2024/08/30 by Ntalampekos, Dimitrios · 1 citation
#30C62 #Complex Variables (math.CV) #FOS: Mathematics #Metric Geometry (math.MG) #Primary 30L10 #Secondary 30C35
paper · doi:10.48550/arxiv.2408.17174
We characterize conformally removable sets in the plane with the aid of the recent developments in the theory of metric surfaces. We prove that a compact set in the plane is S-removable if and only if there exists a quasiconformal map from the plane onto a metric surface that maps the given set to a set of linear measure zero. The statement fails if we consider maps into the plane rather than metric surfaces. Moreover, we prove that a set is S-removable (resp. CH-removable) if and only if every homeomorphism from the plane onto a metric surface (resp. reciprocal metric surface) that is quasiconformal in the complement of the given set is quasiconformal everywhere.