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| Surfaces with constant mean curvature arise in one of the fundamental problem in differential geometry: the isoperimetric problem. When you try to minimize the area of a surface which enclose a given volume, your solution must have constant mean curvature. When considering surface with boundary it becomes the classical soap bubble problem that every one has already tested. But mathematically, less is known on the space of solutions. For instance, from the point of view of uniqueness, if the boundary is a circle we even don't know if the only two solutions are the spherical caps. However, from the point of view of existence, since the 80's and the work of Brezis and Coron, we know that for a given curve there exist at least two solutions up to assume that the mean curvature is small enough. In this talk I will describe more precisely the space of solutions, we will notably focus on their asymptotic behaviour when the mean curvature goes to zero. Under suitable condition we are able to avoid bubbling phenomena. The main tool is asymptotic analysis for the CMC equation in conformal coordinates which is know to be of critical growth. |
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