  In this paper we study the functional $\SW_{\lambda_1,\lambda_2}$, which is
the the sum of the Willmore energy, $\lambda_1$-weighted surface area, and
$\lambda_2$-weighted volume, for surfaces immersed in $\R^3$. This coincides
with the Helfrich functional with zero `spontaneous curvature'. Our main result
is a complete classification of all smooth immersed critical points of the
functional with $\lambda_1\ge0$ and small $L^2$ norm of tracefree curvature. In
particular we prove the non-existence of critical points of the functional for
which the surface area and enclosed volume are positively weighted.
