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Camillo De Lellis - Center manifolds and regularity of area-minimizing currents (Part 3)


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Camillo De Lellis - Center manifolds and regularity of area-minimizing currents (Part 3)

A celebrated theorem of Almgren shows that every integer rectifiable current which minimizes (locally) the area is a smooth submanifold except for a singular set of codimension at most 2. Almgren’s theorem is sharp in codimension higher than 1, because holomorphic subvarieties of Cn are area-minimizing. In fact the typical singularity of a 2-dimensional area-minimizing current is modelled by branch points of holomorphic curves. These singularities are rather difficult to analyze because they might be very high order phenomena.

 

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