On dynamically convex contact manifolds and filtered symplectic homology (Q6536659)

This paper studies the topological properties of Liouville fillable contact manifolds that admit a dynamically convex contact form.\N\NHere a contact form \(\alpha\) on a (\(2n-1\))-dimensional contact manifold \((\Sigma,\xi)\) with \(c_1(\xi)=0\) is called dynamically convex if the (lower) Conley-Zehnder index of each contractible Reeb orbit is bounded from below by \(n+1\), see [\textit{H. Hofer} et al., Ann. Math. (2) 148, No. 1, 197--289 (1998; Zbl 0944.37031)].\N\NIn the case \(n\geq 3\) and \(\Sigma\) being simply connected and dynamically convex and fillable by a flexible Weinstein manifold \(W\) [\textit{K. Cieliebak} and \textit{Y. Eliashberg}, From Stein to Weinstein and back. Symplectic geometry of affine complex manifolds. Providence, RI: American Mathematical Society (AMS) (2012; Zbl 1262.32026)], the authors show that \((\Sigma,\xi)\) is contactomorphic to the standard sphere and \(W\) is Stein homotopy equivalent to the standard ball.\N\NIn the second part of the paper the authors prove a certain periodicity result for the filtered symplectic homology of topologically simple Liouville domains with vanishing symplectic homology and a boundary with periodic, dynamically convex Reeb flow. Further they characterise such Reeb flows in terms of Gutt-Hutchings capacities [\textit{J. Gutt} and \textit{M. Hutchings}, Algebr. Geom. Topol. 18, No. 6, 3537--3600 (2018; Zbl 1411.53062)].