In this article we prove that the Weinstein conjecture holds for contact manifolds (Σ, ξ) for which Cont0(Σ, ξ) is non-orderable in the sense of Eliashberg-Polterovich [EP00]. More precisely, we establish a link between orderable and hypertight contact manifolds. In addition, we prove for certain contact manifolds a conjecture by Sandon [San13] on the existence of translated points in the non-degenerate case.
We prove an inequality between the L ∞ -norm of the contact Hamiltonian of a positive loop of contactomorphims and the minimal Reeb period. This implies that there are no small positive loops on hypertight or Liouville fillable contact manifolds. Non-existence of small positive loops for overtwisted 3-manifolds was proved by Casals-Presas-Sandon in [CPS16].As corollaries of the inequality we deduce various results. E.g. we prove that certain periodic Reeb flows are the unique minimizers of the L ∞ -norm. Moreover, we establish L ∞ -type contact systolic inequalities in the presence of a positive loop.
H−holomorphic curves are solutions of a specific modification of the pseudoholomorphic curve equation in symplectizations involving a harmonic 1−form as perturbation term. In this paper we compactify the moduli space of H−holomorphic curves with a priori bounds on the harmonic 1−forms.
Contents4 Discussion on conformal period
We prove two theorems on the removal of singularities on the boundary of a pseudo-holomorphic curve. In one theorem, we need no apriori assumption on the area of the curve. The proof uses a doubling argument with the goal of converting curves with boundary to curves without boundary. Our method is new and geometric and it does not need Sobolev spaces and PDEs. 1 arXiv:1210.4324v1 [math.SG] 16 Oct 2012 Definition 2.4. A submanifold W of (M, J) is said to be totally real if dim(W ) = 1 2 dim(M ) and J(T p W ) ∩ T p W = {0} for all p ∈ W . Definition 2.5. Suppose (S, J 1 ) and (N, J 2 ) are manifolds with almost complex structures J 1 and J 2 . A C 1 map h : S → N is J-holomorphic or pseudo-holomorphic if the derivative of h is complex linear with respect to J 1 and J 2 , i.e.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.