Mappings of finite distortion on metric surfaces

Abstract

We investigate basic properties of mappings of finite distortion \(f:X \rightarrow \mathbb {R}^2\), 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 \(f:X \rightarrow \mathbb {R}^2\) 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.