Constant mean curvature surfaces with Delaunay ends

Abstract

In this paper we shall present a construction of complete surfaces M in R3 with finitely many ends and finite topology, and with nonzero constant mean curvature (CMC). This construction is parallel to the well-known original construction by Kapouleas [5], but we feel that ours is somewhat simpler analytically, and controls the resulting geometry more closely. On the other hand, the surfaces we construct have a rather different, 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 [10] 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 [18]. 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 (which are called unduloids) interpolate between the cylinder and an infinite string of spheres arranged along a common axis. The family continues beyond this, but the elements now are immersed (and are called nodoids). The role of Delaunay surfaces in the theory of complete CMC surfaces is analogous to the role of catenoids (and planes) in the study of complete minimal surfaces of finite total curvature. For example, just as any complete minimal surface with two ends must be a catenoid [19], it was proved by Meeks [14] and Korevaar, Kusner and Solomon [8] that any Alexandrov embedded constant mean curvature surface with at most two ends is necessarily a Delaunay surface. A rather more remarkable theorem, paralleling the fact that any end of a complete minimal surface of finite total curvature must be asymptotic to a catenoid or a plane, is the fact that any embedded end of a CMC surface must be asymptotic to one of these rotationally symmetric Delaunay surfaces (and in particular, must be cylindrically bounded).