Convex hypersurface theory in contact topology

• DOI: 10.48550/arxiv.1907.06025
• Publication Date: 2019-01-01
• Authors: Honda, Ko; Huang, Yang
• Keywords: Mathematics - Symplectic Geometry; FOS: Mathematics; Symplectic Geometry (math.SG); 0101 mathematics; 01 natural sciences

V1: 93 pages, 27 figures; V2: Corrected a minor mistake in section 12 and a few typos; V3: Significant revision following referees' feedback, a mistake pointed out by Yasha Eliashberg, Fran\c{c}ois-Simon Fauteux-Chapleau and Dishant Pancholi, and a proof of Lemma 6.3.1 by Emmanuel Giroux<br/>We lay the foundations of convex hypersurface theory (CHT) in contact topology, extending the work of Giroux in dimension three. Specifically, we prove that any closed hypersurface in a contact manifold can be $C^0$-approximated by a convex one. We also prove that a $C^0$-generic family of mutually disjoint closed hypersurfaces parametrized by $t \in [0,1]$ is convex except at finitely many times $t_1, \dots, t_N$, and that crossing each $t_i$ corresponds to a bypass attachment. As applications of CHT, we prove the existence of compatible (relative) open book decompositions for contact manifolds and an existence h-principle for codimension 2 contact submanifolds.<br/>