Looking for flexibility in higher-dimensional contact manifolds

Olga Plamenevskaya

For 3-dimensional contact manifolds, a classical result of Eliashberg describes a large family of “flexible” contact structures, whose properties are completely determined by the algebraic topology information. These contact structures are characterized by the presence of an “overtwisted disk”. In higher dimensions, a class of flexible contact structures is yet to be found, although some conjectural generalizations of an “overtwisted piece” exist. We show that in presence of such a piece, Legendrian knots become flexible, and contact structures exhibit certain flexibility phenomena. Our results are based on an important h-principle for certain Legendrian knots discovered by Murphy and Cieliebak-Eliashberg results on flexible Weinstein manifolds. (This work was done a while ago jointly with M. Murphy, K. Niederkruger, and A. Stipsicz.)

http://scgp.stonybrook.edu/video_portal/video.php?id=1585

Simons- Program: Moduli Spaces of Pseudo-holomorphic curves and their applications to Symplectic Topology