On constant mean curvature 1-immersions of surfaces into hyperbolic 3-manifolds.

The work of Bryant [7] revealed striking analogies between constant mean curvature (CMC) 1-immersions of surfaces into the hyperbolic space H3 (Bryant surfaces) and minimal immer- sions into the euclidean space E3. 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 g ≥ 2) here we pur- sue the existence and uniqueness of (CMC) 1-immersions of S into hyperbolic 3-manifolds. It has been shown in [22] that, for |c| < 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| → 1−. However, the passage to the limit can be prevented by possible blow-up phenomena, and at the limit (|c| → 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 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 g − 1) blow-up points can occur. In this case, for any 1 ≤ ν ≤ g − 1, we identify in the ν-secant variety of Z the appropriate replacement of Z (relative to ν = 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 g = 3 (see Theorem 5), and provide relevant contributions for arbitrary genus.