Curves with stable or semistable normal bundle on Fano hypersurfaces

For every $n\geq 3, g\geq 1$ and all large enough $e$ depending on $n,g$, there exist curves of genus $g$, degree $e$ in a general hypersurface of degree $n$ in $\mathbb P^n$, or in $\mathbb P^n$ itself, whose whose normal bundle $N$ is stable, as is any sufficiently general full-rank subsheaf of $N$. For $g=1$, $N$ is semi-stable. On general hypersurface of degree $d< n$ in $\mathbb P^n$, such that a certain arithmetical condition on $d,n, g$ holds, there exists an arithmetical progression of $e$ values so that curves of degree $e$ and genus $g$ with semistable normal bundle exist. Previous results were restricted to certain cases with ambient space $\P^n$