2023/09/27 by Meier, Damaris, Rajala, Kai · 2 citations
#30F10 (Secondary) #30L10 (Primary) 30C65 #Complex Variables (math.CV) #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2309.15615
We investigate basic properties of mappings of finite distortion f:X → ℝ2, where X is any metric surface, i.e., metric space homeomorphic to a planar domain with locally finite 2-dimensional Hausdorff measure. We introduce lower gradients, which complement the upper gradients of Heinonen and Koskela, to study the distortion of non-homeomorphic maps on metric spaces. We extend the Iwaniec-Šverák theorem to metric surfaces: a non-constant f:X → ℝ2 with locally square integrable upper gradient and locally integrable distortion is continuous, open and discrete. We also extend the Hencl-Koskela theorem by showing that if f is moreover injective then f-1 is a Sobolev map.