  Bernardeau et al. (1997), using perturbation theory, showed that the skewness
of the large-scale lensing-convergence, or projected mass density, could be
used to constrain $\Omega_m$, the matter content of the universe. On the other
hand, deep weak-lensing field surveys in the near future will likely measure
the convergence on small angular scales (< 10 arcmin.), where the signal will
be dominated by highly nonlinear fluctuations. We develop a method to compute
the small-scale convergence skewness, using a prescription for the highly
nonlinear three-point function developed by Scoccimarro and Frieman (1998).
This method gives predictions that agree well with existing results from
ray-tracing N-body simulations, but is significantly faster, allowing the
exploration of a large number of models. We demonstrate that the small-scale
convergence skewness is insensitive to the shape and normalization of the
primordial (CDM-type) power spectrum, making it dependent almost entirely on
the cosmological energy contents, through their influence on the global
geometrical distances and fluctuation growth rate. Moreover, nonlinear
clustering appears to enhance the differences between predictions of the
convergence skewness for a range of models. Hence, in addition to constraining
$\Omega_m$, the small-scale convergence skewness from future deep several-
degree-wide surveys can be used to differentiate between curvature dominated
and cosmological constant ($\Lambda$) dominated models, as well as to constrain
the equation of state of a quintessence component, thereby distinguishing
$\Lambda$ from quintessence as well. Finally, our method can be easily
generalized to other measures such as aperture mass statistics.
