On the Moduli Space of (CMC) 1-immersions of a Closed Surface Into Hyperbolic 3-Manifolds.

Abstract Constant Mean Curvature (CMC) immersions of a closed surface S into hyperbolic 3-manifolds emerged by the work of Uhlenbeck in connection with irreducible representations of the fundamental group into the Mobious group. Moreover, Bryant revealed a bi-holomorphic (cousin) relation between (CMC) 1-immersions of surfaces into the hyperbolic 3-space (Bryant surfaces) and minimal immersions into the Euclidian 3-space. In this note, we survey recent results concerning the existence and uniqueness of (CMC) 1-immersions of a closed surface into hyperbolic 3-manifolds, labelled by Dolbeault co-homology classes. While, (CMC) c-immersions of a surface S (closed, orientable, with genus $$g \ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> ) into hyperbolic 3- manifolds are always available when $$|c| &lt; 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>c</mml:mi> <mml:mo>|</mml:mo> <mml:mo>&lt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> (and described in terms of the tangent bundle of the Teichmueller space of S) we find that (CMC) 1-immersions can be attained only as limits of (CMC) c-immersions for $$|c| \rightarrow 1.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>c</mml:mi> <mml:mo>|</mml:mo> <mml:mo>→</mml:mo> <mml:mn>1</mml:mn> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> However, the passage to the limit can be prevented by possible blow-up phenomena, so that (after scaling) we end up with a (CMC) 1-immersion with (finitely many) conical singularities, consistently with the presence of smooth ends in Bryant surfaces. We see how to encompass the blow-up situation in terms of suitable orthogonality conditions, involving the image Z of the Kodaira map for genus $$g = 2,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> and the $$(g - 1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>g</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -secant variety of Z , for genus $$g = 3.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> Consequently, we can ensure the passage to the limit under an appropriate generic condition (sharp for genus $$g = 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> ), yielding to (CMC) 1-immersions into suitable (germs) of hyperbolic 3-manifolds.