Symbolic dynamics for flows in high dimensions, extending previous work.
problem Coding flows with positive speed in high dimensions.
method Construct symbolic dynamics for flows with positive speed in any dimension.
result Extended symbolic dynamics to flows in high dimensions, including homoclinic classes.
The paper proves conditions for non-uniform expansion in partially hyperbolic systems.
problem Conditions for non-uniform expansion in partially hyperbolic systems.
method Analysis of Lyapunov exponents and dominated splittings.
result Existence of physical SRB measure under specific conditions.
We introduce the concept of hereditarily non uniformly perfect sets, compact sets for which no compact subset is uniformly perfect, and compare them with the following: Hausdorff dimension zero sets, logarithmic capacity zero sets, Lebesgue 2-dimensional measure zero sets, and porous sets. In particular, we give an exa…
We develop a theory of `non-uniformly local' tent spaces on metric measure spaces. As our main result, we give a remarkably simple proof of the atomic decomposition.
A new classifier improves one-class predictions on unevenly sampled data.
problem Non-uniformly sampled data affects one-class classifier performance.
method Dynamic decision boundary based on minimum spanning tree.
result Proves effectiveness and robustness compared to state-of-the-art classifiers.
New method uses SDEs for accurate non-uniformly sampled time series analysis.
problem Characterizing non-uniformly sampled time series with high accuracy.
method Stochastic Differential Equations (SDEs) for modeling, incremental estimation, and model truncation.
result Increased accuracy in characterizing non-uniformly sampled time series.
Anosov flows in hyperbolic 3-manifolds are quasigeodesic if not R-covered.
problem Characterizing Anosov flows in hyperbolic 3-manifolds.
method Analyzing the properties of Anosov flows and foliations.
result Anosov flows in hyperbolic 3-manifolds are quasigeodesic if not R-covered.
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.
Paper studies flow on hyperbolic surfaces to match boundary lengths.
problem Matching boundary lengths of hyperbolic surfaces.
method Combinatorial Yamabe flow on hyperbolic bordered surfaces.
result Flow converges exponentially to a surface with equal boundary lengths.
We consider compact convex hypersurfaces contracting by functions of their curvature. Under the mean curvature flow, uniformly convex smooth initial hypersurfaces evolve to remain smooth and uniformly convex, and contract to points after finite time. The same holds if the initial data is only weakly convex or non-smoot…
Constructs Anosov flows in hyperbolic 3-manifolds, disproving a conjecture.
problem Proving existence of infinitely many distinct Anosov flows in certain 4-manifolds.
method Using Cannon-Thurston maps and pseudo-Anosov quasigeodesic flows.
result Infinitely many distinct Anosov flows in some 4-manifolds.
A discrete method approximates hyperbolic curvature flow in the plane.
problem Modeling wave phenomena in solid-liquid interfaces.
method Semidiscrete finite difference method for hyperbolic curvature flow.
result Error bounds for natural discrete norms are proven.
Combinatorial Ricci flow finds hyperbolic metrics on 3-manifolds.
problem Finding complete hyperbolic metrics on cusped 3-manifolds.
method Analogue of surface and compact 3-manifold flows, minimizing co-volume, extending through singularities.
result Existence of complete hyperbolic metric is equivalent to flow convergence.
Topological entropy decreases strictly along Ricci flow near hyperbolic metrics.
problem Understanding entropy changes in flows near hyperbolic metrics.
method Analysis of geodesic flow on Riemannian manifolds with variable negative curvature.
result Topological entropy strictly decreases along normalized Ricci flow near hyperbolic metrics.
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.
Ancient solutions found for mean curvature flows in hyperbolic spaces.
problem Mean curvature flows of isoparametric submanifolds in hyperbolic spaces.
method Study mean curvature flows of isoparametric submanifolds in hyperbolic spaces.
result Ancient solutions always exist for these flows.
Study of flows on 7D manifolds with holomorphic properties.
problem Characterizing transversely holomorphic partially hyperbolic flows.
method Analyzing flows with biholomorphic holonomy pseudo-group, proving properties under integrable subcenter distribution.
result Flow projects to a transversely holomorphic Anosov flow on a 5D manifold.
The paper solves fractional combinatorial flows for prescribed hyperbolic bordered surfaces.
problem Finding hyperbolic bordered surfaces with prescribed boundary lengths.
method Fractional combinatorial Calabi flow and generalized combinatorial Yamabe flow.
result The flows converge to a hyperbolic surface with prescribed boundary lengths.
In this paper, we investigate two hyperbolic flows obtained by adding forcing terms in direction of the position vector to the hyperbolic mean curvature flows in \cite{klw,hdl}. For the first hyperbolic flow, as in \cite{klw}, by using support function, we reduce it to a hyperbolic Monge-Ampeˋre equation …
Classifies self-similar curve shortening flows in hyperbolic 2-space.
problem Classifying self-similar curve shortening flows in hyperbolic 2-space.
method Analyzes and classifies solutions in hyperbolic 2-space.
result Completes the classification of self-similar curve shortening flows in constant curvature model spaces in 2-dimensions.
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 on stability of mean curvature flow in hyperbolic space.
problem Stability of volume preserving mean curvature flow in hyperbolic space.
method Analysis of initial conditions and flow behavior in hyperbolic space.
result The flow converges exponentially to an umbilical sphere under certain conditions.
Generalizes Anosov flows to partially hyperbolic diffeomorphisms.
problem Classifying partially hyperbolic diffeomorphisms.
method Introducing collapsed Anosov flows and self orbit equivalences.
result All examples in Bonatti et al. belong to the collapsed Anosov flow class.
In this paper we introduce the hyperbolic mean curvature flow and prove that the corresponding system of partial differential equations are strictly hyperbolic, and based on this, we show that this flow admits a unique short-time smooth solution and possesses the nonlinear stability defined on the Euclidean space with …
New translations defined; curve shortening flow solved in hyperbolic plane.
problem Solving curve shortening flow in hyperbolic geometry.
method Introduced new translations, solved equations, analyzed ancient solutions.
result Explicit solutions and area estimates for ancient solutions.
Anosov flows found on many hyperbolic 3-manifolds.
problem Existence of Anosov flows on hyperbolic 3-manifolds.
method Explicit construction of Anosov flows on fibered hyperbolic 3-manifolds.
result Positive density of fibered hyperbolic manifolds carrying Anosov flows.
We consider the evolution of the asymptotically hyperbolic mass under the curvature-normalized Ricci flow of asymptotically hyperbolic, conformally compactifiable manifolds. In contrast to asymptotically flat manifolds, for which ADM mass is constant during Ricci flow, we show that the mass of an asymptotically hyperbo…
Inverse mean curvature flow converges to a disk in hyperbolic space.
problem Understanding flow behavior in hyperbolic geometry.
method Inverse mean curvature flow with free boundary on geodesic spheres.
result Flow converges to a totally geodesic disk.
We prove global existence of instantaneously complete Yamabe flows on hyperbolic space of arbitrary dimension m≥3 starting from any smooth, conformally hyperbolic initial metric. We do not require initial completeness or curvature bounds. With the same methods, we show rigidity of hyperbolic space under the Yamabe…
Ricci flow stability on manifolds with bounded geometry ensures convergence to hyperbolic metrics.
problem Stability and convergence of Ricci flow on manifolds with bounded geometry.
method Continuous dependence on initial conditions, sectoriality of Ricci-DeTurck flow generator, and Hölder norm analysis.
result Ricci flow converges to hyperbolic metrics under certain conditions.
In this paper we introduce and study a new kind of hyperbolic geometric flows --dissipative hyperbolic geometric flow. This kind of flow is defined by a system of quasilinear wave equations with dissipative terms. Some interesting exact solutions are given, in particular, a new concept-- hyperbolic Ricci soliton is int…
In this paper, the author has considered the hyperbolic Kahler-Ricci flow introduced by Kong and Liu [11], that is, the hyperbolic version of the famous Kahler-Ricci flow. The author has explained the derivation of the equation and calculated the evolutions of various quantities associated to the equation including the…
The paper examines flows that preserve area and length in hyperbolic geometry.
problem Preserving area and length in hyperbolic geometry.
method Inverse curvature flows for convex curves in hyperbolic plane.
result The flows converge to geodesic circles under certain conditions.
Hyperbolic solitons on trans-Sasakian space forms and their submanifolds
problem Characterizing hyperbolic solitons on trans-Sasakian space forms and their submanifolds
method Introducing hyperbolic ∗−Ricci solitons and hyperbolic Ricci-Yamabe solitons result Characterizing the nature of hyperbolic solitons on trans-Sasakian space forms and their submanifolds
Geometric inequalities for static convex domains in hyperbolic space proved.
problem Proving geometric inequalities for static convex domains in hyperbolic space.
method Using static convexity of flow hypersurfaces, new inequalities are derived.
result New family of geometric inequalities for static convex domains in hyperbolic space.
Study of flow in hyperbolic space with capillary boundary.
problem Optimizing hypersurfaces with capillary boundary in hyperbolic space.
method Mean curvature flow with capillary boundary, preserving volume and energy.
result Flow converges to a truncated umbilical hypersurface.
Anosov flow found in specific partially hyperbolic systems.
problem Characterizing partially hyperbolic diffeomorphisms with center foliation.
method Analyzing transitive dynamically coherent systems with one-dimensional center foliation.
result Discretized Anosov flow found in systems satisfying f(W)=W for center leaves. In this paper we establish the short-time existence and uniqueness theorem for hyperbolic geometric flow, and prove the nonlinear stability of hyperbolic geometric flow defined on the Euclidean space with dimension larger than 4. Wave equations satisfied by the curvatures are derived. The relation of hypergeometric flo…
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.
Paper improves estimates for discrete Laplace in hyperbolic geometry.
problem Establishing compactness for discrete Laplace in hyperbolic geometry.
method Explicit estimates for discrete Laplace based on Glickenstein-Thomas formulation.
result New proofs of long time existence for Calabi flows in hyperbolic geometry.
New normalizing flows in hyperbolic space improve posterior modeling for hierarchical data.
problem Limited flexibility of existing normalizing flows in Euclidean space for hierarchical data.
method Elevated normalizing flows to hyperbolic spaces using coupling transforms and Wrapped Hyperboloid Coupling.
result Improved performance on density estimation and hierarchical graph data.
New flow expands hypersurfaces in hyperbolic space, showing round limiting shape for certain powers.
problem Understanding the limiting shape of hypersurfaces expanding in hyperbolic space.
method Introduced shifted inverse curvature flow with positive power p for a smooth curvature function. result For 0<p≤1, limiting shape is always round as maximal existence time is approached. CDC-FM improves generative model quality-generalization tradeoff by regularizing with geometry-aware noise.
problem Tradeoff between high sample quality and memorization in deep generative models.
method Introduces Carré du champ flow matching (CDC-FM) that replaces homogeneous noise with anisotropic Gaussian noise capturing latent data manifold geometry.
result CDC-FM consistently offers better quality-generalization tradeoff across diverse datasets and architectures.
Study shows partial hyperbolicity leads to Anosov dynamics in 3-manifolds.
problem Understanding dynamics in hyperbolic 3-manifolds and Seifert manifolds.
method Classification of partially hyperbolic diffeomorphisms and pseudo-Anosov dynamics.
result Complete classification of partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds and Seifert manifolds.
The paper connects geodesic flows, hyperbolic geodesics, and stable ergodicity.
problem Understanding conditions for geodesic flows to be Anosov and ergodic.
method Analyzing Finsler and Riemannian metrics on surfaces, using recent results.
result Geodesic flows on surfaces are C2 stably ergodic if and only if they are Anosov. Study connects flow dynamics to 3D geometry via surface intersections.
problem Relating flow dynamics to geometric properties of 3-manifolds.
method Relates pseudo-Anosov flow dynamics to hyperbolic geometry via curve graphs.
result Established a link between flow invariants and geometric features of 3-manifolds.
The paper solves curvature problems on infinite hyperbolic surfaces.
problem Prescribed curvature flow on hyperbolic surfaces with infinite topological type.
method Introduced a prescribed curvature flow adapted to infinite cellular decompositions.
result Established well-posedness of the flow and proved convergence under certain conditions.
New 3D shapes found without certain flows.
problem Finding 3D shapes without specific flows.
method Using foliations and pseudo-Anosov flows, analyzing cusped hyperbolic 3-manifolds.
result First examples of 3D shapes without veering triangulations.