Isothermic coordinates on sewing surfaces

In the paper we investigated the question about existence and uniqueness of isothermic coordinates on sewing surfaces in R𝑚. Such surfaces is special case of irregular surfaces. We obtained the analog of the famous theorem of V.M. Miklukov (2004) for such surfaces. The result of this paper. Theorem 2. Let 𝒳12 be a pasting together of the pair of the surfaces = = (𝐺𝑖, 𝑓𝑖), (𝑖 = 1, 2) and Γ𝑖 = is quasistraight line. Let '12 : Γ1 → Γ2 be a sewing function. Assume that '12 is quasimonotone function and that 𝑃𝑖(𝑥(𝑖)) = 𝐸𝑖(𝑥(𝑖)) + 𝐺𝑖(𝑥(𝑖)) √︀𝐸𝑖(𝑥(𝑖))𝐺𝑖(𝑥(𝑖)) - 𝐹2 (𝑥(𝑖)), = 1, 2, is 𝑊1,2 loc,Γ𝑖-majorized functions in 𝐺𝑖. There exist isothermic coordinates = ( 1, 2) ∈ 𝐵(𝑂,𝑅), > 1 on 𝒳12. These coordinates are determined uniquely by choice of correspondence ←→ 𝑂, ←→ Ξ, where either the 𝑎, ∈ ∪ Γ𝑖(𝑖 = 1, 2) and 𝑎 ̸= 𝑏, or ∈ 𝐺1, ∈ 𝐺2 and 𝑎 ̸= '12(𝑏).