The global structure of constant mean curvature surfaces

Abstract

In a recent ground-breaking paper Kapouleas constructed this century's first examples of complete, finite topology, properly embedded surfaces with (non-zero) constant mean curvature [Ka l ] . Geometrically these surfaces are quite easy to visualize, as we explain below.The purpose of the present paper is to characterize the extent to which Kapouleas ' examples are typical of all embedded constant mean curvature surfaces. First, and throughout this paper, we scale space so that the (constant) mean curvature (the trace of the second fundamental form) of our surface L" is identically one, and we call such a surface an MC1 surface. The only complete embedded MC1 surfaces of finite topology known before Kapouleas ' work arise from the 1parameter family of axially-symmetric (and periodic) examples found by Delaunay in 1841: Using the maximum axial radius R of such a surface as a parameter, this family begins with the cylinder (R = 1), and evolves through embedded, periodic surfaces until the family degenerates to a linear chain of touching spheres (R = 2); the Delaunay family continues for all R > 2 as periodic immersions. Kapouleas ' construction of embedded surfaces begins with a piecewise linear graph in R 3, having the property that edges are either finite segments (with length a multiple of 4) or half-infinite rays, and with the further property that at each vertex some linear combination of their direction vectors, with positive coefficients, adds to zero. (See Fig. 1.) He constructs an approximate solution to the problem by centering radius R = 2 spheres at all vertices, and connecting these smoothly by Delaunay segments having R < 2 (but near 2) with axes along the graph-edges. (Since the periods of the Delaunay surfaces are slightly longer than 4, he must assume also a "flexibility" condition on the graph, to guarantee that such a gluing