The conformal theory of Alexandrov embedded constant mean curvature surfaces in \(\mathbb R^3\).

*(English)*Zbl 1101.53006
Hoffman, David (ed.), Global theory of minimal surfaces. Proceedings of the Clay Mathematics Institute 2001 summer school, Berkeley, CA, USA, June 25–July 27, 2001. Providence, RI: American Mathematical Society (AMS). Cambridge, MA: Clay Mathematics Institute (ISBN 0-8218-3587-4/pbk). Clay Mathematics Proceedings 2, 525-559 (2005).

The authors study complete Alexandrov embedded constant mean curvature surfaces of finite topology in the Euclidean three-space. The condition that a surface is Alexandrov embedded means that the embedding is extended to a proper immersion of a three-manifold whose boundary is the given surface. Any such surface is conformally equivalent to the complement of finitely many points in a compact Riemann surface. Given a genus \(g\) of a Riemann surface and a number of points \(k\), the constant mean curvature surfaces in study which are conformally equivalent to a surface of genus \(g\) with \(k\) punctured points form the moduli space \({\mathcal M}_{g,k}\).

This paper mostly concerns the structure of the moduli spaces \({\mathcal M}_{g,k}\) and consists of two parts. We recall that a surface \(\Sigma \in {\mathcal M}_{g,k}\) is called nondegenerate if it has no non-trivial Jacoby field which decays at all ends of the surface. The authors prove that for a nondegenerate surface \(\Sigma \in {\mathcal M}_{g,k}\) there exists a one-parameter family of surfaces in \({\mathcal M}_{g,k+1}\) which are obtained by gluing half-Delaunay surfaces with sufficiently small necksizes. In the second part it is proved that the moduli space \({\mathcal M}_{g,k}\) is really analytic (before it was known that it is locally really analytic) and that the forgetful map \[ {\mathcal M}_{g,k} \to {\mathcal T}_{g,k} \] which corresponds to any surface its conformal class (where \({\mathcal T}_{g,k}\) is the Teichmüller space of conformal structures on a compact surface of genus \(g\) with \(k\) punctures) is surjective for \(g=0\). The proof of the surjectivity of the forgetful map is based on the construction from the first part of the paper. A different proof of this result was obtained by R. Kusner [Clay Math. Proc. 2, 585–596 (2005; Zbl 1103.53030)].

For the entire collection see [Zbl 1078.53002].

This paper mostly concerns the structure of the moduli spaces \({\mathcal M}_{g,k}\) and consists of two parts. We recall that a surface \(\Sigma \in {\mathcal M}_{g,k}\) is called nondegenerate if it has no non-trivial Jacoby field which decays at all ends of the surface. The authors prove that for a nondegenerate surface \(\Sigma \in {\mathcal M}_{g,k}\) there exists a one-parameter family of surfaces in \({\mathcal M}_{g,k+1}\) which are obtained by gluing half-Delaunay surfaces with sufficiently small necksizes. In the second part it is proved that the moduli space \({\mathcal M}_{g,k}\) is really analytic (before it was known that it is locally really analytic) and that the forgetful map \[ {\mathcal M}_{g,k} \to {\mathcal T}_{g,k} \] which corresponds to any surface its conformal class (where \({\mathcal T}_{g,k}\) is the Teichmüller space of conformal structures on a compact surface of genus \(g\) with \(k\) punctures) is surjective for \(g=0\). The proof of the surjectivity of the forgetful map is based on the construction from the first part of the paper. A different proof of this result was obtained by R. Kusner [Clay Math. Proc. 2, 585–596 (2005; Zbl 1103.53030)].

For the entire collection see [Zbl 1078.53002].

Reviewer: Iskander A. Taimanov (Novosibirsk)