Study examines tangential real hypersurfaces on Hermite-like manifolds.
problem Characterizing real hypersurfaces on Hermite-like manifolds.
method Introduced tangential real hypersurfaces and derived main identities.
result Discussed contact metric structures in K-contact and cosymplectic cases.
Study real hypersurfaces in complex space forms for an inequality involving a contact invariant.
problem Understanding real hypersurfaces in complex space forms and their properties.
method Investigating real hypersurfaces that achieve equality in a specific inequality involving a contact invariant.
result Characterized real hypersurfaces in complex space forms achieving the equality in the inequality.
Study examines homology of contact CR-submanifolds in complex Euclidean space.
problem Investigating the homology of contact CR-submanifolds.
method Analyzes stable integral currents and homology groups of contact CR-submanifolds in Cm. result Nonexistence of stable integral currents and vanishing theorems for homology groups.
A contact hypersurface in a Kaehler manifold is a real hypersurface for which the induced almost contact metric structure determines a contact structure. We carry out a systematic study of contact hypersurfaces in Kaehler manifolds. We then apply these general results to obtain classifications of contact hypersurfaces …
New proof classifies contact real hypersurfaces in complex hyperbolic quadric.
problem Classifying contact real hypersurfaces with constant mean curvature.
method New proof using tubes and horospheres.
result Contact real hypersurfaces are congruent to tubes or horospheres.
Paper proves stability of pseudo-Einstein contact form existence.
problem Stability of pseudo-Einstein contact form existence under deformations.
method Deformations of real hypersurfaces in complex manifolds.
result Existence of pseudo-Einstein contact form is preserved under deformations.
New examples of real hypersurfaces found in complex hyperbolic quadrics.
problem Existence of specific types of real hypersurfaces in complex hyperbolic quadrics.
method Construction of a one-parameter family of homogeneous Hopf hypersurfaces.
result First known examples of real hypersurfaces with integrable maximal complex subbundle in irreducible Kahler manifolds.
New realizations prove all candidates are Ricci solitons.
problem Understanding contact metric manifolds as Ricci soliton real hypersurfaces.
method Analyzing Lie groups with dimension greater than three and proving Ricci soliton properties.
result All candidates can be realized as homogeneous real hypersurfaces in noncompact real two-plane Grassmannians.
Study on Hopf hypersurfaces in complex Grassmannians, proving constant Reeb curvature.
problem Proving properties of Hopf hypersurfaces in complex Grassmannians.
method Analyzing real hypersurfaces, proving nonexistence of certain foliations, and classifying hypersurfaces.
result Constant Reeb curvature for Hopf hypersurfaces in complex Grassmannians of rank two.
Study on null hypersurfaces in complex contact manifolds.
problem Characterizing null hypersurfaces in indefinite complex contact manifolds.
method Proved classification results for various null hypersurfaces and characterized the ambient space.
result The ambient complex contact manifold must have constant GH-sectional curvature of −3 for certain null hypersurfaces. The paper classifies A-principal Hopf hypersurfaces in complex quadrics.
problem Characterizing A-principal Hopf hypersurfaces in complex quadrics. method Analyzing the singularities of the unit normal vector field and using geometric properties.
result A A-principal Hopf hypersurface in Qm is an open part of a tube around a totally geodesic Qm+1 in Qm. The paper constructs complex structures on hypersurfaces in hyperkahler manifolds.
problem No specific problem stated; focuses on construction of structures.
method Construction of complex metric structures on hypersurfaces in hyperkahler manifolds.
result The construction of complex structures on hypersurfaces in hyperkahler manifolds.
Study new Hopf real hypersurfaces in indefinite complex projective space.
problem Problems posed by H.~Anciaux and K.~Panagiotidou on non-degenerate real hypersurfaces.
method Changed point of view, constructed new families, obtained rigidity results.
result Classified η-umbilical real hypersurfaces and characterized Killing Reeb vector field. Let M be a real hypersurface of a complex space form with almost contact metric structure (φ,ξ,η,g). In this paper, we study real hypersurfaces in a complex space form whose structure Jacobi operator Rξ=R(⋅,ξ)ξ is ξ-parallel. In particular, we prove that the condition ∇ξRξ=0 characterizes the …
Study classifies real hypersurfaces in complex projective spaces based on Lie derivatives and structure Jacobi operator properties.
problem Classifying real hypersurfaces based on Lie derivatives and structure Jacobi operator properties.
method Defined a tensor field RξT(k) from structure Jacobi operator Rξ and Lie derivative, and studied its symmetry and skew-symmetry. result Obtained classifications of real hypersurfaces for which RξT(k) is either symmetric or skew symmetric. New types of null hypersurfaces found in Sasakian space-forms.
problem Identifying new structures in Sasakian space-forms.
method Defined and analyzed contact screen conformal and umbilic null hypersurfaces.
result Proved these hypersurfaces exist in Sasakian space forms with specific curvature.
Simplified proof of Honda-Huang's contact convexity result.
problem Contact convexity in high dimensions
method Simplified proof of Honda-Huang's main result
result Main result from Honda-Huang's paper simplified and presented
Generalizes Giroux's result to higher dimensions and applies to contact manifolds.
problem Characterizing convex surfaces in contact manifolds.
method Extending Giroux's result to arbitrary dimensions and applying to specific hypersurfaces.
result Closed hypersurfaces are C∞-close to convex hypersurfaces. Paper introduces new hypersurface types on statistical manifolds.
problem Understanding new hypersurface types on statistical manifolds.
method Introducing and analyzing new classes of hypersurfaces.
result Obtained properties and relations on specific hypersurface types.
There are considered 4-dimensional pseudo-Riemannian spaces with inner products of signature (3,1) and (2,2). The objects of investigation are space-like and time-like hyperspheres in the respective cases. These hypersurfaces are equipped with almost contact B-metric structures. The constructed manifolds are characteri…
Study on contact type hypersurfaces in 4-space, proving no Brieskorn spheres can embed.
problem Constraints on the topology of 3-manifolds as contact type hypersurfaces in R4. method Obstruction derived from Heegaard Floer homology.
result No Brieskorn homology sphere can be embedded in R4. Study shows non-umbilical qc hypersurfaces in hyper-Kähler manifolds have specific structures.
problem Characterizing non-umbilical quaternionic contact hypersurfaces in hyper-Kähler manifolds.
method Analyzes properties of quaternionic contact hypersurfaces in hyper-Kähler manifolds, focusing on those that are not totally umbilical.
result Non-umbilical qc hypersurfaces in hyper-Kähler manifolds have induced qc structures locally qc homothetic to the standard 3-Sasakian sphere.
Study of intrinsic sub-Laplacian for hypersurfaces in contact sub-Riemannian manifolds.
problem Characterizing the intrinsic sub-Laplacian for hypersurfaces in contact sub-Riemannian manifolds.
method Construction and analysis of the intrinsic sub-Laplacian using Riemannian approximations and stochastic processes.
result The intrinsic sub-Laplacian is stochastically complete, ensuring the process does not hit characteristic points.
A type of almost contact hypersurfaces with Norden metric of a Kaehler manifold with Norden metric is considered. The curvature tensor and the special sectional curvatures are characterized. The canonical connection on such manifolds is studied and the form of the corresponding Kaehler curvature tensor is obtained. Som…
The study explores conditions for realizing lens spaces as boundaries of Milnor fibers of hypersurface singularities.
problem Realizing other lens spaces as boundaries of Milnor fibers of hypersurface singularities.
method Examining necessary conditions for the realization of (L(p,q),ξ) as boundaries of Milnor fibers. result Obtained a series of necessary conditions for realizing other lens spaces as boundaries of Milnor fibers.
In this paper, we study the invariant and noninvariant hypersurfaces of (1,1,1) almost contact manifolds, Lorentzian almost paracontact manifolds and Lorentzian para-Sasakian manifolds, respectively. We show that a noninvariant hypersurface of an (1,1,1) almost contact manifold admits an almost product structure. We in…
Study on real hypersurfaces in complex quadric with special connections and operators.
problem Classifying real hypersurfaces in complex quadric for vanishing tensor fields.
method Defined k-th generalized Tanaka-Webster connections and associated operators, then classified hypersurfaces. result Identified real hypersurfaces where certain tensor fields vanish, focusing on structure Lie operator.
Study lightlike hypersurfaces in a specific type of manifold.
problem Characterize lightlike hypersurfaces in indefinite almost contact metric-manifolds.
method Analyze the position of the structure vector field and classify hypersurfaces into two types.
result Prove there are only two types of lightlike hypersurfaces: ascreen and inascreen.
We describe explicitly all quaternionic contact hypersurfaces (qc-hypersurfaces) in the flat quaternion space $\Hnn$ and the quaternion projective space. We show that up to a quaternionic affine transformation a qc-hypersurface in $\Hnn$ is contained in one of the three qc-hyperquadrics in $\Hnn$. Moreover, we show tha…
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. Paper characterizes umbilical hypersurfaces using a generalized overdetermined problem.
problem Characterizing umbilical hypersurfaces in space forms.
method Using a Serrin-type partially overdetermined problem with inhomogeneous Robin boundary condition.
result Any contact angle θ ∈ (0, π) can be achieved, generalizing previous results.
Paper proves inequality for capillary hypersurfaces with new proof.
problem Proving a Heintze-Karcher type inequality for hypersurfaces with capillary boundary.
method Using a mixed boundary value problem in Reilly type formula to establish the inequality.
result New proof of Alexandrov type theorem for capillary hypersurfaces.
This article describes the following results which relate to each other; i) convergence of high dimensional contact structure to codimension one foliation with Reeb component, ii) relation between Nil-type and Sol-type contact submanifolds of S^5, iii) definition of convex Thurston-Bennequin inequality, and iv) general…
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 prove that every quaternionic-contact structure can be embedded in a quaternionic manifold and define a second fundamental form for a such embedding.
The purpose of this paper is to introduce Liouville hypersurfaces in contact manifolds, which generalize ribbons of Legendrian graphs and pages of supporting open books. Liouville hypersurfaces are used to define a gluing operation for contact manifolds called the Liouville connect sum. Performing this operation on a c…
Extends Moutard quadric concept to higher dimensions.
problem Higher order contact of quadrics with surfaces in 3D.
method Extension to hypersurfaces in arbitrary dimensions.
result Extension of Moutard quadric concept to higher dimensions.
Study contact geometry of energy hypersurface in symmetric 3-body problem on S^2.
problem Contact geometry of energy hypersurface in symmetric 3-body problem on S^2.
method Contact geometry analysis, Moser regularization, Weinstein conjecture.
result Components of energy hypersurface are contactomorphic to R^3 or its connected sum for specific energies.
The paper proves inequalities for convex capillary hypersurfaces in a half-space.
problem Proving inequalities for convex capillary hypersurfaces in a half-space.
method Locally constrained inverse curvature flow with spherical cap convergence.
result Proves a complete family of Alexandrov-Fenchel inequalities for convex capillary hypersurfaces.
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.
Conditions for hypersurfaces of generalized Kähler manifolds to have specific structures are established.
problem Conditions for hypersurfaces of generalized Kähler manifolds to have specific structures.
method Established conditions for induced generalized metric F structure to be a generalized CRFK structure.
result Characterization of binormality and examples of binormal structures.
We study stable immersed capillary hypersurfaces in a domain B which is either a half-space or a slab in the Euclidean space Rn+1. We prove that such a hypersurface Σ is rotationally symmetric in the following cases: (1) n=2, B is a slab and Σ has genus zero, (2) n≥2, $\mathc…
In this note we define the notion of collarable slices of Lagrangian submanifolds. Those are slices of Lagrangian submanifolds which can be isotoped through Lagrangian submanifolds to a cylinder over a Legendrian embedding near a contact hypersurface. Such a notion arises naturally when studying intersections of Lagran…
The shape operator and induced almost contact structure of nearly Kahler hypersurfaces do not commute.
problem Characterizing hypersurfaces in nearly Kahler spaces.
method Construction of twistor spaces and analysis of shape operators and induced structures.
result The shape operator and induced almost contact structure of nearly Kahler hypersurfaces do not commute.
We show that any open subset of a contact manifold of dimension greater than three contains a certain non-convex hypersurface violating the Thurston-Bennequin inequality.
Study stable capillary hypersurfaces with planar boundaries in half-spaces and domains.
problem Characterize stable capillary hypersurfaces with planar boundaries in bounded domains.
method Analyzes hypersurfaces in half-spaces and domains bounded by hyperplanes, proving conditions for stability and shape.
result Stable hypersurfaces in certain domains are spherical caps or pieces of spheres.
Study anisotropic flow for capillary hypersurfaces, proving new inequalities.
problem Anisotropic capillary hypersurfaces and their properties.
method Anisotropic volume-preserving mean curvature flow, new approach for strictly convex initial hypersurfaces.
result Established new Alexandrov-Fenchel inequalities for strictly convex anisotropic capillary hypersurfaces.
Constructs minimal capillary cones with specific symmetry and proves their existence and uniqueness.
problem Proves existence and uniqueness of minimal capillary cones with bi-orthogonal symmetry.
method Solves a nonlinear free boundary equation parametrized by the contact angle and uses monotonicity properties.
result Demonstrates that minimizing capillary hypersurfaces can have singularities in codimension 7.