Constant mean curvature surfaces with Delaunay ends

In this paper we shall present a construction of Alexandrov-embedded complete surfaces M in R with nitely many ends and nite topology, and with nonzero constant mean curvature (CMC). This construction is parallel to the well-known original construction by Kapouleas [3], but we feel that ours somewhat simpler analytically, and controls the resulting geometry more closely. On the other hand, the surfaces we construct have a rather di erent, and usually simpler, geometry than those of Kapouleas; in particular, all of the surfaces constructed here are noncompact, so we do not obtain any of his immersed compact examples. The method we use here closely parallels the one we developed recently [8] to study the very closely related problem of constructing Yamabe metrics on the sphere with k isolated singular points, just as Kapouleas' construction parallels the earlier construction of singular Yamabe metrics by Schoen [15]. The original examples of noncompact CMC surfaces were those in the one-parameter family of rotationally invariant surfaces discovered by Delaunay in 1841 [2]. One extreme element of this family is the cylinder; the `Delaunay surfaces' are periodic, and the embedded members of this family interpolate between the cylinder and an in nite string of spheres arranged along a common axis. The family continues beyond this, but the elements now are immersed, and we shall not consider them here. We are mostly interested in surfaces which are Alexandrov embedded. This condition, by de nition, means that although the surface may be immersed, the immersion extends to one of the solid handlebody bounded by the surface. The CMC surfaces we construct here have Alexandrov embedded ends; if the minimal k-noids out of which they are built (as we describe below) are Alexandrov embedded, then the whole surfaces satisfy this condition. The rôle of Delaunay surfaces in the theory of complete CMC surfaces is analogous to the rôle of catenoids (and planes) in the study of complete minimal surfaces of nite total curvature. For example, just as any complete minimal surface with two ends must be a catenoid [16], it was proved by Meeks [11] and Korevaar, Kusner and Solomon