  For smooth mappings of the unit disc into the oriented Grassmannian manifold
$\mathbb G_{n,2}$, H\'elein (2002) conjectured the global existence of Coulomb
frames with bounded conformal factor provided the integral of $|\boldsymbol
A|^2$, the squared-length of the second fundamental form, is less than
$\gamma_n=8\pi$. It has since been shown that the optimal bounds on the
integral of $|\boldsymbol A|^2$ that guarantee this result are: $\gamma_3 =
8\pi$ and $\gamma_n = 4\pi$ for $n \geq 4$. For isothermal immersions, this
hypothesis is equivalent to saying the integral of the sum of the squares of
the principal curvatures is less than $\gamma_n$. The goal here is to prove
that when $n=3$ the same conclusion holds under weaker hypotheses. In
particular, it holds for isothermal immersions when $|\boldsymbol A|$ is
square-integrable and the integral of $|K|$, $K$ the Gauss curvature, is less
than $4\pi$. Since $2|K| \leq |\boldsymbol A|^2$ this implies the known result
for isothermal immersions, but $|K|$ may be small when $|\boldsymbol A|^2$ is
large. That the result under the weaker hypothesis is sharp is shown by
Enneper's surface and stereographic projections. The method, which is purely
analytic, is then extended to investigate the case when the length of the
second fundamental form is square-integrable.
