Generic uniqueness for the Plateau problem
speaker: Gianmarco Caldini (Università di Trento)abstract: Given a boundary curve in \mathbb{R}3, uniqueness of the minimizing surface is known only under very restrictive assumptions and many examples have been provided of curves admitting different minimizers. In the late seventies, Morgan proved under specific conditions that almost every curve is the boundary of a unique area-minimizing surface, showing that uniqueness is a \emph{generic} property. Relying on a recent boundary regularity theorem proved by De Lellis, De Philippis, Hirsch and Massaccesi, I will explain how it is possible to extend Morgan's result to full generality, \emph{i.e.} to general ambient manifolds of any dimension, to any codimension and with no assumptions on the geometry of the boundary. The argument of the proof also delivers generic absence of two-sided boundary points and generic multiplicity one of area-minimizing integral currents. If time permits, I will also explain how to prove generic uniqueness for a Plateau-type problem studied in optimal transport, providing a strategy that may have a number of other applications. The talk is based on a joint study with A. Marchese, A. Merlo and S. Steinbruechel.
timetable:
Wed 20 Sep, 17:20 - 18:20, Aula Dini
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