  In this paper, we study the existence and classification problems of
left-symmetric superalgebras on special linear Lie superalgebras
${\mathfrak{sl}}(m|n)$ with $m\neq n$. The main three results of this paper
are: (i) a complete classification of the left-symmetric superalgebras on
${\mathfrak{sl}}(2|1)$, (ii) ${\mathfrak{sl}}(m|1)$ does not admit
left-symmetric superalgebras for $m \geq 3$, and (iii) ${\mathfrak{sl}}(m+1|m)$
admits a left-symmetric superalgebra for every $m \geq 1$. To prove these
results we combine existing results on the existence and classification of
left-symmetric algebras on the Lie algebras ${\mathfrak{gl}}_m$ with a detailed
analysis of small representations of the Lie superalgebras
${\mathfrak{sl}}(m|1)$. We also conjecture that ${\mathfrak{sl}}(m|n)$ admits
left-symmetric superalgebras if and only if $m = n+1$.
