Restrictions on rational surfaces lying in very general hypersurfaces

We study rational surfaces on very general Fano hypersurfaces in $\mathbb{P}^n$, with an eye toward unirationality. We prove that given any fixed family of rational surfaces, a very general hypersurface of degree $d$ sufficiently close to $n$ and $n$ sufficiently large will admit no maps from surfaces in that family. In particular, this shows that for such hypersurfaces, any rational curve in the space of rational curves must meet the boundary. We also prove that for any fixed ratio $\alpha$, a very general hypersurface in $\mathbb{P}^n$ of degree $d$ sufficiently close to $n$ will admit no maps from a surface satisfying $H^2 \geq \alpha HK$, where $H$ is the pullback of the hyperplane class from $\mathbb{P}^n$ and $K$ is the canonical bundle on the surface.