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Abstract: In the first part of this lecture, we talk about compact stable CMC surfaces with free boundary in compact convex domains of the Euclidean three-space. We prove that every compact stable CMC surface with free boundary in a convex domain with boundary geometry sufficiently close to the round two-sphere is topologically a disk or an annulus. This fact, together with a previous one due to Ros and Vergasta, implies a complete classification of these surfaces in the unit Euclidean three-ball. In the last part of the lecture, we deal with a characterization of the equatorial disk and the critical catenoid in the unit Euclidean three-ball based on a pinching condition involving its second fundamental form and support function. This last part is a joint work with Lucas Ambrozio.
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