Title
Interior singularities of area-minimizing currents with high codimension
Abstract
Integral currents are a weak generalization of smooth oriented manifolds with boundary and provide a natural setting in which to study the Plateau problem: ‘What are the surfaces of least m-dimensional area that span a given (m-1)-dimensional boundary?’
Unfortunately, the weak nature of integral currents permit the formation of singularities. The problem of determining the size and structure of the set of interior singularities of an area-minimizer in this setting has been studied by many since the 1960s, with ground-breaking contributions from De Giorgi, Almgren, Federer, Simon, Bombieri, Guisti and others. The codimension one case is easier to handle and has been reasonably well understood. However, in the higher codimension case, the state of the art in higher dimensions is still the (m-2)-Hausdorff dimension bound on the singular set due to Almgren, the (originally 1700 page) proof of which has since been simplified by De Lellis-Spadaro.
In this talk I will review the key features of Almgren’s approach and discuss how to strengthen this to an upper Minkowski dimension estimate. I will also explain why the techniques of the proof give an insight into the problem of establishing rectifiability for the singular set.
This is work in preparation based on unpublished ideas of Camillo De Lellis (IAS).
Access the event online
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