Related papers: An overtwisted convex hypersurface in higher dimen…

We investigate the relationships between the Lipschitz outer geometry and the embedded topological type of a hypersurface germ in $(\mathbb C^n,0)$. It is well known that the Lipschitz outer geometry of a complex plane curve germ determines…

We give a possible generalization of Lutz twist to all dimensions. This reproves the fact that every contact manifold can be given a non-fillable contact structure and also shows great flexibility in the manifolds that can be realized as…

We show the existence of tight contact structures on infinitely many hyperbolic three-manifolds obtained via Dehn surgeries along sections of hyperbolic surface bundles over circle.

Liebmann's Theorem asserts that a compact, connected, convex surface with constant mean curvature (CMC) in the Euclidean space must be a totally umbilical sphere. In this article we extend Liebmann's result to hypersurfaces with boundary.…

We introduce a modification procedure for Engel structures that is reminiscent of the Lutz twist in 3-dimensional Contact Topology. This notion allows us to define what an Engel overtwisted disc is, and to prove a complete h-principle for…

We construct symplectic submanifolds of symplectic manifolds with contact border. The boundary of such submanifolds is shown to be a contact submanifold of the contact border. We also give a topological characterization of the constructed…

The existence of closed hypersurfaces of prescribed curvature in globally hyperbolic Lorentzian manifolds is proved provided there are barriers.

It has been proposed that equilibrium thermodynamics is described on Legendre submanifolds in contact geometry. It is shown in this paper that Legendre submanifolds embedded in a contact manifold can be expressed as attractors in phase…

We study Legendrian embeddings of a compact Legendrian submanifold $L$ sitting in a closed contact manifold $(M,\xi)$ whose contact structure is supported by a (contact) open book $\mathcal{OB}$ on $M$. We prove that if $\mathcal{OB}$ has…

We study compatible contact structures of fibered, positively-twisted graph multilinks in the 3-sphere and prove that the contact structure of such a multilink is tight if and only if the orientations of its link components are all…

We propose a proof of the characterization of spherical hypersurfaces as the only strictly pseudoconvex hypersurfaces which have contracting germs of diffeomorphisms. Unlike previous proofs, our approach does not use Chern-Moser theory. It…

We investigate families of Legendrian submanifolds of 1-jet spaces by developing and applying a theory of families of generating family homologies. This theory allows us to detect an infinite family of loops of Legendrian n-spheres embedded…

We show that in Cartan-Hadamard manifolds $M^n$, $n\geq 3$, closed infinitesimally convex hypersurfaces $\Gamma$ bound convex flat regions, if curvature of $M^n$ vanishes on tangent planes of $\Gamma$. This encompasses…

We obtain a sub-Riemannian version of the classical Gauss-Bonnet theorem. We consider subsurfaces of a three dimensional contact sub-Riemannian manifolds, and using a family of taming Riemannian metric, we obtain a pure sub-Riemannian…

We prove several results on approximation and interpolation of holomorphic Legendrian curves in convex domains in $\mathbb{C}^{2n+1}$, $n \geq 2$, with the standard contact structure. Namely, we show that such a curve, defined on a compact…

We give a combinatorial description of the Legendrian contact homology algebra associated to a Legendrian link in $S^1\times S^2$ or any connected sum $\#^k(S^1\times S^2)$, viewed as the contact boundary of the Weinstein manifold obtained…

We prove a result which establishes restrictions on the pseudoholomorphic curves which can exist in a stable Hamiltonian manifold in the presence of certain $\mathbb{R}$-invariant foliations of the symplectization by holomorphic…

We prove two weighted geometric inequalities that hold for strictly mean convex and star-shaped hypersurfaces in Euclidean space. The first one involves the weighted area and the area of the hypersurface and also the volume of the region…

Let M be a hyperbolic 3-manifold with nonempty totally geodesic boundary. We prove that there are upper and lower bounds on the diameter of the skinning map of M that depend only on the volume of the hyperbolic structure with totally…

Reflections from hypersurfaces act by symplectomorphisms on the space of oriented lines with respect to the canonical symplectic form. We consider an arbitrary $C^{\infty}$-smooth hypersurface $\gamma\subset\mathbb R^{n+1}$ that is either a…