Non-convex Hypersurfaces And Robust Heterodimensional Cycles

A closed hypersurface in a contact manifold is convex if it admits a transverse contact vector field. A closed hypersurface is robustly non-convex if it cannot be smoothly approximated by a convex hypersurface. The existence of such hypersurfaces was a longstanding open problem, resolved recently by myself using tools from partially hyperbolic dynamics.

In this talk, I will explain the relationship between robust non-convexity and the presence of a structure called a  heterodimensional cycle in the characteristic foliation of the hypersurface. As an application, one can prove that any closed hypersurface in any contact manifold can be perturbed in the C0

-topology to a robustly non-convex hypersurface. This substantially generalizes and simplifies my previous work. This talk covers joint work with my PhD student Michael Huang (USC).