Explicit Homotopies of Liouville Domains

This dissertation contains results that contribute to and use the theory of convex hypersurfaces in contact manifolds. First, we generalize a $3$-dimensional convexity criterion result of Giroux \cite{giroux1991convexite}. Specifically, we show that the criterion holds in contact manifolds of arbitrary dimension. As an application, we show that a particular closed hypersurface introduced by A. Mori \cite{mori2011convex} is $C^{\infty}$-close to a convex hypersurface. Second, inspired by the techniques of Honda and Huang in \cite{honda2019convex}, we develop explicit local operations that may be applied to Liouville domains with the goal of simplifying the dynamics of the Liouville vector field. As an application, we show that Mitsumatsu's well-known Liouville-but-not-Weinstein domains are stably Weinstein, answering a question posed by Huang \cite{huang2019dynamical}