Regularity of area-minimizing surfaces in higher codimension: old and new
Regularity of area-minimizing surfaces in higher codimension: old and new
The theory of integral currents, developed by Federer and Fleming in the 60s, gives a powerful framework to solve the Plateau's problem in every dimension and codimension. The interior and boundary regularity theory for the codimensionone case isÌý rather well understood, thanks to the work of several mathematiciansÌýin the 60es, 70es and 80es.
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In codimension higher than one the phenomenon of branching causes insteadÌývery serious problems. A celebrated monograph of Almgren provided a far-reachingÌýinterior regularity theorem. However, his original (typewritten) manuscriptÌýwas more than 1700 pages long. Four years ago, in a series of works with EmanueleÌýSpadaroÌý we have given a substantially shorter and simpler version of Almgren'sÌýtheory, building upon large portions of his program. This has allowed us to goÌýbeyond Almgren's result in several directions: in this talk I will discuss someÌýof the most recent achievements.