Isothermic coordinates on sewing surfaces

A. N. Kondrashov

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 6 (2016), pp. 70-80

Résumé

In the paper we investigated of question about existence and uniqueness of isothermic coordinates on sewing surfaces in $\mathbb{R}^m$. The 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 $\mathcal{X}_{12}$ be a pasting together of the pair of the surfaces $\mathcal{X}_i=(G_i,f_i)$, $(i=1,2)$ and $\Gamma_i=\partial G_i$ is quasistraight line. Let be a $\varphi_{12}:\Gamma_{1}\to\Gamma_{2}$ is sewing function. Assume that $\varphi_{12}$ is quasimonotone function and that $$P_i(x^{(i)})=\frac{E_i(x^{(i)})+G_i(x^{(i)})}{\sqrt{E_i(x^{(i)})G_i(x^{(i)} )-F_i^2(x^{(i)})}}, \ i=1,2,$$ is $W^{1,2}_{\mathrm{loc},\Gamma_i}$-majorized functions in $G_i$. There is exist isothermic coordinates $\xi=(\xi_1,\xi_2) \in B(O,R)$, $R>1$ on $\mathcal{X}_{12}$. These coordinates are determined uniquely by choice of correspondence $a\longleftrightarrow O$, $b\longleftrightarrow \Xi$, where either the $a, b\in G_{i}\cup\Gamma_{i}(i=1,2)$ and $a\ne b$, or $a\in G_1$, $b\in G_2$ and $a\ne\varphi_{12}(b)$.

Détail

Keywords: isothermic coordinates, sewing surfaces, sewing functions, quasisymmetric function, $W^{1,2}_{\mathrm{loc},\Gamma}$-majorized functions, quasistraight line.