Nearly geodesic surfaces are filling

Let $M$ be a closed hyperbolic $3$-manifold. A homotopy class $[S]$ of surfaces in $M$ is filling if any representative cuts $M$ into components contractible in $M$. We prove that there exist $\epsilon_0, g_0>0$ such that every homotopy class of $(1+\epsilon)$-quasi-Fuchsian surfaces with $0<\epsilon\leq \epsilon_0$ or totally geodesic surfaces of genus $\geq g_0$ in $M$ is filling. As a corollary, except for at most finitely many totally geodesic surfaces, embedded incompressible quasi-Fuchsian surfaces in $M$ have constants bounded below by $1+\epsilon_0$. This also gives a gap theorem for embedded minimal surfaces. Each of these surfaces separates any pair of distinct points at the sphere of infinity. Crucial tools include the rigidity results of Mozes-Shah, Ratner, and Shah. This work is inspired by a question of Wu and Xue whether random geodesics on random hyperbolic surfaces are filling.