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We introduce the spherical Plateau problem, a variational problem in homologyclasses of quotients of the infinite dimensional separable Hilbert sphere. Westart with a review of the Ambrosio-Kirchheim theory of metric currents, andthe barycenter map method developed by Besson-Courtois-Gallot. Then, we outlinethe following applications: 1. the intrinsic uniqueness of spherical Plateausolutions for negatively curved, locally symmetric, closed oriented manifolds,2. the intrinsic uniqueness of spherical Plateau solutions for all3-dimensional closed oriented manifolds, 3. the construction of higherdimensional analogues of hyperbolic Dehn fillings. We also propose some openquestions.

Metric currents, barycenter maps and the spherical Plateau problem