Abstract for the talk on 01.04.2022 (11:00 h)Arbeitsgemeinschaft ANGEWANDTE ANALYSIS
Anna Skorobogatova (Princeton University)
Interior regularity of area-minimizing currents with high codimension
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?’
However, the weak nature of integral currents permit the formation of singularities. The problem of determining the size and structure of the interior singular set of an area-minimizer in this setting has been studied by many since the 1960s, with many ground-breaking contributions. The codimension one case is significantly easier to handle, but in the higher codimension case, much less progress has been made since the celebrated (m-2)-Hausdorff dimension bound on the singular set due to Almgren, the proof of which has since been simplified by De Lellis and Spadaro.
In this talk I will review the key features of the proof of Almgren/De Lellis-Spadaro and discuss how to strengthen this to an upper Minkowski dimension estimate. I will also discuss some work in progress with Camillo De Lellis (IAS) towards establishing rectifiability of the singular set in high codimension.