The work of Bryant [7] revealed striking analogies between constant mean curvature (CMC) 1-immersions of surfaces into the hyperbolic space \(\mathbb {H}^3\) (Bryant surfaces) and minimal immersions into the euclidean space \(\mathbb {E}^3.\) Ever since, the role of (CMC) 1-immersions in hyperbolic geometry has been widely explored, see e.g. [44] and references therein. In account of [16, 53] and after [48], for a given surface S (closed, orientable and of genus \(\mathfrak {g}\ge 2\) ) here we pursue the existence and uniqueness of (CMC) 1-immersions of S into hyperbolic 3-manifolds. It has been shown in [22] that, for \(\vert c \vert <1\) , the moduli space of (CMC) c-immersions of S into hyperbolic 3-manifolds can be parametrised by elements of the tangent bundle of the Teichmüller space of the surface S. In turn in [48] it was pointed out that (CMC) 1-immersions enter as "critical" objects, in the sense that they can be attained only as limits of (CMC) c-immersions, as \(|c| \rightarrow 1^-.\) However, the passage to the limit can be prevented by possible blow-up phenomena, and at the limit ( \(|c| \rightarrow 1^-\) ) we could end up at best with an immersed surface having conical singularities supported at finitely many points (the blow-up points). If the genus \(\mathfrak {g}=2\) then blow up can occur at a single point, and in [48] it was shown how it could be prevented and the passage to the limit ensured in terms of the image Z of Kodaira map given in (2.30). In this note we show that actually blow-up can occur only at one of the six Weierstrass points of the surface. Thus, in Theorem 1 and Theorem 4 we establish existence and uniqueness results under a sufficient "compactness" condition, which in fact turns out to be also necessary, as shown in [51]. In addition we analyze the case of higher genus, where multiple (up to \(\mathfrak {g}- 1\) ) blow-up points can occur. In this case, for any \(1 \le \nu \le \mathfrak {g}- 1,\) we identify in the \(\nu \) -secant variety of Z the appropriate replacement of Z (relative to \(\nu =1\) ), see Proposition 2.1. Moreover, in Theorem 3 we improve in a substantial way the asymptotic analysis of [48], which concerns only the case of "blow-up" with minimal mass. As a consequence, we cover the case of genus \(\mathfrak {g}=3\) (see Theorem 5), and provide relevant contributions for arbitrary genus.
