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arXiv · 2309.15615

Mappings of finite distortion on metric surfaces

Damaris Meier·Kai Rajala
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Abstract

We investigate basic properties of mappings of finite distortion $f:X \to \mathbb{R}^2$, 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-\v{S}ver\'ak theorem to metric surfaces: a non-constant $f:X \to \mathbb{R}^2$ 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.

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Damaris Meier, Kai Rajala. 2023-09-27. Mappings of finite distortion on metric surfaces. https://arxiv.org/abs/2309.15615

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