A-natural variatiorial problem in Geometry is the Isoperimetric problem which asks which surface "has the least area among the surfaces enclosing a fixed volume. The answer is, of course, the round sphere, and a proof Of thishas been known for a long time. A more general question is to determine the surfaces whose area is critical under deformations which keep the enclosed volume unchanged. Such surfaces are often called soap bubbles because a soap filmor, more generally, a fluid' interface — in equilibrium betweentwo regions of different pressure and subject only to the forces induced by this pressure and the surface tension, has area which is critical under deformations which keep the enclosed volume unchanged. The differential equation charac-terizing such surfaces locally is H = constant 4= 0, where ti is the mean curvature. We adopt the abbreviations from now oh "CMC surface" to stand for "closed constant mean curvature immersed smooth surface in E3", and "CMC immersion" for the corresponding immersion.
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