Minimal surfaces, constant mean curvature surfaces, and the isoperimetric problem

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... We propose to study the isoperimetric problem for symmetric spaces M = G/K of non-compact type. There are three reasons to believe that this is a natural setting: Firstly, symmetric spaces of non-compact type provide enough homogeneity in order to avoid minimizing sequences that disappear at infinity. Secondly, they are Hadamard manifolds, and this may be the basis for ruling out discontinuous changes of the shape of the extremal domains W when the volume v varies. The latter phenomenon is observed e.g. in complex-projective spaces. Finally, the ring of G-invariant differential operators on a symmetric space is commutative, and thus there is a rich supply of spherical functions that obey interesting non-linear differential identities; hence we may hope to obtain explicit descriptions of the isoperimetric domains W.