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Recent advances in the global theory of constant mean curvature surfaces. (English) Zbl 1075.53008
Bahri, Abbas (ed.) et al., Noncompact problems at the intersection of geometry, analysis, and topology. Proceedings of the conference on noncompact variational problems and general relativity held in honor of Haim Brezis and Felix Browder at Rutgers University, New Brunswick, NJ, USA, October 14–18, 2001. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3635-8/pbk). Contemporary Mathematics 350, 179-199 (2004).
This is a survey of recent progress in the global theory of constant mean curvature (CMC) surfaces. It is restricted to nonminimal constant mean curvature surfaces and to the analytic approach of the gluing method, but the author carefully explains in the introduction the context of the restrictions within the larger theory of CMC surfaces. Also, it is restricted to Alexandrov embedded CMC surfaces, and again the author explains the motivations for this restriction. Delaunay surfaces and the balancing formula and Delaunay asymptotics of CMC ends are described. A nice intuitive explanation for the gluing method is provided: Imagine making a CMC surface with a finite number of asymptotically Delaunay ends and a compact piece in the middle – the gluing method is about how to fill in the compact piece in the middle. The associated Jacobi operator is described, and it is crucial in the gluing method.
The essential notion of nondegeneracy of CMC surfaces is described, in both the narrow and broad senses, and interesting questions about the degenerate case are raised. The author gives a list of six different ways in which the gluing method has been applied, which this reviewer found particularly enlightening. Toward the end of the paper, a description of the moduli space \(M_{g,k}\) of Alexandrov embedded CMC surfaces of genus \(g\) with \(k\) ends is given. In particular, near nondegenerate surfaces in the moduli space, it is explained that the moduli space has dimension \(3k\), which is interestingly independent of the genus \(g\). Other subjects that are touched upon: CMC surfaces in other asymptotically Euclidean \(3\)-manifolds, and connections to solutions of the singular Yamabe problem.
For the entire collection see [Zbl 1052.58001].

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
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