arXiv research

Study on contact type hypersurfaces in 4-space, proving no Brieskorn spheres can embed.

Paper introduces new hypersurface types on statistical manifolds.

Study lightlike hypersurfaces in a specific type of manifold.

Paper proves inequality for capillary hypersurfaces with new proof.

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…

Study contact geometry of energy hypersurface in symmetric 3-body problem on S^2.

Study on contact Hamiltonian functions for singular contact structures.

Paper characterizes umbilical hypersurfaces using a generalized overdetermined problem.

Study on null hypersurfaces in complex contact manifolds.

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 examples of real hypersurfaces found in complex hyperbolic quadrics.

Study examines tangential real hypersurfaces on Hermite-like manifolds.

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…

Study real hypersurfaces in complex space forms for an inequality involving a contact invariant.

A closed contact manifold is called Besse when all its Reeb orbits are closed, and Zoll when they have the same minimal period. In this paper, we provide a characterization of Besse contact forms for convex contact spheres and Riemannian unit tangent bundles in terms of $S^1$-equivariant spectral invariants. Furthermor…

New Minkowski inequality for capillary surfaces in half-space.

The paper proves a Willmore-type inequality for unbounded convex sets.

Study anisotropic flow for capillary hypersurfaces, proving new inequalities.

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 $R_ξ=R(\cdot,ξ)ξ$ is $ξ$-parallel. In particular, we prove that the condition $\nabla_ξ R_ξ=0$ characterizes the …

Study examines homology of contact CR-submanifolds in complex Euclidean space.

Simplified proof of Honda-Huang's contact convexity result.

Study of intrinsic sub-Laplacian for hypersurfaces in contact sub-Riemannian manifolds.

A real hypersurface in the complex quadric $Q^m=SO_{m+2}/SO_mSO_2$ is said to be $\mathfrak A$-principal if its unit normal vector field is singular of type $\mathfrak A$-principal everywhere. In this paper, we show that a $\mathfrak A$-principal Hopf hypersurface in $Q^m$, $m\geq3$ is an open part of a tube around a t…

Study classifies real hypersurfaces in complex projective spaces based on Lie derivatives and structure Jacobi operator properties.

In this paper, we construct complex metric structures on complex hypersurfaces in hyperkahler manifolds. This construction is that in contact geometry.

We obtain several results for (iterated) planar contact manifolds in higher dimensions: (1) Iterated planar contact manifolds are not weakly symplectically semi-fillable. This generalizes a 3-dimensional result of Etnyre to a higher-dimensional setting. (2) They do not arise as nonseparating weak contact-type hypersurf…

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…

Symmetric hypersurfaces and boundaries in R^n+1 with group actions.

We introduce two classes of null hypersurfaces of an indefinite Sasakian manifold, $(\overline{M}, \overlineφ,ζ, η)$, tangent to the characteristic vector field $ζ$, called; {\it contact screen conformal} and {\it contact screen umbilic} null hypersurfaces. These hypersurfaces come in to fill the existing gap in screen…

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…

Paper proves short-term existence of fractional mean curvature flow.

Each hypersurface of a nearly Kähler manifold is naturally equipped with two tensor fields of $(1,1)$-type, namely the shape operator $A$ and the induced almost contact structure $φ$. In this paper, we show that, in the homogeneous NK $\mathbb{S}^6$ a hypersurface satisfies the condition $Aφ+φA=0$ if and only if it is …

We give examples of compact symplectic manifolds with disconnected contact type boundary in dimension $4n$ for any $n\geq 1$. The example is given by a subset of the tangent bundle of a compact quotient of the complex hyperbolic space endowed with the canonical symplectic form plus a generalized magnetic field and its …

We generalize a result of Giroux which says that a closed surface in a contact $3$-manifold with Morse-Smale characteristic foliation is convex. Specifically, we show that the result holds in contact manifolds of arbitrary dimension. As an application, we show that a particular closed hypersurface introduced by A. Mori…

We study Weyl structures on lightlikes hypersurfaces endowed with a conformal structure of certain type and specific screen distribution: the Weyl screen structures. We investigate various differential geometric properties of Einstein-Weyl screen structures on lightlike hypersurfaces and show that, for ambiant Lorentzi…

New definition of stable $(r+1)$-th capillary hypersurfaces proposed.

We show that any compact quaternionic contact (qc) hypersurfaces in a hyper-Kähler manifold which is not totally umbilical has an induced qc structure, locally qc homothetic to the standard 3-Sasakian sphere. We also show that any nowhere umbilical qc hypersurface in a hyper-Kähler manifold is endowed with an involutiv…

We explore a natural generalization of systolic geometry to Finsler metrics and optical hypersurfaces with special emphasis on its relation to the Mahler conjecture and the geometry of numbers. In particular, we show that if an optical hypersurface of contact type in the cotangent bundle of the 2-dimensional torus encl…

A long-standing conjecture in Hamiltonian Dynamics states that the Reeb flow of any convex hypersurface in $\mathbb{R}^{2n}$ 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…

Shows smoothness of varifolds with specific boundary angles.

The study shows knots from 3-braids cannot be concordant to a specific Legendrian unknot.

We prove that every quaternionic-contact structure can be embedded in a quaternionic manifold and define a second fundamental form for a such embedding.

Extends Moutard quadric concept to higher dimensions.

It is known that the lens space $L(2n,1)$ supports a virtually overtwisted contact structure arising as the boundary of the Milnor fiber of a complex hypersurface singularity. In this article we study the problem of realizing other $(L(p,q),ξ)$ in such a way, obtaining a series of necessary conditions for this to happe…

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…

We study the geometry of multidimensional scalar $2^{nd}$ order PDEs (i.e. PDEs with $n$ independent variables) with one unknown function, viewed as hypersurfaces $\mathcal{E}$ in the Lagrangian Grassmann bundle $M^{(1)}$ over a $(2n+1)$-dimensional contact manifold $(M,\mathcal{C})$. We develop the theory of character…

Develops new approach to recover CR structures from their Levi foliations.

Constructs CMC hypersurfaces in S^4 from piecewise-smooth unions of spheres.