(CMC) 1-immersions of surfaces into hyperbolic 3-manifolds

TL;DR

This paper studies constant mean curvature 1-immersions of surfaces into hyperbolic 3-manifolds, analyzing their existence, singularities, and limits as the mean curvature approaches 1, with results applicable to surfaces of any genus.

Contribution

It extends the analysis of CMC 1-immersions to surfaces of arbitrary genus, providing conditions for existence, uniqueness, and handling blow-up phenomena.

Findings

CMC 1-immersions are limits of CMC c-immersions as c approaches 1 from below.

Introduces an orthogonality condition to manage blow-up phenomena.

Establishes existence and uniqueness results under generic conditions.

Abstract

Constant Mean Curvature (CMC) 1-immersions of surfaces into hyperbolic 3-manifolds are natural and yet rather curious objects in hyperbolic geometry with interesting applications. Firstly, Bryant revealed surprising relations between (CMC) $1$-immersions of surfaces into $\mathbb H^3$ (Bryant surfaces) and (cousins) minimal immersions into $\mathbb E^3.$ In addition, the interest to (CMC) immersions of a surface $S$ (closed, orientable, with genus $\mathfrak{g} \geq2$) into hyperbolic 3-manifolds was motivated by Uhlenbeck in connection to irreducible representations of the fundamental group $\pi_{1}(S)$ into $PSL(2,\mathbb{C}).$ However a (CMC) 1-immersed compact surface is likely to develop singularities (punctures at finitely many points), and indeed in our analysis the prescribed value 1 of the mean curvature enters as a "critical" parameter. In fact, Huang-Lucia-Tarantello showed that (CMC) $c$-immersions of $S$ into hyperbolic 3-manifolds exist for $|c | <1$ and are parametrized by elements of the tangent bundle of the Teichmueller space of $S.$ More importantly, (CMC) $1$-immersions are attained only as "limits" for $|c| \to 1^-$ . In general the passage to the limit can be prevented by possible blow-up phenomena captured in terms of the Kodaira map and its suitable extension respectively for genus $\mathfrak{g}=2$ and $\mathfrak{g}=3.$ Here we handle the case of surfaces of any genus. In Theorem , we are able to encompass the blow up situation in terms of an appropriate "orthogonality" condition. Subsequently, we can provide the existence and uniqueness of (CMC) 1-immersions under an appropriate "generic" condition, see Theorem 2.