On CMC-immersions of a closed surface into hyperbolic 3-manifolds

Gabriella Tarantello; Stefano Trapani

doi:10.3934/dcds.2026078

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2026, Volume 55: 64-92. Doi: 10.3934/dcds.2026078

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On CMC-immersions of a closed surface into hyperbolic 3-manifolds

Gabriella Tarantello^, and
Stefano Trapani

Università di Roma TorVergata, Via della Ricerca Scientifica 1, 00133, Roma, Italy

^*Corresponding author: Gabriella Tarantello
^*Corresponding author: Gabriella Tarantello

Received: February 20, 2026

Early access: April 7, 2026

Published online: April 7, 2026

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Abstract

We survey recent results about the existence and uniqueness of Constant Mean Curvature (CMC) immersions of a closed orientable surface S (with genus $ \mathfrak{g} \geq 2 $) into hyperbolic 3-manifolds. Such immersions were considered by Uhlenbeck in view of their connection to irreducible representations of the fundamental group into the Möbious group. The value $ c = 1 $ of the mean curvature plays a special role in this context as Bryant pointed out an unexpected bi-holomorphic (cousin) correspondence between (CMC) 1-immersions of surfaces into the hyperbolic 3-space (Bryant surfaces) and minimal immersions into the Euclidean 3-space. In fact, when $ |c| < 1 $ then (CMC) c-immersions of S into hyperbolic 3-manifolds are always available and labelled by the tangent bundle of the Teichmüller space of S. Thus, we may find (CMC) 1-immersions as limits of such (CMC) c-immersions, for $ |c| \to 1^- $. However, the passage to the limit can be prevented by possible blow-up phenomena. We see how to encompass the blow-up situation in terms of suitable orthogonality conditions, which identify analytic sub-varieties of positive co-dimension. Consequently, we can ensure the passage to the limit under an appropriate generic condition (sharp for genus $ \mathfrak{g} = 2 $), yielding (CMC) 1-immersions into suitable (germs) of hyperbolic 3-manifolds.

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Mathematics Subject Classification: 35J50, 35J61, 53C42, 32G15, 30F60.

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