arXiv subjects
Publications and source records attributed to Richard Siefring.
We exhibit transverse knot types on the standard contact $3$-sphere that cannot be realized as periodic Reeb orbits of a dynamically convex contact form. In particular, such transverse knot types do not arise as closed characteristics of strictly convex energy levels on a four dimensional symplectic vector space.
We establish some new existence results for global surfaces of section of dynamically convex Reeb flows on the three-sphere. These sections often have genus, and are the result of a combination of pseudo-holomorphic curve methods with some elementary ergodic methods.
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 hypersurfaces. This result has applications in the first author's work on algebraic torsion in higher dimensional contact manifolds.
It's known from from work of Hofer, Wysocki, and Zehnder [1996] and Bourgeois [2002] that in a contact manifold equipped with either a nondegenerate or Morse-Bott contact form, a finite-energy pseudoholomorphic curve will be asymptotic at each of its non removable punctures to a single periodic orbit of the Reeb vector field and that the convergence is exponential. We provide examples here to show that this need not be the case if the contact form is degenerate. More specifically, we show that on any contact manifold $(M, ξ)$ with cooriented contact structure one can choose a contact form $λ$ with $\kerλ=ξ$ and a compatible complex structure $J$ on $ξ$ so that for the associated $\mathbb{R}$-invariant almost complex structure $\tilde J$ on $\mathbb{R}\times M$ there exist families of embedded finite-energy $\tilde J$-holomorphic cylinders and planes having embedded tori as limit sets.
We study the intersection theory of punctured pseudoholomorphic curves in $4$-dimensional symplectic cobordisms. We first study the local intersection properties of such curves at the punctures. We then use this to develop topological controls on the intersection number of two curves. We also prove an adjunction formula which gives a topological condition that will guarantee a curve in a given homotopy class is embedded, extending previous work of Hutchings. We then turn our attention to curves in the symplectization $\mathbb{R}\times M$ of a $3$-manifold $M$ admitting a stable Hamiltonian structure. We investigate controls on intersections of the projections of curves to the $3$-manifold, and we present conditions that will guarantee the projection of a curve to the $3$-manifold is an embedding. Finally we consider an application concerning pseudoholomorphic curves in manifolds admitting a certain class of holomorphic open book decomposition, and an application concerning the existence of generalized pseudoholomorphic curves.
We consider a $3$-manifold $M$ equipped with nondegenerate contact form $λ$ and compatible almost complex structure $J$. We show that if the data $(M, λ, J)$ admits a stable finite energy foliation, then for a generic choice of distinct points $p$, $q\in M$, the manifold $M'$ formed by taking the connected sum at $p$ and $q$ admits a nondegenerate contact form $λ'$ and compatible almost complex structure $J'$ so that the data $(M', λ', J')$ also admits a stable finite energy foliation. Along the way, we develop some general theory for the study of finite energy foliations.
Withdrawn by author due to copyright transfer. See: http://www3.interscience.wiley.com/cgi-bin/abstract/116836954/ABSTRACT