Addendum proves properties of contact stationary Legendrian surfaces in S^5.
problem Characterize contact stationary Legendrian surfaces in S^5.
method Analyzes properties of surfaces with specific curvature conditions.
result Totally umbilical contact stationary Legendrian surfaces are totally geodesic.
A contact stationary Legendrian submanifold of S2n+1 is a Legendrian submanifold whose volume is stationary under contact deformations. The simplest contact stationary Legendrian submanifold (actually minimal and Legendrian) is the real, equatorial n-sphere S0. This paper develops a method for constructing co…
Establishes a boundary maximum principle for varifolds with fixed contact angle.
problem Boundary behavior of varifolds with contact angle constraints.
method Maximum principle for stationary pairs of varifolds with fixed contact angle condition.
result Boundary maximum principle proven for stationary varifolds.
Let (M5,α,gα,J) be a 5-dimensional Sasakian Einstein manifold with contact 1-form α, associated metric gα and almost complex structure J and L a contact stationary Legendrian surface in M5. We will prove that L satisfies the following equation \begin{eqnarray}\label{equ} -Δ^νH+(K-1)H=0, \end{eqnarray}…
The paper classifies Willmore Legendrian surfaces in S^5 and studies their properties.
problem Classifying and understanding Willmore Legendrian surfaces in S^5.
method Using an equality from Luo's work, the authors relate Willmore Legendrian surfaces to contact stationary Legendrian surfaces and prove classification results.
result Classification of Willmore Legendrian spheres in S^5 and integral inequality for Willmore Legendrian surfaces.
Constructs solutions to Einstein-Maxwell-current system using Sasakian manifolds.
problem Solving the Einstein-Maxwell-Current system with inhomogeneous charged particle density.
method Using Sasakian manifolds to specify magnetic field and electric current.
result Solutions with arbitrary function describing charged particle density and curvature.
This paper analyzes the configurations of shapes that shows a spacelike liquid drop in Minkowski space deposited over a spacelike plane Π. We assume the presence of a uniform gravity field directed toward Π and that the volume of the drop is prescribed. Our interest are the liquid drops that are critical points of …
The paper studies mean curvature flow with contact angles in high-dimensional cylinders.
problem Mean curvature flow with prescribed contact angles in a high-dimensional cylinder.
method Derives uniform-in-time gradient bounds and presents a trichotomy result for asymptotic behavior.
result The solution converges to a translating solution with positive speed when a specific condition is met.
We prove that there exist solutions for a non-parametric capillary problem in a wide class of Riemannian manifolds endowed with a Killing vector field. In other terms, we prove the existence of Killing graphs with prescribed mean curvature and prescribed contact angle along its boundary. These results may be useful for…
Paper studies rigid properties of closed CSL submanifolds in unit spheres.
problem Understanding geometric properties of closed CSL submanifolds in unit spheres.
method Analyzes the rigidity of closed CSL submanifolds in the unit sphere using geometric methods.
result Closed CSL submanifolds in S5 with specific properties are either totally geodesic or flat minimal Legendrian tori. Formula proves almost monotonicity for H-minimal surfaces in Heisenberg group.
problem Analyzing H-minimal Legendrian surfaces in Heisenberg group.
method Proved an almost monotonicity formula.
result Deduced a Bernstein-Liouville type theorem.
The Navier-Stokes equations on certain manifolds can perform universal computation.
problem Computational universality in viscous fluids.
method Cosymplectic geometry and harmonic 1-forms.
result Stationary Navier-Stokes solutions exhibit Turing completeness.
New techniques analyze steady fluid flows on non-compact manifolds.
problem Analyzing steady fluid flows on non-compact manifolds with cylindrical ends.
method New techniques using b-calculus and b-symplectic structures. result New proofs and descriptions of b-symplectic structures on singular sets of the Bernoulli function. New examples show contact structures can be isotopic but not contact-isotopic.
problem Identifying contact structures that are isotopic but not contact-isotopic.
method Constructing specific contactomorphisms in all dimensions.
result Infinitely many examples of contactomorphisms that are isotopic but not contact-isotopic.
The study explores embedding closed contact manifolds in higher dimensions.
problem Iso-contact embeddings of closed contact manifolds in co-dimension 2.
method Analyzes conditions for iso-contact embeddings and applies results to specific cases.
result Closed contact 3-manifolds without 2-torsion in their cohomology iso-contact embed in the standard contact 5-sphere.
Study on contact Hamiltonian functions for singular contact structures.
problem Understanding infinitesimal contact transformations on singular contact structures.
method Showed injectivity and provided an explicit local formula for the inverse map.
result Explicit local formula for the inverse map when contact structure has singularities of the first type.
Contact round surgery proves existence of contact structures on 3-manifolds.
problem Existence of contact structures on 3-manifolds.
method Contact round surgery and Lutz twist/Giroux torsion operations.
result Any contact structure on a closed orientable 3-manifold is constructed from the standard contact structure on the 3-sphere via contact round surgeries.
The paper defines and studies contact surgery numbers for contact 3-manifolds.
problem Understanding the minimal number of components of a surgery link describing a contact 3-manifold.
method Defined and studied various versions of contact surgery numbers, relating them to other invariants and computing specific cases.
result There exist infinitely many non-isotopic contact structures on certain manifolds that cannot be obtained by a single rational contact surgery from the standard tight contact 3-sphere.
Study weak quasi contact metric manifolds to generalize K-contact and Sasakian manifolds criteria.
problem Generalize K-contact and Sasakian manifolds criteria using weak quasi contact metric manifolds.
method Study weak quasi contact metric manifolds and generalize theorems for K-contact and Sasakian manifolds.
result Provide new criterions for K-contact and Sasakian manifolds in terms of curvature tensor and geometric objects.
Simplified construction of contact triad connection for analysis.
problem Analyzing contact instanton equation on contact triads.
method Characterization via almost contact structure and construction using almost contact moving frame.
result Simpler and more canonical construction of contact triad connection.
The paper examines conditions for contact surgeries on rational homology 3-spheres.
problem Conditions for contact surgeries on rational homology 3-spheres.
method Analyzes sufficient conditions for contact surgeries using Legendrian knots and links.
result Provides sufficient conditions for surgeries to have vanishing contact invariants or to be overtwisted.
We study the contact equivalence problem for toric contact structures on S3-bundles over S2. That is, given two toric contact structures, one can ask the question: when are they equivalent as contact structures while inequivalent as toric contact structures? In general this appears to be a difficult problem. To f…
Contact surgeries transform manifolds, and all admit symplectic caps.
problem Transforming contact manifolds using surgeries.
method Contact surgeries on isotropic and coisotropic spheres.
result All closed oriented contact manifolds admit symplectic caps.
Introduces contact dual pairs using line bundles.
problem Understanding contact geometry and Jacobi structures.
method Line bundle approach to contact and Jacobi geometry.
result Characteristic Leaf Correspondence Theorem for contact dual pairs.
New contact structures on folded sums of contact mapping tori are tight under certain conditions.
problem Understanding tight contact structures on folded sums of contact mapping tori.
method Alternative bundle-theoretical construction and gluing process near the fold.
result Folded contact structures on folded sums of contact mapping tori are tight under specific conditions.
Abstract: Survey on contact submanifolds.
problem Understanding contact submanifolds.
method None specified, survey of existing knowledge.
result Discussion of various contact submanifolds.
Embeds all contact 3-manifolds into specific 5-manifolds.
problem Embedding all contact 3-manifolds into fixed contact 5-manifolds.
method Spun embeddings and Lefschetz fibrations.
result Embeds all contact 3-manifolds into a Stein fillable contact structure on the twisted S3-bundle over S2 and a unique overtwisted contact structure on S3imesS2. Study explores weak generalized K-contact structures in contact metric spaces.
problem Exploring weak generalized K-contact structures in contact metric spaces.
method Introducing a weak (κ,μ) condition and proving existence of K-contact and (κ,μ=2)-structures. result Existence of K-contact and (κ,μ=2)-structures under certain conditions on the Boeckx invariant. 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.
Local flatness theorem for paraquaternionic contact structures.
problem Local flatness of paraquaternionic contact manifolds.
method Defined paraquaternionic contact conformal curvature tensor and showed local flatness condition.
result Paraquaternionic contact conformal curvature vanishing implies local flatness.
New examples show contact invariants can be non-zero even with half Giroux torsion.
problem Understanding contact invariants and their obstructions in 3-manifolds.
method Use bordered contact invariants and innermost contact structures.
result Found closed contact 3-manifolds with non-vanishing contact invariants.
Study of cosmetic contact surgeries on knots, excluding some cases.
problem Determining cosmetic contact surgeries on transverse knots.
method Analyzing contact surgeries and their outcomes on transverse knots.
result Exclude non-trivial cosmetic contact surgeries for certain transverse knots.
We consider certain type of fiber bundles with odd dimensional compact contact base, exact symplectic fibers, and the structure group contained in the group of exact symplectomorphisms of the fiber. We call such fibrations "contact symplectic fibrations". By a result of Hajduk-Walczak, some of these admit contact struc…
The paper finds minimal contact triangulations for 3-manifolds, proving a linear bound on vertex count.
problem Finding the minimum number of vertices in contact triangulations for 3-manifolds.
method Explicit examples and linear growth bound analysis for overtwisted contact structures.
result The number of vertices for minimal contact triangulations grows at most linearly with respect to the d3 invariant. Survey of contact homology for contact manifolds.
problem Understanding contact manifolds through homology.
method Holomorphic curves invariants.
result Useful for students and young mathematicians.
New result on contact surgeries and overtwistedness.
problem When contact (+1/n)-surgery is overtwisted.
method Analyzes contact surgeries and provides counterexamples.
result Provides conditions for overtwisted contact surgeries.
The paper connects complex contact structures to specific types of almost contact 3-structures.
problem Understanding the relationship between complex contact structures and almost contact 3-structures.
method Proving that every complex contact structure gives rise to a distinguished almost contact metric 3-structure.
result The paper provides new examples of manifolds with specific contact and almost contact structures.
The paper constructs contact-hyperbolic manifolds with large automorphism groups.
problem Finding contact-hyperbolic manifolds with large automorphism groups.
method Holomorphic contact structures, pseudometrics, and symplectic quotients.
result Explicit examples of contact-hyperbolic contact manifolds are constructed.
Study contact structures on specific manifolds, proving infinite non-isotopic structures and tight/overtwisted classifications.
problem Classifying contact structures with specific contact surgery numbers on Brieskorn spheres and lens spaces.
method Algorithm for computing Euler class from rational contact surgery, Legendrian knot classification, and analysis of contact structures.
result Infinitely many non-isotopic contact structures on Brieskorn spheres and lens spaces cannot be obtained by a single rational contact surgery.
Contact round surgeries on (S3,ξst) help in constructing and understanding contact 3-manifolds.
problem Constructing contact 3-manifolds using Legendrian surgeries.
method Introducing contact round surgeries of indices 1 and 2, and associating them with surgery diagrams.
result Every closed connected contact 3-manifold can be obtained by a sequence of contact round surgeries on Legendrian knots in (S3,ξst). Develops k-contact geometry theory for field theories.
problem Analyse field theories using k-contact geometry.
method Distributions maximally non-integrable with k commuting Lie symmetries.
result Established k-contact distributions and their relationships.
We consider contact elements in the sutured Floer homology of solid tori with longitudinal sutures, as part of the (1+1)-dimensional topological quantum field theory defined by Honda--Kazez--Matić in \cite{HKM08}. The Z2 SFH of these solid tori forms a "categorification of Pascal's triangle", and contact structur…
Contact surgeries yield algebraically overtwisted manifolds.
problem Understanding algebraically overtwisted contact manifolds through surgeries.
method Contact (+1)-surgeries on Legendrian spheres in flexibly fillable contact manifolds. result Yielding algebraically overtwisted manifolds when the Legendrian's homology class is not annihilated.
New transverse links solve contact manifold problems.
problem Contact manifold structures.
method Transverse links in 3-sphere contact structures.
result All contact 3-manifolds are contact branched covers over a transverse link.
Contact forms on 3-manifolds can have very large systolic ratios.
problem Understanding the systolic ratios of contact forms on 3-manifolds.
method Analyzing the systolic ratio of contact forms defined by co-orientable contact structures.
result Contact forms on any closed 3-manifold can have arbitrarily large systolic ratios.
A contact projective structure is a contact path geometry the paths of which are among the geodesics of some affine connection. In the manner of T.Y. Thomas there is associated to each contact projective structure an ambient affine connection on a symplectic manifold with one-dimensional fibers over the contact manifol…
Generalizes Tulczyjew triples for contact manifolds in Hamiltonian and Lagrangian formalisms.
problem Tackles the need for a geometric tool in contact manifolds.
method Introduces a generalized Tulczyjew triple for contact manifolds.
result Contact Hamiltonians and Lagrangians as sections of line bundles determine dynamics on contact phase space.
The study finds tight contact structures without fillings in high dimensions.
problem Finding tight contact structures that cannot be filled by symplectic forms.
method Construction of specific contact structures on manifolds of various dimensions.
result Existence of tight contact structures without fillings in all dimensions n≥3 and for n=2 under certain conditions.