Conformal Structures and Necksizes of Embedded Constant Mean Curvature Surfaces

Details Conformal Structures and Necksizes of Embedded Constant Mean Curvature Surfaces Conformal Structures and Necksizes of Embedded Constant Mean Curvature Surfaces

2002-07-19

arxiv.org/abs/math/0207160v1

Mathematics

Differential Geometry

10 pages, 2 figures, 2001 MSRI/Clay workshop on global theory of minimal surfaces

Scientific paper

Let M = M_{g,k} denote the space of properly (Alexandrov) embedded constant mean curvature (CMC) surfaces of genus g with k (labeled) ends, modulo rigid motions, endowed with the real analytic structure described in [kmp]. Let $P = P_{g,k} = r_{g,k} \times R_+^k$ 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 r_{g,k}. Then the parabolic classifying map, Phi: M --> 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.

Affiliated with

Also associated with

No associations

Rating

Conformal Structures and Necksizes of Embedded Constant Mean Curvature Surfaces does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Conformal Structures and Necksizes of Embedded Constant Mean Curvature Surfaces, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Conformal Structures and Necksizes of Embedded Constant Mean Curvature Surfaces will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-397396