Complete constant mean curvature surfaces in Euclidean three-space. (English) Zbl 0699.53007

Kapouleas, Nicolaos

Ann. Math. (2) 131, No. 2, 239-330 (1990).

This paper is one of the major breakthroughs in the theory of constant mean curvature (CMC) surfaces. The idea is the construction of such surfaces by attaching together spheres and Delaunay surfaces along a graph to obtain a surface with \(H\) close to 1. The important main result of the paper is the perturbation theory for obtaining the desired CMC surfaces by normal variation. The author obtains complete solutions for surfaces of prescribed genus \(> 1\) and number of ends \(> 0\). The methods also allow the construction of closed examples, see [N. Kapouleas, Compact constant mean curvature surfaces in Euclidean three-space, J. Differ. Geom. 33, No. 3, 683-715 (1991, Zbl 0727.53063)].

Reviewer: D. Ferus

MSC:

53A10	Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53C42	Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)

Keywords:

inverse function theorem; spectrum of the Laplacian; Delaunay surfaces; balancing condition; constant mean curvature

Citations:

Zbl 0727.53063