Study on real hypersurfaces in Kaehler manifolds with isometric Reeb flow.
problem Characterizing real hypersurfaces with isometric Reeb flow.
method Investigation of structure and classification in Kaehler manifolds.
result Classification of real hypersurfaces in irreducible Hermitian symmetric spaces.
The study classifies real hypersurfaces in complex hyperbolic quadrics with isometric Reeb flow.
problem Classifying real hypersurfaces with isometric Reeb flow in complex hyperbolic quadrics.
method Classification based on the properties of the hypersurfaces and their embeddings.
result The existence and properties of real hypersurfaces with isometric Reeb flow are classified, leading to the non-existence in odd-dimensional cases.
We classify real hypersurfaces with isometric Reeb flow in the complex quadrics Q^m for m > 2. We show that m is even, say m = 2k, and any such hypersurface is an open part of a tube around a k-dimensional complex projective space CP^k which is embedded canonically in Q^{2k} as a totally geodesic complex submanifold. A…
The paper simplifies proofs and characterizes contact structures in 3D.
problem Contact structures induced by geodesic vector fields in 3D.
method New proofs and characterizations of contact structures.
result Contact structures in 3D are universally tight under certain conditions.
We classify all of real hypersurfaces M with Reeb invariant shape operator in complex hyperbolic two-plane Grassmannians SU2,m/S(U2⋅Um), m≥2. Then it becomes a tube over a totally geodesic SU2,m−1/S(U2⋅Um−1) in SU2,m/S(U2⋅Um) or a horosphere whose center at infinity is …
Explains how Reeb dynamics connects to topology.
problem Understanding the relationship between Reeb dynamics and topology.
method Contact cuts, lifting group actions, and other construction methods.
result Methods for constructing Reeb flows with finite periodic orbits.
Entropy rigidity for Finsler flows but collapse for Reeb flows.
problem Entropy behavior of Reeb and Finsler flows on contact manifolds.
method Analysis of topological entropy for Reeb and Finsler flows.
result Uniform positive lower bound for Finsler flows but arbitrarily small topological entropy for Reeb flows.
Study obstructions to conformally Anosov Reeb flows on 3-manifolds.
problem Existence of conformally Anosov Reeb flows on 3-manifolds.
method Geometric and topological obstructions, Riemannian conditions.
result Obtained obstructions and conditions for the existence of conformally Anosov Reeb flows.
Invites study of contact structures and Reeb flows dynamics.
problem Relating dynamics of two Reeb flows of the same contact structure.
method Gathers results and poses many questions and conjectures.
result Many new questions and conjectures posed.
Survey on finding global surfaces of section for Reeb flows.
problem Finding global surfaces of section for Reeb flows.
method Symplectic Topology methods.
result Existence of closed geodesics and sharp systolic inequalities.
Reeb flow made transverse to foliations without invariant measures.
problem Making Reeb flow transverse to foliations without invariant measures.
method Leafwise Brownian motion to construct transverse measures.
result Reeb flow has no contractible orbits when transverse to foliations without invariant measures.
We show, using standard results in length spectrum rigidity and symplectic homology, that if the unit tangent bundles of two compact surfaces of negative curvature are exact symplectomorphic, then the underlying surfaces are isometric, and hence the disk bundles are symplectomorphic smoothly up to the boundaries. Motiv…
New contact structures detected by contact homology.
problem Detecting pseudo-Anosov flows in contact structures.
method Introducing pseudo-Anosov contact structures and using contact homology.
result Contact homology detects pseudo-Anosov flows and contact structures properties.
Study k-almost Ricci solitons on contact metric manifolds.
problem Characterize k-almost Ricci solitons on contact metric manifolds.
method Prove isometric properties and extend results for k-almost gradient Ricci solitons and k-almost Ricci solitons.
result Compact K-contact metric manifolds that are k-almost gradient Ricci solitons are isometric to a unit sphere.
New flow category for contact manifolds from Reeb orbits.
problem No direct problem stated; focuses on new construction.
method Adapting Kuranishi charts to contact setting, associating flow category based on Reeb orbits and pseudo-holomorphic buildings.
result Lifts contact homology and associates flow bimodule to exact symplectic cobordisms.
The paper adapts results for Reeb flows and Hamiltonian flows, showing all orbits are closed have identical periods.
problem Adapting results for Reeb flows and Hamiltonian flows with closed orbits.
method Adapting results from Geodesic circle foliations to Reeb and Hamiltonian flows.
result All orbits on connected contact manifolds with closed orbits have identical periods.
New insights into Reeb dynamics from Finsler geometry.
problem Understanding periodic orbits in Reeb flows.
method Contact geometry, Hamiltonian circle actions, surgery construction.
result Compact manifolds with arbitrarily large numbers of periodic Reeb orbits.
3D contact manifolds have optimal higher systolic ratios.
problem Optimizing higher systolic ratios in 3D contact manifolds.
method Proving Besse contact forms maximize certain ratios.
result Besse contact forms are local maximizers of higher systolic ratios.
The paper characterizes Besse and Zoll Reeb flows on specific manifolds.
problem Characterizing Besse and Zoll Reeb flows on different manifolds.
method Using spectral invariants and Ekeland-Hofer capacities.
result Characterizations of Besse and Zoll Reeb flows for specific manifolds.
The study finds sufficient conditions for Reeb flows to have genus zero global surfaces of section.
problem Finding conditions for Reeb flows to have genus zero global surfaces of section.
method Analyzes linking assumptions on periodic orbits and ambient contact geometry.
result Reveals sufficient conditions for Reeb flows to have genus zero global surfaces of section.
Study on periodic orbits in contact geometry and Finsler geodesics.
problem Existence and non-existence of periodic Reeb orbits on contact manifolds.
method Interpretation of Finsler geodesics as Reeb flows and use of open books.
result Existence of periodic Reeb orbits on contact manifolds with suitable open books.
Deformations of the Reeb flow of a Sasakian manifold as transversely Kähler flows may not admit compatible Sasakian metrics anymore. We show that the triviality of the (0,2)-component of the basic Euler class characterizes the existence of compatible Sasakian metrics for given small deformations of the Reeb flow as tra…
Study eta invariant remainder on contact manifolds, improving previous results.
problem Eta invariant remainder in metric contact manifolds.
method Analyzes remainder term in semiclassical limit, using volumes of recurrence sets of Reeb flow.
result Improves remainder term for Anosov Reeb flows and certain elliptic flows.
The study finds at least two closed orbits for Reeb flows on certain contact manifolds.
problem Existence of at least two closed orbits for Reeb flows on contact manifolds.
method Analysis of equivariant symplectic homology and contact finite quotients.
result Existence of at least two geometrically distinct closed orbits under specified conditions.
Paper proves S-stability of foliations on flow-spines with transverse Reeb flow.
problem Characterizing S-stable foliations on flow-spines with transverse Reeb flow.
method Introduced S-stability for foliations on branched simple polyhedrons and proved stability for 1-forms with dβ>0. result Proved the number of simple tangency points of an S-stable foliation on a flow-spine is at least 2.
Explicitly determined foliated cohomology of affine Reeb flow on Hopf manifold.
problem Determine the foliated cohomology of the affine Reeb flow on the Hopf manifold.
method Explicit calculation and analysis of the cohomology space and its dual.
result The space HF1(M) contains obstructions to solving the cohomological equation. The paper studies Cotton solitons on specific geometric manifolds.
problem Analyzing Cotton solitons in almost Kenmotsu 3-h-manifolds. method Examined potential vector fields and their relationship with the Reeb vector field.
result Steady Cotton solitons on non-Kenmotsu manifolds are locally isometric to H2(−4)imesR. Characterizes Anosov flows in 3D using symplectic and contact geometry.
problem Understanding Anosov flows in 3D.
method Purely contact and symplectic geometric methods.
result Characterization of Anosov flows based on Reeb flows and underlying (bi)-contact structures.
Characterizes Anosov flows via contact geometry.
problem Understanding Anosov 3-flows through contact geometry.
method Investigates interactions with Reeb dynamics and proves a technical theorem.
result Space of adapted geometries homotopy equivalent to Anosov flows.
Study shows magnetic trajectories in Berger spheres are homogeneous.
problem Homogeneity of contact magnetic trajectories in Berger spheres.
method Proved every contact magnetic trajectory is a product of a homogeneous geodesic and a charged Reeb flow.
result Contact magnetic trajectories in Berger spheres are homogeneous.
Homogeneous magnetic trajectories in a special linear group proven.
problem Proving homogeneity of magnetic trajectories in a specific group.
method Using contact magnetic curves and geodesics.
result Every contact magnetic trajectory is a product of a homogeneous geodesic and a charged Reeb flow.
The paper characterizes contact 3-manifolds with closed Reeb orbits.
problem Characterizing contact 3-manifolds with closed Reeb orbits.
method Analyzing the action spectrum and minimal periods of Reeb orbits.
result A contact form with an action spectrum of rank 1 is uniquely determined by the minimal periods of its closed Reeb orbits.
Geodesics spiral around Reeb orbits in 3D contact manifolds.
problem Understanding geodesics in sub-Riemannian geometry.
method Normal form along Reeb orbits due to Melrose.
result Sub-Riemannian geodesics spiral around Reeb orbits in both phase and configuration spaces.
Proves stability of geodesic flows on closed surfaces.
problem Stability of geodesic flows on closed surfaces.
method Generic Riemannian metrics and Reeb flows.
result Proves C2-stability conjecture for geodesic flows. We give necessary and sufficient conditions for a closed connected co-orientable contact 3-manifold (M,ξ) to be a standard lens space based on assumptions on the Reeb flow associated to a defining contact form. Our methods also provide rational global surfaces of section for nondegenerate Reeb flows on $(L(p,q),ξ_{…
We investigate the equivariant cohomology of the natural torus action on a K-contact manifold and its relation to the topology of the Reeb flow. Using the contact moment map, we show that the equivariant cohomology of this action is Cohen-Macaulay, which is a generalization of equivariant formality for torus actions wi…
The study classifies contact metric manifolds based on Ricci-Yamabe solitons.
problem Classifying contact metric manifolds based on Ricci-Yamabe solitons.
method Analyzing specific types of solitons in contact metric manifolds.
result Contact metric manifolds are classified based on the properties of Ricci-Yamabe solitons.
The abstract proves that certain Reeb vector fields on 3-manifolds have Birkhoff sections.
problem Existence of Birkhoff sections for Reeb vector fields on 3-manifolds.
method Showed existence of Birkhoff sections for Reeb vector fields satisfying Kupka-Smale condition.
result Reeb vector fields on closed 3-manifolds with Kupka-Smale condition admit Birkhoff sections.
We introduce the notion of contact Ricci flow associated with the Reeb vector field. Using it, we give a simple proof of the Poincare conjecture.
A long-standing conjecture in Hamiltonian Dynamics states that the Reeb flow of any convex hypersurface in R2n carries an elliptic closed orbit. Two important contributions toward its proof were given by Ekeland in 1986 and Dell'Antonio-D'Onofrio-Ekeland in 1995 proving this for convex hypersurfaces satis…
We study compatible toric Sasaki metrics with constant scalar curvature on co-oriented compact toric contact manifolds of Reeb type of dimension at least 5. These metrics come in rays of transversal homothety due to the possible rescaling of the Reeb vector fields. We prove that there exist Reeb vector fields for which…
In this paper, we prove (1): for any closed contact three-manifold with a C∞-generic contact form, the union of periodic Reeb orbits is dense, (2): for any closed surface with a C∞-generic Riemannian metric, the union of closed geodesics is dense. The key observation is C∞-closing lemma for 3D R…
Study shows how to realize Ricci curvature as Reeb vector field for contact 3-manifolds.
problem When can a function be realized as Ricci curvature of a Reeb vector field?
method Topological tools to show realization, resolving singularities depend on contact topology.
result Every admissible function can be realized as Ricci curvature for a singular metric away from a measure zero set.
The study proves properties of spectral selectors for contact manifolds and applies them to contact big fibers and geodesics.
problem Properties of spectral selectors for contact manifolds.
method Algebraic properties of spectral selectors for strongly orderable contact manifolds.
result Established contact big fiber theorem and constructed norms on contactomorphism group universal cover.
The study connects ECH capacities to Anosov flows, proving infinite capacities and obstructions.
problem Understanding ECH capacities and their relation to Anosov flows.
method Relating ECH capacities to Anosov flows dynamics, proving infinite capacities and obstructions.
result ECH capacities are infinite for many symplectic 4-manifolds, including cotangent disk bundles over surfaces of genus at least two.
New surgery method preserves Anosov flow properties using bi-contact geometry.
problem Defining a new type of surgery on Anosov flows.
method Bi-contact geometry and Reeb dynamics.
result Necessary and sufficient condition for generating contact Anosov flows.
Isomorphic cosymplectomorphism groups imply diffeomorphic manifolds.
problem Determining when cosymplectic manifolds are diffeomorphic based on their cosymplectomorphism groups.
method Characterized Reeb flow, used to descend isomorphism to symplectic base manifolds, preserved monodromy class ensuring bundle equivalence.
result Isomorphic cosymplectomorphism groups imply diffeomorphic manifolds.
Study of intersections in Hamiltonian orbits on cotangent bundles.
problem Understanding intersections of projected Hamiltonian orbits in cotangent bundles.
method Generic submersive level set analysis, multi-jet transversality theorem.
result Projected Hamiltonian orbits have discrete intersections, which can be perturbed away under certain conditions.