Krzysztof Wysocki | Searcharxiv

arXiv2021

Genus zero global surfaces of section for Reeb flows and a result of Birkhoff

We exhibit sufficient conditions for a finite collection of periodic orbits of a Reeb flow on a closed $3$-manifold to bound a positive global surface of section with genus zero. These conditions turn out to be $C^\infty$-generically necessary. Moreover, they involve linking assumptions on periodic orbits with Conley-Zehnder index ranging in a finite set determined by the ambient contact geometry. As an application, we reprove and generalize a classical result of Birkhoff on the existence of annulus-like global surfaces of section for geodesic flows on positively curved two-spheres.

math.DS↗

arXiv2017

Polyfold and Fredholm Theory

This is a reference volume on polyfold and Fredholm theory.

math.FA↗

arXiv2007

Branch structure of J-holomorphic curves near periodic orbits of a contact manifold

Let $M$ be a three-dimensional contact manifold and $ψ:D\setminus\{0\}\to M\times{\Bbb R}$ a finite-energy pseudoholomorphic map from a punctured disc in ${\Bbb C}$, that is asymptotic to a periodic orbit of the Reeb vector field. This article examines conditions under which smooth coordinates may be defined in a tubular neighbourhood of the orbit such that $ψ$ resembles a holomorphic curve, invoking comparison with the theory of topological linking of plane complex algebroid curves near a singularity. Examples of this behaviour which are studied in some detail include pseudoholomorphic maps into ${\Bbb E}_{p,q}\times{\Bbb R}$, where ${\Bbb E}_{p,q}$ denotes a rational ellipsoid with contact structure induced by the complex structure of the ambient ${\Bbb C}^{2}$. Contact structures arising from non-standard circle-fibrations of the three-sphere are also examined.

math.CV↗