Conformal structures and necksizes of embedded constant mean curvature surfaces. (English) Zbl 1103.53030

Kusner, Rob

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, 585-596 (2005).

Author’s abstract: Let \({\mathcal M}={\mathcal M}_{g,k}\) denote the space of properly embedded (or Alexandrov immersed) constant mean curvature (CMC) surfaces of genus \(g\) with \(k\) (labeled) ends, modulo rigid motions, endowed with the real analytic structure described in [R. Kusner, R. Mazzeo and D. Pollack, Geom. Funct. Anal. 6, No. 1, 120–137 (1996; Zbl 0966.58005)]. Let \({\mathcal P}={\mathcal P}_{g,k}={\mathcal R}_{g,k} \times \mathbb R^k_+\) be the space of parabolic structures over Riemann surfaces of genus \(g\) with \(k\) (marked) punctures, the real analytic structure coming from the \(3g-3+k\) local complex analytic coordinates on the Riemann moduli space \({\mathcal R}_{g,k}.\) Then the parabolic classifying map, \(\Phi : {\mathcal M} \rightarrow {\mathcal P},\) which assigns to a CMC surface its induced conformal structure and asymptotic necksizes, is a proper, real analytic map. It follows that \(\Phi \) is closed and in particular has closed image. For genus \(g=0\), this can be used to show that every conformal type of multiply punctured Riemann sphere occurs as a CMC surface, and \(-\) under a nondegeneracy hypothesis \(-\) that \(\Phi \) has a well-defined (mod 2) degree. This degree vanishes, so generically an even number of CMC surfaces realize any given conformal structure and asymptotic necksizes. In case of \(k=3\) compare with [K. Grosse-Brauckmann, R. B. Kusner and J. M. Sullivan, Proc. Natl. Acad. Sci. USA 97, No. 26, 14067–14068 (2000; Zbl 0980.53011) and J. Reine Angew. Math. 564, 35–61 (2003; Zbl 1058.53005)].
For the entire collection see [Zbl 1078.53002].

Reviewer: Erich Hoy (Friedberg)

MSC:

53C42	Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A10	Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53A07	Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
30F99	Riemann surfaces

Keywords:

constant mean curvature surfaces; conformal structure; ends

Citations:

Zbl 0966.58005; Zbl 0980.53011; Zbl 1058.53005