Conformal mappings of manifolds of bounded curvature that preserve curvature of sets (Q2276769)

A homeomorphism f: (M\({}_ 1,\rho_ 1)\to (M_ 2,\rho_ 2)\) between two differentiable surfaces \(M_ 1\) and \(M_ 2\) with intrinsic metrics \(\rho_ 1\) and \(\rho_ 2\) of bounded curvature (in the sense of A. D. Aleksandrov) is called a conformal map, iff f preserves the angles between intersecting curves. In the present note, the author proves the following theorem on conformal mappings of surfaces: Let \(f: (M_ 1,\rho_ 1)\to (M_ 2,\rho_ 2)\) be a conformal surface map, \(D_ 1\subset M_ 1\) an open subset whose closure is homeomorphic to a disc, \(D_ 2:=f(D_ 1),\) and assume that f preserves the curvature of \(D_ 1\). Then there are isothermal coordinates in \(D_ 1\) and \(D_ 2\) such that the restricted map \(f: D_ 1\to D_ 2\) is induced, via those coordinates, by the exponential of a harmonic map H(z) on the complex plane. Moreover, if \((x,y)\in D_ 1\times D_ 1,\) \(x\neq y\) and y converges to x (with respect to \(\rho_ 1)\), then \[ \lim_{y\to x}\frac{\rho_ 2(f(x),f(y))}{\rho_ 1(x,y)}=e^{\cdot H(\phi_ 1(x))} \] where \(\phi_ 1\) denotes the isothermal coordinate mapping on \(D_ 1\). This theorem generalizes the classical result of D. E. Menchoff (1926), which states that conformal mappings between plane domains are analytic. The method of proof consists of a combination of Menchoff's result and an earlier theorem by \textit{Yu. G. Reshetnyak} [cf. Isothermal coordinates in manifolds of bounded curvature. I, Sib. Mat. Zh. 1, 88-116 (1960; Zbl 0108.338)], which guarantees the existence of appropriate isothermal coordinates under the assumptions made. At the end of his paper, the author points out that his theorem provides proofs for his previous results announced in a research report from about ten years ago [cf. the author, Sov. Math. Dokl. 20, 1068-1070 (1979); translation from Dokl. Akad. Nauk SSSR 248, 796-798 (1979; Zbl 0441.53045)].