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Isoperimetric domains in the Riemannian product of a circle with a simply conneted space form and applications to free boundary problems. (English) Zbl 0956.53049
Overview of the main results: The isoperimetric domains in \(S^1(r) \times {\mathbb{R}}^n\), \(2 \leq n \leq 7\), are (i) geodesic balls for small values of volume, and (ii) tubes around closed geodesics \(S^1(r) \times {\text{point}}\) for large values of volume. For \(n \geq 9\) there are minimizers whose boundaries are generated by unduloids. In \(S^1(r) \times H^2\) (i) balls and (ii) tubes occur depending on the value of volume. In \(S^1(r) \times S^2\) (i) balls, (ii) tubes and (iii) sections bounded by totally geodesic 2-spheres can happen, but there are also cases in which tubes (ii) are not solutions or slices (iii) are not solutions depending on the radius \(r\). Furthermore the authors treat stable drops between two parallel hyperplanes in \({\mathbb{R}}^{n+1}\), \(2 \leq n \leq 7\). The methods used are symmetry, ODE analysis and stability of CMC hypersurfaces.

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53C65 Integral geometry
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