Metric currents, barycenter maps and the spherical Plateau problem

We first review the theory of integral currents in metric spaces due to Ambrosio-Kirchheim and others, and the barycenter map method developed by Besson-Courtois-Gallot in their work on the entropy rigidity problem. We then apply those elements to the study of the spherical Plateau problem, a volume minimization problem in quotients of the Hilbert sphere. We outline the proofs of the intrinsic uniqueness of spherical Plateau solutions for locally symmetric closed oriented manifolds of rank 1, as well as for 3-dimensional closed oriented manifolds, and the construction of analogues of hyperbolic Dehn fillings in higher dimensions. We raise some open questions.Comment: v2: Content restructured, title changed. Some results from v1 were improved and will appear elsewher

Avances en Matemática Discreta en Andalucía. V Encuentro andaluz de Matemática Discreta. La Línea de la Concepción (Cádiz), 4-5 de julio de 2007

V Encuentro andaluz de Matemática Discreta. La Línea de la Concepción (Cádiz), 4-5 de julio de 200