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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$g \ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>) into hyperbolic 3- manifolds are always available when <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$|c| < 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>c</mi> <mo stretchy="false">|</mo> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> (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 <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$|c| \rightarrow 1.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>c</mi> <mo stretchy="false">|</mo> <mo stretchy="false">→</mo> <mn>1</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$g = 2,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> and the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$(g - 1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>g</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-secant variety of Z,  for genus <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$g = 3.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>=</mo> <mn>3</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> Consequently, we can ensure the passage to the limit under an appropriate generic condition (sharp for genus <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$g = 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>), yielding to (CMC) 1-immersions into suitable (germs) of hyperbolic 3-manifolds.

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

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2026年3月31日

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Gabriella Tarantello,
Stefano Trapani

摘要

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 $g \ge 2$ $g \geq 2$ ) into hyperbolic 3- manifolds are always available when $|c| < 1$ $| c | < 1$ (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 $|c| \rightarrow 1.$ $| c | \to 1 .$ 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 $g = 2,$ $g = 2,$ and the $(g - 1)$ $(g - 1)$ -secant variety of Z, for genus $g = 3.$ $g = 3 .$ Consequently, we can ensure the passage to the limit under an appropriate generic condition (sharp for genus $g = 2$ $g = 2$ ), yielding to (CMC) 1-immersions into suitable (germs) of hyperbolic 3-manifolds.