The topology and geometry of embedded surfaces of constant mean curvature

where Ai(p) and A2(p) are the principal curvatures of M at p. When H is constant, M is called a surface of constant mean curvature. A surface is said to have finite type if it is homeomorphic to a closed surface with a finite number of points removed. An important problem in classical differential geometry is the classification of properly embedded finite type surfaces M of constant mean curvature in R . If M is a closed embedded surface of constant mean curvature, then it follows from Alexandrov [1] that M must be a round sphere. The classical examples of properly embedded surfaces with zero mean curvature are the plane, the helicoid and the catenoid. Surfaces of zero mean curvature are usually called minimal surfaces. The remaining classical examples of properly embedded surfaces of constant mean curvature were found by Delaunay [4]. The Delaunay surfaces are surfaces of revolution. Recently Hoffman and Meeks [6, 7] have found examples of properly embedded minimal surfaces which are homeomorphic to closed surfaces of positive genus with 3 points removed. Callahan, Hoffman and Meeks [3] have found other examples with more ends. An annular end E of & properly embedded surface in R 3 is a properly embedded annulus E in M where E is homeomorphic to S x [0,1). When M has finite type, then every end of M is annular. Hoffman and Meeks have developed a theory to deal with global problems concerning the geometry of properly embedded minimal surfaces M and, in particular, they show that most annular ends of M converge at infinity in R to a flat plane or to the end of a catenoid. Recently N. Kapouleas [8] in his thesis has shown that for every positive integer k > 2, there exists a properly embedded surface M& of finite type with nonzero mean curvature and with k ends. He also has constructed highergenus examples. As in the case of minimal surfaces, the annular ends of a properly embedded surface of nonzero constant mean curvature have a special geometry and play an important role in global theorems.