Mappings of finite distortion on metric surfaces

We investigate basic properties of mappings of finite distortion f:X → R² f : X → 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 → R² f : X → 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 f - 1 is a Sobolev map.