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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.

##### MSC:
 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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