  Motivated by the work of Bryant on constant mean curvature (CMC)
$1$-immersions of surfaces into the hyperbolic space H^3 and after the results
of Tarantello (2023), we pursue a possible parametrization for the moduli space
of (CMC) 1-immersions of a surface S (closed, orientable and of genus >1) into
hyperbolic 3-manifolds. Those immersions enter as "critical" object in our
analysis. In fact, they can be attained only as limits of the (CMC)
c-immersions (as c tends to 1), obtained in Huang-Lucia-Tarantello (2022), for
|c|<1. However, such passage to the limit can be prevented by possible blow-up
phenomena, so that the pullback metrics of the (CMC) c-immersions may yield (at
the limit) to a singular metric with conical singularities at finitely many
points (the blow-up points). In case of genus g=2, blow up can occur only at a
single point, and in Tarantello (2023) it was shown how it could be prevented
and the passage to the limit ensured in terms of the Kodaira map. In this note
we sharpen this result and for genus g=2, we obtaina condition (we believe
sharp) which involves only the Kodaira map on the six Weierstrass points. In
addition we tackle the case of higher genus, where multiple blow-up points
occur. In this case, we need to identify a suitable replacement of the Kodaira
map, now defined on the space of non-zero effective divisors. More importantly,
we need to improve in a substantial way the asymptotic analysis of Tarantello
(2023) limited to the case of "blow-up" with minimal mass. In this direction we
give a contribution which best applies to the case of genus g=3, but also
provides a relevant step and a convincing indication on what should happen in
the general case.
