(CMC) 1-immersions of surfaces into hyperbolic 3-manifolds
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2025
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| author | Tarantello, Gabriella Trapani, Stefano |
| author_facet | Tarantello, Gabriella Trapani, Stefano |
| contents | 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 $π_{1}(S)$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_11894 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | (CMC) 1-immersions of surfaces into hyperbolic 3-manifolds Tarantello, Gabriella Trapani, Stefano Differential Geometry Analysis of PDEs 35J50, 35J61, 53C42, 32G15, 30F60 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 $π_{1}(S)$ 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. |
| title | (CMC) 1-immersions of surfaces into hyperbolic 3-manifolds |
| topic | Differential Geometry Analysis of PDEs 35J50, 35J61, 53C42, 32G15, 30F60 |
| url | https://arxiv.org/abs/2506.11894 |