Moduli space theory for constant mean curvature surfaces immersed in space-forms. (English) Zbl 1136.53048

Gonçalves, Alexandre; Uhlenbeck, Karen

Commun. Anal. Geom. 15, No. 2, 299-305 (2007).

The authors study surfaces with constant mean curvature \(c\) in a \(3\)-dimensional real space form of constant sectional curvature \(k\). They are particularly interested in the case when \(k +c^2\) is strictly negative. The authors want to determine solutions of the Gauss and Codazzi equation for a fixed Riemannian surface. In order to do so, they take a fixed Riemann surface with initial constant curvature metric \(g\) (the actually induced metric on the constant mean curvature surface will be conformally equivalent to \(g\)) and a holomorphic quadratic differential, which is transformed into a \((-1,1)\) form using the metric. The main result states that, given a Riemann surface and a \((-1,1)\) cohomology class, if \(k +c^2\) is negative, there exists a unique solution of the Gauss-Codazzi equations.

Reviewer: Luc Vrancken (Valenciennes)

MSC:

53C42	Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C21	Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C24	Rigidity results

Keywords:

Constant mean curvature; moduli space