Applications of compactness results for harmonic maps to stable constant mean curvature surfaces
We prove the compactness of the space of compact volume-preserving stable surfaces with genus \(g\geq 2\) immersed in quotients of \(\mathbb{R}^3\) by discrete subgroups of translations. The main tool in the proof is a compactness result for harmonic maps due to \textit{J. Sacks} and \textit{K. Uhlenbeck} [Trans. Am. Math. Soc. 271, 639-652 (1982; Zbl 0527.58008)] which can be applied to the harmonic Gauss maps of the immersions. Our results allow us to describe the isoperimetric domains in \(\mathbb{T}^2 \times \mathbb{R}\), where \(\mathbb{T}^2\) is a flat two-dimensional torus with injectivity radius equal to 1 and large area.
scientific article; zbMATH DE number 1121727