  We consider the full three-dimensional Ginzburg-Landau model of
superconductivity with applied magnetic field, in the regime where the
intensity of the applied field is close to the "first critical field" $H_{c_1}$
at which vortex filaments appear, and in the asymptotics of a small inverse
Ginzburg-Landau parameter $\varepsilon$. This onset of vorticity is directly
related to an "isoflux problem" on curves (finding a curve that maximizes the
ratio of a magnetic flux by its length), whose study was initiated in [Rom\'an,
C. On the First Critical Field in the Three Dimensional Ginzburg-Landau Model
of Superconductivity. Commun. Math. Phys. 367, 317-349 (2019).
https://doi.org/10.1007/s00220-019-03306-w] and which we continue here. By
assuming a nondegeneracy condition for this isoflux problem, which we show
holds at least for instance in the case of a ball, we prove that if the
intensity of the applied field remains below $H_{c_1}+ C \log |\log
\varepsilon|$, the total vorticity remains bounded independently of
$\varepsilon$, with vortex lines concentrating near the maximizer of the
isoflux problem, thus extending to the three-dimensional setting a
two-dimensional result of [Sandier, E., Serfaty, S. Ginzburg-Landau minimizers
near the first critical field have bounded vorticity. Cal Var 17, 17-28 (2003).
https://doi.org/10.1007/s00526-002-0158-9]. We finish by showing an improved
estimate on the value of $H_{c_1}$ in some specific simple geometries.
