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**The structure of complete embedded surfaces with constant mean curvature.**
*(English)*
Zbl 0726.53007

From the introduction: We consider complete, properly embedded surfaces \(\Sigma \subset\mathbb{R}^3\) which are of finite topological type and have constant nonzero mean curvature. Besides the round sphere the simplest such \(\Sigma\) are the noncompact, periodic surfaces of revolution discovered by Delaunay. The main result of this paper is that for each annular end \(A\subset \Sigma\) there is a Delaunay surface \(D\subset\mathbb{R}^3\) to which A converges exponentially as \(| x| \to \infty\). This means \(\Sigma\) itself is conformally a compact Riemann surface having finitely many punctures. As a preliminary step we prove that Delaunay surfaces are the only two-ended \(\Sigma\). We also derive and use a “balancing formula”. It implies, for example, that each end of \(\Sigma\) has a “weight vector” parallel to the axis of its limiting Delaunay surface and that the sum of these weights is 0.

Two recent papers stimulated this research. In [ibid. 27, No. 3, 539–552 (1988; Zbl 0617.53010)] W. Meeks III proved that any annular end of \(\Sigma\) is contained in a solid half-cylinder of some finite radius, that no one-ended \(\Sigma\) exists and that two-ended \(\Sigma\) are contained in solid cylinders. In [Ann. Math. (2) 131, No. 2, 239–330 (1990; Zbl 0699.53007)] N. Kapouleas constructed a wealth of (immersed and embedded) constant mean curvature surfaces \(\Sigma\) by solving an elliptic singular perturbation problem. His examples all have asymptotically Delaunay ends. Also, to construct suitable initial surfaces for his perturbation technique he required an approximate “balancing condition” (implied by our balancing formula) to hold for his configuration.

Two recent papers stimulated this research. In [ibid. 27, No. 3, 539–552 (1988; Zbl 0617.53010)] W. Meeks III proved that any annular end of \(\Sigma\) is contained in a solid half-cylinder of some finite radius, that no one-ended \(\Sigma\) exists and that two-ended \(\Sigma\) are contained in solid cylinders. In [Ann. Math. (2) 131, No. 2, 239–330 (1990; Zbl 0699.53007)] N. Kapouleas constructed a wealth of (immersed and embedded) constant mean curvature surfaces \(\Sigma\) by solving an elliptic singular perturbation problem. His examples all have asymptotically Delaunay ends. Also, to construct suitable initial surfaces for his perturbation technique he required an approximate “balancing condition” (implied by our balancing formula) to hold for his configuration.