  We consider self-repelling elastic manifolds with a domain $[-N,N]^d \cap
\mathbb{Z}^d$, that take values in $\mathbb{R}^D$. Our main result states that
when the dimension of the domain is $d=2$ and the dimension of the range is
$D=1$, the effective radius $R_N$ of the manifold is approximately $N^{4/3}$.
This verifies the conjecture of Kantor, Kardar and Nelson [7]. Our results for
the case where $d \geq 3$ and $D <d$ give a lower bound on $R_N$ of order
$N^{\frac{1}{D} \left(d-\frac{2(d-D)}{D+2} \right)}$ and an upper bound
proportional to $N^{\frac{d}{2}+\frac{d-D}{D+2}}$. These results imply that
self-repelling elastic manifolds with a low dimensional range undergo a
significantly stronger stretching than in the case where $d=D$, which was
studied by the authors in [10].
