Study calculates volume of small sub-Riemannian balls in 3D contact manifolds.
problem Computing the volume of small sub-Riemannian balls in 3D contact manifolds.
method Asymptotic expansion and geometric invariants of the sub-Riemannian structure.
result Expressed first geometric coefficients in terms of sub-Riemannian structure invariants.
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.
The paper shows that certain bundles have unique volumes.
problem Volume rigidity of specific circle bundles.
method Proving volume rigidity for principal circle bundles over complex projective spaces.
result Principal circle bundles are volume rigid among K-contact manifolds. Bi-contact surgery operations can be applied to Anosov flows.
problem Characterizing Anosov flows and their properties.
method Metric and contact geometric characterizations, Liouville geometry, Reeb dynamics.
result Bi-contact surgery operations can be applied to Anosov flows.
Inequalities found in contact and symplectic geometry.
problem Finding inequalities in contact and symplectic geometry.
method Proving inequalities for Zoll contact and odd-symplectic forms.
result Proves a local systolic-diastolic inequality for Zoll contact and odd-symplectic forms.
A rigidity theorem for smooth Legendrian self-shrinkers is proven.
problem Understanding the structure of Legendrian self-shrinkers.
method Estimating weighted volume to prove optimal volume growth.
result Rigidity theorem for entire smooth Legendrian self-shrinkers.
We prove existence of regions minimizing perimeter under a volume constraint in contact sub-Riemannian manifolds such that their quotient by the group of contact transformations preserving the sub-Riemannian metric is compact.
New bound on ECH sub-leading asymptotics.
problem Understanding ECH capacities in detail.
method Analyzing sub-leading asymptotics of ECH spectrum.
result New bound on sub-leading asymptotics of ECH capacities.
We prove a Bishop volume comparison theorem and a Laplacian comparison theorem for three dimensional contact subriemannian manifolds with symmetry.
Contact forms with large systolic ratios found in arbitrary dimensions.
problem Finding contact forms with large systolic ratios in arbitrary dimensions.
method Generalized plug construction and Liouville open books.
result Every co-orientable contact structure admits a contact form with arbitrarily large systolic ratio.
We study the general structure of the AdS_5/CFT_4 correspondence in type IIB string theory from the perspective of generalized geometry. We begin by defining a notion of "generalized Sasakian geometry," which consists of a contact structure together with a differential system for three symplectic forms on the four-dime…
New systolic inequality for 3D contact forms on Seifert bundles.
problem Bounding the shortest Reeb orbit period in terms of contact volume.
method Proved a general systolic inequality for S1-invariant contact forms on Seifert bundles.
result Validated systolic inequality on Seifert bundles with non-zero Euler number.
Corrects a lemma in a 2009 paper about contact homology and Floer homology.
problem An error in a lemma about contact homology and Floer homology.
method None, as it is a correction of an existing lemma.
result Corrects an error in a previously published lemma.
Study compatible and associated metrics for contact-symplectic structures, showing geodesic integral curves and minimal leaf properties.
problem Characteristics foliations of metric contact-symplectic structures.
method Analysis of compatible and associated metrics, study of geodesic integral curves, and minimal leaf properties.
result Integral curves of the Reeb vector field are geodesics for any compatible metric, and associated metrics share a common volume element.
To any smooth compact manifold M endowed with a contact structure H and partially integrable almost CR structure J, we prove the existence and uniqueness, modulo high-order error terms and diffeomorphism action, of an approximately Einstein ACH (asymptotically complex hyperbolic) metric g on M×(−1,0). W…
A contact pair on a manifold always admits an associated metric for which the two characteristic contact foliations are orthogonal. We show that all these metrics have the same volume element. We also prove that the leaves of the characteristic foliations are minimal with respect to these metrics. We give an example wh…
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.
The paper explores how vector fields relate to volume in geometric contexts.
problem Existence of nondiffeomorphic contact forms with identical Reeb vector fields.
method Analyzes geodesible vector fields and their associated Euler classes, applying topological and geometric theorems.
result Proves the Gauss-Bonnet and Poincaré-Hopf theorems for 2D orbifolds using geodesible vector fields.
Study on mean Euler characteristic of Gorenstein toric contact manifolds.
problem Calculating the mean Euler characteristic of Gorenstein toric contact manifolds.
method Using the relationship between mean Euler characteristic and the normalized volume of the toric diagram, and applying results from Batyrev and Dais.
result Twice the mean Euler characteristic of a Gorenstein toric contact manifold equals the Euler characteristic of any crepant toric symplectic filling.
Finite-volume Ricci solitons with constant-length potential are trivial.
problem Characterizing non-compact Ricci solitons with specific properties.
method Analyzing conditions on scalar curvature and potential field length.
result Non-compact Ricci solitons with constant-length potential are trivial.
Let M be a compact orientable Seifered fibered 3-manifold without a boundary, and α an S1-invariant contact form on M. In a suitable adapted Riemannian metric to α, we provide a bound for the volume Vol(M) and the curvature, which implies the universal tightness of the contact structure ξ=kerα.
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…
This paper improves energy estimates for Seiberg-Witten Floer generators.
problem Recovering the volume of contact 3-manifolds using ECH capacities.
method Stronger estimates on the energy of min-max Seiberg-Witten Floer generators.
result Directly proves the ECH capacities recover volume theorem.
A taut contact sphere on a 3-manifold is a linear 2-sphere of contact forms, all defining the same volume form. In the present paper we completely determine the moduli of taut contact spheres on compact left-quotients of SU(2) (the only closed manifolds admitting such structures). We also show that the moduli space of …
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.
We introduce a new geometric structure on differentiable manifolds. A \textit{Contact} \textit{Pair}on a manifold M is a pair (α,η) of Pfaffian forms of constant classes 2k+1 and 2h+1 respectively such that α∧dαk∧η∧dηh is a volume form. Both forms have a characteristic foliation whose …
We show the existence of a weak bi-invariant symmetric nondegenerate 2-form on the contact diffeomorphisms group Dθ of a contact Riemannian manifold (M,g,θ) and study its properties. We describe the Euler's equation on a Lie algebra of group Dθ and calculate the sectional curvature of $\math…
We study curvature dimension inequalities for the sub-Laplacian on contact Riemannian manifolds. This new curvature dimension condition is then used to obtain: 1) Geometric conditions ensuring the compactness of the underlying manifold (Bonnet-Myers type results); 2) Volume estimates of metric balls; 3) Gradient bounds…
On a sub-Riemannian manifold we define two type of Laplacians. The \emph{macroscopic Laplacian} Δω, as the divergence of the horizontal gradient, once a volume ω is fixed, and the \emph{microscopic Laplacian}, as the operator associated with a sequence of geodesic random walks. We consider a general class of rando…
In 3D, Zoll contact forms locally maximize systolic ratio.
problem Maximizing systolic ratio in 3D contact geometry.
method Proving Zoll forms locally maximize the systolic ratio in C3-topology. result Every Zoll form admits a C3-neighborhood where systolic ratio is locally maximized. Study on stability of 3D sessile drops, identifying degenerate kernel.
problem Linear stability of three-dimensional sessile drops with a free contact line.
method Derived constrained second variation, formulated Jacobi problem, combined geometric and Fourier analysis.
result Kernel of the constrained Jacobi operator is exactly the space of horizontal translations under pressure-volume nondegeneracy.
The paper extends Q-prime curvature to ACHE manifolds and computes renormalized volumes.
problem Computing Q-curvature and renormalized volumes on ACHE manifolds. method Generalizing Q-prime curvature to ACHE manifolds and using scattering theory. result The integral of the Q-prime curvature defines an invariant of ACHE manifolds. New definition of stable (r+1)-th capillary hypersurfaces proposed.
problem Stability of capillary hypersurfaces in different geometries.
method Defining stable (r+1)-th capillary hypersurfaces as smooth local minimizers of a new energy functional under volume-preserving and contact angle-preserving variations. result Generalization of stability results to (r+1)-th capillary hypersurfaces. In this paper we consider surfaces of class C1 with continuous prescribed mean curvature in a three-dimensional contact sub-Riemannian manifold and prove that their characteristic curves are of class C2. This regularity result also holds for critical points of the sub-Riemannian perimeter under a volume constrain…
The study connects contact forms and Ruelle invariant in convex domains.
problem Understanding the relationship between contact forms and Ruelle invariant in convex domains.
method Using the extrinsic curvature and Ruelle invariant, the authors prove bounds and construct counterexamples.
result First examples of dynamically convex contact 3-spheres not strictly contactomorphic to convex boundaries.
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…
Proves Steiner and tube formulae for 3D contact sub-Riemannian surfaces.
problem Calculating surface properties in complex geometric structures.
method Develops a local Steiner formula for regular surfaces in 3D contact sub-Riemannian manifolds.
result Establishes a formula for surface expansion in arbitrary regions of contact sub-Riemannian manifolds.
A long-standing open problem in systolic geometry asks whether a Riemannian metric on the real projective space whose volume equals that of the canonical metric, but is not isometric to it, must necessarily carry a periodic geodesic of length smaller than π. A contact-geometric reformulation of systolic geometry and th…
Geodesic vector fields on flat 3-manifolds are related to contact structures.
problem Understanding geodesic vector fields on flat 3-manifolds.
method Analyzing geodesic and Reeb vector fields on flat 3-manifolds.
result Geodesic vector fields on closed flat 3-manifolds are Reeb vector fields of contact forms.
Study finds critical points of volume functionals on Sasaki manifolds.
problem Finding Kähler-Einstein metrics on Sasaki manifolds.
method Revisited moment polytopes, applied to volume minimization.
result Transverse coupled Kähler-Einstein metrics found as critical points.
Study on flows of G2-structures on contact Calabi-Yau 7-manifolds.
problem Analyzing the behavior of G2-structures under Laplacian and Hitchin flows.
method Investigation of Laplacian and Hitchin flows on contact Calabi-Yau 7-manifolds.
result Ancient solutions of the Laplacian flow with finite time Type I singularity and immortal solutions of the Laplacian coflow with infinite time Type IIb singularity.
BPt is a Python library for ML with neuroimaging data.
problem Analyzing large neuroimaging datasets using machine learning.
method Unified framework of ML tools for tabulated and neuroimaging data.
result Unified ML tools for neuroimaging and tabulated data.
The paper proves a Willmore-type inequality for unbounded convex sets.
problem Proving a Willmore-type inequality for unbounded convex sets.
method Analytical proof involving hypersurfaces, contact angle conditions, and asymptotic volume ratio.
result The Willmore-type inequality holds for unbounded closed convex sets with certain conditions.
The paper confirms a specific type of Sasakian manifold's structure.
problem Characterizing Sasakian manifolds with nonnegative transverse bisectional curvature.
method Analyzing the Sasakian analogue of Yau's uniformization conjecture.
result 5-dimensional Sasakian manifolds with positive transverse bisectional curvature are CR-biholomorphic to the standard Heisenberg group.
The study connects electromagnetic structures to Legendrian fields on the 3-sphere.
problem Understanding the topology of stable electromagnetic structures.
method Connecting null solutions to Maxwell's equations with Legendrian fields on the 3-sphere.
result Any (possibly knotted) toroidal surface can be realized as a magnetic surface of a null solution, implying stability.
Paper proves geodesics are evenly distributed on surfaces.
problem Existence of equidistributed closed geodesics on surfaces.
method Volume property of embedded contact homology, local variational constructions, and transversality arguments.
result Equidistribution of nondegenerate closed geodesics for generic metrics on closed surfaces.
We conjecture formulae of the colored superpolynomials for a class of twist knots Kp where p denotes the number of full twists. The validity of the formulae is checked by applying differentials and taking special limits. Using the formulae, we compute both the classical and quantum super-A-polynomial for the twist k…
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 …