Abstract
In this chapter we focus the case that the boundary curve is a circle. In previous chapters, we have obtained some results that prove that the surface is umbilical under the assumption of embeddedness. In this chapter we shall consider that the topology of the surface is the simplest one, say, a disk. We shall prove that if a cmc disk immersed in \(\mathbb{R}^{3}\) spans a circle and its area is less than or equal to of a spherical cap with the same mean curvature and boundary, then the surface is, indeed, a spherical cap. Also, we shall prove that under the assumption of stability, a cmc disk spanning a circle must be umbilical, given a partial answer of the boundary version of the Barbosa-do Carmo theorem. Finally, we end the chapter obtaining estimates of the area of a cmc disk with circular boundary.
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López, R. (2013). Constant Mean Curvature Disks with Circular Boundary. In: Constant Mean Curvature Surfaces with Boundary. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39626-7_7
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DOI: https://doi.org/10.1007/978-3-642-39626-7_7
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