructed as obstructions to the convergence of the Kähler-Ricci flow on Fano manifolds, following earlier constructions of Kohn, Siu, and Nadel, and using the recent estimates of Kolodziej and Perelman
The paper studies ideal flows of closed curves, classifying critical points and proving flow behavior.
problem Analyzing the generalised ideal flow of closed planar curves.
method Completely classifies critical points and proves properties of the m-ideal flow. result For m>1, the m-ideal flow of closed curves converges to a round multiply-covered circle. Proves unique hyperbolic metric for 3-manifolds with ideal triangulation.
problem Proving a unique hyperbolic metric for 3-manifolds with specific triangulations.
method Combining combinatorial Ricci flow with ideal triangulation for pseudo 3-manifolds.
result Extended Ricci flow converges to the hyperbolic metric exponentially fast.
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.
The purpose of this paper is to calculate the support of the multiplier ideal sheaves derived from the Kähler-Ricci flow on certain toric Fano manifolds with large symmetry. The early idea of this paper has already been in Appendix of \cite{futaki-sano0711}.
S. K. Donaldson asked whether the lower bound of the Calabi functional is achieved by a sequence the normalized Donaldson-Futaki invariants. We answer the question for the Ricci curvature formalism, in place of the scalar curvature. The principle is that the stability indicator is optimized by the multiplier ideal shea…
On certain del Pezzo surfaces with large automorphism groups, it is shown that the solution to the Kähler-Ricci flow with a certain initial value converges in C∞-norm exponentially fast to a Kähler-Einstein metric. The proof is based on the method of multiplier ideal sheaves.
Paper proves Luo's conjecture for 3D triangulated manifolds.
problem Finding hyperbolic metrics on compact 3-manifolds with boundary.
method Introduced and extended combinatorial Ricci flow to handle singularities.
result Proved Luo's conjecture affirmatively for ideal triangulations.
In this note we construct Nadel multiplier ideal sheaves using the Ricci flow on Fano manifolds. This extends a result of Phong, Sesum and Sturm. These sheaves, like their counterparts constructed by Nadel for the continuity method, can be used to obtain an existence criterion for Kahler-Einstein metrics.
Classifies pseudo-Anosov flows on 3-manifolds up to orbit equivalence.
problem Classifying pseudo-Anosov flows on 3-manifolds up to orbit equivalence.
method Generalized Anosov-like actions on bifoliated planes, ideal boundary analysis.
result Pseudo-Anosov flows on 3-manifolds are determined up to orbit equivalence by their ideal boundary actions.
The paper studies the combinatorial p-th Calabi flow for finite and infinite circle patterns.
problem Establishing convergence and long-time existence of the combinatorial p-th Calabi flow.
method Combinatorial p-th Calabi flow for finite and infinite ideal circle patterns.
result Sharp criterion for convergence in finite case and long-time existence in infinite case for p≥2. The paper proves a combinatorial Ricci flow converges to hyperbolic structures on certain 3-manifolds.
problem Proving convergence of combinatorial Ricci flow to hyperbolic structures.
method Combinatorial Ricci flow on closed pseudo 3-manifolds with specific edge valences.
result Existence and uniqueness of a complete hyperbolic metric with totally geodesic boundary.
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.
This paper contains a generalization of the convex ideal case of the Thurston-Andreev theorem when the genus is greater than 1. The heart of the paper concerns taking formal angle data on a surface and ``conformally flowing'' this formal angle data to uniquely associated uniform angle data. This flow turns out to be th…
New criteria for ideal circle patterns on surfaces.
problem Determining when a surface supports ideal circle patterns.
method Introducing a character L(D,Φ) and using combinatorial Ricci flows. result Simpler and more easily verifiable criteria for ideal circle patterns.
The article confirms Thurston's conjecture for a specific class of 3-manifolds using combinatorial Ricci flow.
problem Thurston's triangulation conjecture for hyperbolic 3-manifolds.
method Combinatorial Ricci flow (CRF) with specific conditions and techniques to handle intrinsic difficulties.
result A class of 3-manifolds admits a unique complete hyperbolic metric with totally geodesic boundary.
Paper extends port-Hamiltonian model to include internal energy for compressible and incompressible flow.
problem Modeling fluid flow dynamics with internal energy and constraints.
method Derived port-Hamiltonian model using interconnection maps and added internal energy and constraint forces.
result Model accurately represents both compressible and incompressible fluid flow.
Paper develops a new fluid flow model with energy exchange through boundaries.
problem Modeling ideal fluid flow with energy exchange through boundaries.
method Port-Hamiltonian model based on Stokes-Dirac structures.
result Wide range of fluid dynamical systems can be achieved with this model.
We study the blowup behavior at infinity of the normalized Kahler-Ricci flow on a Fano manifold which does not admit Kahler-Einstein metrics. We prove an estimate for the Kahler potential away from a multiplier ideal subscheme, which implies that the volume forms along the flow converge to zero locally uniformly away f…
By proving precisely which singularity index lists arise from the pair of invariant foliations for a pseudo-Anosov surface homeomorphism, Masur and Smillie determined a Teichmüller flow invariant stratification of the space of quadratic differentials. In this final paper of a three-paper series, we give a first step to…
We consider the evolution of a compact segment of an analytic curve on the unit tangent bundle of a finite volume hyperbolic n-manifold under the geodesic flow. Suppose that the curve is not contained in a stable leaf of the flow. It is shown that under the geodesic flow, the normalized parameter measure on the curve…
The study identifies conjugate and cut points in ideal fluid motion configurations.
problem Understanding stability and re-convergence of fluid configurations.
method Existence and non-existence of conjugate points in specific fluid configurations, using geometric and physical analysis.
result Existence of conjugate points in Kolmogorov flows and non-existence in Arnold steady states.
Paper finds sparse representation of functions using inverse scale space flow.
problem Finding sparse representation of L2 functions. method Inverse scale space flow to minimize L2 loss. result Convergence to optimal solution in ideal and noisy cases.
A large class of variational equations for geometric objects is studied. The results imply conformal monotonicity and Liouville theorems for steady, polytropic, ideal flow, and the regularity of weak solutions to generalized Yang-Mills and Born-Infeld systems.
Investigates fluid flow perturbations using geometric theory.
problem Analyzing linear perturbations in non-equilibrium fluid flows.
method Uses second order variations of the action and Jacobi fields.
result Demonstrates numerical simulations of perturbation dynamics.
The paper applies combinatorial Ricci flows to prove hyperbolic structures on 3-manifolds.
problem Proving the existence of hyperbolic structures on 3-manifolds with cusps.
method Combinatorial Ricci curvature flow methods to study pseudo 3-manifolds and ideal triangulations.
result The extended Ricci flow converges to a decorated hyperbolic polyhedral metric if and only if there exists a zero Ricci curvature metric.
In this paper we use a gradient flow to deform closed planar curves to curves with least variation of geodesic curvature in the L2 sense. Given a smooth initial curve we show that the solution to the flow exists for all time and, provided the length of the evolving curve remains bounded, smoothly converges to a mult…
The paper explores the geometric properties of fluid flows and their symmetries.
problem Understanding the geometric properties of fluid flows and their symmetries.
method Analyzing the Euler equation and its relation to geodesic flows on groupoids of multiphase diffeomorphisms.
result Generalized flows, multiphase fluids, and vortex sheets are all geodesics on certain groupoids of multiphase diffeomorphisms.
The paper introduces combinatorial Calabi flows to find hyperbolic metrics on surfaces with boundary.
problem Finding hyperbolic metrics on surfaces with totally geodesic boundaries of given lengths.
method Introducing combinatorial Calabi flows and proving their long time existence and global convergence.
result Proves the long time existence and global convergence of combinatorial Calabi flow on surfaces with boundary.
Paper introduces VDE, a variance-reduced determinant estimator.
problem Estimating determinants with low variance and efficiency.
method Combines variational inference and spherical normalizing flows.
result VDE achieves zero variance in ideal cases, requiring only one sample.
It is known that principal orbits of Hermann actions on a symmetric space of non-compact type are curvature-adapted isoparametric submanifolds having no focal point of non-Euclidean type on the ideal boundary of the ambient symmetric space. In this paper, we investigate the mean curvature flows for such a curvature-ada…
Harmonic maps from complex plane to hyperbolic space constructed using heat flow.
problem Constructing harmonic maps from complex plane to hyperbolic space.
method Heat flow method to construct harmonic maps.
result Harmonic maps are unique once the principal part of their Hopf differential is prescribed.
We study the Riemannian geometry of 3D axisymmetric ideal fluids. We prove that the L2 exponential map on the group of volume-preserving diffeomorphisms of a 3-manifold is Fredholm along axisymmetric flows with sufficiently small swirl. Along the way, we define the notions of axisymmetric and swirl-free diffeomorp…
This paper uses combinatorial Ricci flow to tackle Thurston's triangulation conjecture.
problem Thurston's triangulation conjecture for hyperbolic 3-manifolds.
method Combinatorial Ricci flow approach to prove convergence and geometric decompositions.
result Combinatorial Ricci flow converges if and only if the triangulation is geometric.
In this paper, we study the Ricci flow of solvmanifolds whose Lie algebra has an abelian ideal of codimension one, by using the bracket flow. We prove that solutions to the Ricci flow are immortal, the omega-limit of bracket flow solutions is a single point, and that for any sequence of times there exists a subsequence…
The paper studies a flow on complex Lie groups, showing convergence to solitons.
problem The study of curvature flows on complex Lie groups.
method Positive Hermitian curvature flow on left-invariant metrics.
result The flow converges to solitons in both nilpotent and almost-abelian cases.
The maximum hyperbolic polyhedron volume is found to be the rectification of its skeleton.
problem Finding the maximum volume of hyperbolic polyhedra with given combinatorics.
method Applying a volume-increasing flow to any hyperbolic polyhedron, handling degeneracies carefully.
result The supremum volume is always the volume of the rectification of the 1-skeleton.
Review of diffusion models for SBI in non-ideal data scenarios.
problem Inference of parameters from complex simulation outputs with intractable likelihoods.
method Diffusion models for likelihood-free inference, addressing model misspecification, unstructured observations, and missing data.
result Improved robustness and efficiency in SBI methods for non-ideal data scenarios.